ABSTRACT
Title of dissertation: PROBING FUNDAMENTAL PHYSICS
WITH GRAVITATIONAL WAVES FROM
INSPIRALING BINARY SYSTEMS
Noah Sennett
Doctor of Philosophy, 2021
Dissertation directed by: Professor Alessandra Buonanno
Department of Physics
The mergers of black holes and/or neutron stars in binary systems produce the
most extreme gravitational environments in the local universe. The first direct detec-
tions of gravitational waves by Advanced LIGO and Virgo provide unprecedented
observational access to the highly dynamical, strong-curvature regime of gravity.
These measurements allow us to test Einstein?s theory of General Relativity in this
extreme regime. This thesis examines how the gravitational-wave signal produced
during the inspiral?the earliest phase of a binary?s coalescence?can better inform
our understanding of the fundamental nature of gravity.
My work addressing this topic is comprised of two major components. First, I
examine the behavior of binary black-hole and neutron-star systems in various possi-
ble extensions of General Relativity, constructing analytic models of their orbital mo-
tion and gravitational waveform?their gravitational-wave signature?during their
inspiral. The majority of alternative theories I consider modify General Relativ-
ity by introducing a new scalar component of gravity. In many of these theories,
standard perturbative techniques are used to model the inspiral of binary systems.
However, I also examine in depth the non-perturbative phenomenon of scalariza-
tion for which such methods fail. I show that this phenomenon occurs due to a
second-order phase transition in the strong-gravity regime and develop an analytic
framework to model the effect across a range of alternative theories of gravity.
The other component of this thesis is the development of a statistical infras-
tructure suitable for testing General Relativity using gravitational-wave observa-
tions. I adopt a more flexible and modular approach than existing alternatives,
allowing this infrastructure to be immediately applied with a wide range of wave-
form models. In work done in conjunction with the LIGO Scientific and Virgo
Collaborations, I use this statistical framework to place bounds on phenomenologi-
cal deviations from General Relativity using the binary black-hole and neutron-star
events detected during LIGO?s first and second observing runs?no evidence for
deviations from Relativity is found.
These two research directions outlined above are complementary; the type of
statistical inference discussed here requires models for the gravitational-wave signal
produced by inspiraling systems that allow for deviations from General Relativity,
and the analytic models I construct are suitable for this task. In this thesis, I carry
out the complete procedure of building and employing analytic models of gravita-
tional waveforms to place constraints on specific alternative theories of gravity with
observations by LIGO and Virgo.
PROBING FUNDAMENTAL PHYSICS WITH GRAVITATIONAL
WAVES FROM INSPIRALING BINARY SYSTEMS
by
Noah Sennett
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Professor Alessandra Buonanno, Chair/Advisor
Professor Theodore Jacobson
Professor Julie McEnery
Professor Peter Shawhan
Professor Raman Sundrum
?c Copyright by
Noah Sennett
2021

Preface
The following several hundred pages comprise my Ph.D. dissertation, the cul-
mination of over half a decade of scientific research. Compiling work on this scale
is a sensible means of justifying one?s qualification for an advanced degree to a
committee (and the institution they represent) that could not directly observe the
completion of said work. Yet this monolithic focus on one individual?s research is
in many way at odds with the pace and practice of modern science, where work is
often finished on much shorter timescales and in collaboration with several others.
The work I present in this dissertation is original and my own but in many cases
has been published previously with several co-authors in peer-reviewed journals.
But, as this dissertation is intended as a measure of my scientific achievement, in
this preface I explicitly delineate the portions of work done with collaborators to
which I was the primary contributor. This preface is not intended as a summary
of the material covered in these chapters; such an overview is reserved until after
a pedagogical introduction to the scientific topics at hand in Section 1.4. Chapters
to which I am the sole contributor employ the pronouns I/my, whereas those that
include work done with collaborators use we/our.
Chapter 1 provides an introduction to the scientific topics covered in this thesis,
a review of relevant literature, and an overview of the remainder of the dissertation.
I was the sole author of this chapter.
Chapter 2 contains work published in Ref. [1]. In this work, I was the primary
contributor to the analysis of non-perturbative phenomena in Einstein-Maxwell-
dilaton gravity (Section 2.2.4). Additionally, I provided the discussion on the
ii
prospects of constraining this theory with gravitational-wave observations of bi-
nary black holes by analyzing bounds on dipole energy flux (Section 2.3.2) and the
estimated fraction of signal-to-noise ratio these deviations from general relativity
would provide for ground-based detectors (Section 2.3.4).
Chapter 3 contains work published in Ref. [2]. I contributed significantly to
every component of this chapter. At many points in the chapter, extended calcu-
lations were carried out independently by myself and collaborator Sylvain Marsat
and then compared as a means of verifying the results; as such, he and I contributed
approximately equally to the work.
Chapter 4 contains work published in Ref. [3]. I was the primary contributor
to and author of all of the work in this chapter.
Chapter 5 contains work published in Ref. [4]. This work serves as a reca-
pitulation, reinterpretation, and extension of many of the results presented in the
previous chapter (and Ref. [3]). As such, I was the primary contributor and was
primarily responsible for all text and figures in the work.
Chapter 6 contains work published in Ref. [5]. Though the overarching ideas
contained in the work arose through discussion between all the co-authors, I was the
primary contributor to all results, text, and figures presented in the chapter with
the exception of the discussion of critical exponents of phase transitions of compact
objects in alternative theories of gravity (Section 6.4.2).
Chapter 7 contains work published in Ref. [6]. This work represents a collabo-
ration between co-authors from the pulsar timing and gravitational-wave communi-
ties; my primary contribution was to the components of the publication focused on
iii
gravitational-wave observations. Specifically, I was the primary contributor to the
discussion of projected constraints on dynamical scalarization given pulsar-timing
observations at the time of publication (Section 7.4.2).
Chapter 8 contains work published in Ref. [7]. Though my co-authors had car-
ried out analogous calculations in previous work, and thus offered invaluable insight
in carrying out the calculations here, I was primarily responsible for the results and
discussion presented in this chapter. The only component of the work to which I was
not the primary contributor was the discussion of waveform approximants (Section
8.6.1, specifically Figure 8.4).
Chapter 9 contains work published in Refs. [8?10] as well as unpublished
work [11]. While the results presented in this chapter have appeared elsewhere,
the discussion given in this chapter is restricted to results to which I was the pri-
mary contributor, with the following qualifications. In certain places, e.g. Figure
9.1, additional results from Refs. [9, 10] are included to provide comparison between
others? work and my own; the discussion in the text explicitly distinguishes between
these results and those to which I contributed directly. Additionally, the investi-
gation done to place constraints on higher-order curvature corrections from binary
black hole observations (Section 9.5) was split primarily between myself and col-
laborator Richard Brito. We each made approximately equal contributions to these
results?e.g. I produced results for the left panel of Figure 9.5 while he produced
results for the right. However, I was primarily responsible for all text and figures
presented in this chapter.
Chapter 10 serves as a conclusion for the dissertation, providing a recapitula-
iv
tion of important results from the previous chapters as well as directions for future
work. I was the sole author for this chapter.
v
Acknowledgments
First and foremost, I would like to thank my advisor, Alessandra Buonanno,
for her help and guidance throughout my graduate career. She provided me with
ideas and advice on several challenging and timely projects while still allowing me
the flexibility to carve out my own niche in our burgeoning field. She has taught
me the care and patience needed to produce my highest-quality work and helped
me bolster the impact of such work on the scientific community at large. And,
simply put, this dissertation would not exist if not for her unwavering confidence
and support.
I would like to thank all of my co-authors of the papers that comprise this
dissertation: Richard Brito, Alessandra Buonanno, La?szlo? Gergely, Victor Gor-
benko, Tanja Hinderer, Maximiliano Isi, Mohammed Khalil, Michael Kramer, Syl-
vain Marsat, Serguei Ossokine, Bangalore Sathyaprakash, Leonardo Senatore, Lijing
Shao, Jan Steinhoff, Justin Vines, and Norbert Wex. I am grateful to the LIGO
Scientific and Virgo Collaborations, specifically the Testing General Relativity sub-
group, for facilitating much of the work presented here.
I owe my gratitude to that entire Astrophysical and Cosmological Relativity
division of the Albert Einstein Institute, comprised of too many wonderful scientists
and friends for me to list here; I?ve learned more about our field through discussions
over lunch and coffee than I ever could on my own. There are a few people I would
like to individually recognize: Sylvain Marsat, for his countless lessons in post-
Newtonian theory and using Mathematica to its fullest; Jan Steinhoff, for revealing
vi
the wonderful theoretical basis behind waveform modeling and our many invaluable
conversations on testing relativity without relying on phenomenonlogy; and Vivien
Raymond, for introducing me to parameter estimation with LAL and constantly
helping when my runs would invariably crash. I would also like to heartfully thank
the staff and international office at the AEI for their invaluable help in getting
situated upon moving from the US.
I would like to thank my friends and peers at UMD?Bill, Kyle, Seokjin, and
Yidan?for sharing the ups and downs of the start of graduate school, and I would
like especially to thank my friends and fellow students at the AEI?Andrea, Daniel,
Kyohei, Mohammed, Nils, Riccardo, and Roberto?for helping transform my stay
in Germany from an extended expedition into a home away from home.
And lastly, I would like to thank my parents for their constant support, acting
as a solid foundation and sounding board throughout the trials and tribulations of
my Ph.D.
vii
Table of Contents
Preface ii
Acknowledgements vi
List of Tables xiv
List of Figures xv
List of Abbreviations xviii
1 Introduction 1
1.1 Einstein?s theory of general relativity . . . . . . . . . . . . . . . . . . 1
1.2 Gravitational-wave science with compact binaries . . . . . . . . . . . 7
1.2.1 Generation by compact binary coalescence . . . . . . . . . . . 7
1.2.1.1 Early inspiral . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1.2 Late inspiral and merger . . . . . . . . . . . . . . . . 12
1.2.1.3 Ringdown . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.2 Observation with ground-based detectors . . . . . . . . . . . . 17
1.2.2.1 Laser-interferometric detectors . . . . . . . . . . . . 17
1.2.2.2 Bayesian inference with gravitational waves . . . . . 21
1.3 Testing general relativity with gravitational waves . . . . . . . . . . . 28
1.3.1 Motivations for modifying general relativity . . . . . . . . . . 28
1.3.2 The menagerie of modified gravity . . . . . . . . . . . . . . . . 30
1.3.2.1 Additional fields . . . . . . . . . . . . . . . . . . . . 31
1.3.2.2 Broken diffeomorphism invariance . . . . . . . . . . . 36
1.3.2.3 Non-locality . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.3 Tests of gravity with gravitational waves . . . . . . . . . . . . 39
1.3.3.1 Testing various facets of gravitational wave signals . 39
1.3.3.2 Designing gravitational-wave tests of general relativity 42
1.4 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.4.1 Post-Newtonian and effective-one-body descriptions of binary
black holes in Einstein-Maxwell-dilaton gravity . . . . . . . . 47
1.4.2 Post-Newtonian waveforms in massless scalar-tensor theories . 49
viii
1.4.3 Dynamical scalarization as a non-perturbative phenomenon . . 50
1.4.4 Dynamical scalarization as a second-order phase transition . . 53
1.4.5 Theory-agnostic modeling of dynamical scalarization . . . . . 56
1.4.6 Projected constraints on scalarization by combining pulsar-
timing and gravitational-wave observations . . . . . . . . . . . 58
1.4.7 Tidal signature of boson stars in binary systems . . . . . . . . 61
1.4.8 Parameterized tests of general relativity with generic frequency-
domain waveform models . . . . . . . . . . . . . . . . . . . . . 64
2 Hairy binary black holes in Einstein-Maxwell-dilaton theory and their effective-
one-body description 69
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2 Einstein-Maxwell-dilaton theory . . . . . . . . . . . . . . . . . . . . . 74
2.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2.2 Black-hole solution . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2.3 Dynamics of a test black-hole in a background black-hole space-
time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.4 Compact objects in Einstein-Maxwell-dilaton and scalar-tensor
theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.3 Post-Newtonian approximation in Einstein-Maxwell-dilaton theory . . 88
2.3.1 Two-body dynamics . . . . . . . . . . . . . . . . . . . . . . . 88
2.3.2 Gravitational energy flux . . . . . . . . . . . . . . . . . . . . . 94
2.3.3 Gravitational-wave phase in the stationary-phase approximation100
2.3.4 Number of useful gravitational-wave cycles . . . . . . . . . . . 107
2.4 Effective-one-body framework . . . . . . . . . . . . . . . . . . . . . . 109
2.4.1 Effective-one-body Hamiltonian in Garfinkle-Horowitz-Strominger
gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.4.2 Effective-one-body Hamiltonian in Schwarzschild gauge . . . . 118
2.4.3 Comparison of two effective-one-body Hamiltonians in Einstein-
Maxwell-dilaton theory . . . . . . . . . . . . . . . . . . . . . . 122
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3 Gravitational waveforms in scalar-tensor gravity at 2PN relative order 132
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.2 Gravitational waves in scalar-tensor gravity . . . . . . . . . . . . . . 137
3.2.1 Binary systems of compact objects . . . . . . . . . . . . . . . 137
3.2.2 Generic constraints on scalar-tensor gravity . . . . . . . . . . 140
3.2.3 Detector response . . . . . . . . . . . . . . . . . . . . . . . . . 144
3.3 Dynamics for quasi-circular orbits . . . . . . . . . . . . . . . . . . . . 146
3.4 Radiative coordinates and hereditary contributions . . . . . . . . . . 149
3.4.1 Radiative coordinates . . . . . . . . . . . . . . . . . . . . . . . 150
3.4.2 Tail contributions . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.4.3 Memory contributions . . . . . . . . . . . . . . . . . . . . . . 153
3.5 Balance equation and phase evolution . . . . . . . . . . . . . . . . . . 155
3.5.1 The dipole-driven regime . . . . . . . . . . . . . . . . . . . . . 158
ix
3.5.2 The quadrupole-driven regime . . . . . . . . . . . . . . . . . . 159
3.6 Spin-weighted spherical modes of the waveform . . . . . . . . . . . . 162
3.7 Stationary phase approximation . . . . . . . . . . . . . . . . . . . . . 167
3.7.1 The dipole-driven regime . . . . . . . . . . . . . . . . . . . . . 169
3.7.2 The quadrupole-driven regime . . . . . . . . . . . . . . . . . . 170
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
3.9 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4 Modeling dynamical scalarization with a resummed post-Newtonian expan-
sion 175
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.2 Non-perturbative phenomena in scalar-tensor gravity . . . . . . . . . 179
4.2.1 Spontaneous scalarization: single neutron star . . . . . . . . . 183
4.2.2 Dynamical scalarization: neutron-star binaries . . . . . . . . . 186
4.3 The post-Dickean expansion . . . . . . . . . . . . . . . . . . . . . . . 191
4.3.1 Action and field equations . . . . . . . . . . . . . . . . . . . . 191
4.3.2 Relaxed field equations . . . . . . . . . . . . . . . . . . . . . . 193
4.4 Structure of the near-zone fields . . . . . . . . . . . . . . . . . . . . . 196
4.5 Two-body Equations of motion . . . . . . . . . . . . . . . . . . . . . 200
4.5.1 Newtonian order . . . . . . . . . . . . . . . . . . . . . . . . . 200
4.5.2 Post-Dickean order . . . . . . . . . . . . . . . . . . . . . . . . 203
4.6 Structure of the far-zone fields . . . . . . . . . . . . . . . . . . . . . . 205
4.6.1 Near-zone contribution to the scalar field . . . . . . . . . . . . 206
4.6.2 Radiation-zone contribution to the scalar field . . . . . . . . . 208
4.7 Two-body Scalar mass . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.8 Validity of the Post-Dickean expansion . . . . . . . . . . . . . . . . . 217
4.8.1 Quasi-equilibrium configurations . . . . . . . . . . . . . . . . . 219
4.8.2 Earlier analytic models . . . . . . . . . . . . . . . . . . . . . . 223
4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
4.10 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
5 Effective action model of dynamically scalarizing binary neutron stars 230
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.2 Nonperturbative phenomena in scalar-tensor gravity . . . . . . . . . . 233
5.2.1 Spontaneous scalarization . . . . . . . . . . . . . . . . . . . . 236
5.2.2 Dynamical and induced scalarization . . . . . . . . . . . . . . 238
5.3 Effective action with a dynamical scalar charge . . . . . . . . . . . . 241
5.3.1 The effective point-particle action . . . . . . . . . . . . . . . . 244
5.3.2 Dynamics of a binary system . . . . . . . . . . . . . . . . . . 248
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.4.1 Computing c(i) . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.4.2 Comparison against previous models . . . . . . . . . . . . . . 255
5.5 Dynamical scalarization as a phase transition . . . . . . . . . . . . . 261
5.5.1 Spontaneous scalarization of an isolated body . . . . . . . . . 263
5.5.2 Dynamical scalarization of equal-mass binaries . . . . . . . . . 264
x
5.5.3 Scalarization of unequal-mass binaries . . . . . . . . . . . . . 266
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6 Theory-agnostic modeling of dynamical scalarization in compact binaries 272
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.2 Linear mode instability and effective action close to critical point . . 278
6.3 Matching strong-field physics into black-hole solutions . . . . . . . . . 284
6.4 Modeling strong-gravity effects within the theory-agnostic framework 289
6.4.1 Spontaneous scalarization . . . . . . . . . . . . . . . . . . . . 290
6.4.2 Critical exponents of phase transition in gravity . . . . . . . . 292
6.4.3 Dynamical scalarization . . . . . . . . . . . . . . . . . . . . . 295
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
6.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
7 Constraining nonperturbative strong-field effects in scalar-tensor gravity by
combining pulsar timing and laser-interferometer gravitational-wave detec-
tors 304
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
7.2 Nonperturbative strong-field phenomena in scalar-tensor gravity . . . 307
7.3 Constraints from binary pulsars . . . . . . . . . . . . . . . . . . . . . 312
7.4 Projected sensitivities for laser-interferometer gravitational-wave de-
tectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
7.4.1 Dipole radiation for binary neutron-star inspirals . . . . . . . 325
7.4.2 Dynamical scalarization . . . . . . . . . . . . . . . . . . . . . 336
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
7.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
8 Distinguishing boson stars from black holes and neutron stars from tidal
interactions in inspiraling binary systems 349
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
8.2 Boson star basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
8.3 Tidal perturbations of spherically-symmetric boson stars . . . . . . . 358
8.3.1 Background configuration . . . . . . . . . . . . . . . . . . . . 358
8.3.2 Tidal perturbations . . . . . . . . . . . . . . . . . . . . . . . . 360
8.3.3 Extracting the tidal deformability . . . . . . . . . . . . . . . . 362
8.4 Solving the background and perturbation equations . . . . . . . . . . 364
8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
8.5.1 Massive Boson Stars . . . . . . . . . . . . . . . . . . . . . . . 369
8.5.2 Solitonic boson stars . . . . . . . . . . . . . . . . . . . . . . . 372
8.5.3 Fits for the relation between M and ? . . . . . . . . . . . . . 374
8.6 Prospective constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 375
8.6.1 Estimating the precision of tidal deformability measurements . 375
8.6.2 Distinguishability with a single deformability measurement . . 383
8.6.3 Distinguishability with a pair of deformability measurements . 386
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8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
8.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
9 Parameterized tests of general relativity with generic frequency-domain wave-
form models 398
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
9.2 The FTA construction of generalized waveform models . . . . . . . . . 402
9.3 Theory-agnostic inspiral tests with LIGO/Virgo events . . . . . . . . 405
9.4 Constraints on Jordan-Fierz-Brans-Dicke gravity from GW170817 . . 410
9.4.1 Gravitational-wave signature of Jordan-Fierz-Brans-Dicke grav-
ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
9.4.1.1 Isolated neutron star solutions in Jordan-Fierz-Brans-
Dicke gravity . . . . . . . . . . . . . . . . . . . . . . 414
9.4.1.2 Polynomial fits of the neutron star scalar charge . . . 416
9.4.2 Constraining ?0 with GW170817 . . . . . . . . . . . . . . . . 418
9.5 Constraints on higher-order curvature corrections from GW151226
and GW170608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
9.5.1 Gravitational-wave signature of higher-order curvature correc-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
9.5.1.1 Conservative dynamics . . . . . . . . . . . . . . . . . 426
9.5.1.2 Dissipative dynamics . . . . . . . . . . . . . . . . . . 427
9.5.1.3 Gravitational waveform in the stationary phase ap-
proximation . . . . . . . . . . . . . . . . . . . . . . . 428
9.5.1.4 On the validity of the post-Newtonian waveform model429
9.5.2 Constraining d? with GW151226 and GW170608 . . . . . . . 430
10 Conclusions and future work 436
A The 1PN two-body Lagrangian in Einstein-Maxwell-dilaton theory 442
A.1 Expanding the metric and connection coefficients . . . . . . . . . . . 442
A.2 Expanding the action . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
A.3 The field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
A.4 The 1PN Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
B Energy flux to next-to-leading PN order in Einstein-Maxwell-dilaton theory450
B.1 Scalar energy flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
B.2 Vector energy flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
B.3 Tensor energy flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
B.4 Energy flux for circular orbits . . . . . . . . . . . . . . . . . . . . . . 464
C Translating between scalar-tensor theory notations 466
D Explicit formulas for the Fourier-domain phasing of waveforms in scalar-
tensor theories 468
E Post-Dickean expansion of mass and scalar charge 478
xii
F Two-body potentials at post-Dickean order 480
G A closer look at the dynamical scalarization model of Palenzuela et. al. 482
H Effective action for compact objects away from critical point 488
I Numerical calculation of the effective point-particle action V (q) 493
J Ingredients for the Fisher matrix analysis 495
K Dynamical scalarization in ultra-relativistic binary neutron stars 498
xiii
List of Tables
2.1 Effective-one-body Hamiltonians in Einstein-Maxwell-dilaton theory . 124
3.1 Scalar-tensor parameters . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.2 Weak-field constraints on scalar-tensor parameters . . . . . . . . . . . 142
4.1 Resummation schemes of the post-Dickean expansion . . . . . . . . . 193
4.2 Orbital and gravitational-wave frequency of dynamical scalarization . 222
7.1 Parameters of binary pulsar used to constrain scalar-tensor theories . 313
7.2 Constraints on scalar-tensor parameters from binary-pulsar observa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
7.3 Limits from binary-pulsar observations on number of gravitational-
wave cycles in binary neutron stars in scalar-tensor theories . . . . . . 326
7.4 Dynamical scalarization frequency of equal mass binary neutron stars 337
7.5 Dynamical scalarization of binary neutron stars given binary pulsar
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
9.1 One-dimensional polynomial fits of neutron star coupling in Jordan-
Fierz-Brans-Dicke gravity . . . . . . . . . . . . . . . . . . . . . . . . 416
9.2 Multivariate polynomial fits of neutron star coupling in Jordan-Fierz-
Brans-Dicke gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
E.1 Resummed component of mass and scalar charge for various resum-
mation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
xiv
List of Figures
1.1 Theory-specific/agnostic and top-down/bottom classification of the-
sis contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2 Projected constraints on Einstein-Maxwell-dilaton gravity . . . . . . . 49
1.3 Scalar charge of dynamically scalarized binary neutron star . . . . . . 52
1.4 Hamiltonian of dynamically scalarized binary neutron star . . . . . . 54
1.5 Scalarization phase space for binary black holes in Einstein-Maxwell-
scalar gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.6 Projected constraints on scalar dipole . . . . . . . . . . . . . . . . . . 60
1.7 Tidal deformability of boson stars . . . . . . . . . . . . . . . . . . . . 62
1.8 Constraints on higher-order curvature corrections . . . . . . . . . . . 67
2.1 Scalar charge of isolated black hole in Einstein-Maxwell-dilaton gravity 80
2.2 Scalar charge of test black hole in Schwarzschild background in Einstein-
Maxwell-dilaton gravity . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3 Validity of post-Newtonian expansion in various alternative theories
of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4 Scalar charge of binary black hole with equal electric charge . . . . . 92
2.5 Scalar charge of binary black hole with equal mass . . . . . . . . . . . 93
2.6 Energy flux of binary black holes in Einstein-Maxwell-dilaton gravity 96
2.7 Energy flux of binary black holes in Einstein-Maxwell-dilaton gravity 97
2.8 Projected constraints on Einstein-Maxwell-dilaton gravity . . . . . . . 98
2.9 Phase difference between Einstein-Maxwell-dilaton gravity and Gen-
eral Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.10 Number of useful cycles in Einstein-Maxwell-dilaton gravity . . . . . 110
2.11 Effective-one-body binding energy in Einstein-Maxwell-dilaton gravity 123
2.12 Innermost stable circular orbit of effective spacetime in Einstein-
Maxwell-dilaton gravity . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.1 Scalar charge of isolated neutron star in scalar-tensor theories . . . . 184
4.2 Analytic approximations of scalar-tensor theories . . . . . . . . . . . 186
4.3 Breakdown of post-Newtonian expansion at onset of dynamical scalar-
ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
xv
4.4 Scalar mass of dynamically scalarized binary neutron star . . . . . . . 218
4.5 Resummed scalar charge of binary neutron star . . . . . . . . . . . . 223
4.6 Comparison of models of dynamical scalarization . . . . . . . . . . . 224
5.1 Effective point-particle potential for neutron stars in scalar-tensor
theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.2 Dynamical scalarization of binary neutron stars . . . . . . . . . . . . 256
5.3 Binding energy of dynamically scalarized binary neutron stars . . . . 259
5.4 Hamiltonian of dynamically scalarized binary neutron star . . . . . . 265
5.5 Binding energy of binary neutron stars undergoing spontaneous, in-
duced, and dynamical scalarization . . . . . . . . . . . . . . . . . . . 266
6.1 Theory-agnostic approach to model scalarizing compact objects . . . 280
6.2 Effective point-particle potential for black holes in Einstein-Maxwell-
scalar theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
6.3 Critical behavior of black-hole scalar charge near the onset of scalar-
ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
6.4 Scalar charge of dynamically scalarized black holes in Einstein-Maxwell-
scalar theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
6.5 Scalarization phase space of binary black holes in Einstein-Maxwell-
scalar theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
7.1 Spontaneous scalarization in scalar-tensor theories . . . . . . . . . . . 309
7.2 Neutron-star mass and radius for several equations of state . . . . . . 314
7.3 Posterior distributions of scalar-tensor parameters from binary-pulsar
observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.4 Constraints on scalar-tensor parameters from binary-pulsar observa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
7.5 Allowed scalar coupling of isolated neutron stars given binary-pulsar
observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
7.6 Noise spectral density of ground-based gravitational-wave detectors . 330
7.7 Sensitivity of ground-based gravitational-wave detectors to scalar dipole
in binary neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . 331
7.8 Correlations between binary neutron-star parameters . . . . . . . . . 332
7.9 Projected upper limits on scalar-tensor parameters from gravitational-
wave observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
7.10 Dynamical scalarization of binary neutron stars given binary pulsar
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
8.1 Boson star tidal perturbations . . . . . . . . . . . . . . . . . . . . . . 367
8.2 Tidal deformability of massive boson stars . . . . . . . . . . . . . . . 369
8.3 Tidal deformability of solitonic boson stars . . . . . . . . . . . . . . . 372
8.4 Phase difference between tidal waveform models . . . . . . . . . . . . 379
8.5 Projected tidal-parameter measurement uncertainty for binary black
holes with Advanced LIGO . . . . . . . . . . . . . . . . . . . . . . . . 385
xvi
8.6 Projected tidal-parameter measurement uncertainty for binary neu-
tron stars with Advanced LIGO . . . . . . . . . . . . . . . . . . . . . 387
8.7 Discriminating neutron stars from boson stars with projected mea-
surement precision of Advanced LIGO . . . . . . . . . . . . . . . . . 390
8.8 Discriminating black holes from boson stars with projected measure-
ment precision of current and planned gravitational-wave detectors . 393
9.1 Posterior distributions of deviations from post-Newtonian predictions
in binary black hole and neutron star events observed by LIGO and
Virgo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
9.2 Upper bounds on deviations from post-Newtonian predictions in bi-
nary black hole and neutron star events observed by LIGO and Virgo 408
9.3 Prior distributions on Jordan-Fierz-Brans-Dicke parameter . . . . . . 420
9.4 Posterior distributions on Jordan-Fierz-Brans-Dicke parameter from
GW170817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
9.5 Independent constraints on higher-order curvature corrections from
GW151226 and GW170608 . . . . . . . . . . . . . . . . . . . . . . . . 433
9.6 Combined constraints on higher-order curvature corrections from both
GW151226 and GW170608 . . . . . . . . . . . . . . . . . . . . . . . . 434
G.1 Ambiguities in the dynamical scalarization model of Palenzuela et al.
due to derivatives of scalar charge . . . . . . . . . . . . . . . . . . . . 483
G.2 Ambiguities in the dynamical scalarization model of Palenzuela et al.
due to post-Newtonian corrections to feedback loop . . . . . . . . . . 485
K.1 Dynamical scalarization of neutron stars with ultra-relativistic cores . 499
xvii
List of Abbreviations
ADM Arnowitt-Deser-Misner
BBH Binary black hole
BH Black hole
BNS Binary neutron star
BS Boson star
DD Dipole-driven
EA Einstein-?ther
EFT Effective field theory
EMd Einstein-Maxwell-dilaton gravity
EMS Einstein-Maxwell-Scalar gravity
EOB Effective-one-body
ESGB Einstein-Scalar-Gauss-Bonnet gravity
GR General relativity
GW Gravitational wave
JFBD Jordan-Fierz-Brans-Dicke gravity
NS Neutron star
PN Post-Newtonian
QD Quadrupole-driven
SPA Stationary phase approximation
ST Scalar-tensor
xviii
Chapter 1: Introduction
1.1 Einstein?s theory of general relativity
Formulated by Albert Einstein over a century ago [12, 13], general relativity
(GR) remains one of the most successful physical theories of the modern era. At
its core, the theory embodies the equivalence principle?the notion that a local ob-
server cannot distinguish the effects of gravity from the ?fictitious forces? expected
in a non-inertial frame of reference. Mathematically, this stipulation requires that
(locally) the equations that describe the laws of physics including gravity can be
reduced to their non-gravitational analogs through an appropriate choice of coordi-
nate system. The underlying structure of such a coordinate system was established
by Minkowski a decade prior [14]: space and time come fused together as a four-
dimensional spacetime that is covariant under Lorentz transformations. Combining
these two ideas, the natural canvas for a relativistic theory of gravity is curved
spacetime, represented by a differentiable manifold M equipped with a metric g??
and the unique torsion-free, affine connection compatible with that metric. Grav-
ity is described entirely by the curvature of this connection, as encapsulated by
1
the Riemann tensor R 1???? . At any point in spacetime, one can construct normal
coordinates in which (locally) the metric reduces to that of Minkowski spacetime
??? = diag(?1, 1, 1, 1), and thus all laws of physics appear as they would in special
relativity, i.e. in their form absent gravity.
In GR, the metric g?? dynamically responds to matter (or energy) as described
by the Einstein field equations
R?? ?
1
g??R + ?g?? = 8?T?? , (1.1)
2
where R ???? ? g R???? is the Ricci tensor, R ? g??R?? is the Ricci scalar, ? is
a cosmological constant, and T?? is the stress-energy tensor of the matter in the
spacetime, which I collectively represent by the field(s) ?. As most concisely put by
Misner, Thorne, and Wheeler, these differential equations determine the manner in
which ?matter tells space how to curve [and] space tells matter how to move,? [15].
The latter statement can be more clearly seen using the Bianchi identity in conjunc-
tion with Eq. (1.1) to show
? T ??? = 0, (1.2)
which, when T?? describes a point particle minimally coupled to gravity, reduces to
the geodesic equation for that particle?s worldline. Solutions to the Einstein field
1Throughout this thesis, I adopt the sign conventions of Ref. [15] for the Riemann tensor and
its associated covariant derivatives ??. Additionally, unless explicitly stated otherwise, I work in
geometric units where G = c = 1. Greek indices represent temporal and spatial components (e.g.
? = 0, 1, ...) whereas Latin indices represent only spatial components (e.g. i = 1, 2, ...).
2
equations are local extrema of the Einstein-Hilbert action
?
1 ?
S[g 4?? , ?] = d x ?g (R? 2?) + Sm[g?? , ?], (1.3)
16?
where Sm is the action of the matter ? on the curved spacetime specified by g?? ,
which determines the stress-energy tensor via
??2 ?SmT?? = ? . (1.4)g ?g??
For example, in the aforementioned case of a point particle of mass m minimally
coupled to the metric, the appropriate matter action and stress-energy are
? ? ?m
Sm = ?m d? ?g u?u??? , T?? = ?? d? u?u??
(4)(??(?)? x?), (1.5)
g
where u? ? d??/d? is the four-velocity of the particle?s worldline ??, parameterized
by an arbitrary affine parameter ? .
Compared to its non-relativistic predecessors, GR represents a massive repu-
diation in our understanding of the nature of gravity. Accompanying this paradigm
shift, the theory predicts novel phenomena absent in Newtonian gravity. This thesis
primarily contends with two such predictions: black holes (BHs) and gravitational
waves (GWs). I provide a brief introduction to each topic in the remainder of this
subsection.
First predicted only one year after the advent of GR [16], BHs are localized
regions of curvature. The defining feature of BH spacetimes is the presence of an
event horizon. This hypersurface divides the spacetime into interior and exterior
regions such that future-oriented timelike and lightlike curves originating in the for-
mer cannot cross into the latter. Physically, the event horizon represents a one-way
3
boundary for the flow of information; any matter or radiation that enters the BH
interior can never escape. Black holes primarily form through the gravitational col-
lapse of matter. Whether a collapsing matter configuration will form a BH depends
primarily on its compactness, i.e. the ratio of its mass to its (linear) size; though still
unproven, the Hoop conjecture [17] posits that a BH forms only when matter of mass
M fits within a spherical shell with the Schwarzschild radius RS = 2M . In our local
universe, the densities needed to form a BH can only occur during the final stage of
the evolution of very massive stars. However, in the very early universe, critically
over-dense regions could have also collapsed to form primordial BHs that may have
survived to the present [18?20]. Current observations point to the existence of astro-
physical BHs of two distinct varieties, distinguished by their mass: stellar-mass BHs
(100M . M . 102M) and supermassive BHs (105M . M . 109M 2). This
thesis focuses on the former class, as they represent viable sources for ground-based
GW detectors.
A number of properties of BHs predicted in GR offer compelling targets for
both theoretical and observational study. For example, BH spacetimes in GR ap-
pear only to come in a relatively limited variety?this property is often categorized
as the ?no-hair conjecture.? This property posits that all isolated BHs coupled to
electromagnetism are uniquely specified by their mass, spin, and electric charge;
contrast this scenario to that of composite systems of charged baryons, which can
differ by any of an infinite number of independent multipole moments. This prop-
2The latter class is sometimes further subdivided into ?massive? and ?supermassive? BHs; this
distinction is not needed for this thesis, so I categorize those two classes together.
4
erty has also been shown to hold for BHs coupled to other fields [21?23] (though not
all [24]). Another intriguing property of BHs in GR is their connection to space-
time singularities. Penrose famously showed that BHs formed through gravitational
collapse will generically host a singularity in their interior region, provided that the
imploding matter obeys the null energy condition [25]. The presence of such a sin-
gularity indicates that GR is non-predictive in such a region and highlights the need
for new physical insight to complete the theory.
Besides BHs, another of the earliest predictions of GR was the existence of
GWs [26, 27]?wavelike perturbations of the metric that propagate energy and mo-
mentum at the speed of light. Their basic structure can be seen by considering
gravity linearized around a flat background, wherein the metric takes the form
g?? = ??? + h?? with |h?? |  1. Inserting this ansatz into Eq. (1.1) and drop-
ping terms non-linear in the metric perturbation h?? reveals that its trace-reverse
h? ? h ? 1? ????? ?? ?? h?? obeys a wave equation provided that one works in the2
Lorenz gauge ? h???? = 0:
h??? = ?16?T?? , (1.6)
where  is the flat-space D?Alambertian operator. In vacuum, one can use gauge
symmetries to eliminate all but two independent components of this tensor, which
correspond to the two polarizations of GWs predicted in GR. In the appropriate
gauge, these non-trivial components are spatial (h = 0), transverse (?ij0? ?ihjk = 0)
and traceless (?ijhij = 0). These physical components can be distilled from h??? with
5
an appropriate projection operator that depends on the direction of propagation n?
hTTij =?
kl
ij (n?)h?kl, (1.7)
where the superscript denotes the ?transverse-traceless? gauge described above; a
complete expression for ? klij is given in Eq. (1.39) of Ref. [28].
The central theme of this thesis is to motivate and improve tests of the funda-
mental nature of gravity using GWs. I focus on GWs produced by compact binary
systems, i.e. binaries comprised of objects with compactness C = M/R & 10?2; the
only known astrophysical objects that surpass this threshold are BHs and neutron
stars (NSs). In the remainder of this chapter, I outline the phenomenology of such
GWs and discuss the techniques used to model these predictions. I outline how
these waveform models are employed to detect GWs and to infer the properties of
the signals? sources and the underlying fundamental physics that govern their be-
havior. Next, I discuss motivations for and examples of modified theories of gravity,
highlighting the potential imprint of such modifications in GW signals observable
with detectors. I describe various considerations involved in formulating tests of
gravity with GW observations. Finally, I end with a more detailed overview of the
work contained in this thesis.
6
1.2 Gravitational-wave science with compact binaries
1.2.1 Generation by compact binary coalescence
The GWs produced by coalescing compact binary systems can be divided into
three phases: inspiral, merger, and ringdown. In this section, I provide a brief
overview of each of these phases, highlighting the techniques used to study the two-
body problem in each regime. This thesis primarily focuses on the inspiral behavior
of binary systems, and so I discuss that topic in the greatest detail. However, I
include an abbreviated review of the other phases as a foundation for discussions of
inspiral-merger-ringdown (IMR) models that synthesize information from all regimes
and also to establish the broader context for the inspiral results presented here.
1.2.1.1 Early inspiral
The inspiral phase of a binary system begins with the two bodies well-separated.
The binary is characterized by three independent distance scales: the size of each
body R1 and R2 and their separation r. For compact objects, to which I restrict my
attention, the bodies? masses satisfy mi ? Ri; for convenience, I label the bodies
such that m1 ? m2. The inspiral begins in the regime in which there is a clear hi-
erarchy between these various scales, meaning that m2  m1 and/or m1  r. The
gravitational radiation produced by the binary during the early inspiral can differ
significantly between these two cases. Accordingly, the computational techniques
used to predict this radiation differ as well. Analytic approaches to the two-body
7
problem attempt to solve the field equations (1.1) perturbatively in the appropri-
ate small parameter: q ? m2/m1 or  ? M/r, respectively, where M ? m1 + m2.
The former approach relies on computing the so-called gravitational self-force of the
smaller body moving in a background spacetime associated with the larger body;
see Ref. [29] and references therein for a pedagogical review of this approach. The
latter approach is known as the post-Newtonian (PN) approximation, which serves
as the foundation for much of the modeling efforts described in the thesis. I outline
a few basic results derived using the PN approximation below, but refer the reader
to Ref. [30] for a detailed review of the subject.
Within the PN approach, the motion of the binary is solved order-by-order in
a power series expansion in the small parameter  around the Newtonian prediction.
Equivalently, because I consider bound orbits, this is a low-velocity expansion with
 ? v2. The quadrupole formula, computed first in Einstein?s original paper on
GWs [26] , describes the leading-order PN prediction for the GWs observed far
from a compact source. Despite its simplicity, this result offers key insights into the
gravitational radiation from compact binaries that remain true at higher PN orders.
I provide a brief derivation below.
Working within linearized gravity (a valid assumption when working at leading
PN order), Eq. (1.6) can be?formally inverted leaving( ) ?
h? (x?) =4 d4x? ? x0 ? x?0 ? | ? ?| Tij(x )ij ? x x | ? ?| ,x x (1.8)
4
= d3x? T ? ?ij(tret + x ? n?,x ),
d
where I have used that at distances d ? |x|  r far from the binary, the denominator
8
can be approximated by |x? x?| = d? x? ? n? +O(r2/d) and defined the unit vector
n? ? x/d and the retarded time t ? x0ret ? d. Then, one can Taylor expand the
integrand in Eq. (1.8) to
(x? ? n?)2
Tij(tret + x
? ? n?,x?) =Tij(t ,x?) + (x? ? n?) ? T (t ,x? 2 ?ret 0 ij ret ) + ?0Tij(tret,x ) + . . . ,2
(1.9)
The most relevant time scale of the binary?s motion is its orbital period ?orb = r/v;
thus we see that each time derivative above introduces a power of v to the final
result, and so Eq. (1.9) corresponds to a PN expansion. Keeping only the first
term, I insert Eq. (1.9) into Eq. (1.8). Because I am working within the regime
of linearized gravity, the non-linearities in Eq. (1.2) can be neglected, leaving the
identity ? T ??? = 0; by repeatedly employing this identity, one can integrate Eq. (1.8)
by parts to show that
2 ( )
h? 3ij = M?ij(tret) +O(v ) , (1.10)
? d
where M ? d3x? T 00ij xixj is the quadrupole moment of the source. Finally, the
physical (non-gauge) components representing the GWs are extracted by applying
the operator ? klij to Eq. (1.10). Note that by simplifying Eq. (1.2) as done above, I
have implicitly assumed that the sources are weakly self-gravitating, i.e. the gravita-
tional field does not comprise a major component of their stress-energy. Given that
I wish to study the behavior of compact objects, whose self-gravity is non-negligible,
this assumption may seem unfounded. However, the property of effacement in GR
guarantees that the strong internal gravitational fields of the compact bodies influ-
ence the dynamics of a binary only at comparatively high PN order: at 2PN for
9
spinning bodies and 5PN for non-spinning bodies [31].
I highlight a few important consequences of Eq. (1.10). First, the dominant
component of GWs arises from quadrupolar radiation, i.e. radiation produced by a
time-varying quadrupole moment. Though the derivation above was done only at
leading PN order, in fact, monopolar and dipolar radiation (which dimensional ar-
guments would suggest enter at lower order) are absent in GR even when non-linear
gravitational interactions are considered. These dissipative channels are protected
due to the diffeomorphism invariance of the theory and the fact that gravity is medi-
ated solely by the metric; these conditions guarantee that GWs behave as massless,
spin-2 gauge fields, which cannot be be excited by a purely monopolar or dipolar
source (see Secs. 2.2.3 and 3.3.1 of Ref. [28] for more detail). However, as we shall
see in Section 1.3, many possible extensions of GR violate these two conditions; the
additional dissipative channels available in these theories can dramatically affect the
rate at which compact binaries coalesce.
A second key observation is that GWs produced during the inspiral are di-
rectly linked to the motion of the binary. Using two copies of Eq. (1.4) as the
stress-energy for a binary system, one sees that the quadrupole moment oscillates
at twice the orbital frequency. Hence, for binaries on circular orbits, the dominant
GW frequency is precisely two times the orbital frequency. However, the GW sig-
nal produced by binaries following eccentric orbits is comprised of a more complex
spectrum of Fourier modes. Throughout this thesis, I restrict my attention to sys-
tems on non-eccentric orbits. This assumption is valid for stellar-mass binaries that
form in isolation, as they are expected to have radiated nearly all eccentricity before
10
reaching orbital frequencies observable with ground-based GW detectors [28, 32].
However, binaries that form through dynamical processes, such as by dynamical
capture in dense stellar environments [33] or through three-body interactions like
the Kozai-Lidov mechanism [34, 35], may not have time to circularize before be-
coming observable with ground-based detectors; in these scenarios, the binaries can
retain eccentricity up to e . 10?1 upon entering the frequency band of Advanced
LIGO/Virgo. Work to detect and measure such eccentric systems is underway [36],
but I do not pursue it further in this dissertation.
Without eccentricity, the inspiral evolution of a binary is well-described by a
circular orbit whose radius slowly decreases?such a binary is said to follow quasi-
circular orbits. Using the quadrupole formula (1.10) and dimensional analysis, one
can formulate this statement more precisely by comparing the orbital period ?orb to
the radiation-reaction timescale ?RR, which is the timescale over which the dissipa-
tion of energy through GWs alters the circular orbit. One can approximate ?RR as
the ratio of the binding energy E ? Mv2 and the rate at which energy is emitted
by the system dE/dt. The energy flux F propagated by gravitational radiation is
encapsulated in the Landau-Lifshitz pseudotensor [37]; in the transverse-traceless
gauge, this is given by F ? h?TT ij 10ij h?TT ? v /d2. Integrating the flux over a sphere of
size d, one finds ? 8RR ? E/(dE/dt) ?M/v , which differs from the orbital timescale
?orb ? r/v ?M/v3 by a factor of v?5. Thus, during the early inspiral, where v  1
by assumption, the evolution of the binary is said to be (approximately) adiabatic.
Extending the GW prediction (1.10) to higher PN order requires utilizing
the hierarchy of timescales in the problem. In addition to the the orbital and
11
radiation-reaction timescales discussed above, spinning bodies will also precess on
the timescale ?prec, with ?orb  ?prec  ?RR. One begins by determining the or-
bital dynamics of the binary, working on short enough timescales that precession
and radiation-reaction enter as sub-leading effects; I carry out this procedure in the
context of two modified theories of gravity in Chapters 2 and 4. Inserting this so-
lution into the stress-energy tensor, one then computes the gravitational radiation
at the given PN order (see Chapter 2 for an example). The precessional dynamics,
which I do not consider in this thesis, are then determined by averaging the pre-
vious solutions over the orbital timescale, while still treating radiation-reaction as
subdominant. Finally, by relating the energy carried away by this outgoing radia-
tion to a commensurate shift in the energy of the orbit, one determines the rate at
which the quasi-circular orbits shrink. All of these pieces are combined to produce
the most accurate prediction for the GWs at a given PN order; I carry out these
last steps in Chapters 2 and 3 for two alternative theories of gravity.
1.2.1.2 Late inspiral and merger
As the orbit of a compact binary shrinks, the perturbative PN approximation
becomes increasingly inaccurate. Ultimately, the description of the system as two
separate bodies breaks down, and one must contend with the non-linearities in
the field equations (1.1) in a non-perturbative way. In this subsection, I discuss
the phenomenology at the end of the inspiral and introduce the effective-one-body
(EOB) formalism [38, 39], a resummation of the PN approximation that extends its
12
validity further into the non-linear regime. Finally, I outline how numerical relativity
(NR) is used to compute the GW signal in the highly non-linear regime, and how
these predictions can be incorporated into the previous analytic approaches.
To better understand the behavior of a binary as the PN approximation breaks
down, I first consider the low mass-ratio limit q  1. To zeroth order in this small
parameter, the smaller mass M2 simply evolves along geodesics of the spacetime
determined by M1. For simplicity, let us assume that the larger mass is a BH.
Then, the quasi-circular orbits that characterize the PN inspiral can only extend up
to the innermost stable circular orbit (ISCO) of the background spacetime. For a
Schwarzschild background, the last (stable) circular orbit has a radius of rISCO =
6M1 ? 6M ; if the larger BH has spin, the ISCO radius decreases (increases) for
prograde (retrograde) orbits of the smaller body. After passing through this orbit,
the smaller body plunges into the larger BH on a geodesic with non-negligible radial
velocity. This evolution is quite distinct from the earlier quasi-circular orbits, in
which the velocity was primarily tangential; thus the passage through the ISCO and
the start of the plunge offers a clear indication that the inspiral has ended.
The EOB formalism uses results derived in the PN regime to extend the test-
particle limit used above to non-zero values of q. The overarching aim of this
approach is to map the dynamics of a binary to that of a test body moving in
an effective background spacetime determined by the binary?s characteristics. This
goal is realized by mapping the Hamiltonians and radiation-reaction forces (which
describe the conservative and dissipative sectors of the system, respectively) for test
particle motion to a resummation of PN description of the two-body problem.
13
For demonstrative purposes, I simply focus on the conservative sector and ig-
nore the effect of spins. Schematically, one maps the real and effective spacetimes by
identifying conserved action angle variables in the orbits of each. This identification,
in turn, determines a mapping between the 2-body Hamiltonian Hreal describing the
PN motion of two bodies and the Hamiltonian Heff describing the motion of a test
particle moving through the effective spacetime, i.e. Heff = f(Hreal). The energy
map, first derived in Ref. [38] is given by
Heff ?
[ ]
? Hreal ?M ? Hreal ?M
= 1 + , (1.11)
? ? 2 ?
where ? = M1M2/M
2 is the reduced mass and ? = ?/M is the symmetric mass
ratio, or equivalently, ? ( )
Hreal ?
Heff ? ?
M = M 1 + 2? ?M. (1.12)
?
The effective spacetime is constructed by first considering generic deformations
to a background Kerr spacetime (or Schwarzschild for non-spinning binaries). From
this ansatz for the effective geometry, one constructs the suitable effective Hamil-
tonian Heff that describes test-body motion in that spacetime. Then, one identifies
a suitable choice of effective spacetime by requiring that Heff satisfy Eq. (1.12) for
an appropriate PN estimate of Hreal. I carry out this procedure in Chapter 2 in
the context of a modified theory of gravity. Note that the knowledge of the real
Hamiltonian is incomplete, e.g. the expression is only known up to some finite PN
order, which leaves some flexibility in how the effective spacetime is defined. Often-
times, one leverages this ignorance by calibrating the model to include information
from other sources, such as from gravitational self-force calculations [40?43] or from
14
NR [44, 45]. This extra tuning, particularly from NR simulations, allows the ana-
lytic EOB formalism to track the evolution of a binary through merger, where highly
non-linear interactions arise.
Even discounting its impact on the calibration of EOB models, the advent
of NR simulations of binary systems over the last decade and a half has played a
crucial role in furthering our understanding of compact binary mergers. Numerical
relativity is based on the 3+1 decomposition of spacetime first described within the
Arnowitt-Deser-Misner (ADM) formalism [46, 47], which divides the Einstein equa-
tions into a set of four constraint equations, to be solved on a spacelike hypersurface,
and 12 first order evolution equations, which conserve the constraint equations, i.e.
if the constraint equations are satisfied on one spacelike hypersurface, then they are
also satisfied on nearby slices. However, the first successful numerical evolutions of
BBHs was not achieved until much later [48?50] using multiple different formula-
tions of the Einstein equations. Today, several NR codes actively produce numerical
evolutions of BBHs and BNSs and extract the GW signals they generate, using var-
ious formulations of the Einstein equations, numerical schemes, and computational
techniques; see Refs. [51, 52] and references therein for details. However, such sim-
ulations can be very computationally expensive, taking months to finish even when
parallelized over several hundred cores on supercomputers. Thus, analytic mod-
els remain necessary as a fast and efficient supplement to numerical simulations
to detect GW events with LIGO/Virgo and infer properties of their corresponding
sources.
15
1.2.1.3 Ringdown
After merger, the system continues to emit GWs as it relaxes into its final
equilibrium configuration. Sufficiently long after the merger, the system is well-
described by perturbations of the background solution to (1.1) corresponding to the
final remnant object. When the remnant is just given by a vacuum configuration,
e.g. after the merger of a stellar-mass binary black hole (BBH) or neutron-star black-
hole (NSBH) binary, the system is completely described by tensor perturbations on
a Kerr background
g = g(Kerr)?? ?? + h?? , (1.13)
with h??  1. I focus on this simple case, but similar phenomenology can occur after
the formation of a hypermassive NS from a binary neutron star (BNS) merger [53,
54], though in that case, the system is described by perturbations to both the metric
and matter fields.
Working within the linear regime, that is ignoring any backreaction of h?? on
(Kerr)
g?? , and assuming physical boundary conditions, the late-time behavior of h?? is
given by a sum of quasinormal modes whose time-dependence takes the form ei?nt
with a complex frequency ?n [55, 56]. Thus, any distant observer would measure
a GW signal composed of damped sinusoids. Fully non-linear NR simulations of
merging BHs confirm that in astrophysical systems, the post-merger GW signal is
indeed well reproduced by a sum of exponentially decaying sinusoids.
16
1.2.2 Observation with ground-based detectors
In the following subsections, I provide an overview of laser-interferometric GW
detectors and how they are used to measure GW. Other techniques for detecting
GWs have been devised (e.g. Weber bars [57?59] and atom interferometry [60?62]),
but I focus only on laser interferometry for the remainder of the thesis.
1.2.2.1 Laser-interferometric detectors
The measurement of GWs involves tracking the motion of freely-falling test
masses, which, absent any external forces, simply follow geodesics of spacetime.
Incident GWs, which are wave-like perturbations of curvature, will induce devia-
tions to the background geodesic motion of the masses. The equivalence principle
guarantees that the response of a single test mass cannot be disentangled from the
gauge freedom in defining a reference frame; however, the relative motion of a sys-
tem of (separated) test masses does provide a signature for GWs. In the limit that
the separation of the masses is much smaller than the curvature scale, the relative
separation ??(?) of the two bodies is given by the geodesic deviation equation
( )
T?? (T ?? ??) = R? T ?T ??? +O ?2? ? ??? (1.14)
where T ? = d??/d? is the four-velocity of one of the bodies. For weak perturbations
(0)
?R???? that vary over different timescales than the background R???? (e.g., a static
background curvature produced by the Earth and an incident GW with only non-
zero frequency content), one sees that the change in relative separation of two test
17
masses initially at rest scales as ?? ? h ?(0), where h  1 is the magnitude of
the perturbation and ?(0) is the separation of the background geodesics. Thus, the
success of a GW detector relies on its ability to resolve small distance variations
between test masses located at long relative distances.
Laser interferometry is an ideal technique for achieving this goal. In its sim-
plest form, a phase-coherent beam of light (e.g., from a laser) is passed through a
beam splitter, and the subsequent beams are directed through two orthogonal arms.
The beams are reflected off mirrors at the end of each arm, and then are recombined
back at the source. The difference in the path length for each beam (modulo the
wavelength of the light) determines the intensity of the combined light, allowing
for relative distance measurements on the sub-wavelength scale. Absent any other
forces, the mirrors in each arm follow geodesics, and so a GW passing through the
detector changes the proper length of each cavity and thus the intensity of the re-
combined beam. In reality, other external forces are applied to the mirrors; some,
like the vertical force to suspend the mirrors in the cavities, do not impact the re-
sponse of the interferometer to an incident GW, whereas others, such as those from
radiation pressure, thermal and seismic fluctuations carried through the suspension
system, or time-varying (Newtonian) gravitational gradients from the external en-
vironment, are more difficult to disentangle from GWs, and thus serve as sources of
noise in the detector.
The LIGO [63] and Virgo [64] instruments refine this basic principle using
more complex optical systems; I outline a few such improvements, but direct the
interested reader to Refs. [63?66] for more detail. Rather than simply recombining
18
the laser light after a single transit through the detector, Fabry-Perot cavities are
used for the arms, allowing one to increase the effective path length of the photons
by ? O(100), with an equivalent increase in sensitivity to GWs. A similar technique
is used to increase the effective power of the laser by another factor of ? O(100); a
power recycling mirror is coupled to the system to capture light emitted from the
beam splitter back to the laser. Increasing the laser power helps to reduce quantum
shot noise in the system, described below.
Experimental noise ultimately determines the sensitivity of a detector, and
consequently the quantity and quality of the GW science for which it can be used.
In the remainder of this section, I outline the major sources of noise in the LIGO
and Virgo instruments. Each source of noise impacts the frequency-dependent re-
sponse of the detector differently, and thus techniques to mitigate noise over some
frequency bands may worsen sensitivity at other frequencies. The calibration of
the detectors involves balancing the various sources of noise to construct the most
effective instrument for the desired science. I discuss two broad categories of noise
sources below: optical read-out noise and displacement noise [28]. The former class
of noise arises from the quantum nature of the optical system, whereas the latter
represents classical disturbances that impact the motion of the test masses.
Optical read-out noise in an interferometer is comprised of quantum shot noise
and radiation-pressure noise. Shot noise occurs due to the quantized nature of light;
because the beam traveling through the detector arms is composed of a finite number
of photons, the intensity of light measured by the interferometer fluctuates follow-
ing Poisson counting statistics. This noise impacts the detector?s sensitivity at high
19
frequencies, and can be mitigated by increasing the power of laser light. The pho-
tons in the interferometer exert radiation pressure on the test masses; because the
intensity of photons incident on the test masses fluctuates, so too does this pres-
sure, inducing another source of stochastic noise in the detector. Radiation pressure
noise impacts the low-frequency sensitivity of the detector, and (conversely to shot
noise) can be mitigated by decreasing the amount of laser light in the interferometer.
Thus, a tradeoff between these two sources of noise must be made in the design of
an interferometric GW detector
The two dominant sources of displacement noise are seismic and thermal fluc-
tuations. At very low frequencies, the dominant source of noise comes from seismic
activity, both from anthropogenic (e.g. nearby technicians, cars, etc.) and natural
sources (e.g. surface waves in the Earth?s crust). Some components of this noise
can be mitigated through the complex isolation systems [63], but others, such as
the time-varying gravity-gradients [67] are unavoidable. Thermal fluctuations in
the test masses and the systems used to suspend them also cause small displace-
ments in the faces of the mirrors, which in turn induce small displacements in the
effective length of the detector arms. The impact of thermal noise on the detec-
tor?s sensitivity decreases with frequency, but does so at a lesser rate than seismic
and radiation pressure noise; thus thermal noise plays the most significant role at
moderate frequencies [63].
20
1.2.2.2 Bayesian inference with gravitational waves
Laser interferometers are designed to resolve the minute relative distance fluc-
tuations caused by a passing GW. However, as discussed above, instrumental and
environmental noise are unavoidable; typically, noise has a much larger impact on
the detector output than any incident GW. Fortunately, the characteristics of both
the noise and underlying signal are well understood; this knowledge is leveraged
to extract the signal from noise. In this section, I outline the strategies used by
the LIGO and Virgo collaborations to detect GWs and subsequently estimate the
properties of their sources.
For concreteness, I consider the time-dependent output of a single detector
d(t) ? h(t) + n(t) where h(t) arises from an incident GW and n(t) from noise;
typically, |h(t)|  |n(t)|. Note that h(t) is a contraction of the GW hij(t ? n? ? x)
(traveling in the n? direction) and a detector tensor Dij(n?), which encapsulates the
directional dependence of the detector?s sensitivity [28]. In anticipation of work to
come, it is often convenient to project the incident GW into a polarization basis. In
the limit that the wavelength of the GW is much larger than the detector, one can
drop the spatial dependence of hij measured at the detector, and thus decompose
the signal into
?
hij(t) = hA(t)e
A
ij(n?), (1.15)
A
where {eAij} are a complete set of polarization tensors. Then the detector response
21
is given by
?
h(t) = FA(n?)hA(t), (1.16)
A
where FA(n?) ? DijeAij(n?) is the antenna pattern of the detector, which represents
the directional-dependence of the detector?s sensitivity to each GW polarization.
The noise in the LIGO and Virgo detectors is approximately stationary, mean-
ing its statistical properties (e.g. mean, variance, etc.) are nearly constant over time.
Without loss of generality, I can assume the mean of the noise ?n(t)? = 0, where
?. . .? represents an ensemble average of noise realizations or, assuming ergodicity, an
average over a long time interval. In the Fourier domain, the correlation function
for a stationary stochastic process takes a particularly simple form
? 1n??(f)n?(f ?)? = ?(f ? f ?)Sn(f), (1.17)
2
where Sn(f) is the noise spectral density of the detector; due to this simplicity, it
is often more convenient to work in the Fourier domain. The noise is also approxi-
mately Gaussian, meaning that the probability of a particular noise realization n is
given by
[ ]
P (n) = N 1exp ? (n|n) , (1.18)
2
where N is an overall normalization and I have(introduced the inn? ? )
er product
? ?
| ? a? (f)b?(f)
? a??(f)b?(f)
(a b) df = 4 Re df . (1.19)
?? Sn(f)/2 0 Sn(f)
Matched filtering is one approach adopted by the LIGO and Virgo teams to
identify the presence of a GW produced by a coalescing binary system in the data
22
stream d(t).3 This technique involves applying a filter to d(t) that amplifies the
presence of the GW h(t) relative to the noise n(t). The optimal (Wiener) filter for
such a task is given in the Fourier domain by [28]
h?(f)
W?h(f) = , (1.20)
Sn(f)
where the subscript highlights the dependence of the filter on h. This choice max-
imizes the (averaged) measured signal-to-noise ratio of the signal ???, where ? is
defined for a generic filter W as
( ?? ??? df s?(f)W? ?(f)? ? ) . (1.21)? 1/2
df 1?? Sn(f)|W? (f)|22
Using the filter in Eq. (1.20) gives the optimal signal-to-noise ratio SNR ? (h|h)1/2.
For stationary, Gaussian noise, ?2 follows a ?2 distribution with two degrees of
freedom whose mean is determined by the optimal SNR of the underlying GW.
Thus, for a particular noise realization n(t), the magnitude of ? allows one to de-
termine the statistical significance of the presence of a non-zero GW in the data.
Larger ? corresponds to a higher statistical significance, and thus lower false-alarm
probability?the probability of such a value of ? being achieved only through noise.
A segment of data is said to contain a GW provided that ? exceeds some prede-
termined threshold, which corresponds to a desired maximum false-alarm rate. In
practice, the noise in an actual detector is non-stationary and non-Gaussian, and so
?2 does not follow a ?2 distribution and serves as a poor test statistic for determining
3Other, more model-independent strategies are also employed to detect GWs, particularly for
short signals. These include the coherent WaveBurst [68, 69] and omicron-LALInference-Burst [70]
pipelines and BayesWave [71] searches.
23
the presence of a GW in the data. More sophisticated test statistics are used in its
place to mitigate these non-ideal aspects of the noise. However, I restrict my atten-
tion to the idealized case here, noting that the same overall strategy can be applied
using test statistics better adapted to the noise. Additional statistical significance is
gained by folding into the analysis the fact that a true GW signal should be observ-
able coincidentally across a network of detectors; I ignore this added complication
here.
Gravitational-wave detection requires one to assess whether a GW is present
in a stretch of data. Unlike in the discussion above, the precise form of h (and thus
optimal choice of Wh) is not known ahead of time. To circumvent this problem,
one approach instead uses a large bank of template waveforms {h?(f ;?)}, each with
different intrinsic parameters (masses, spins, etc.) and/or extrinsic parameters (dis-
tance, sky location, etc.), represented by ?. The signal-to-noise ratio ? is computed
using each waveform in the template bank, representing a goodness-of-fit of the
waveform to the data. Using a finite template bank reduces the signal-to-noise ratio
(and thus the statistical significance of a detection) compared to when the optimal
matched filter Wh is used; this reduction can be mitigated using a sufficiently dense
bank of templates. Consider the case where the incident GW is given by h?(f ; ??),
whereas the best-fit waveform in the template bank is h?(f ;?). Then ?? computed
using the optimal filter is related to the maximum ? recovered with the template
24
bank by ? ?
? ?? ? [ (h(??)|h(?)) ?? ?? 1 ] , (1.22)1/2?
(h(??)|h(??))(h(?)|h(?))
where the second term in parentheses is called the mismatch between the two wave-
forms. LIGO and Virgo use template banks constructed such that any possible
waveform in the parameter space of its search has a mismatch of less than 3% with
respect to at least one template; for more detail see Refs. [72?74]. However, one
can also appreciate from Eq. (1.22) the importance of accurate waveform models for
GW detection: inaccuracies in waveform models will further reduce the recovered
signal-to-noise ratio.
The matched-filtering strategies outlined above are optimized to quickly de-
termine whether a stretch of data d contains a GW. However, they do not guarantee
that parameters ? of the best-fit template accurately match the parameters ?? of the
true underlying signal. Instead, once a GW is detected, parameter estimation is
carried out within a more comprehensive Bayesian framework. The ultimate goal
of this procedure is to estimate the posterior probability of the parameters of the
signal given the observed data p(?|d); this probability is given by Bayes? theorem as
| p(?)p(d|?)p(? d) = , (1.23)
p(d)
where p(d|?) is the likelihood of the data given the parameters, p(?) is the prior
probability on the model parameters, and p(d) is the evidence for the data. I detail
the significance of each of these terms below.
Again assuming stationary Gaussian noise, the likelihood can be inferred di-
rectly from Eq. (1.18): if the total signal is modeled by d = h(?) + n, then this
25
equation becomes [ ]
1
p(d|?) = N exp ? (d? h(?)|d? h(?)) . (1.24)
2
Note that this is the likelihood corresponding to detection in a single interferometer,
but under the assumption that the noise in each detector in a network is uncorrelated
with the others?, the joint likelihood is simply the product of the likelihood in each
detector.
The prior probability represents the a priori expectation for the distribution
of the model parameters. The particular choice of prior distribution becomes less
relevant if the likelihood is much more sharply peaked with respect to ?; in this limit,
the prior can be treated as approximately constant, and thus only contributes to the
normalization of the posterior. This situation arises for GW signals with high SNR:
if the data corresponds to a loud GW signal with parameters ??, i.e. d = h(??) + n,
then the likelihood take[s the approximate form
1 ( )]
p(d|?) = N exp ? ?ij(? ? ??)i(? ? ??)j ? 1 +O((SNR)?1 , (1.25)
( ) 2
where ? = ?h ?hij i | j is the Fisher information matrix evaluated at ??. The pos-?? ??
terior probab?ility distribution is well approximated by the likelihood provided that
|??p(??)| ? | ??1ij |  1, i.e. the prior varies more slowly than the likelihood near
its peak.
Finally, the evidence p(d) does not meaningfully inform the posterior probability?
given the prior and the likelihood, the evidence simply represents the overall normal-
ization of the posterior distribution (which must integrate to unity to be a proper
probability distribution). However, the evidence is a key component in Bayesian
26
model selection: given two competing hypotheses (or models) H1 and H2, the odds
ratio between the hypotheses is given by
OH p(H1|d) p(H1) p(d|H1)1H = H | = H , (1.26)2 p( 2 d) p( 2) p(d|H2)
where the last term is the Bayes factor, i.e. the ratio of the evidence for each model.
In principle, the posterior distribution can be computed directly from Eq. (1.23)
(up to normalization) over the full parameter space of ? given a particular set of
priors, waveform model h(?) and data stream d. In practice, this direct approach is
infeasible due to the high dimensionality of the parameter space and the computa-
tional expense of evaluating the likelihood function Eq. (1.24). Instead, stochastic
sampling methods are used to efficiently and adaptively choose points in the pa-
rameter space for which to evaluate the likelihood and reconstruct the posterior
distribution. The two methods employed in this thesis are Markov chain Monte
Carlo (MCMC) and nested sampling. The MCMC methods employ the Metropolis-
Hastings algorithm [75, 76]: a Markov chain of points in parameter space is con-
structed through a sequence of jump proposals such that the stationary distribution
of the chain matches the posterior probability distribution p(?|d). The acceptance
or rejection of each proposal (which determines whether the next element of the
chain is the proposed point or current point, respectively) is decided randomly with
a probability related to the relative posterior probability of the proposed and current
points. Nested sampling uses the algorithm of Skilling [77] to estimate a sequence
of likelihood contours and the prior volume within each by picking points in pa-
rameter space with progressively higher likelihood. Then, the posterior distribution
27
is reconstructed by re-sampling this sequence of points according to their posterior
probabilities. In conjunction with the LSC Algorithm Library (LAL), I use the
LALInference [78] software package to perform parameter estimation with either of
the sampling techniques mentioned above.
1.3 Testing general relativity with gravitational waves
One of the primary goals of GW science is to probe the fundamental nature of
gravity and test whether it is best described by GR. I begin this section by presenting
theoretical motivations for undertaking this endeavor. I then discuss the form in
which modifications to GR could appear, both at the fundamental level (e.g. in the
action) and as differences in GW phenomenology. Finally, I outline some strategies
used to place constraints on deviations from GR with GWs.
1.3.1 Motivations for modifying general relativity
General relativity has passed every experimental test of the past hundred
years [79]. These tests convincingly validate the predictions of GR over scales of
1?m . ` . 1AU. Yet, there remain several compelling reasons to continue to
explore alternative theories that modify GR on different scales.
As discussed in Sec. 1.1, singularities naturally arise in GR. In addition to
the case of gravitational collapse mentioned earlier, singularities also occur in the
early-time limit of many cosmological models. Singularities point to a deficiency
in the theory; GR becomes non-predictive for the causal future of a singularity.
28
However, by appropriately modifying the Einstein-Hilbert action, one can resolve
the interior singularity of BHs [80] or the early-time singularity in cosmological
spacetimes [81, 82].
Ignorance of the fundamental physics underpinning current cosmological mod-
els provides another reason to consider modifications to GR. The ?CDM model re-
mains the most successful cosmological model to date, yet it posits that the majority
of the energy in the Universe today is comprised of dark energy and dark matter.
While many proposals beyond the Standard Model have been put forth to explain
these dark sectors, none have been confirmed to date. Moreover, the ?CDM model
seems to require very precise fine-tuning: the value of the cosmological constant ?
consistent with cosmological observations differs from the vacuum energy expected
via dimensional analysis by 120 orders of magnitude. As an alternative to extend-
ing the Standard Model, modified theories of gravity can also explain some of the
phenomena for which these dark sectors are invoked.
Finally, GR is a strictly classical theory, whereas a complete theory of every-
thing will presumably also incorporate quantum effects into gravity. In principle,
deviations from GR could persist even as one takes the classical limit of such a theory.
For example, in string theory?the most well-known quantum-gravity theory?the
scalar dilaton naturally accompanies the tensor graviton; this additional field could
(in principle) play a role in the classical limit of the theory. Reversing this line of
reasoning, classical modifications to GR could potentially make the theory easier to
quantize. For example, adding higher-order curvature terms to GR makes the the-
ory renormalizable [83]; though this property is not strictly necessary to construct
29
a quantum theory of gravity, it would make the quantization of the classical theory
much more straightforward.
In this thesis, I primarily focus on potential deviations from relativity in the
?strong-field? regime, defined as where the (dimensionless) Newtonian potential
?Newt ? M/R and (dimensionful) curvature scale ? ? M/R3 are both large. Cor-
rections in this regime can potentially alleviate each of the issues discussed above,
either directly (e.g. to resolve singularities or improve renormalizability) or indi-
rectly (e.g. via corrections to behavior on cosmological scales induced by deviations
on much shorter distance scales).
1.3.2 The menagerie of modified gravity
As outlined in Sec. 1.1, GR directly embodies the equivalence principle. How-
ever, it is not the only possible theory of gravity to do so; the same fundamen-
tal tenants are encoded into all metric theories of gravity?theories in which the
non-gravitational fields ? couple minimally to the metric. Yet, out of all possible
metric theories, GR remains exceptionally simple. This sentiment is made more pre-
cise through Lovelock?s theorem [84, 85]: the Einstein field equations are the only
second-order field equations that respect the equivalence principle and whose ac-
tion is diffeomorphism-invariant, local, and dependent only on the four-dimensional
metric.
Lovelock?s theorem singles out GR as the unique metric theory that satisfies
all of the aforementioned conditions. Conversely, all alternative theories of gravity
30
must violate at least one of these requirements. From this perspective, each of the
conditions of Lovelock?s theorem can be viewed as a fundamental pillar on which GR
is based; if any of these pillars are removed, GR is no longer the only possible gravity
theory. In the remainder of this subsection, I discuss examples of alternative theories
that are allowed by relaxing each of these conditions in turn, focusing specifically on
the modifications to GR that impact the GW signal observable with ground-based
detectors. However, I restrict my attention to metric theories (which satisfy the
equivalence principle by construction) and ignore the possibility of field equations
containing derivatives higher than second order, as these would generally manifest
an Ostrogradsky instability [86]. The remaining pillars that one can relax are: (i)
gravity is mediated solely by the metric, (ii) gravity is diffeomorphism invariant, (iii)
gravity is local, and (iv) spacetime is four-dimensional. For brevity, I discuss only
one or two alternative theories of gravity for each condition to highlight the new GW
phenomenology that can occur outside of GR; for a more representative survey of
the landscape of modified gravity, see Refs. [79, 87?89]. Because the phenomenology
manifested in higher-dimensional theories (iv) can be found in the other alternatives
I present, I will not discuss this final type of modification in detail.
1.3.2.1 Additional fields
Perhaps the simplest way to extend GR is to include additional fields that
mediate gravity. In a metric theory, these new fields cannot couple directly to
matter, as this would prevent the universal coupling of the non-gravitational sector
31
to the metric. The prototypical example of such an extension are scalar-tensor (ST)
theories, which include a scalar field non-minimally coupled to the metric. The
action in such theories ?is given by
1 ?
SST[g
4
?? , ?, ?] = d x ?g e? (R? ?(?)g????????? U(?)) + Sm[g?? , ?],
16?
(1.27)
where I have restricted my attention to the case of zero cosmological constant (which
I continue to do for the remainder of the thesis). Written this way, the scalar field
takes the same form as a dilaton, which arises in many theories of quantum gravity,
e.g. string theory. ST theories are used to incorporate modifications to GR in both
the strong- and weak-gravity regimes. I work within the former class of theories
in several places in this thesis; see Chapters 3-7. For weak-field (i.e. cosmological)
modifications, ST theories are often ex?pressed instead as f(R) theories, whose action
1 ?
Sf(R)[g?? , ?] = d
4x ?gf(R) + Sm[g?? , ?]. (1.28)
16?
can be mapped to Eq. (1.27) via e? = f ?(R), ?(?) = 0, and U(?) = e?R? f(R).
The class of theories described by Eq. (1.27) is actually a very restricted subset
of viable scalar extensions of GR. Horndeski theories [90] cover all possible alterna-
tive theories with a single scalar field whose field equations are second order, so as
to avoid Ostragradsky instabilities. More recently, an even larger classes of ?beyond
Horndeski? theories have been established [91?93] whose equations of motion are
higher than second order, but that remain free of ghosts.
The GWs produced from compact binary systems in ST theories differ from
those produced in GR in a number of ways. First, the orbital evolution of a binary
32
is impacted by the introduction of the new scalar degree of freedom. The most
dramatic change occurs when the Compton wavelength of the scalar is significantly
larger than the separation of the binary; in this case, the binary can emit dipole
radiation, sourced by a time-varying scalar dipole moment. Recall that dipole radi-
ation is prohibited in GR and, from dimensional analysis, expected to scale with a
weaker power of 1/c (i.e. enter at lower PN order) than quadrupole radiation. Thus,
binaries with non-zero scalar dipole moment can emit significantly more energy and
thus evolve more rapidly than the equivalent system in GR, particularly at large
orbital separations. If the Compton wavelength of the scalar is much smaller than
the orbital separation, then the field will be effectively screened away through a
Yukawa-like potential [94] such that the dynamics much more closely resemble GR
(e.g. no dipole radiation will be emitted). Nevertheless, even in this case, the struc-
ture of the compact objects in the binary can significantly differ from GR, which
can offer another signature of deviations; for example, in massive ST theories, the
mass [95] and tidal deformability [96] of neutron stars can differ from the equivalent
star in GR, which would be imprinted in the GWs produced.
Besides modifications to their source, GWs in ST theories also contain addi-
tional polarization content. Beyond the two transverse-traceless polarizations modes
found in GR, ST theories predict a third transverse breathing-mode that is isotropic
in the plane orthogonal to its wave vector; this new mode corresponds to a helicity-0
state of the graviton. Massive ST theories contain an additional fourth, longitudi-
nal polarization [97]. With just a single detector, the polarization content of a
GW signal is degenerate with the wave?s direction of propagation (or equivalently,
33
the relative orientation of the detector and the source), and thus cannot be deter-
mined. However, with a network of N non-coaligned detectors, one can, in principle,
discriminate between N polarization states of the incident wave.
Scalar-tensor theories are a standard first example of modified gravity because
they predict a wide variety a phenomena that are relatively easy to compute. How-
ever, no-hair theorems guarantee that isolated BHs are no different in ST theories
than in GR (although there may be certain astrophysical contexts in which these
theorems can be circumvented [98]). Yet, there exist other extensions of gravity in
which BHs do behave differently than in GR. For completeness, I discuss one such
example: Einstein-?ther (EA) theory. This theory allows for breaking of the local
boost symmetry of GR (i.e. violations of Lorentz invariance) by introducing a unit
vector u? known as the ?ther field that represents a preferred timelike direction.
This is a particularly compelling alternative theory because it has recently been
shown to have a well-posed Cauchy formulation [99].
The action for EA theory is given by
?
1 ?
S 4EA = d x ?g [R + L ?u + ?(u u? + 1)] + Sm[g?? , ?], (1.29)
16?
where
L ??K?? ? u?? u?u ?? ? ? , (1.30)
K?? ?c g??g + c ???? + c ???? ? c u? ??? 1 ?? 2 ? ? 3 ? ? 4 u g??, (1.31)
and u? is constrained to be a unit timelike vector through the Lagrange multiplier
?. If u? is hypersurface orthogonal, then it can be rewritten as the normal vector
34
?
to a preferred time-foliation ? of the spacetime, i.e. u? = ???/ (??????), where
? = const are spacelike hypersurfaces. In this special case, EA theory coincides with
Khronometric theory [100], which is the low-energy limit of Hor?ava gravity [101],
a proposed theory of quantum gravity more amenable to standard quantum field
theory techniques than GR.
As with ST theories, the introduction of a new field in EA theory impacts
the evolution of compact binary systems and the propagation of the GWs that they
emit. There exist five propagating modes in EA theory: the two standard tensor
modes found in GR, two transverse vector modes and a single scalar mode [102]. The
existence of scalar and vector modes allows for the emission of dipole radiation. The
amount of dipole radiation emitted by a binary is determined by the relative motion
of the binary and the ?ther field and the structure of each body. In the limit that
each body moves slowly relative to the ?ther and at leading PN order, the emission
of scalar and vector radiation is determined by the so-called ?sensitivity? of each
body, which depends on the composition of that object. While the sensitivities
of spherically symmetric NSs are known [103], the analogous computation for BHs
has not been undertaken in EA theory4. Nevertheless, spherically symmetric BH
solutions have been constructed in EA theory [105], and their sensitivity is known
to be non-zero [106].
4Note that the calculation of BH sensitivities has been recently computed in Khronometric
theory [104], which corresponds to EA theory for a restricted set of ?ther field configurations.
35
1.3.2.2 Broken diffeomorphism invariance
Massive gravity is a commonly studied alternative to GR that violates diffeo-
morphism invariance. A fully non-linear theory of massive gravity free of ghosts was
derived fairly recently by de Rham, Gabadadze, and Tolley (dRGT) [107, 108]. The
action for dRGT massive gravity is[given schematically by? ]? ?41 4 1S 2 ?dRGT[g?? , ?] = d x ?g R + mg ?nLn(K ?) + Sm[g?? , ?], (1.32)16? 2
n=0
where ?n are constant coefficients, mg is the mass of the graviton, Ln[X?? ] is a
?
function containing n contractions of X, and K? ? ?? ? g??? ? f?? where f?? is a
non-dynamical, reference metric; see Ref. [108] for more detail. The appearance of
the reference metric breaks the diffeomorphism invariance of the theory.
A massive graviton has five possible polarization states, as is expected for a
massive, spin-2 particle. In addition to the two helicity-2 transverse traceless modes
also present in GR, the theory predicts a helicity-0 transverse breathing mode and
two spin-1 longitudinal modes [109]. However, the two helicity-1 modes do not
couple directly to matter, and thus are not expected to be produced through bi-
nary coalescence, and the remaining helicity-0 mode is suppressed by Vainshtein
screening [110] in the presence of matter (e.g. within the dark matter halo of a
galaxy) [109]; hence, GWs observed on Earth will typically be of only the polar-
izations expected in GR. However, the presence of non-zero graviton mass causes
these GWs to propagate subluminally. This modification of the dispersion relation
induces a frequency-dependent shift in the overall phase of the incident GW [111].
36
Constraining this deviation from the signal predicted in GR allows one to constrain
the mass of the graviton using GWs; I discuss this type of test in more detail below.
1.3.2.3 Non-locality
Adding non-local corrections to GR can both help resolve singularities that
appear in solutions of collapsing matter or evolving cosmologies and potentially ex-
plain the accelerated expansion of our universe. For illustrative purposes, I consider
the theory proposed in Ref. [112], where non-local corrections appear over long dis-
tance scales; these terms could arise as corrections from quantum gravity in the
infrared regime. The action for this theory is given by
? [ ]
1 ?4 ? ? 1
(
2 ? )1 2Snon-local = d x g R m R 2 R , (1.33)
16? 6
where m represents the mass scale of new physics. Cosmological solutions in this
theory have been constructed with self-accelerating expansion in which the effective
dark energy is generated by these non-local terms [112]. These solutions are consis-
tent with current Solar System and cosmological observations [113], indicating that
this theory provides a compelling alternative to the current ?CDM paradigm.
The primary GW signature of this theory is associated with the propagation
of GWs. The non-local terms in Eq. (1.33) dynamically generate a mass for the
conformal mode of the metric; however this mode is non-radiative, and does not
comprise part of the observable GW signal generated in this theory. In fact, unlike
the massive gravity theory discussed in the previous subsection, the only propagating
degrees of freedom in this non-local theory are two spin-2 modes that propagate at
37
the speed of light (as in GR) [114]. However, unlike GR, the damping of these modes
as they propagate over cosmological distances differs from that in GR [115].
The amplitude of GWs is inversely proportional to the luminosity distance to
the source dL; this quantity reduces to the distance d in Eq. (1.10) in the special
case that the background spacetime is Minkowskian. However, in more generic
cosmological backgrounds, the luminosity distance will depend on the redshift of the
signal z observed today due to the cosmological evolution of the Universe. In GR,
the luminosity distance for GWs takes the same familiar form as for electromagnetic
waves; for a spatially flat, Friedmann?Lemaitre?Robertson?Walker universe with an
evolving scale factor a(t), I have
?
1 + z z ? dz??0dL(z) = , (1.34)
H0 ? (1 + z?)40 R + ?M(1 + z?)3 + ?DE(z)
where H0 ? a?0/a0 is the Hubble constant (measured today), ?R, ?M , ?DE are the
density of radiation, matter, and dark energy, respectively, and ? 20 ? 3H0/(8?) is the
critical density needed to ensure a flat spatial metric. However, in the non-local ex-
tension of GR considered here, the luminosity distance for GWs no longer takes this
form, and so discrepancies can emerge between the luminosity distance recovered
with the electromagnetic signal dEML and with the GW signal d
GW
L from the same
source. This type of analysis has been carried out using the coincident electromag-
netic and GW observations of GW170817 [116]; however, only weak bounds on this
theory could be placed due to the poor measurement of H0 from those observations.
38
1.3.3 Tests of gravity with gravitational waves
1.3.3.1 Testing various facets of gravitational wave signals
The previous subsection outlined the range of phenomenology that can arise in
modified theories of gravity. In this section, I discuss how to formulate tests of GR
given these predictions. I detail three classes of tests, corresponding to various types
of deviations from GR that could be imprinted on the GW signal from a compact
binary inspiral. I first consider deviations to the GW generation by the system,
e.g. modifications to orbital motion of the binary. Next, I formulate tests of GW
propagation. Finally, I discuss tests of the GW polarizations present in an emitted
signal. Though many alternative theories of gravity predict deviations from GR in
more than one of these categories, it is often convenient to consider these various
phenomena independently.
As we have seen in the previous sections, the inspiral portion of the an ob-
served GW signal contains a direct imprint of the orbital motion of the binary that
sources it. Accordingly, deviations to the binary?s evolution due to modifications
to GR impact the inspiral portion of the GW. To measure (or constrain) these de-
viations, one performs Bayesian inference employing the techniques of Sec. 1.2.2.2.
This approach relies on a waveform model to reproduce the observed GW signal;
however, to capture potential deviations from GR, this model must be ?generalized?
to include parameterized deviations away from the GR prediction. The additional
modifications are controlled through a set of deviation parameters {???n}. Though
39
there are several possible ways in which to introduce parameterized deviations to
a GR signal, a common choice for tests of the inspiral evolution is to consider in-
dependent modifications of each series coefficient in a PN-expanded prediction for
the signal [73, 117?123]?in this case, each ???n represents the extent of deviation
from the GR prediction at a particular PN order. This particular approach for
constructing generalized waveform models can be easily related to the predictions
of many alternative theories of gravity; provided that an alternative theory admits
a low-velocity expansion and that deviations from GR are small, then the signal
predicted in this alternative theory can be mapped to the model outlined above.
Then, using Bayesian inference, one attempts to measure/constrain the deviation
parameters in this model for a given GW observation to determine whether that
signal is consistent with the predictions of GR (i.e. consistent with no deviations
in the waveform model). Chapter 9 focuses on this type of test using observations
made by LIGO and Virgo.
Tests of GW propagation can be designed in much the same way as tests
of GW generation: modifications to the propagation of GWs are introduced to
a GR waveform model through a set of parameterized deviations. For example,
Refs. [9, 10, 111, 122, 124?126] considered corrections to the graviton dispersion
relation in GR, which, in the Fourier domain, is simply E2 = p2, where E and p are
the temporal and spatial components of the wavevector of a propagating plane wave.
These references considered deviations that were polynomial in |p|; as before the
deviation parameters represent the magnitude of the correction at each polynomial
order. Modifications to the dispersion relation modify the phase evolution of the
40
GW signal (see Ref. [126]), and these deviations can be constrained in a similar
manner to deviations to GW generation discussed above. Other types of tests of GW
propagation are possible with coincident observations of electromagnetic and GW
signals from a single source. For example, simply computing the difference in time
of arrival between ?-rays and GWs produced by the BNS coalescence GW170817
provided strong constraints on many alternative theories of gravity [127?134].
Tests of GW polarizations differ significantly from those of GW generation or
propagation. They rely on the fact that the directional sensitivity of a given detector
differs for each possible polarization. The directional dependence of the sensitivity
of interferometer I to a GW of polarization A incident from a particular position
(A)
in the sky ? can be expressed as an antenna pattern F(I) (?). With only a single
detector, the polarization and amplitude of a GW signal are completely degenerate;
it requires a network of detectors to disentangle these quantities. If the sky location
of the source of GWs is known (e.g. from the a coincident electromagnetic obser-
vation), then one approach to identify non-GR polarization content in the signal
employs so-called null streams [135]. With a network of three or more independent
detectors, one can project out the tensorial (GR) modes present in the data stream
from each interferometer via straightforward algebraic manipulation. Then, if any
coherent signal remains across the resulting null streams, this indicates the presence
of non-GR polarization content in the signal. An alternative approach uses Bayesian
model selection (as described in Section 1.2.2.2) to differentiate between different
polarizations [136]. One considers the relative odds that an observed GW can be
described by polarization A versus polarization B by computing the Bayesian evi-
41
dence in support of the following hypotheses: the GW signal at detector I is given
(A) (A)ij (B) (B)ij
by h(I) = F(I) hij or the GW signal at detector I is given by h(I) = F(I) hij.
Unlike the null stream approach, this method requires some waveform model hij
for the incident GW; in Refs. [9, 122], this analysis was conducted using waveforms
constructed with a morphology-independent sum of sine-Gaussians [71, 137] to con-
strain the polarization content in observed GW signals. However, note that, at
present, the hypotheses used in this test are somewhat artificial; GW sources gener-
ate multiple polarizations of GWs in most modified theories of gravity, whereas the
hypotheses assumed here assume the entire signal is of only one polarization.
1.3.3.2 Designing gravitational-wave tests of general relativity
Tests of GR are commonly classified into one of two contrasting categories:
theory-specific and theory-agnostic tests [88] (or equivalently, direct and parame-
terized tests [89]). These two types of tests differ conceptually in the underlying
assumptions they require. Theory-specific tests are based on the supposition that
a specific theory (or set of theories) correctly describes the classical behavior of
gravity, and then use GW observations to better inform our understanding of these
theories. In contrast, theory-agnostic tests do not rely wholly on any particular the-
ory of gravity, but rather consider the predictions in a parameterized neighborhood
of theories around GR.
The choice of theory-specific or theory-agnostic approach guides the ultimate
design of GW tests (such as those outlined in the previous subsection) one performs.
42
For concreteness, let us consider a specific example: suppose I wish to test the inspi-
ral dynamics (GW generation) in a observed signal produced by a coalescing BNS.
A theory-specific test would consider a specific class of theories?for example, ST
theories?in which the GW signal from such a system would differ from the predic-
tions of GR. The dominant non-GR effect in such theories arise from the emission
of dipole radiation, which contributes a -1PN correction to the phase evolution of
the frequency-domain GW signal. The magnitude of this effect is governed by a
set of theory parameters, which, in the case of ST theories, simply characterize the
functions ?(?) and U(?) in Eq. (1.27). In fact, if I assume that U(?) = 0, then the
dominant non-GR effect is determined only by the constant ?0?the first term in
the PN expansion of ?(?) = ?0 + ?
?
0? + . . .. Thus, a theory-specific test could use
a waveform model generalized to incorporate this -1PN effect (as characterized by
?0) in hopes of measuring this fundamental parameter of the assumed theory. Or,
one could use Bayesian model selection to compare the relative odds that this ST
theory is favored/disfavored versus a GR model.
In contrast, a theory-agnostic test of similar deviations could be used to con-
strain generic -1PN deviations to the GW signal produced by a BNS in GR. Again,
one must use a waveform model generalized from a GR prediction to include the
possibility of such a -1PN effect. However, unlike in the theory-specific case, the
magnitude of this correction would not be regulated by any fundamental constant of
a theory; instead, one could simply introduce a generic deviation parameter ??? that
represents a free coefficient that dictates the size of this -1PN term. One could then
employ many of the same statistical techniques in hopes of measuring/constraining
43
???. The information inferred on this deviation parameter offers a measure of con-
sistency between the observed data and the predictions of GR.
As this example illustrates, the actual procedure for carrying out theory-
specific and theory-agnostic tests can be remarkably similar. And yet, the statistical
questions addressed in each of the outlined tests are fundamentally different. Within
a Bayesian framework, this distinction is inherently manifested in the choice of prior
distribution [p(?) in Eq. (1.23)] one employs. In fact, as long there exists a bijection
relating ?0 to ???, one can convert the posterior distribution measured on either
quantity (in the theory-specific or theory-agnostic test outlined above, respectively)
to a posterior on the other; then all differences in the two results stem from the
corresponding choices of priors.
A complementary dichotomy distinguishing different types of GW tests of GR
are the notions of top-down and bottom-up designs [88].5 A top-down approach be-
gins from some fundamental physical principle, e.g. an action or set of symmetries,
and then translates this foundation to a set of observables that can be tested against
data. Conversely, a bottom-up approach starts from some phenomenological model
of observables, then, through comparison against data, infers properties about the
underlying physics at play. The two example tests laid out above would be classi-
fied as top-down (starting from a particular action) and bottom-up (starting from
a phenomenological feature in the GW signal), respectively. However, these two
5Reference [88] presents theory-specific and top-down tests as one and the same, and similarly
with theory-agnostic and bottom-up tests. As laid out in this subsection, I believe that two
categorizations are largely orthogonal, not redundant.
44
Top-Down
9.4 2 4 5 3
9.5
6
Theory-Specific 7 Theory-Agnostic
8
9.3
Bottom-Up
Figure 1.1: Schematic representation of how the contents of each chap-
ter could be categorized as theory-specific vs. theory-agnostic and top-
down vs. bottom-up tests of GR. For chapters for which additional work
is required to formulate a test of GR (e.g. additional PN computations
are needed), I assume the corresponding missing pieces can be completed
with similar techniques as existing work.
45
dichotomies are independent, e.g. one can design a top-down, theory-agnostic test.
To help guide the reader, I present a schematic representation in Fig. 1.1 of how
the various chapters of this thesis (discussed in detail below) would be classified
according to these two categorizations.
1.4 Overview of this thesis
The remainder of the thesis is organized so as to loosely follow the spec-
trum from theory-specific to theory-agnostic tests of gravity using GW observations
of binary inspirals. I begin in Chapters 2 and 3 with perturbative calculations
of waveform models in particular classes of alternative theories of gravity, which
could serve as the foundation for theory-specific tests [1, 2]. In Chapters 4-8, I
turn to models of specific phenomenology that can occur in a range of potential
modifications of GR. Chapters 4-6 contend with a particular phenomenon known
as scalarization that occurs in several alternative theories, culminating in a theory-
agnostic analytic model of the phenomenon [3?5]. Chapter 7 demonstrates how such
a model could be employed to test GR, focusing on a specific alternative theory of
gravity [6]. Chapter 8 examines the potential to test gravity with an even more
widespread phenomenon?the tidal deformations of the compact bodies in inspiral-
ing binaries?using various boson star models as benchmarks to determine whether
exotic compact objects could be distinguished from BHs or NSs [7]. Chapter 9 con-
tends with an even broader range of phenomenology, developing an infrastructure
to allow parameterized deviations to be incorporated into the inspiral phase of a
46
generic waveform model and then using this construction to perform various tests
of GR with existing GW observations by LIGO and Virgo. This chapter discusses
theory-agnostic tests performed as part of the LIGO Scientific Collaboration [9, 10]
as well as tests of specific alternative theories of gravity [11, 138]. Finally, Chapter
10 provides some concluding remarks and possible directions for future work.
I briefly summarize the findings documented in each chapter below, highlight-
ing certain specific results and figures that reappear later.
1.4.1 Post-Newtonian and effective-one-body descriptions of binary
black holes in Einstein-Maxwell-dilaton gravity
In Chapter 2, I study the inspiral of BBHs in Einstein-Maxwell-dilaton grav-
ity (EMd) [1]. In this theory, gravity contains an additional scalar component (a
dilaton), which is sourced by electromagnetism. Due to its simplicity, this is a conve-
nient toy model for generic modified theories in which the no-hair theorems of GR no
longer hold: the theory is characterized by a single parameter a, which determines
the coupling strength between the dilaton and Maxwell field. In EMd specifically,
electrically charged BHs also host an additional scalar charge absent in GR. In this
work, I examine BBHs with electric (and scalar) charge, and contrast their behavior
with electrically charged binaries in GR. Though astrophysical BHs are expected
to be electrically neutral6, EMd serves as a useful proxy for more generic no-hair
violations. This work provides a foundation for modeling and testing more generic
6Certain dark matter proposals have been made that would allow BHs to attain significant
electric charge [139?144], which would allow for viable tests of EMd.
47
such phenomena using GW detections of BBHs.
Working within the PN approximation, I compute the dynamics and energy
flux of BBHs at next-to-leading order. I use these results to compute ready-to-use
Fourier-domain waveforms for the inspiral in EMd. The key feature that distin-
guishes charged BBHs in EMd from their counterparts in GR is the emission of
scalar dipole radiation (discussed in Sec. 1.2). I study in detail the impact of this
additional energy flux to the gravitational waveform, relative to both charged and
uncharged BBHs in GR. Figure 1.2 depicts the projected constraints that can be set
on the EMd coupling parameter a given a GW measurement of dipole flux consis-
tent with the current best dipole constraint from X-ray binary systems [145]. Here I
parameterize the dipole emission as a dipole-flux ratio B relative to the quadrupolar
flux predicted in GR FGR
(
F = F 1 +Bv?
)
2
GR , (1.35)
Current measurements of low-mass X-ray binaries provide a constraint of |B| .
2?10?3 [145] and combined bounds from BBH events observed during the first and
second observing runs of LIGO and Virgo constrain |B| . 3 ? 10?3 [10], so I use
|B| . 10?3 as a rough benchmark for my analysis.
As discussed in Section 1.2, the PN approximation becomes increasingly in-
accurate towards the end of the inspiral of BBHs, and more sophisticated tools are
required to model the waveform (and thus test the nature of gravity) beyond this
phase. Following the analogous approach used in GR, I construct an EOB formal-
ism for binaries in EMd to resum the PN results discussed above. This extends
48
50
40
30
20
10
0
0.0 0.5 1.0 1.5
Figure 1.2: Projected constraints on EMd from dipole flux measure-
ments with GW detectors. Allowed values of EMd coupling a consistent
with a dipole flux ratio of |B| ? 10?3 are indicated by the shaded regions
as a function of mass-weighted total electric charge. Colors indicate var-
ious possible electric dipoles consistent with the bound on B. Figure
taken from Ref. [1].
the validity of the EMd model to a more relativistic regime and more accurately
reproduces the test-particle limit. I explore the gauge freedom inherent to any EOB
construction, examining two possible generalizations of the gauge used in GR models
for EMd. This discussion is invaluable as one explores potential EOB formulations
of more generic alternative theories of gravity.
1.4.2 Post-Newtonian waveforms in massless scalar-tensor theories
In Chapter 3, I consider the inspiral dynamics in a broader class of scalar
extensions of GR [2]. I focus on a popular class of ST theories in which an additional
massless scalar field couples non-minimally to the metric [146, 147]. Unlike in EMd,
49
all matter couples (indirectly) to this scalar field, but isolated BHs do not because of
no-hair theorems [23, 148]. Thus, the class of theories considered in this chapter can
be tested through the GW observation of BNSs or neutron-star black-hole systems.
Using previously derived PN results [149?151], in this chapter I compute ready-
to-use frequency-domain inspiral waveforms for non-spinning, non-eccentric binary
systems. I use the most accurate (highest PN order) results known at the time
of writing, which correspond to 2PN relative order. As with many scalar exten-
sions of GR, the primary observational signatures of this theory arise from the
emission of scalar-dipole radiation. I discuss in detail two regimes of the inspiral:
the dipole-driven regime, which occurs at very low-frequencies where dipole radi-
ation dominates quadrupolar radiation, and the quadrupole-driven regime, which
occurs later in the inspiral where quadrupolar radiation is dominant. Given current
observational constraints, ground-based and space-based GW detectors are likely
only sensitive to binaries in the latter stage, but binary pulsar measurements could
potentially probe the former.
1.4.3 Dynamical scalarization as a non-perturbative phenomenon
Scalarization of compact objects (BHs and NSs) arises from spontaneous sym-
metry breaking of an additional scalar component of gravity [4, 152]. Scalariza-
tion can significantly alter the evolution of a binary system, hastening the inspiral
and leading to a dramatically different GW signal. Spontaneous scalarization?the
scalarization of a single, isolated object?has been found in several scalar exten-
50
sions of GR [147, 153?168]. In contrast, dynamical scalarization?scalarization that
occurs during the coalescence of a binary system?has been demonstrated only for
BNSs in ST theories [169?172]. Chapters 4 and 5 explicitly focus on these theories
known to exhibit dynamical scalarization, while Chapter 6 examines new theories
in which the phenomenon is likely to occur.
In Chapter 4, I show that dynamical scalarization is a non-perturbative phe-
nomenon in the sense that its appearance coincides with the breakdown of the PN
approximation [3]. A similar result was known already for spontaneous scalariza-
tion [153]; thus, my work places the two phenomena on equal footing. I develop a
diagnostic to test the convergence of the PN expansion in the context of scalariza-
tion, which in turn, can be used to estimate the onset of dynamical scalarization.
To model dynamical scalarization, I construct a semi-analytic method that
resums the PN expansion at the level of the action?I denote this resummation as
the post-Dickean (PD) expansion. I perform this resummation by introducing a new
dynamical variable to the Lagrangian of the theory that tracks the non-perturbative
growth of the scalar field near each compact body. Working at next-to-leading order
(1PD, analogous to 1PN), I compute the early-inspiral dynamics of BNSs before and
after dynamical scalarization.
An example of such a result is shown in Fig. 1.3; I plot the scalar mass MS?a
measure of scalarization?of a BNS as a function of orbital and GW frequency. The
blue and green curves depict two models of dynamical scalarization constructed in
Chapter 4. I compare these semi-analytic results against previous phenomenological
models [173] (black with pink shading) and quasi-equilibrium configuration (QE)
51
fGW [Hz] fGW [Hz]
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200
10-5 ?? ? ? ? ? ? ?
? ?
?? 1PD (
? ? ? ? ?( ) (?? )) 10-5 ? ? ? ? ? ? ??(m RJ1PD m( , F ?RE), F(?? )) ?
10-6 ? Palenzuela et al 10
-6
? 1PN MS+2PN EOM
10-7 ???
QE
10-7
?? ??
10-8 10- ?
?
8
???? ??? ?
101 101 
100 100 
10-1 10-1
10-2 10-2
0.00 0.01 0.02 ?/ 0.03 0.04 0.05 0.00 0.01 0.02 ?/ 0.03 0.04 0.05GM c3 GM c3
Figure 1.3: Scalar mass Ms as a function of orbital frequency ? and
gravitational wave frequency fGW for a BNS that undergoes dynamical
scalarization. The blue and green curves correspond to two PD formu-
lations constructed in Chapter 4. These semi-analytic models reproduce
the numerical quasi-equilibrium (QE) configuration calculations of [174]
(dashed black) to within a few percent; the fractional error is shown in
the bottom panel. The PD models achieve comparable accuracy to the
phenomenological model of Ref. [173] (solid black curve with estimated
systematic uncertainty in pink). Because dynamical scalarization is a
non-perturbative phenomenon, it cannot be captured using only the PN
approximation, whose prediction is shown in red. Figure taken from
Ref. [3].
calculations (dashed black) made with NR [174]. The red dot-dashed curve shows
the PN prediction for the scalar mass of the system; the discrepancy between this
perturbative prediction and the QE results confirms that dynamical scalarization is
a non-perturbative phenomenon.
52
Fractional Error MS [M?]
Fractional Error MS [M?]
1.4.4 Dynamical scalarization as a second-order phase transition
Building on the observation in Chapter 4 that the onset of dynamical scalar-
ization coincides with the loss of analyticity of the PN expansion, in Chapter 5, I
aim to explain the physical origin behind the breakdown of the standard perturba-
tive approach. I show explicitly that dynamical scalarization is a second-order phase
transition [4]. Given this perspective, the breakdown of the PN expansion found
previously is unsurprising. Recall that the PN formalism provides an approximate
solution to the Einstein equation expanded perturbatively around a Newtonian so-
lution; in the context of the two-body problem, the evolution of a binary system is
formulated in terms of its behavior at very large separations (the Newtonian limit).
However, because dynamical scalarization is a second-order phase transition, the
order parameters that describe a binary system exhibit discontinuous derivatives at
the onset of scalarization. The PN expansion is blind to phenomenology that arises
spontaneously in the relativistic regime (e.g. a new scalarized phase), and thus can-
not sensibly reproduce dynamical scalarization. Note that despite the similarities
between spontaneous and dynamical scalarization, only the latter phenomenon poses
such an obstacle to the standard PN approach; in contrast, spontaneous scalariza-
tion remains present in a binary system even at very large separations, and thus can
be integrated seamlessly into the PN formalism.
Besides providing a better conceptual understanding of the phenomenon, the
connection between dynamical scalarization and phase transitions offers some in-
sight as to how to best construct analytic models of binary systems that undergo
53
Figure 1.4: The Hamiltonian H of an equal-mass BNS (vertical axis)
(2)
as function of scalar charge Q and effective parameter ceff = c
(2) ? 1/r.
(2)
When ceff becomes negative, stable equilibrium scalarized configurations
(Q 6= 0) emerge and unscalarized configurations (Q = 0) become un-
stable. The bottom lower plane shows the phase space of equilibrium
configurations, with the critical point separating the unscalarized and
scalarized phases indicated with the red point. Both H and Q are given
(2)
in units of M, whereas ceff is shown with units of M
?1
 . Figure adapted
from Ref. [4].
scalarization. In Chapter 5, I construct such a model for binaries in the ST theories
considered in Chapter 4. Like the approach adopted in Chapter 4, I introduce new
dynamical variables that track the scalarization of each body during the evolution
of the inspiral. Unlike the previous approach, which incorporated a large degree of
freedom in how these variables were defined, the construction in Chapter 5 explicitly
uses the scalar charge Q of each body as dynamical variables; these quantities are
a natural choice, as they also serve as order parameters characterizing the second-
order phase transition of dynamical scalarization.
54
With this approach, I construct an analytic model of dynamical scalarization
at leading (Newtonian) order. I compute the conservative dynamics of a binary sys-
tem at this order and calculate the Hamiltonian that governs its motion. This mea-
sure of energy allows us, for the first time, to estimate the stability of dynamically
scalarized systems and draw connections to the Landau model of phase transitions.
For example, Fig. 1.4 depicts the appearance of a Mexican-hat shaped potential
(characteristic of a second-order phase transition) during dynamical scalarization of
a binary system. In this diagram, the translucent sheet depicts the Hamiltonian
(vertical axis) of a BNS as a function of scalar charge Q and an effective parameter
(2)
ceff , defined as the difference between a coefficient c
(2) that depends on the structure
(2)
of the NSs and the inverse of their separation 1/r. At large distances (positive ceff ),
the energy is minimized in an unscalarized configuration (Q = 0). However, below
(2)
a critical separation (ceff = 0), the local minima shift to scalarized configurations
(Q =6 0), while the unscalarized configuration becomes unstable. These two phases
are indicated with the green and blue points in Fig. 1.4, respectively, with the crit-
ical point at which scalarization begins marked in red. The equilibrium solutions
are projected onto the bottom plane to depict the phase space of binary configu-
rations; the bifurcation of stable solutions at the critical point demonstrates the
discontinuous first derivative of the order parameter Q expected in a second-order
phase transition.
55
1.4.5 Theory-agnostic modeling of dynamical scalarization
Chapters 4 and 5 examined the conceptual and technical challenges of con-
structing an analytic model of dynamical scalarization, yet restricted their scope to
a narrow class of ST theories [153] in which scalarization has been previously been
shown to occur [169?172]. However, the recent discovery of spontaneously scalarized
BH solutions in other scalar extensions of GR [158?168] has renewed interest in such
non-perturbative phenomena in a broader context. Chapter 6 examines scalarization
(spontaneous and dynamical) from a theory-agnostic perspective that encompasses
all theories in which the phenomena are known to occur [5]. I show that any the-
ory that admits spontaneous scalarization must also admit dynamical scalarization.
Additionally, I establish a robust parameterization that completely characterizes
scalarization at Newtonian order that is suitable for searches for deviations from
GR with GW observations.
In keeping with this theory-agnostic direction, I begin by identifying the key
criteria needed for scalarization to occur around a compact object in any modified
theory of gravity: (i) the theory must admit a GR solution, i.e. one in which
the scalar field vanishes and (ii) this unscalarized solution must be unstable to
scalar perturbations. Working in the regime close to the critical point at which
the first scalar mode instability arises (i.e. near the onset of scalarization), I use
effective field theory (EFT) techniques to construct an effective point-particle action
describing each compact object in a binary system. Using this ansatz, I compute
the conservative dynamics governing the inspiral of binary systems at Newtonian
56
300 DS SS
200
100
0
300 DS SS
200
100
0
300 DS SS
200
100
0
0.2 0.4 0.6 0.8 1.0
Figure 1.5: The phase space of scalarization that occurs for BBHs in
EMS gravity. The parameter ? modulates the coupling strength between
the Maxwell and scalar fields; three choice are depicted here. Black holes
with substantial electric charge will spontaneously scalarize (SS) across
all orbital/GW frequencies, shown here with light shading. Below the
threshold for spontaneous scalarization, dynamical scalarization (DS)
can still occur, shown with darker shading. The frequency at which dy-
namical scalarization occurs is greater for less electrically-charged BBHs,
as depicted with the dark boundary of the shaded region. Figure adapted
from Ref. [5].
order; this calculation closely resembles the results of Chapter 5.
I demonstrate how this model can be applied to a generic alternative theory
using Einstein-Maxwell-scalar (EMS) gravity as a proxy. In EMS theory, BHs with
sufficient electric charge can spontaneously manifest scalar charge [158]. Starting
from sequences of static, spherically-symmetric solutions, I use this EFT approach
to reconstruct the effective action governing the dynamics of the critical scalar mode
that leads to scalarization; I find close agreement with independent calculations of
57
the mode dynamics from linear perturbation analyses of GR solutions. Then, I use
this effective action to explicitly predict the onset of dynamical scalarization across
the parameter space of EMS theory, shown in Fig. 1.5. This result represents the first
prediction of dynamical scalarization in a BBH, and because it is made for a theory
whose formulation in NR has already been established [158], numerical confirmation
would be quite straightforward. Though substantial effort is expended exploring the
full phase space of scalarization in EMS theory, I emphasize throughout Chapter
6 how this same procedure can be straightforwardly applied to other alternative
theories of gravity.
1.4.6 Projected constraints on scalarization by combining pulsar-
timing and gravitational-wave observations
Though GW detectors offer the best hope of detecting phenomenology like
dynamical scalarization, the best constraints on many theories that manifest such
effects come from radio observations of binary pulsars. In Chapter 7, I investigate the
constraints that pulsar timing can set on a popular ST theory that admits sponta-
neous and dynamical scalarization in NSs and determine the degree to which future
GW observations can improve these constraints [6]. Finally, I examine whether the
constraints set on this ST theory rule out the possibility of detecting dynamical
scalarization with GWs; I find that current bounds on the theory are unable to do
so.
I consider the specific class of ST theories introduced in Ref. [153], which was
58
also used in Chapters 4 and 5. The strongest bounds on this theory come from
binary NS-white dwarf (WD) systems. The relativistic densities required to trigger
scalarization can be found in NSs but not in WDs, and thus NS-WD systems can
potentially host larger scalar dipoles (and thus greater dipole radiation) than BNSs.
I place constraints on the ST theories of [153] using timing measurements of five
NS-WD systems; prior to this work, constraints had only been set using individual
binary pulsars.
Additionally, in this work, I examine the dependence of these constraints on
the NS equation of state (EOS), i.e. the internal structure of the NS. The constraints
that can be set from a single binary pulsar depend crucially on this unknown nuclear
physics; fortunately, by combining observations from several pulsars, this variance
can be mitigated. Figure 1.6 highlights some of the constraints set using this method:
I plot with colored lines the upper bounds on the dimensionless scalar dipole |??| of
a hypothetical BNS consistent with current binary pulsar measurements for different
possible EOSs. The system considered here has a fixed secondary mass of mA =
1.25M while the primary mass is allowed to vary.
After establishing the most stringent (at the time of writing) constraints from
binary pulsars, I examine the degree to which future GW detections could improve
these bounds. Working in the limit of large SNR, for which the posterior distribution
is well-approximated by Eq. (1.25), I estimate the precision with which current?
Advanced LIGO (aLIGO)?and planned?Cosmic Explorer (CE) and Einstein Tele-
scope (ET)?ground-based GW detectors can measure the scalar dipole of a BNS at
a luminosity distance of 200 Mpc. The bounds achievable with these detectors are
59
AP3 H4 MS2 WFF1
AP4 MPA1 PAL1 WFF2
ENG MS0 SLy4
10-1
aLIGO : 10Hz
10-2
CE : 5Hz
10-3
ET : 1Hz
10-4
1.3 1.6 1.9 2.2 2.5 2.8
mB/M?
Figure 1.6: (Colored lines): Upper bound on dimensionless scalar dipole
|??| at a 90% confidence level from binary pulsar observations as a
function of primary NS mass for several possible EOSs?the secondary
NS has a fixed mass of mA = 1.25M and the same EOS as the primary.
(Dashed lines): Projected statistical uncertainty for |??| measured for
a BNS at a luminosity distance of 200 Mpc for current and planned
ground-based GW detectors. Figure taken from Ref. [6].
depicted with dashed lines in Fig. 1.6. Because the theoretical values of scalar dipole
constrained by binary pulsars (colored lines) exceed the statistical uncertainty ex-
pected in such a measurement with GWs (dashed lines), I can conclude that current
detectors could potentially extend bounds for a narrow range of mass configurations,
but next-generation detectors will offer much broader improvements. Furthermore,
I investigate whether dynamical scalarization could be observed by GW detectors
given the theory constraints set by binary pulsars; detecting this phenomenon would
provide smoking-gun evidence in support of deviations from GR. Using an identi-
fication criterion established in the literature [175], I use the model constructed in
Chapter 4 to demonstrate several possible binary configurations for which dynamical
scalarization could potentially be detected with Advanced LIGO, thereby validating
60
max|??|  or  ?(|??|)
continued study and future searches for the phenomenon.
1.4.7 Tidal signature of boson stars in binary systems
In Chapter 8, I shift my attention from phenomenology that can only occur in
scalar extensions of GR to that which occurs in much broader contexts [7]. Specif-
ically, I explore how tidal interactions between inspiraling bodies can be used to
probe new physics. These interactions are directly linked to the structure of the
compact objects in the binary, and thus may be able to distinguish BHs and NSs
from other exotic alternatives in binary systems. During the adiabatic inspiral, the
dominant tidal interaction is characterized entirely by the tidal deformability ? of
each body, which describes the linear response of a spherical body?s quadrupole
moment Qij to an externally imposed quadrupolar tidal field Eij, i.e.
Q 5ij = M ?Eij, (1.36)
where the body?s mass M has been factored out so that ? is dimensionless.
In this chapter, I investigate whether tidal effects in a GW signal can differ-
entiate BHs and NSs from boson stars (BSs), a compelling toy model for exotic
compact objects. Boson stars are simply self-gravitating clouds of (classical) scalar
field. They are an exemplary class of exotic compact objects because they have been
shown to form dynamically through gravitational collapse and, by tuning the mass
? of the fundamental boson field, can be constructed with any astrophysical mass.
Various families of BSs have been postulated, each differentiated by the type of
scalar self-interaction they consider. In this thesis, I consider two common choices:
61
104
103
102
0 1 2 3 4
102
101
100
0 2 4 6 8 10 12
Figure 1.7: Tidal deformability ? for massive (top panel) and solitonic
(bottom panel) BSs as a function of mass M , rescaled by boson mass ?.
Colors represent various choices of self-interaction strength, given by ?
and ?0 for massive and solitonic BSs, respectively. Figure adapted from
Ref. [7].
62
massive BSs, characterized by a quartic interaction V (?) = ?2|?|2 + ?|?|4/2, and
solitonic BSs, with the interaction potential V (?) = ?2|?|2(1? 2|?|2/?20)2.
I begin by computing the tidal deformability of each family of BSs across a
broad range of parameter space. This procedure involves first constructing numer-
ical solutions for spherically-symmetric scalar configurations, and then solving the
first-order perturbation equations describing static, tidal deformations of these back-
grounds. Finally, the quantities of interest, i.e. the mass M and tidal deformability
?, are extracted from the asymptotic behavior of each of these solutions. Figure 1.7
presents the results of these calculations, in which the tidal deformability is shown
as a function of mass for various choice of self-interaction strength (depicted with
different colors) for massive (top panel) and solitonic (bottom panel) BSs.
I next turn to the question of whether GW detectors can measure tidal de-
formabilities with enough precision to differentiate these classes of BSs from BHs
or NSs. Similar to the approach in Chapter 7, I use the Fisher information matrix,
in conjunction with Eq. (1.25), to estimate the statistical uncertainty expected for
these tidal parameters for canonical systems with aLIGO, ET, and CE. I outline
two strategies for differentiating binary BHs and NSs from binary BSs with a GW
inspiral, the first relying on only a measurement of the tidal deformability of a sin-
gle body and the second combining measurements of each body?s deformability. I
find that massive BSs can be distinguished from BHs with current GW detectors,
but that future detectors would be needed to differentiate solitonic BSs and BHs.
Matters are slightly more complicated for the case of NSs, as they have their own
inherent (non-zero) tidal deformability; however even using current detectors, I find
63
that one can differentiate between BNSs and binary BSs.
1.4.8 Parameterized tests of general relativity with generic frequency-
domain waveform models
Chapter 9 introduces new tools to enable tests of GR with GW inspirals across
a variety of contexts. Here I focus on parameterized tests, that is, statistical tests
that aim to measure (or constrain) a set of ?deviation parameters? that quantify
discrepancies in an observed GW signal and the predictions of GR. These types
of tests rely on ?generalized? waveform models?models in which the parameter
space of a baseline GR waveform is extended to include potential deviations from
GR; these modifications are controlled by a set of deviation parameters. For ex-
ample, in a theory-specific test, these new parameters could represent the physical
constants, masses, couplings, etc. that define a particular theory that one aims to
measure. In contrast, for a theory-agnostic test, these parameters would describe
the phenomenological features in the waveform one wishes to constrain.
The chapter describes a new infrastructure for constructing generalized wave-
forms from any non-precessing7, frequency-domain GR baseline model. Given its
flexibility, this new framework is denoted as the flexible theory-agnostic (FTA) ap-
proach. Prior to the introduction of the FTA framework, parameterized tests of
GW inspirals only employed the small subset of waveform models with known, an-
7The restriction to non-processing waveform models is made for simplicity; extending the ap-
proach documented here to precessing models is relatively straightforward and represents a possible
direction for future work.
64
alytic representations in the frequency-domain (e.g. Refs. [73, 118, 120, 122, 176]).
With this new tool, I am now able to extend these tests to new waveform models,
and determine the systematic biases that arise due to the choice of underlying GR
waveform.
I begin Chapter 9 by detailing the construction of the FTA infrastructure and
the statistical framework in which it can be employed. Then, I use this approach
to perform three different tests of GR using GW observations made during the first
and second observing runs of LIGO and Virgo. Unlike in previous chapters, here
I carry out Bayesian inference using the full suite of statistical tools outlined in
Sec. 1.2.2.2.
First, as part of the LIGO Scientific Collaboration, I place theory-agnostic
constraints on phenomenological deviations to the inspiral phase from the catalog
of BBH [10] and BNS [9] observations made by LIGO and Virgo. All tests that I
perform indicate consistency with the predictions of GR. Additionally, I find good
agreement between the bounds recovered using a generalized waveform constructed
with the FTA infrastructure and those constructed with an earlier approach [120, 176]
with a different baseline GR model. This agreement suggests that no significant
systematic errors arise from the choice of waveform model, thereby solidifying the
claim that no deviations from GR are present.
Next, I use the FTA framework to place constraints on Jordan-Fierz-Brans-
Dicke gravity (JFBD) [177?179] from the BNS observation GW170817 [11]. This
ST theory represents one of the oldest and most well-known alternative relativistic
theories of gravity. Although the constraints I place are ultimately not competitive
65
with existing bounds from Solar System experiments, this analysis provides a useful
benchmark for GW tests of GR that can be easily understood by a large segment
of the physics community. Jordan-Fierz-Brans-Dicke gravity is characterized by a
single parameter ?0, which represents the coupling of the scalar field to test parti-
cles, with ?0 = 0 corresponding to GR.
8 I perform two complementary analyses on
GW170817 to constrain ?0: the first using a waveform model generalized to include
phenomenological deviations from GR (akin to that used in the theory-agnostic
tests described above) and the second using a generalized waveform model adapted
specifically to JFBD. Taking the more conservative bounds from these two analyses,
I find that the observed BNS can set the constraint |? | . 4? 10?10 ; for comparison
the strongest bounds from weak-field tests are |?0| ? 4? 10?3 [180].
Finally, I examine the constraints that can be placed on certain higher-order
curvature corrections that generically arise in many possible extensions of GR [138].
For definiteness, I examine the particular model proposed in [181], which uses EFT
techniques to construct the generic extension to GR?subject to a number of rea-
sonable assumptions?whose effects are potentially observable by ground-based GW
detectors. Starting from the results of [181], I compute the leading-order PN correc-
tions to a frequency-domain GR waveform predicted in this theory. These deviations
are parameterized by the energy scale ? (or its equivalent distance scale d? ? ??1)
at which higher-order curvature corrections appear. I use the FTA framework to con-
struct a generalized waveform model including these effects, and then use Bayesian
8For historical reasons, this parameter is often expressed instead as ?BD ? 12??2 30 ? 2 , where
GR is recovered in the limit that ?BD ??.
66
0
?5
?10
?15
?20
GW151226
?25 GW170608
? Combined30
75 100 125 150 175 200
d?[km]
Figure 1.8: The logarithm of the Bayes factor (BF) between the compet-
ing hypotheses that higher-order curvature corrections do (non-GR) or
do not (GR) arise at a fixed scale distance d?. Dashed and dot-dashed
lines depict the BFs recovered using GW151226 and GW170608, respec-
tively, and the solid line indicates the BF achieved by combining both
events. By imposing a conservative significance threshold to select one
hypothesis over the other | log BF ? 5| (shown with a horizontal gray
line), I translate these results into a bound on d?: I rule out the appear-
ance of higher-order curvature corrections at scales of ? 70 ? 200 km.
Figure adapted from Ref. [138].
inference to constrain the energy scales at which new physics could appear. My
results are summarized in Fig. 1.8: I plot the Bayes factor (BF) [the last term
in Eq. (1.26)] between the competing hypotheses that higher-order curvature cor-
rections do/do not arise at a given distance scale d?. The dashed and dot-dashed
curves respectively depict the BFs recovered using BBH events GW151226 [182] and
GW170608 [183]?the two lowest-mass (longest) BBH events observed during the
first two observing runs of LIGO and Virgo; the solid curves show the BFs achieved
by combining information from both events. The magnitude of the logarithm of the
BF indicates how strongly one hypothesis is preferred over the other. By setting
a conservative significance threshold of | log BF| ? 5, I can translate these result
into bounds on d?. I find that these GW events can rule out the appearance of
67
log BFnon?GRGR
higher-order curvature corrections arising on distance scales of ? 70? 200 km.
68
Chapter 2: Hairy binary black holes in Einstein-Maxwell-dilaton the-
ory and their effective-one-body description
Authors: Mohammed Khalil, Noah Sennett, Jan Steinhoff, Justin Vines, and
Alessandra Buonanno1
Abstract: In General Relativity and many modified theories of gravity, iso-
lated black holes (BHs) cannot source massless scalar fields. Einstein-Maxwell-
dilaton (EMd) theory is an exception: through couplings both to electromagnetism
and (non-minimally) to gravity, a massless scalar field can be generated by an elec-
trically charged BH. In this work, we analytically model the dynamics of bina-
ries comprised of such scalar-charged (?hairy?) BHs. While BHs are not expected
to have substantial electric charge within the Standard Model of particle physics,
nearly-extremally charged BHs could occur in models of minicharged dark matter
and dark photons. We begin by studying the test-body limit for a binary BH in EMd
theory, and we argue that only very compact binaries of nearly-extremally charged
BHs can manifest non-perturbative phenomena similar to those found in certain
scalar-tensor theories. Then, we use the post-Newtonian approximation to study
the dynamics of binary BHs with arbitrary mass ratios. We derive the equations
1Originally published as Phys. Rev. D98, 104010 (2018).
69
governing the conservative and dissipative sectors of the dynamics at next-to-leading
order, use our results to compute the Fourier-domain gravitational waveform in the
stationary-phase approximation, and compute the number of useful cycles measur-
able by the Advanced LIGO detector. Finally, we construct two effective-one-body
(EOB) Hamiltonians for binary BHs in EMd theory: one that reproduces the ex-
act test-body limit and another whose construction more closely resembles similar
models in General Relativity, and thus could be more easily integrated into existing
EOB waveform models used in the data analysis of gravitational-wave events by the
LIGO and Virgo collaborations.
2.1 Introduction
The first observations of gravitational waves (GWs) from coalescing binary
black holes (BHs) [123, 182?185] and neutron stars [186] offer unprecedented oppor-
tunities to test the highly dynamical, strong-field regime of General Relativity (GR)
[79, 89, 187]. Leveraging the extraordinary precision of GW detectors to test grav-
ity requires waveform models that incorporate potential deviations from GR. One
can construct such models in a theory-independent way by considering phenomeno-
logical deviations to waveform models in GR and then constraining the magnitude
of these corrections, see, e.g., the constructions of [117?120]. Such an approach
has been used by the LIGO and Virgo collaborations to test GR with binary BHs
[73, 122, 123]. Alternatively, one can compute the waveform produced in a particu-
lar alternative theory, which can then be used to measure directly the fundamental
70
quantities that define that modified theory of gravity [79].
Here, we adopt the latter approach, focusing on the dynamics of binary BHs
in Einstein-Maxwell-dilaton (EMd) theory. This theory originated as a low-energy
limit of string theory [188, 189]. In EMd theory, a scalar field (the dilaton) couples to
a vector field (the photon) such that BHs with electric charge also source the scalar;
the BH develops a scalar charge, or hair. It has been shown that in GR (and some
scalar extensions) isolated BHs cannot carry such a charge [23, 148]; these results
are often referred to as ?no-hair theorems.? Analytic solutions exist in EMd theory
for spherically symmetric BHs parameterized by the dilaton coupling constant a [see
Eqs. (2.1) and (2.2) below for the action]. For a = 0, the theory reduces to Einstein-
Maxwell (EM) theory and the BH solution is the Reissner-Nordstro?m metric. For
a = 1, the solution corresponds to the low energy limit of heterotic string theory.
?
For a = 3, the solution corresponds to Kaluza-Klein BHs [190], and an analytic
solution for charged spinning BHs in EMd theory is only known for that value of a
[191].
In the absence of electric charge, isolated BHs in EMd theory behave as in
GR. Within the Standard Model, astrophysical BHs are expected to be electrically
neutral; however, there exist various theoretical mechanisms beyond the Standard
Model that would allow BHs to accumulate non-negligible charge. For a BH with
charge Q and mass M to accrete a particle with the same-sign charge q and mass m,
gravitational attraction between the two bodies must overpower their electrostatic
repulsion, i.e., q Q . mM , or equivalently Q/M . m/q.2 Furthermore, a charged
2Throughout this work, we use geometric units, in which G = c = 4?0 = 1, where G is the
71
BH will neutralize via spontaneous pair production [192] or interactions with astro-
physical plasmas [193] over timescales that grow with the mass-to-charge ratio of
the available fundamental particles. For electrons, the dimensionless mass-to-charge
ratio m /q ? 10?22e e severely limits the charge that BHs can develop through ac-
cretion, and guarantees that any BH charged through other means will discharge
quickly. However, particles with much larger mass-to-charge ratios are predicted in
models of minicharged dark matter [139?141] and would allow BHs to acquire and
retain a much larger electric charge [194]. Similarly, models in which dark mat-
ter is charged under a hidden U(1) gauge field [142?144], a ?dark photon,? would
allow for BHs to develop significant hidden charge, provided that the ratio of the
dark-matter particle?s mass to its (hidden) charge is sufficiently large [194]. These
two types of dark matter models are consistent with laboratory experiments and
cosmological observations [195?197]; current constraints restrict the new particles?
mass to 1 GeV . m . 10 TeV [144] and its charge to . 10?14(m/GeV)qe [198] (see
also Fig. 1 in Ref. [194]).
The dynamical evolution of binary BHs in EMd theory has been studied in
various contexts. Numerical-relativity simulations of single and binary BHs were
performed in Ref. [199]. The authors considered small electric charges and found
that the resulting gravitational waveforms are difficult to distinguish from those
in GR. Numerical-relativity simulations of the collision of charged BHs with large
electric charges in EM theory were performed in Refs. [200, 201], where it was found
that a significant fraction of the energy is carried away by electromagnetic radiation.
bare gravitational constant.
72
In this work, we compute the conservative and dissipative dynamics of a bi-
nary BH, and the resulting gravitational waveform, in EMd theory, to first order
in the (weak-field and slow-motion) post-Newtonian (PN) approximation. We also
construct an effective-one-body (EOB) Hamiltonian description [38, 39] of the con-
servative dynamics, which provides an analytical resummation of the PN dynamics
to exactly recover the test-body limit. In late 2017, the 1PN Lagrangian for a two-
body system in EMd theory was derived independently in Ref. [202] using a method
different from our own. In that work, the author also discussed an abrupt transition
in the scalar charge of a BH as the external scalar field is varied. However, we show
here that this transition occurs only in binaries composed of nearly-extremal charged
BHs and only near the end of their coalescence. Although extremally charged BHs
are excluded when restricting to the Standard Model of particle physics, they are still
viable in minicharged dark matter and dark photons models, as we have discussed
above.
The paper is structured as follows. In Sec. 2.2, we study the behavior of a
small BH in the background of a much more massive companion. By exploring
the response of this test BH to its external environment, we discuss whether non-
perturbative, strong-field phenomena, akin to those seen in binary neutron stars in
scalar-tensor (ST) theories, can occur in binary BHs in EMd theory. In Sec. 2.3,
we use the PN approximation to study the dynamics of a binary system with an
arbitrary mass ratio. We derive the two-body 1PN Lagrangian and Hamiltonian
(with details relegated to Appendix A) and calculate the scalar charge of the two
bodies. Further, we derive (with details in Appendix B) the next-to-leading order
73
PN scalar, vector, and tensor energy fluxes emitted by the binary. Restricting
our attention to quasi-circular orbits, we compute the Fourier-domain gravitational
waveform at next-to-leading-order using the stationary-phase approximation. In
Sec. 2.4, we work out an EOB description of the PN Hamiltonian in EMd theory.
We construct two EOB Hamiltonians: one based on the exact BH solution, and the
other based on an approximation to that solution. The former is more physical in
the strong-gravity regime because it exactly reproduces the dynamics in the test-
body limit; the latter uses the same gauge as EOB models in GR, and thus would
be easier to integrate into existing data-analysis infrastructure. We compare these
two EOB Hamiltonians by calculating the binding energy and the innermost stable
circular orbit to determine the region of the parameter space in which they agree.
Finally, we present some concluding remarks in Sec. 2.5.
2.2 Einstein-Maxwell-dilaton theory
2.2.1 Setup
We consider a generalization of EMd theory presented in Refs. [188, 189] in
the Jor?dan frame??g? ( )
S = d4x e?2a? R? + (6a2 ? 2)g???????????? F ????F + Sm(g??? , A?, ?),
16?
(2.1)
where ? is a scalar field (the dilaton), a is the dilaton coupling constant, F?? ?
???A? ? ???A? is the electromagnetic field tensor, and tildes signify quantities in
the Jordan frame. We also include some matter fields ?, which couple minimally
74
to g??? and, through some fundamental electric charge, to A?; we represent this
total matter action schematically with Sm. By construction, electrically neutral,
non-self-gravitating matter configurations will follow geodesics of g??? , and thus this
theory respects the weak equivalence principle. However, self-gravitating systems
are bound (in part) through non-linear interactions of the scalar field. The back-
reaction of the scalar field on the metric exerts an additional force on such systems,
causing them to no longer follow geodesics; thus, this theory violates the strong
equivalence principle.
The Einstein frame provides a more convenient representation of EMd theory.
Performing the conformal transformation g = A?2?? (?)g??? with A = ea?, the action
becom?es ??g ( )
S = d4x R? 2g??? ?? ?? e?2a?F F ??? ? ?? + Sm(A2(?)g?? , A?, ?), (2.2)
16?
where g?? is the Einstein-frame metric. Here, we primarily work in the Einstein
frame, but occasionally use quantities in the Jordan frame, denoted with tildes. For
a discussion of the equivalence between the two frames see Ref. [203].
For the matter action Sm, we adopt the approach introduced by Eardley [204],
in which each body is treated as a delta function and the dependence on the scalar
field is incorporated into the masses. For charged monopolar point particles, neglect-
ing dipoles/spins and higher multipoles, the matter action in the Einstein frame can
be written as [147]
?? [ ? ]
Sm = ? dt mA(?) ?g?? v?Av?A ? qAA?v?A , (2.3)
A
where mA(?) is the field-dependent mass of particle A, qA is the electric charge,
75
v? ? u?/u0A A A where u?A is its four-velocity, and the fields are evaluated at the particle?s
location. The mass in the Einstein frame m(?) is related to the mass in the Jordan-
Fierz frame m?(?) by
m(?) = A(?)m?(?), (2.4)
where m?(?) is generally not a constant except for bodies with negligible self-gravity.
In most cases, a closed-form expression for the field-dependent mass m(?)
cannot be found. Instead, one expands the mass about the external/background
value ?0 of the scalar field ? ? ( )
d lnm(?)? 1 d2 lnm(?) ? 1
lnm(?) = lnm(?0) + ?? ??+ ? ??2 +O , (2.5)d? ? 2 d?2 ?? c60 0
where ?? ? ?? ?0. The mass expansion can be parameterized in terms of
d lnm(?) d?(?)
?(?) ? , ?(?) ? , (2.6)
d? d?
where ? is referred to as the (dimensionless) scalar charge. With these parameters,
the mass expansion can be [written as ( )]
1 1
m(?) = m 1 + ???+ (?2 + ?)??2 +O , (2.7)
2 c6
where the field-dependent mass is denoted by the Gothic script m, while the mass
evaluated at the background value of the scalar field is denoted by m. We also drop
the dependence of the parameters on the background value to simplify the notation,
i.e., ? ? ?(?0), and ? ? ?(?0). For the field-dependent parameters, we always
explicitly write ?(?) and ?(?). The expression for ?(?) depends on the structure
of the body; for static BHs, it depends only on the charge-to-mass ratio, whereas
for baryonic matter, it also depends on the body?s composition.
76
We note that Eq. (2.3) together with the expansion of the mass (2.7) provide
a systematic construction of an effective source or action for an extended object
in a PN expansion. We neglect couplings to derivatives of the field, which would
correspond to dipole/spin and higher multipole interactions. Due to invariance
under gauge transformations A? ? A?+??, the charges qA must be constant; they
cannot depend on the scalar field like the masses.
2.2.2 Black-hole solution
The metric for an electrically-charged non-rotating BH in EMd theory is given
by [188, 189]
ds2 = ?A(r)dt2 +B(r)dr2 + r2C(r)d?2, (2.8)
with
( r )( r ) 1?a2+ ? 1+a2
A(r) = 1? 1? , (2.9a)
r r
1
B(r) = ( , ) (2.9b)A(r) 2r 2a? 1+a2
C(r) = 1? , (2.9c)
r
where the constants r+ and r? are given in terms of the Arnowitt-Deser-Misner
mass M and electric charge Q by ( )
r+ 1? a2 r?
M = + , (2.10)
2 1 + a2 2
2 r+r?Q = e?2a?0 . (2.11)
1 + a2
The constant r+ corresponds to the outer horizon, and r? corresponds to the inner
horizon. The surface area of the horizon (entropy of the BH) is proportional to
77
r2+C(r+). Here, we refer to the metric (2.8) as the GHS metric, after Garfinkle,
Horowitz and Strominger who found the solution in that form in Ref. [189].
The electromagnetic four-potential A?, for an electrically-charged BH, is given
by
Q
A0(r) = ? e2a?0 , Ai(r) = 0 , (2.12)
r
and the scalar field ? is given by
a ( r )?
?(r) = ?0 + ln 1? . (2.13)
1 + a2 r
While we consider only electric charges here, we note that the solution for a magnet-
ically charged BH can be obtained from the above solution via the duality rotation
that sends F 1 ?2a? ?? 3?? ? e ?? F?? and ?? ??. In addition to the electric charge,2
BHs in EMd theory can acquire scalar charge, also called dilaton charge, defined by
[189] ?
? 1D d2?????, (2.14)
4?
where the integral is over a two-sphere at spatial infinity, leading to
a
D = r? . (2.15)
1 + a2
Far from the BH, we have ?(r) ' ?0?D/r+O(1/r2), which means that D acts as
the monopole charge sourcing the scalar field.
3 The results of Sec. 2.2 hold also for magnetic charges if we flip the sign of ?. However, the
PN and EOB results in the following sections would change in non-trivial ways for the magnetic
case, since the BH?s F?? is given by F?? = Qm sin ? with a magnetic charge Qm, as opposed to
Ftr = Q/r
2 with an electric charge Q (all other components being zero in each case).
78
The constants r+ and r? can be expressed in terms of the mass and the dilaton
charge, or the mass and electric charge, as
1 + a2
r? = D
a
1 + a2 ( ? )
= ? M ? M
2 ? (1? a2)Q2e2a?0 , (2.16a)
1 a2
2
r+ = 2M ?
1? a
? Da
= M + M2 ? (1? a2)Q2e2a?0 . (2.16b)
Expressing quantities in terms of the dilaton charge D, rather than the electric
charge Q, makes most equations simpler as it avoids the square root. Therefore, in
most of the equations below, we use D instead of Q. The relation between Q and
D can be read off from Eq. (2.16a), or Eq. (2.16b),
2
Q2 e2a?
2M ? 1? a0 = D D2 . (2.17)
a a2
The maximum electric charge of the BH occurs when r+ = r?, which leads to
?
Q a?0 2max e = 1 + a M. (2.18)
Hence, for nonzero values of a, an EMd BH can be more charged than an extremal
Reissner-Nordstro?m BH with the same mass. Since the dilaton charge is related to
the electric charge via Eq. (2.17), the maximum electric charge (2.18) corresponds
to the maximum dilaton charge Dmax = aM .
Without loss of generality, we set the background scalar field to zero, i.e.,
?0 = 0. To recover the dependence on ?0, one can simply rescale all electric charges
by the factor ea?0 , and add the constant ?0 to the scalar field.
4 We also consider
4 To see why this is true, consider the action (2.2) with the transformation Q ? Qea?0 and
79
only non-negative values of a since the action (2.2) is invariant under a ? ?a and
?? ??, so the predictions for negative dilaton couplings are given by changing the
sign of the scalar field.
1.0 10
0.8 8
0.6 6
0.4 4
0.2 2
0.0 - - 04 2 0 2 4 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
Figure 2.1: ?(?) for a = 1 with different charge-to-mass ratios (left),
and for different values of a with q = 0.95 qmax (right).
2.2.3 Dynamics of a test black-hole in a background black-hole space-
time
Before turning to the dynamics of a generic two-BH system in EMd theory, it
will be useful to study the test-body limit of such a system, i.e., the limit in which
one body?s mass is negligible compared to the other?s. In EM theory (without the
? ? ? + ?0. The vacuum part of the action is symmetric under that transformation, and in
the matter action (2.3), the mass m(?) is parameterized in terms of the difference ? ? ?0. The
electromagnetic part of the matter action is more subtle; it depends on qv?A? ? Qqe2a?0/r,
and hence, one can absorb a factor of ea?0 into each of the two charges. However, since A0 =
?Qe2a?0/r, the transformation Q? Qea?0 , ?? ?+ ?0 is not valid in equations that depend on
A?; one first needs to express A? in terms of the charges before performing that transformation.
80
1.0 10
0.8 8
0.6 6
0.4 4
0.2 2
0.0 0
1 2 3 4 5 1 2 3 4 5
Figure 2.2: ?(r) for a = 1 with different charge-to-mass ratios (left),
and for different values of a?(right). In both plots, the charge of the
large BH is extremal Q = 1 + a2M , and r is scaled by the horizon
radius, which is given by Eq. (2.16b). For a = 1, the horizon radius is
2M independently of the charge or the coupling constant.
dilaton), the test-body limit of a charged BH corresponds simply to a monopolar
point-mass with constant mass and constant charge. In EMd theory, however, a
BH?s mass must retain a dependence on the dilaton field even as its size goes to
zero. In the zero-size limit, we can use the local value of the (background) dilaton
field ?, at the small BH?s location, to determine its mass m(?) in the same way
that a lone finite-size BH?s mass would be determined by the asymptotic value of
the field (as in the previous subsection). This defines what we mean by a ?test BH?
in EMd theory.5
Let us suppose a test BH with mass m(?), electric charge q, and dilaton charge
d moves in the fixed background spacetime of a larger BH with mass M , electric
5 This is not to be confused with some uses of the phrase ?test body? in the context of ST
theories, where one means a body with negligible self-gravity (unlike a BH), so that the mass in
the Jordan-Fierz frame is constant and the scalar charge is zero.
81
charge Q, and dilaton charge D. The mass of the test BH m(?) depends on the
scalar field ? generated by the larger BH. The expansion of m(?) is given in terms
of the parameters ? and ? by Eq. (2.7), and the scalar field ? is given by Eq. (2.13).
To find how ? and ? depend on the mass and charge of the BH, one needs
to find the dependence of the mass on the scalar field. We can get a differential
equation for m(?) from Eq. (2.16a), or Eq. (2.16b), by identifying the mass M and
charge Q with those of the test BH, i.e., M ? m(?) and Q ? q. The background
value of the scalar field can be identified with the field from the more massive BH
?0 ? ?, and the scalar charge by D ? dm(?)/d?, as was shown by the matching
conditions in Ref. [202]. This leads to the equation
dm(?) a [ ? ]
= ? m(?)? m(?)
2 ? (1? a2)q2e2a? , (2.19)
d? 1 a2
which, as far as we know, has no analytic solution for arbitrary values of a. Never-
theless, we can still obtain an expression for the dimensionless scalar charge, which
is defined by Eq. (2.6), [ ? ]
a q2e2a?
?(?) = 1? 1? (1? a2) , (2.20)
1? a2 m2(?)
and ?? ?a2q2e2a? a2?(?) = ?? 1? ? , (2.21)(1 a2)m2(?) ? ? q2e2a?1 (1 a2)
m2(?)
in agreement with Ref. [202].
It is interesting to note that an exact analytic solution to the differential
equation (2.19) can be found when t?he coupling constant a = 1, that is
1
m(?) = const. + q2e2? . (2.22)
2
82
Since the above expression should give m when ? = 0, the integration constant is
found to be m2 ? 1q2. Hence,
2 ?
1 1
m(?) = m2 ? q2 + q2e2? . (2.23)
2 2
By differentiating m(?), we get the parameters
q2 q2 q4
? = , ? = ? . (2.24)
2m2 m2 2m4
In Fig. 2.1, we plot ?(?) as a function of ?. We see that the test BH?s
?(?) transitions between two values: zero and a. The function ?(?) reaches its
maximum value when the quantity q2 e2a?/m2 approaches 1 + a2, which means that
?
in the Jordan-Fierz frame, the charge q approaches the extremal value 1 + a2m?,
where the mass in the Jordan-Fierz frame m? is given by Eq. (2.4). Changing the
charge-to-mass ratio shifts the curve on the horizontal axis, while changing a changes
the maximum value of ? and determines how quickly this transition occurs.
We emphasize that the scalar field ? generated by the more massive BH is
always negative, as can be seen from Eq. (2.13), so the test BH always descalarizes.
Further, because of the logarithm, the magnitude of ? increases slowly with decreas-
ing separation until r approaches the inner horizon r?, where it diverges. For the
scalar charge of the test BH to change dramatically before merging with its much
larger companion, both BHs must be close to extremally charged. As discussed in
Sec. 2.1, extremally-charged BHs can exist in minicharged dark matter and dark
photon models. If the test BH is not sufficiently charged, its scalar charge is close
to zero when well separated from its companion, and then monotonically decreases
83
toward zero as the binary evolves. The total shift in the scalar field that the test
BH experiences prior to crossing the outer horizon is given by
[ ? ]? ? a 1 D/Dmax?(r+) ?( ) = ln ? ? . (2.25)1 + a2 1 (1 a2)D/2Dmax
Thus, if the large BH is not also sufficiently charged, then the test BH?s scalar charge
does not change dramatically.
In Fig. 2.2, we substitute the expression for the scalar field of the larger BH
?(r) into that for the scalar charge of the test BH ?(?), and plot ?(r) versus the
separation r scaled by the horizon radius. When setting the charge of the large
?
BH to its extremal value, Q = 1 + a2M , we see that the charge of the test BH
also needs to be near extremal for the descalarization transition to occur. Yet, the
transition only occurs very close to the horizon of the background BH. Hence, we
expect this descalarization to drastically affect the GW signature only during the
late inspiral and plunge of a test BH into a more massive BH and only when the
BHs are nearly-extremally charged, when the horizon, the innermost-stable circular
orbit, and the divergence in ? coincide. This result is analogous to extremal Kerr
BHs, where the plunge occurs at significantly smaller separations [205]. However,
a comparable-mass binary does not perform many orbits at small separations due
to stronger radiation reaction, and thus we expect that the transition in the scalar
charge would have a negligible effect on GWs from the inspiral of a comparable-mass
binary.
We note that, while the descalarization transition occurs for near-extremal
BHs, the largest change in the value of ? from infinity until, e.g., r = 2r+ occurs
84
when the electric charge is q/m ? 1, as can be seen in the left panel of Fig. 2.2.
This is due to the slope of ?(?) at the background value of the scalar field ?0 = 0.
So, in order to increase the change in the scalar charge to observe descalarization,
it is important to have a maximal ?(?) ? d?(?)/d?.
2.2.4 Compact objects in Einstein-Maxwell-dilaton and scalar-tensor
theories
Certain ST theories can exhibit non-perturbative phenomena, known as in-
duced or dynamical scalarization, in binary systems of neutron stars [169?172].
Having established how a BH responds to its scalar environment, we now investi-
gate whether such effects could arise in binary BHs in EMd theory. In Ref. [199], the
authors suggested that dynamical and induced scalarization are much less signifi-
cant in EMd theory than in ST theories. In this subsection, we support this claim
using more quantitative arguments by directly comparing the behavior of BHs and
neutron stars in the respective theories.
In Ref. [3], the authors argued that the onset of induced and dynamical scalar-
ization coincide with a breakdown of the PN approximation. Specifically, these non-
perturbative phenomena indicate that the scalar field has grown beyond the validity
of a PN expansion of m, e.g., Eq. (2.7). A useful diagnostic for determining the
onset of such phenomena is to compare the relative size of the coefficients of such a
power series to the small parameter with which one constructs the expansion.
While both EMd theory and ST theories include an additional scalar field,
85
the non-minimal coupling of that field to the Jordan-Fierz (physical) metric can
differ substantially. To facilitate comparisons between these theories, we consider
an expansion of m in GN(?), the parameter that characterizes the gravitational force
felt between two test bodies placed in the scalar background ?. In both EMd theory
and ST theories, this Newton?s ?constan[t? is(given by) ]2
? A2 d logAGN(?) (?) 1 + . (2.26)
d?
We expand m in terms o[f this qu(antity ) ( ) ]
G?G0 ? 2G G0
m(GN) = m 1 + C
N N
1 + C2 + . . . , (2.27)
G0 0N GN
where we have defined
G0N ? [GN(? = 0)], (2.28a)
d logm
C1 ? [ , ] (2.28b)d logGN G =G0N N( )2
? 1 d
2 logm d logm ? d logmC2 + . (2.28c)
2 (d logG 2N) d logGN d logGN
GN=G
0
N
We compare these coefficients for BHs in EMd theory to that of neutron stars
in Brans-Dicke gravity [177?179], defined by the coupling
ABD(?) = e??0?, (2.29)
and theories first considered by Damour and Esposito-Fare?se (DEF) [147, 153]
A ?? ?20 /2DEF(?) = e , (2.30)
in which induced and dynamical scalarization can occur when ?0 is sufficiently neg-
ative. In Fig. 2.3, we plot the ratio C2/C1 for compact objects in the various
86
104
103
102
101
100
10-1 -10-1 -10-4 -10-7 0 10-7 10-4 10-1
Figure 2.3: Ratio of the coefficients C2/C1 defined in Eqs. (2.28b)
and (2.28c) as a function of GN for BHs in EMd theory (solid) and
neutron stars in various ST theories (dashed). Annotated points depict
this ratio at various separations for a test BH with q = 0.99qmax in the
background of a BH with Q = Qmax in EMd theory (r+ refers to the
outer horizon of the background spacetime).
theories. For the ST theories, we consider neutrons stars with m = 1.45M with
the piecewise polytropic fit to the SLy equation of state constructed in Ref. [206].
The solid curve depicts this ratio for BHs in EMd theory with coupling a = 10. By
comparison, this same quantity is shown with red and blue dashed curves for neu-
tron stars in Brans-Dicke gravity with ?0 = 0.03 and in the theory of Damour and
Esposito-Fare?se with ?0 = ?4.4, respectively. Note that by inserting Eq. (2.30) into
Eq. (2.26), one sees that this theory is only defined for GN(?) > G
0
N. For reference,
we indicate with black points the separation at which these values are achieved in
EMd theory when the test BH in placed in the background of an extremally charged
BH; r+ corresponds to the outer horizon of the background BH. We see that the
magnitude of the ratio C2/C1 drastically differs between ST theories that manifest
87
induced and dynamical scalarization (DEF) and EMd theories. This result indicates
that a perturbative expansion of the dynamics has a larger regime of validity, and
that non-perturbative phenomena are less likely to emerge during the coalescence
of binary BHs in EMd theory.
2.3 Post-Newtonian approximation in Einstein-Maxwell-dilaton the-
ory
2.3.1 Two-body dynamics
To go beyond the test-body limit, treating two-body systems with arbitrary
mass ratios, we employ the PN approximation, which is valid in the weak-field, slow-
motion regime [30]. In Appendix A, we derive results for the conservative dynamics
of a binary BH system in EMd theory, at next-to-leading order in the PN expansion,
i.e., at 1PN order. We employ the Fokker action method [207] (see also Ref. [208]),
which has been used to treat the 4PN dynamics in GR [209], and the 2PN [208] and
3PN [210] dynamics in ST theories. We begin by considering the PN expansions of
the EMd action in Eq. (2.2) and the matter action for point particles in Eq. (2.3),
using the mass expansion in terms of the ? and ? parameters from Eq. (2.7). From
the initial full action expanded to 1PN order, we obtain field equations for the scalar
field, the metric potential, and the electromagnetic 4-potential. The Fokker action
is obtained by plugging the (regularized) solutions to the field equations back into
the action, eliminating the field degrees of freedom, yielding an action depending
only on the matter variables. We work in the harmonic gauge g?????? = 0 and the
88
Lorenz gauge ? A?? = 0 throughout. The final result for the two-body Lagrangian
is given by ( )
? 1 2 1 2 q1q2 m1m2L = m1 ?m2 + m1v + m2v + 1 + ?1?2 ?
2 1 2 2 m1m2 r
1 1 q1q2
+ m 4 41v1[+ m2v2 + [v1 ? v2 + (n ? v1)(n ? v2)]8 8 2rm1m2 ]
+ [(3? ?
2 2
1?2)(v1 + v2)? (7? ?1?2)(v1 ? v2)? (1 + ?1?2)(n]? v1)(n ? v2)2r
? m1m2 (1 + 2? ? )(m +m ) +m ?2 21 2 1 2 1 1(?[2 + ?2) +m2?
2
2(?
2
1 + ?1)2r2
q1q2 1 ]
+ ( [m)1(1 + a?1) +m2(1 + a?2)]? m 2 22 2 1q2(1 + a?1) +m2q1(1 + a?2)r 2r
+O 1 , (2.31)
c4
where r ? x1 ? x2 is the separation between the two bodies, and n ? r/r. This
Lagrangian agrees with the one derived by Damour and Esposito-Fare?se [147, 208]
when the Maxwell fields are zero. The standard 1PN Lagrangian in GR is obtained
by setting qi = ?i = ?i = 0, while the Lagrangian in EM theory is obtained when
?i = ?i = 0. Note that, since we use the mass expansion in Eq. (2.7) given in terms
of generic parameters ? and ?, our results are not restricted to BHs in EMd theory,
but are applicable to more generic bodies as well.
During the course of this project, the same 1PN Lagrangian for a two-body
system in EMd theory was derived independently by Julie? in Ref. [202]. While our
results agree, our derivation differs from that of Ref. [202] in some notable respects.
In Ref. [202], the (unexpanded) field equations were directly obtained from the
action (2.2), and then those equations were expanded and solved for the fields. The
primary difference with our derivation is in how Ref. [202] constructed the two-body
Lagrangian: (i) taking (only) the matter action for one body (without the field part
89
of the action, and without the matter action for the other body), which would apply
if the body were a test body in some given fields, (ii) inserting for those fields the
(regularized) solutions to the field equations resulting from the total (two bodies +
fields) action, and (iii) taking the resultant Lagrangian and ?symmetrizing? it with
respect to the two bodies. While this procedure does produce a correct Lagrangian
at 1PN order, it is not justified in general, and it is important to see how the result
can be obtained from a consistent treatment of the full action for the two bodies and
fields. In Ref. [202], it was also found that it is possible to parameterize the 1PN
Lagrangian in EMd theory to have the same structure as the 1PN Lagrangian in
ST theories, which means that many results in ST theories can be directly extended
to EMd theory at 1PN order. We choose not to use that parameterization to make
the dependence on the electric charges more apparent, and because many of our
results are specific to EMd theory, such as calculating the vector energy flux and
developing the EOB Hamiltonians.
The Hamiltonian in the center-of-mass frame can be derived from the La-
grangian using the Legendre transformation [211]
H = v ? p? L, (2.32)
where the relative velocity v ? v1 ? v2 and the center-of-mass momentum
?L
pi = . (2.33)
?vi
90
This leads to the energy
1
E = M + ?v2[?(G12M? 3+ (1? 3?))?v42 r 8 ]
G12M? 3? ?1?2
+ + ? v2[ + ?r?
2
2r 1 + ?1?2 ? q1q2M?
M2? q2
+ (1 + ?1? )
2
2 +X2?
2
2?1 +X ?
2 2
1 1?2 +X1 ](1 + a(?1)2r2 M? )
q21 q1q2 1+X2 (1 + a?2)? 2 (1 + a?1X1 + a?2X2) +O , (2.34)
M? M? c4
where r? = n ?v, and we defined the total mass M , reduced mass ?, symmetric mass
ratio ?, and the mass ratios Xi in terms of the constant masses m1 and m2 by
? ? m1m2 ?M m1 +m2 , ? , ? ? ,
M M
? m1 ? m2X1 , X2 . (2.35)
M M
We also define the coefficient G12 by
G12 ?
q1q2
1 + ?1?2 ? , (2.36)
M?
which reduces to the usual definition in ST theories when the electric charges are
zero. The advantage of including the charges in G12 is that the Newtonian-order
acceleration is simply given by a = ?G12Mn/r2 +O(1/c2).
Expressing the energy in terms of the center-of-mass momentum p ? p1 =
?p2, instead of the velocity, we obtain the Hamil[to(nian ) ]
p2 G12M? 1 p
4 G12M 3? ?1?2
H = M + [? ? (1? 3?) ?
2
? q q + ? p + ?p
2
2? r 8 ?3 2?r 1 + ? ? 1 2 r1 2 M?
M2? q2 q2
+ (1 + ? ? )21 2 +X2?
2
2?1]+X1?
2
(1?2)+X
2
1 (1 + a?1) +X
1
2 (1 + a?2 2)2r M? M?
? q1q2 12 (1 + a?1X1 + a?2X2) +O , (2.37)
M? c4
91
1.00 1.00
0.95 0.95
0.90 0.90
0.85 0.85
0.80 0.80
5 10 15 20 5 10 15 20
Figure 2.4: Scalar charges scaled by their asymptotic value as a function
of the separation r of a binary BH scaled by the total mass. In both plots,
the charge-to-mass ratio q1/m1 = q2/m2 = 1 and the dilaton coupling
a = 1; in the left panel ? = 0.24, while in the right ? = 0.1.
where pr = n ? p.
Next, we examine how the scalar charges of the two bodies change with their
separation. The dilaton charge is given by
dm(?)
D(?) = = m(?)?(?). (2.38)
d?
For the two bodies, the dilaton charge as a function of the separation r has the
expansion
[ 1( ) ( ) ]
D1(r) = m1 ? + (?
2
1 1 + ?1)?1(r) + 3?1?1 + ?
3 + ?? ?2(r) +O 1/c6 ,
2 1 1 1
[ (]2.39a)1( ) ( )
D2(r) = m ?
2
2 2 + (?2 + ?2)?2(r) + 3? ? + ?
3 + ?? ?22 2 2 2 2(r) +O 1/c6 ,2
(2.39b)
where ?? ? d?(?)/d?|? , ?1 is the scalar field at the location of body 1, and ?0 2 is
92
1.00 1.00
0.95 0.95
0.90 0.90
0.85 0.85
0.80 0.80
5 10 15 20 5 10 15 20
Figure 2.5: Scalar charges of a binary BH as a function of r for equal
masses (? = 1/4), dilaton coupling a = 1, and charge-to-mass ratios
q1/m1 = 1, q2/m2 = 1.4 (left) and q1/m1 = 1, q2/m2 = 0.5 (right).
the scalar field at the location of body 2. From the 1PN scalar field in Eq. (A.32),
?2m2 m1m2 ( ) aq1q2
?1(r) = ? + ?2 + ? 21?2(+ ?1?)2 ?r r2 r2aq2
+ 2
1
+ ?2m2(n( ? a2) +O 1/c
6 ,) (2.40a)2r2 2
??1m1 m1m2 2 ? aq1q2?2(r) = + ?1 + ?2?1(+ ? ?r r2 2)1 r2aq2 1
+ 1 ? ?1m1(n ? a1) +O 1/c6 , (2.40b)
2r2 2
where, using a = ?G Mn/r2 +O(1/c212 ) and Eq. (B.19),
m2 ?G12m2
( )
a = a = n+O 21
M r2
m1 G12m1 (
1/c ) , (2.41a)
a2 = ? a = n+O 1/c2 . (2.41b)
M r2
In Fig. 2.4, we plot D1(r) and D2(r) for charge-to-mass ratios q1/m1 =
q2/m2 = 1, dilaton coupling constant a = 1, and symmetric mass ratios ? = 0.24
and ? = 0.1. The curves are plotted until r = 3M because the PN expansion
becomes inaccurate well before that separation. From the figure, we see that the
93
scalar charge of both bodies decreases as the separation decreases, with the charge
of the lighter body decreasing more quickly. Figure 2.5 shows the scalar charge as a
function of the separation for equal masses but with different charge-to-mass ratios.
We keep q1/m1 = 1 while q2/m2 takes the values 1.4 and 0.5. The scalar charge
of the less-charged body decreases more quickly with decreasing separation. These
results are consistent with what was found in the previous section for the scalar
charge of a test BH, but here, we do not see a transition or a divergence near the
horizon.
2.3.2 Gravitational energy flux
From the 1PN expansion, we computed the next-to-leading order scalar, vec-
tor, and tensor energy fluxes for general orbits (see Appendix B for the derivation).
In a 1/c expansion, the leading terms are the scalar and vector dipole fluxes, which
are of order 1/c3, while the leading order tensor flux is of order 1/c5, which is the
same as the next-to-leading order scalar and vector fluxes. We computed the next-
to-leading order tensor flux, which is of order 1/c7, because that is the maximum
level of approximation accessible by use of the 1PN near-field equations. The scalar
and vector dipole fluxes depend on the difference between the charges of the two
bodies. The scalar flux also includes a monopole term that vanishes for circular
orbits.
The total energy flux is the sum of the scalar, vector, and tensor fluxes
F = FS + FV + FT , (2.42)
94
where the expressions for the fluxes through next-to-leading order for general orbits
are given in Appendix B. The fluxes for circular orbits are given by
?2x4 ?2x5 [FS = (?1 ? ?2)2 + 20f (? ? ? )22 2 ? 1 23G1(2 ) 15G12 ] ( )
+5 fS
1
v(2 + f
S
1/r +)16 (X1?
2
2 +X[ 2?1)(+O ), (2.43a)c72 2
F 2?
2x4 q q 2?21 ? 2 x
5 q1 q2
V = + 20f ?
3G2( m m ) 15G2 ?12 1 2 1(2 m)]1 m2 (2 )q1 q2 1
+8 X V2 +X1 + 5 fv2 + f
V
1/r +O , (2.43b)m1 m2
32?2x5 2?2x6 ( )
c7 ( )
F 1T = + fTv4 + fTv2/r + fT1/r2 + 672f2 2 ? +O , (2.43c)5G 912 105G12 c
where the coefficients f are given by Eqs. (B.36), (B.56), (B.79), and (B.85). The
energy flux is expressed in terms of the parameter x defined by
x ? (G12M?)2/3 , (2.44)
where ? is the orbital frequency, which is ?perturbatively gauge-invariant? in the
sense that it remains fixed under coordinate transformations to arbitrary PN order.
In Figs. 2.6 and 2.7, we plot the total energy flux in EMd theory with a = 1
relative to the flux when all charges are zero versus the binary?s gauge-invariant
velocity v = (G12M?)
1/3, i.e., we plot (F ? Fq=0)/Fq=0. For comparison, Fig. 2.6
also includes the energy flux in EM, when scalar charges are zero but not the electric
charges. The plots start at v = (G 1/312M?) = 0.15 which corresponds to a total
mass M = 20M, and a lower GW frequency in the detector of 10 Hz. In the
plots, we used the next-to-leading order scalar and vector fluxes, but only used the
leading Newtonian order tensor flux, because the 1PN energy flux in GR is given by
95
4 EMd
30 EM
25 3
20 2
15
1
10
0
5
0 -1
0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5
Figure 2.6: Energy flux in EMd theory and EM relative to the un-
charged GR flux plotted versus the gauge-invariant velocity for cir-
cular orbits v ? (G M?)1/312 for coupling constant a = 1, for equal
masses, and for charge-to-mass ratio q1/m1 = 1, q2/m2 = 0.8 (left) and
q1/m1 = 1, q2/m2 = ?0.8 (Right).
FGR ? x5 ? const. x6; the minus sign of the second term causes the flux to become
negative at large frequencies.
From the plots, we see that at small frequencies (large separations), the dif-
ference with GR is greater than at larger frequencies because the dipole scalar and
vector fluxes dominate (FS ? x4 while F 5T ? x ). For equal charges, the scalar and
vector dipole fluxes are both zero, which means the total energy flux is the tensor
flux that is proportional to x5. Hence, the next-to-leading order flux in EMd theory
becomes a constant shift to the GR flux, and the relative flux plotted in the figures
becomes a straight line, as can be seen in Fig. 2.7.
In the two panels of Fig. 2.6, we use charge-to-mass ratios q1/m1 = 1, q2/m2 =
0.8 (left) and q1/m1 = 1, q2/m2 = ?0.8 (right). For same-sign charges, at a fixed
frequency, there is a greater difference from GR than for opposite-sign charges and
96
100 0.8, 0.2
10-1
10-2 0.1, 0.01
10-3
- 0.01, 0.0110 4
10-5 0.001, 0.002
0.2 0.3 0.4 0.5
Figure 2.7: Energy flux in EMd theory relative to the uncharged GR
flux for coupling constant a = 1 plotted versus v = (G12M?)
1/3, for
equal masses, and for various charge-to-mass ratios.
also a greater difference between EMd and EM. This is because the energy flux is
inversely proportional to G212 = (1 + ?1?2 ? q1q2/m1m 22) , which is larger when the
electric charges have opposite signs than when they have the same sign. In the right
panel, the plotted curves become negative when F < Fq=0, which occurs because
G12 > 1 for opposite-sign charges, which makes the EMd flux smaller than the GR
flux at some frequency.
In Fig. 2.7, we plot the energy flux for several charge-to-mass ratios. In that
figure, we do not plot the flux in EM theory, because it is almost the same as the
EMd flux for charges q /m . 0.5 since F ? ?2 ? q4/m4i i S i i i , which is much smaller
than FV ? q2i /m2i for small charges. The plot shows the flux for same-sign charges
in a log plot; for small charges . 0.01, the EMd flux decreases significantly and
becomes very close to the GR flux.
The most salient feature that differentiates EMd theory from GR from the
97
50
40
30
20
10
0
0.0 0.5 1.0 1.5
Figure 2.8: Allowed values of EMd coupling a consistent with a dipole
flux constraint of |B| ? 10?3 as a function of mass-weighted total electric
charge. Colors indicate various possible electric dipoles consistent with
the bound on B.
perspective of GW observations is the presence of dipole radiation. At leading
order, the energy flux can be written as
( )
F = F ?1GR 1 +Bx , (2.45)
where FGR is the GR quadrupole flux, and B parameterizes the strength of dipolar
emission, which is given by [ ( ) ]2
5 ? 2 q1 ? q2B = (?1 ?2) + 2 . (2.46)
96 m1 m2
The presence of dipole flux has been constrained in several types of binary
systems. The best constraints on the B come from radio observations of pulsar?
white-dwarf binaries, which lead to the bound |B| . 10?9 [212]. For binaries con-
taining a single BH, the strongest bound comes from low-mass X-ray binaries, in
which the companion is a main-sequence star: |B| . 2 ? 10?3 [145]. To date, no
98
bound has been set from GW observations of binary BHs, but at design sensitivity,
LIGO could set a bound of |B| . 8 ? 10?4 for a GW150914-like event, and LISA
could lower that bound to 10?8 [145].
We wish to understand how well such a bound on dipole radiation in binary
BHs can constrain EMd theory. Given the discussion above, we consider a hypo-
thetical binary BH observation that constrains the dipole flux to |B| . 10?3. The
coupling a that characterizes EMd theory enters the prediction of B through the
dimensionless scalar charges of the two bodies. Equation (2.46) demonstrates that
for a given value of B, the scalar and electric dipoles are degenerate, and thus no
constraint can be set on a directly with only a bound on the dipole flux. However,
if an independent measurement of the total charges could be made ? e.g., through
measurements of the ringdown spectrum of the final remnant?one can potentially
break this degeneracy and constrain EMd theory.
In Fig. 2.8, we show the values of a consistent with |B| ? 10?3 as a function of
mass-weighted total charge |q1/m1 +q2/m2| for various possible values of the electric
dipole |q1/m1?q2/m2|. The maximum allowed electric dipole is achieved in the limit
that a = 0, wherein the scalar charges of the BHs vanish and our bound on the dipole
flux translates directly to the bound on the electric dipole |q1/m1? q2/m2| . 0.098.
Unsurprisingly, we find that the constraint that can be set on a depends primarily
on the magnitude of the electric charges in the binary: for equal-mass systems, the
strongest constraints can be set when the BHs have large, nearly-equal charges, and
the weakest constraints when the BHs have small, opposite charges. We see that
for any realistic constraint on dipole flux, the parameter a is completely unbounded
99
without an independent measurement of the electric charges.
2.3.3 Gravitational-wave phase in the stationary-phase approxima-
tion
Equipped with PN descriptions of the conservative and dissipative sectors of
binary dynamics in EMd theory, we compute a key observable for GW detections:
the Fourier-domain gravitational waveform. We utilize the stationary-phase approx-
imation to perform this calculation, relying on the fact the GW phase evolves much
more rapidly than its amplitude during the adiabatic inspiral along quasi-circular
orbits.
We consider a GW detector a distance R ?GR ? r/v from a binary BH. In
the vicinity of the detector, the metric takes the form
g?? = ??? + h?? , (2.47)
where ??? is the Minkowski metric and h?? contains two propagating, transverse-
traceless polarizations h+ and h?, which comprise the GW produced by the binary.
6
At the fixed distance R, the GW can be decomposed into spin-weighted spherical
6A GW detector also responds to the scalar field through the coupling given in Eq. (2.2). These
scalar waves represent a transverse breathing polarization of perturbations to the Jordan-Fierz
metric. Because standard search techniques are targeted at the transverse-traceless polarizations,
we consider only those gravitational modes in this work. Differentiating between the various polar-
izations of GWs requires a network of detectors; our ability to identify additional GW polarizations
will improve as more ground-based detectors come online.
100
harmonics
? ??
h+ ? ih? = ?2Y`m(?,?)h`m(t), (2.48)
`?2 m=?`
where ?,? are angular coordinates that define the propagation direction from the
source to the detector [30]. We further decompose each mode into an amplitude and
complex phase
h`m(t) = A (t)e
im?(t)
`m , (2.49)
where ?(t) is the orbital phase of the binary.
We compute the Fourier-transform of the GW using
? ?
h?`m(f) = dt h`m(t)e
?2i?ft. (2.50)
??
During the adiabatic inspiral, the amplitude and orbital frequency evolve much more
slowly than the orbital phase, i.e., |A?`m/A`m|  ? and |??|  ?2 for m 6= 0 modes.
Thus, the integral in Eq. (2.50) is highly oscillatory and can be approximated by
expanding the integrand about the time at which the complex phase is stationary.
Using the stationary-phase approximation, the Fourier-domain waveform is then
given by
h?SPA`m (f) = A`m(f)e?i?`m(f)?i?/4, (2.51)
(m) ?? (m)?`m(f) = 2?ftf m?(tf ), (2.52)
A (m) 2?`m(f) = A`m(tf ) , (2.53)(m)
m??(tf )
(m)
where tmf is defined implicitly as the time at which m?(tf ) = 2?f . Following the
notation common in the literature, we employ the binary?s gauge-invariant velocity
101
for circular orbits v ? x1/2 = (G M?)1/312 and introduce a similar notation for the
GW frequency f as v ? (?G Mf)1/3 (m)f 12 . Then, by construction, one finds v(tf ) =
(2/m)1/3vf and can rewrite Eq. (2.52) as( )
1 ??
?`m(f) = m v
3t(v)? ?(v) ?? . (2.54)G12M v=(2/m)1/3vf
From here onwards, we focus only on the dominant ` = |m| = 2 modes and drop the
explicit mode numbers for notational simplicity; because we restrict our attention
to non-spinning systems, the modes obey the symmetry relation
h`m = (?1)`h?`,?m, (2.55)
and thus we can consider only the m = 2 mode without loss of generality.
The orbital phase and frequency are computed using the balance equation
dE
= ?F . (2.56)
dt
From this equation, we deduce
?
1 v E ?(v?)
?(v) =? ? dv?v?3ref ? , (2.57)G12M v F(v?)refv
? dE/dv?t(v) =tref dv? , (2.58)
v F(v?)ref
where ?ref and tref refer to an arbitrary reference point in the evolution of the binary.
Inserting these results into Eq. (2.54), the Fourier-domain phase is given by
?
2 vref ( ) E ?(v)
?(f) = 2?ft 3 3ref ? ?ref + vf ? v F dv. (2.59)G12M v (v)f
The energy flux in terms of x is given by Eq. (2.43a). The energy E is given
by Eq. (2.34), and it can be expressed in terms of x using Eqs. (B.82) and (B.85),
102
which leads to
? [ ( )]
E = ? x 1 + fEx+O 1/c4 , (2.60)
2
where the coefficie(nt fE is given by[ )?1 2 1 + ? 3? ?1?2f 2 2E = G12 + ? q q ? (1 + ?1?2) ?X2?2?1 ?X ?23G2 14 1 + ? ? 1 2 ] 1?212 1 2 M?
q2 q2? 2 q1q2X1 (1 + a? )?X 11 2 (1 + a?2) + 2 (1 + aX1?1 + aX2?2) .
M? M? M?
(2.61)
To evaluate the integral in Eq. (2.59), we need to distinguish between two
regimes, similarly to what was done in Ref. [2]. In one regime, the electric charges
are small and the inspiral is driven by the tensor quadrupole flux. In the other
regime, the electric charges are large and the inspiral is driven by the dipole flux.
For the quadrupole-driven (QD) case, we approximate the integrand in Eq. (2.59)
by
? [ ]E (v) ?
F '
E (v) FS(v) + FV (v)
(v) F 1? . (2.62)T (v) FT (v)
Then, we expand the integrand using the next-to-leading order fluxes. Evaluating
the integral leads to the phase [ ]
QD
QD ? 1 ?
( )
? (f) = 2?ft ? + ?QD + ?2 QDref ref 0 + ? v
2 +O v4 , (2.63)
v5 v2 2
103
with the coefficie{nts ( ) ( ) ( )2
QD ? G12 ? S S V V q1 q2?0 = 96 + 5 f1/r + fv2 + 10 f1/r + f[v2 + 40f? ?4096? ( m)1 m2 ]2
5 ( ) q1 q2
+ 336fE ? 672f? ? fT ? fT ? fT 2 ? + (? ? ? )2
168( ) 1/r2 v2/r v4 1 2m1 m2 }2
q1 q2
+ 16 X2[ +X1 + 20f?(?
2
1 ? ?]2) + 16 (X
2
2?1 +X1?2) , (2.64a)
m(1 m2 )2
QD ? 5G12 q1 q2? 2?2 = 2{ ? + (?1 ? ?2) , (2.64b)7168? m1 m2 [ ( ) ]2
QD ? 5G12 ? q1 q2?2 = 32256fE + 48? 20f 2E ? ? 10fE (?1 ? ?2)1(548288? ) ( m1 m2 )
? [67(2f
T
? + f1/r)2 + f
T T
v2(/r + fv4 ?) 672f? + f
T
1/r2 + f
T T
v2/r + f(v4 ? 336fE)2
? 5 fS S q1 q21/r(+ fv2 + 10 f
V
1/r)+ f
V
v2 + 20f?(?
2
1 ? ?2) ]+ 40f? ?m1 m22
q1 q2
+[16(X2 +X)1 + 16 (X]2?1 +X1?2)
2
m1 m2 }
2
5 q q ( )21 ? 2+ 2 + (?1 ? ?2)2 672f? + fT T T
224 m m 1/r
2 + fv2/r + fv4 ,
1 2
(2.64c)
where the coefficients f are given by Eqs. (B.36), (B.56), (B.79), and (B.85). When
the charges are zero, this phase reduces to the next-to-leading order GR result, i.e.,
?QD0 ? 3/128?, ?QD2 ? 5(743 + 924?)/32256?, and ?QD?2 ? 0.
For the dipole-driven (DD) case, we take the tensor flux at the same order as
the scalar and vector fluxes, i.e., to O(x5). Evaluating the integral in (2.59) leads
to
?DD [ ]
?DD(f) = 2?ft ? ? + 0 1 + ?DDv2ref ref +O(v4) , (2.65)
v3 2
104
where the c[oefficients are given by( ) ]2 ?1
?DD
G12 q1 q2 2
0 = [ 2 ? + (?1 ? ?2)] , (2.66a)? ( m1 m2? )2 ?1 [DD 9 q1 q2 ( ) ( )?2 = 2 ? + (?1 ? ?2)2 96 + 10 fV + fV2 + 5 fS + fS210 m 1/r v1 m2 ( 1/r v )2
? q1 q210(fE ? 2f?)(?1 ? ?2)2 +(16 (X2?1 +X
2
1?)2) ]? 20(fE ? 2f?) +m1 m22
q1q1 q1 q2
+ 80(fE ? 2f?) + 16 X2 +X1 . (2.66b)
m1m2 m1 m2
When we set the electric charges to zero, but keep the scalar charges nonzero, this
result agrees with the (ST) result derived in Ref. [2].
We wish to understand how well a GW signal produced in EMd theory [e.g.
Eq. (2.63)] can be distinguished observationally from a signal in GR. Answering this
question definitively falls beyond the scope of this work. To perform such a study,
one would need to perform a Bayesian hypothesis test on injections of EMd signals
into detectors with realistic noise, comparing the relative evidence that the signal
matches template waveforms in either EMd theory or GR; for examples of such
analyses for other modifications to GR, see Refs. [120, 175, 213?215]. Instead of this
detailed study, we compute two comparatively simple measures of distinguishability:
the difference in total phase, and, in Sec. 2.3.4, the number of ?useful? GW cycles.
To compare the phase calculated in EMd theory with that in GR, we need to
align the waveforms and then compute dephasing from this alignment point. We
choose to do the alignment around the ?merger frequency,? which for simplicity we
choose to be the innermost-stable circular orbit (ISCO) frequency f ?3/2ISCO = 6 /?M
for a Schwarzschild BH. Next, we determine tref and ?ref such that the waveform
105
104
103 0.4, 0.10.1, 0.01
102 0.01, 0.01
101 0.001, 0.002
100
10-1
10-2
10-3
10-4
10-5
10-6
0.15 0.20 0.25 0.30 0.35 0.40
Figure 2.9: Phase difference in radians between EMd theory and GR as
a function of v, computed in the quadrupole-driven regime, for various
charge-to-mass ratios, and for equal masses (? = 1/4).
reaches a local maximum at this point and the phase reaches some fixed value, e.g.,
zero. To satisfy these two conditions, one can choose tref and ?ref such that at fISCO,
d?(f)/df = 0 and ?(f) = 0. For the QD case, this leads to
( )
QD ?10/3 2/3 4/3tref = 108MG12 ( 10G
QD
12 ?0 +G ?
QD + 84?QD12 2 ?
? )2 ,
QD ?7/3 2/3 QD 4/3? = 12 6G 8G ? +G ?QD QDref 12 12 0 12 2 + 60??2 . (2.67)
Similarly, for the DD case, we get
( )
2/3
tDDref = 6M?G?2 DD DD12 ?0 1(8 +G12 ?2 ,
DD 2
)
2/3
?ref = 4 G
?1?DD 9 +G ?DD . (2.68)
3 12 0 12 2
In Fig. 2.9, we plot the difference between the phase calculated in EMd theory
with a = 1 and the phase when all charges are zero, which is the phase in GR
up to 1PN order. For the configurations considered here, v = 0.15 corresponds to
approximately 10 Hz for a 20M system. Because the charges are relatively small,
106
we compute the phase using Eq. (2.63). For systems whose components? charge-
to-mass ratio qi/mi . 0.01, the two waveforms differ by less than one radian over
the frequency range of a ground-based GW detector. The phase difference does not
depend strongly on the value of a; for values of a ? 1000 and charge-to-mass ratios
qi/mi . 10?3 analogous to those considered in Ref. [199], the phase difference agrees
with that shown in Fig. 2.9 within 10%.
2.3.4 Number of useful gravitational-wave cycles
The total number of GW cycles between frequencies fmin and fmax is given by
? fmax df d?
Ntot = , (2.69)
f 2? dfmin
where ? is the gravitational wave phase. The instantaneous number of cycles spent
near some frequency f is defined by multiplying the above integrand by f
? f d?N(f) . (2.70)
2? df
However, GW detectors are not equally sensitive to all parts of the waveform because
the noise spectral density of the detector is frequency dependent. A better proxy
for how observationally different two waveforms are is to compare the number of
?useful? cycles in each. This measure was originally introduced in Ref. [216]. One
computes the total phase accumulated in each frequency bin and then weights this
estimate by the sensitivity of a detector at that frequency. Because the strain
sensitivity of the detector is concentrated in just a window of frequency space, the
result would also depend on the mass of the system. The number of useful cycles is
107
defined by [216]
[? f ] [? ]?1max df fmax df
Nuseful(f) ? w(f)N(f) w(f) , (2.71)
f f fmin fmin
where the weight w(f) ? A2(f)/fSn(f), while A(f) is the GW amplitude, and Sn(f)
is the noise spectral density of the detector. We use the zero-detuned high-power
noise spectral density of Advanced LIGO at design sensitivity [217].
Using the balance equation dE/dt = ?F , and the relation between the GW
phase and orbital frequency ?? = ?, the instantaneous number of cycles in Eq. (2.70)
can be reformulated as
v4 E ?(v)
N(f) = ? , (2.72)
3?MG12 F(v)
which can be computed in the quadrupole-driven regime using Eq. (2.62). For
the GW amplitude, we used the Newtonian order approximation for the transverse-
traceless polarizations A(f) ? v2, since the effect from the amplitude on the number
of cycles is small compared to the phase. We can then calculate numerically the
number of useful cycles using Eq. (2.71).
In Fig. 2.10, we show the relative difference between Nuseful in EMd theory
with a = 1 and the same quantity when all charges are zero (GR to 1PN order).
The number of cycles in EMd theory is less than in GR except for equal charges,
because the leading dipole radiation dominates the Newtonian order corrections to
the binding energy. We find that for systems with qi/mi ? 0.1, the number of useful
cycles in GR and EMd differs by O(1).
The quantity plotted in Fig. 2.10 provides a rough estimate of the observable
size of deviations from GR relative to the overall GW signal strength. We recast
108
this quantity in terms of the optimal signal-to-noise ratio (SNR) of the waveforms,
defined by
? fmax 2
SNR2
|A(f)|
= 4 df . (2.73)
f Sn(f)min
Using Eq. (2.53), this relation can be rewritten as
? fmax
SNR2
df
= 4 w(f)N(f), (2.74)
f fmin
and thus
( ) ( )
| q=0 ? | | (2 q=0 ) ( )? 2 | | | ( )N Nuseful SNR SNR 2useful 2 ?SNR ?SNRq=0 =N 2 q=0 = +O ,SNR SNRuseful SNR
( ) (2.75)
where, ?SNR = SNRq=0 ? SNR is the difference in SNR between signals in GR
and EMd theory. Thus, Fig. 2.10 indicates that corrections arising from the presence
of electric and scalar charges in EMd theory can account for only a few percent of
the total SNR for systems with electric dipole ? 0.1.
2.4 Effective-one-body framework
In this section, we construct two EOB Hamiltonians: one based on the GHS
metric Eq. (2.8), in which the potential C(r) 6= 1, which we call the GHS gauge; the
other is based on an approximation to this metric by making a transformation to a
gauge were the potential C(r) = 1, which we call the Schwarzschild gauge.
The EOB Hamiltonian in the GHS gauge is more physical in the strong-gravity
regime since it exactly reproduces the test-body limit of the two-body dynamics.
109
100
10-1 0.4, 0.1
10-2 0.1, 0.01
10-3
10-4 0.01, 0.01
10-5
0.001, 0.002
10-6
20 40 60 80 100
Figure 2.10: Number of useful cycles versus the total mass for various
charge-to-mass ratios, and for equal masses (? = 1/4). The number of
cycles in EMd theory is less than in GR except for equal charges.
That is, it belongs to a class of Hamiltonians implementing exact solutions to the
field equations for isolated objects/BHs. However, this class of Hamiltonians is very
theory specific ? for example the analytic ST vacuum metric in Refs. [218, 219]
is distinct from the analytic EMd metric when we set the electromagnetic fields to
zero. In addition, many BH solutions in alternative theories do not even have an
analytic solution that can be used. The advantage of using a Hamiltonian based on
the approximate metric in the Schwarzschild gauge, is that it is easier to implement
in data-analysis studies of GWs observed by LIGO and Virgo. One would take the
existing EOB Hamiltonians in GR as a starting point and add EMd corrections in
the same way as, e.g., tidal corrections are added [220]. Within the regime of small
deviations from GR, the two EOB Hamiltonians in EMd theory are expected to
closely agree.
In Refs. [219, 221], the EOB framework was extended to ST theories. In
110
Ref. [219], the motion of a binary BH was mapped to the motion of a test body, such
that the effective metric is a ?-deformation of the ST metric. This approach is similar
to our EOB Hamiltonian in the GHS gauge, but we find a different mapping for the
scalar charge. In Ref. [221], the motion of the binary in ST theory was mapped to
the motion of a test body around an effective BH in GR, but the effective metric
does not reproduce exactly the test-body limit of ST theory. In contrast, whereas
our EOB Hamiltonian in the Schwarzschild gauge is also not exact in the test-body
limit, it still maps the real problem to an effective one in EMd theory (not in GR).
2.4.1 Effective-one-body Hamiltonian in Garfinkle-Horowitz-Strominger
gauge
In the EOB framework, the motion of a binary is mapped to the motion of
a test body in the background of an effective metric. In the effective problem in
EMd theory, we assume that a test body, with mass ? and electric charge q, is
moving in the background of a charged BH with mass M and electric charge Q.
To relate the real two-body problem to the effective one, we impose the following
conditions: (a) M and ? are the total mass and reduced mass of the real description,
i.e., M = m1 +m2 and ? = m1m2/M ; (b) the effective charges Q and q are related
to the real charges by Qq = q1q2, but we do not assume that Q is the total charge;
and (c) the mapping between the real and effective Hamiltonians takes the form
[ ]
HNReff (R,P ) H
NR(r,p) ? HNR(r,p)
= 1 + , (2.76)
? ? 2 ?
111
where the superscript NR means non-relativistic, i.e., HNR = H ?M , and the real
Hamiltonian H is given by Eq. (2.37). The form (2.76) for the ?EOB energy map?
[38] has proven useful in GR up to 4PN order [222], in classical electrodynamics
to 2PN order [223], and in ST gravity to 2PN order [221, 224]. In the first post-
Minkowskian approximation, i.e., to all orders in v/c at linear order in G, it can be
shown to exactly resum the dynamics, producing the arbitrary-mass-ratio two-body
Hamiltonian from the test-body Hamiltonian [224, 225]. For the coordinates in the
effective problem, we use uppercase letters, such as R and P , while for the real
problem, we keep using lowercase letters, such as r and p.
The effective action for the test body is given by
?
Seff = [?m(?) d?eff + qA?dX?] , (2.77)
where ?eff is the proper time of the BH and the effective test-mass m(?) depends
on the scalar field ? generated by the BH, and has the expansion in terms of the
parameters ? and ? as
[
1 ( )]
m(?) = ? 1 + ??+ (?2 + ?)?2 +O 1/c6 . (2.78)
2
Since we do not know, a priori, how the parameters ? and ? of the effective test
body are related to the real problem, we expand the mass in a 1/R expansion
[ ( )]f1 f2
m(R) = ? 1 + + +O 1/c6 (2.79)
R R2
and solve for the unknown coefficients f1 and f2.
We take the effective metric of the background to be a deformation of the EMd
112
metric in the GHS gauge
ds2 2eff = ?d?eff = ?A(R)dT 2 +B(R)dR2 +R2C(R)d?2 , (2.80)
with
( )( ) 1?a2
R R 1+a2+ ?
A(R) = 1? ( 1)? , (2.81a)R R
1 b1
B(R) = ( 1 + , (2.81b)A(R) ) R2a2
R 2? ?
1+a
C(R) = 1 , (2.81c)
R
where R? and R+ are the radii of the inner and outer horizons of the effective BH,
which are given by Eqs. (2.16a) and (2.16b), i.e.,
1 + a2 2
R? = D , R+ = 2M ?
1? a
D . (2.82)
a a
We choose to define R? and R+ by these relations in terms of D, but not in terms of
Q, because the relation between Q and D is deformed by the mapping. We note that
in the above metric?s ansatz, we have added a deformation to B(R) only because,
in EMd theory at 1PN order, the mapping leads to three equations in f1, f2, and
any deformation to the metric. Thus, we can only determine uniquely one unknown
coefficient in the effective metric. So we choose to take that coefficient to be b1, and
assume the possible deformations to A(R) or C(R) to be zero at 1PN order.
The scalar field for a single BH is given by Eq. (2.13); we add a PN deformation
g2/R
2 such that the effective scalar field is given by
( )
a ? R? 1 + a
2 g2
?(R) = ln 1 + . (2.83)
1 + a2 R a R2
113
The electric potential is given by
Q
A0(R) = ? . (2.84)
R
We do not add PN corrections to A0 because those corrections can be absorbed in
the PN corrections to the scalar field or to the relation between D and Q. The
coefficient g2 is not independent of f1 and f2, because the mass expansion can also
be expanded directly in ? [see Eq. (2.7)]
[ ( ) ]
? D? 1 D
2? a 1 1 ( )
m(R) = ? 1 + g 2 2 2 2 6
2 2
?? ? D ? + D ? + D ? +O 1/c .
R R 2a 2 2 2
(2.85)
In what follows, we uniquely solve for the coefficients b1, f1, and f2 by matching the
real Hamiltonian to the effective one by a canonical transformation. Matching the
two mass expansions in Eqs. (2.79) and (2.85) allows us to determine the mapping
for the parameters ? and ?, and for the coefficient g2. The mapping for ? is unique,
but the mapping for ? and g2 is not unique at 1PN order.
To find the effective Hamiltonian, we first find the effective Lagrangian, in the
equatorial plane ? = ?/2, ?
? ?
Leff = qA0 ?
dX dX
m(?)??g?? ,dT dT
= qA0 ?m(?) A(R)?B(R)R?2 ? C(R)R2??2. (2.86)
Then, applying the Legendre transformation Heff = PRR? + P??? ? Leff yields the
effective Hamiltonian ? [ ]
P 2 P 2
Heff = ?qA0 + A(R) m2(?) + ? + R , (2.87)
C(R)R2 B(R)
114
where P? = ?Leff/??? is the angular momentum, and PR = ?Leff/?R? is the radial
momentum.
Before matching the Hamiltonians, we need to apply a canonical transforma-
tion from the real variables, r and p, to the effective ones, R and P . At 1PN order,
this transformation is given by [38]
i i ?G1PN ?G1PNR = r + , Pi = pi ? , (2.88)
?p ?rii
with the generating function
( c )2
G1PN(r,p) = (r ? p) c p21 + , (2.89)
r
where the coefficients c1 and c2 are to be determined by the mapping.
Inserting the expansions of the real and effective Hamiltonians into Eq. (2.76),
and applying the canonical transformation, we obtain the five equations:
2c 21? + ? = 0 , (2.90a)
f1 +M?1?2 = 0 , (2.90b)
M ? c2 + ?+ ??1?2 ?
qQ
+ aD + c 21M? ? ?c1qQ? ?c1f1? = 0 , (2.90c)
M
qQ
b1 + + 2M + 2aD + 4c1?qQ+ 4c
2
1? f1 ? 2c2 ? ?? ??1?2 ? 4c1M?2 = 0 ,
M
(2.90d)
q2Q2
+ q22X1 (1 + a?1) + q
2
1X2 (1 + a?2)? 2?c2 + 4?f1 + 2?c2f1 ? ?f 2M2 1
? ? 2D? D
2
2?f(2 + 2M? + 2aD?? ?D2 + ? + 4M??1?2 + 2M??2a a2 1?)22
? c2 f1+ qQ 2 + 2 ? 2 ? 2a?1X1 ? 2a?2X2 ? 2?1?2 ? 2? ? 2??1?2
M M
+X ?? ?2 +X ?? ?2 + ?2 + 2?22 1 2 1 2 1 ?1? + ?
2?22 1?
2
2 = 0 . (2.90e)
115
Solving these equations respectively for the coefficients c1, f1, c2, b1, and f2 yields
?
c1 = ? , (2.91a)
2?2
f1 = ?M?1?2 , (2.91b)
M? 1
c2 = M + + M??1?2 ?
qQ?
+ aD , (2.91c)
2 2 2?
b1 = 0 , [ (2.91d])
D2 ? D
2
? MD ? aqQD M q
2 q2
f2 = [ aMD?1?2 + +
2
] (1 + a? ) +
1
1 (1 + a? )
2a2
2
2 a ( ? 2 )m2 m1
? 2 ? 1 1M ? ? (? ? )2 ? X ?2? +X ?2 q1q21 2 1 2 2 2 1 1 1?2 ? aM (X1?1 +X2?2) .2 2 ?
(2.91e)
To find the mapping of the scalar charge, we identify the mass expansion in
Eq. (2.79) with the expansion in Eq. (2.85) to give
?D? = f1 , (2.92a)
2
? D ? ? a 2 1g ? D ? + D2?2 12 + D2? = f2 . (2.92b)
2a 2 2 2
Inserting the solution for f1 and f2 gives a unique mapping for ?
M
? = ?1?2 , (2.93)
D
and suggests the following mapping for ?
M2 ( )
? = X 2 22?2?1 +X1?1?2 . (2.94)D2
Further, we take the mapping of the dilaton charge D of the effective BH to be the
sum of the asymptotic value of the scalar charges of the two bodies, i.e.,
D = m1?1 +m2?2 . (2.95)
116
The mapping for ? and ? agrees with what was found in Ref. [219], but the mapping
for D is different. The reason we choose this mapping for D is that it leads to a
simple deformation to the scalar field
?1? a
2 (? 21 ? ?2)
g2 = DM? . (2.96)
2a2 ?1?2
This deformation vanishes in the test-mass-limit ? ? 0, and also when a = 1 or
?1 = ?2. Other choices for D lead to complicated expressions for g2. In obtaining
this result for g2, we use the expression for the electric charge in terms of the scalar
charge, which is valid for BHs only,
q2 2 1? a2i = ? ? ?2 . (2.97)
m2
i
i a a
2 i
This relation follows from Eq. (2.20) after solving for qi in terms of ?i and setting
the scalar field to its asymptotic value.
A convenient mapping for the electric charge is
( )
2 q
2
1 q
2
Q = M + 2 . (2.98)
m1 m2
The reasoning behind this choice is that it is symmetric under the exchange of the
two bodies; it has the correct test-body limit, Q ? q1 when m2/m1 ? 0 with
q2/m2 held constant; and it appears naturally in EM theory as we show in the next
subsection. With that mapping for Q and D, the relation between them is given by
2 2M ? 1? a
2
2 ? 1? a
2
Q = D D (?1 ? ? 2 22) M ? . (2.99)
a a2 a2
One could choose to enforce Eq. (2.17) for generic masses by making a different
choice for D or Q, but this seems to lead to very complicated expressions for them.
117
2.4.2 Effective-one-body Hamiltonian in Schwarzschild gauge
In the EMd metric, the potential C(r) 6= 1, but the standard EOB gauge
is the Schwarzschild gauge C(r) = 1. This is the gauge that was used to derive
the original EOB Hamiltonian [38], which was then improved by calibrating it to
numerical-relativity simulations [226]. Therefore, to profit from the best available
EOB Hamiltonian in GR, we need to construct an EMd-EOB Hamiltonian that is
also in the Schwarzschild gauge.
The EMd metric can be transformed to the Schwarzschild gauge by the coor-
dinate transformation r?2 = r2C(r). However, for arbitrary values of the coupling
constant a, the metric cannot be analytically transformed. Instead, we expand
the EMd metric (2.8) and transform it to get an approximate EMd metric in the
Schwarzschild gauge. We make the c[oordinate trans]formation, valid to 1PN order,
2
2 2 ? 2a r?r? = r 1 ,
(1 + a2)r
a2? r = r? + r? = r? + aD . (2.100)
1 + a2
With that transformation, and inserting the expressions for r? and r+ in terms of
M and Q [Eqs. (2.(16a) and (2.16b))], we ge(t )
2M Q22 ? ? 2 2Mds = 1 + dt + 1 + dr?2 + r?2d?2, (2.101)
r? r?2 r?
which is the same as the Reissner?Nordstro?m metric to 1PN order.
As an ansatz for the effective metric, we assume a metric based on the approx-
imate metric (2.101)
ds2eff = ?A(R)dt2 +B(R)dR2 +R2d?2 , (2.102)
118
with
a1 a2
A(R) = 1 + + + . . . , (2.103a)
R R2
b1
B(R) = 1 + + . . . , (2.103b)
R
and we write the mass expansion[as
f1 f2 ( )]
m(R) = ? 1 + + +O 1/c6 , (2.104)
R R2
where the unknown coefficients a1, a2, b1, f1, and f2 are to be determined by the
mapping. However, the mapping leads to three equations in those five coefficients,
making two of them arbitrary. We choose to take a1 = ?2M and a2 = Q2 so that
the effective metric would agree with the EMd metric in the Schwarzschild gauge
to 1PN order. When we solve for b1, we get b1 = 2M , in agreement with the EMd
approximate metric.
For the effective electric potential, we apply the coordinate transformation
(2.100) with r? = R to get
Q
A0(R) = ? . (2.105)
R + aD
Applying the same transformation to the scalar field, and adding a PN deformation
g /R22 , we obtain [ ]
a 1 + a2 D 1 + a2 g2
?(R) = ln 1? + . (2.106)
1 + a2 a R + aD a R2
The mass expansion in terms of ?, Eq. (2.78), can now be written as an expansion
in 1/R by [ ( )
D? 1 D2? a 1 1 ( ) ]
m(R) = ? 1? + g ?? + D2? + D2?2 + D2? +O 1/c6 .
R R2
2
2a 2 2 2
(2.107)
119
Following the same method used in the previous subsection, the effective
Hamiltonian is given by Eq. (2.87) with the potential C(R) = 1. The relation
between the real and effective Hamiltonians is given by Eq. (2.76), and the canoni-
cal transformation that relates the real and effective variables is given by Eq. (2.88).
Matching the real and effective Hamiltonians, we obtain the five equations:
2c 21? + ? = 0 , (2.108a)
a1 + 2f1 + 2M(1 + ?1?2) = 0 , (2.108b)
2c2 ? a1 ?
qQ
4M + c1a
2 2
1? + 2f1c1? + 2 ? 2?(1 + ?1?2 ? c1qQ) = 0 , (2.108c)
M
b1 ?
qQ
2c2 + 4c1?qQ+ 2a c ?
2
1 1 + 4c f
2
1 1? ? ?? ??1?2 + = 0 , (2.108d)
M
M2?2?2 +M2X ? ? +M2X ?2? ? a ? 2M2 2aqQD1 2 2 2 1 1 1 2 2 ?1?2 + ?
m 2 21q m2q qQ
+ 2 (1 + a?1) +
1 (1 + a?2)? 2a (m1?1 +m2?2)? 2f2 = 0 .
? ? ?
(2.108e)
Solving these equations respectively for the coefficients c1, f1, c2, b1, and f2 yields
c1 = ?
?
, (2.109a)
2?2
f1 = ?M?1?2 , (2.109b)
M? 1 qQ?
c2 = M + + M??1?2 ? , (2.109c)
2 2 2?
b1 = 2M , [ (2.109]d)
a2 aq1q2D q1q M q
2 q22
f2 = ? + ? a (m ? +m ? ) + 2[ 1 1 2 2 ](1 + a?1) +
1 (1 + a?2)
2 ? ? 2 m2 m1
? 2 ? 1 1
( )
M ?1?2 (?
2 2 2
1?2) ? X2?2?1 +X1?1?2 . (2.109e)2 2
Choosing a 22 = Q , so that the effective metric agrees with the EMd metric to
120
1PN order, the above solution for f2 leads to the mapping( )
q2 q2
Q2 = M 1 + 2 . (2.110)
m1 m2
This is because, for the case of EM theory, when we take the parameters ? and ?
in the solution for f2 to be zero, we get f2 = ?a2/2 +M(q2/m + q21 1 2/m2)/2. Hence,
requiring that f2 = 0 in EM theory and that a
2
2 = Q , naturally leads to the charge
map (2.110).
Identifying the mass expansion in Eq. (2.104) with that in Eq. (2.107), leads
to the following mapping for ? and ?
M
? = ?(1?2 , ) (2.111)DM2
? = X ?22 2? +X ?
2
1 1 1?2 , (2.112)D2
which is the same mapping that was found in the previous subsection. Further, tak-
ing the mapping of the dilaton charge to also be given as in the previous subsection
D = m1?1 +m2?2, (2.113)
leads to the astonishingly simple result
g2 = 0. (2.114)
With that mapping for D and Q, the relation between them is given by Eq. (2.99).
Interestingly, the above mappings also lead to a ST EOB Hamiltonian in
Schwarzschild gauge at 1PN order. A 2PN EOB Hamiltonian based on an ex-
act analytic solution for the metric and scalar field can be found in Ref. [219]. The
metric in that work also includes a potential C(R) =6 1, Eq. (II.3) in Ref. [219], and
121
that metric is unrelated to the EMd metric when the electric charges are zero. The
scalar field is given by
[ ]
D a? a
2 ? 2Ma?
?ST = log 1? + ? , (2.115)
a? r 2r2
where a2? = 4(M
2 + D2). The author of Ref. [219] found the same mapping for ?
and ? that we got, but used a different mapping for D (at 2PN order). When we
approximately transform the metric and the scalar field to the Schwarzschild gauge,
in which the potential C(R) = 1, and repeat the same analysis in this section, we
get an EOB Hamiltonian with the same mapping for the scalar charge given in
Eq. (2.113), and with no deformation to the metric or the scalar field to 1PN order.
The point is that the mapping of the scalar charge would be the same in EMd theory
and ST theory, which is another hint that Eq. (2.113) is a good choice at 1PN order.
2.4.3 Comparison of two effective-one-body Hamiltonians in Einstein-
Maxwell-dilaton theory
In this subsection, we compare the two EMd-EOB Hamiltonians with each
other, and also with the EOB Hamiltonian in GR, by calculating the binding energy
and the ISCO. The goal is to investigate the range of parameter space where the
two EMd-EOB Hamiltonians agree.
The mappings of the electric charge, scalar charge, and the parameters ? and
122
0.015 0.015
0.010 0.010
0.005 0.005
0.000 0.000
-0.005 -0.005
-0.010 -0.010
-0.015 -0.015
0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15
0.015
0.01
0.010
0.00
0.005
0.000 -0.01
-0.005 -0.02
-0.010
- -0.030.015
0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15
Figure 2.11: Binding energy EB normalized by the total mass M as
a function of M? for equal masses, ? = 1/4, and for charge-to mass-
ratios q1/m1 = q2/m2 = 0.99, 0.9, 0.5, and q1/m1 = ?q2/m2 = 0.9. To
improve readability, we show the plots only up to the frequency corre-
sponding to R = 1.05RLR or to energy EB/M = 0.015. The point on
each curve indicates the location of the ISCO.
? are the same for the two(EMd-EOB)Hamiltonians, i.e.,
2 q
2 q2
Q = M 1 + 2 , D = m1?1 +m2?2 ,
m1 m2
M M2 ( )
? = ?1?2 , ? = X2?
2
2?1 +X
2
1?1?2 . (2.116)D D2
For the EOB Hamiltonian in the GHS gauge, the effective metric is the GHS met-
ric for ? = 0 [Eqs. (2.80)?(2.82) with b1 = 0]. For the EOB Hamiltonian in the
Schwarzschild gauge, the effective metric agrees with the Reissner-Nordstro?m met-
ric for ? = 0 [Eq. (2.101)]. Other differences between the two Hamiltonians are in
123
the parameters of the mass expansion (2.79), the canonical transformation (2.89),
and the correction to the scalar field [Eqs. (2.83) and (2.106)]. The parameters in
those equations are shown in Table 2.1.
Table 2.1: Difference between the two EOB Hamiltonians in terms of the effective
metric and the parameters of the mass expansion, the canonical transformation, and
the scalar field.
EOB in GHS gauge EOB in Schw gauge
effective metric Eqs. (2.80)?(2.82) Eq. (2.101)
c1 c1 = ??/2?2
c2 Eq. (2.91c) Eq. (2.109c)
f1 f1 = ?M?1?2
f2 Eq. (2.91e) Eq. (2.109e)
g2 Eq. (2.96) g2 = 0
0.13
0.068 0.12
0.11
0.066 0.10
0.09
0.064
0.08
0.062 0.07
-1.0 - 0.060.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
Figure 2.12: Angular frequency at ISCO as a function of the charge-
to-mass ratio q1/m1 from -0.99 to 0.99. In the left panel, q2/m2 =
q1/m1, while in the right, q2/m2 = ?q1/m1. An ISCO frequency of
0.062 corresponds to an ISCO radius ? 6.4M , and a frequency of 0.13
corresponds to radius ? 3.9M .
To find the binding energy from the two EOB Hamiltonians, we start with the
energy map in Eq. (2.76), which gives the relation between the effective Hamiltonian
and the real Hamiltonian. Inverting that relation, we obtain the resummed EOB
124
Hamiltonian ? ( )
Heff
HNREOB = M 1 + 2? ? 1 ?M . (2.117)?
To obtain the binding energy for circular orbits, we set PR = 0, and solve P?R =
??Heff/?R = 0 for the angular momentum P?. However, that equation cannot be
solved analytically because of the non-linearity of the Hamiltonian. Hence, we solve
the equation numerically for P? at specific values of R. Since we want to plot the
binding energy as a function of the orbital frequency ?, we need to calculate the
orbital frequency via
?HEOB ?HEOB ?Heff
? = = . (2.118)
?P? ?Heff ?P?
Then, we calculate the binding energy and orbital frequency as R goes from 100M to
the radius of the light ring. The light ring (or photon orbit) of a (charged) BH metric
in GR is defined as the circular-orbit solution to the geodesic equation of massless
particles. This geodesic equation is actually encoded by our effective Hamiltonian
if we set q = 0 (geodesic motion) and ? = 0 (massless particle). To obtain the light-
ring solution in EMd theory, we hence take the effective Hamiltonian for the case
? = 0 = q, and impose the conditions for circular orbits PR = 0 and P?R = 0. The
latter condition means that we look for an extremum of the effective Hamiltonian,
?
? ?Heff
?
0 = P? = ?R , (2.119)
?R ??=q=PR=0
which is actually a maximum, ?2H 2eff/?R < 0, and the light-ring solution is therefore
unstable. For the Schwarzschild metric in GR, solving this equation for R gives the
125
known value RLR = 3M . For the EMd metric in the GHS gauge
3 aD 1 [ ]
R = M + + 9a2M2 ? 16aMD + 6a3MD + 8D2 ? 1/2LR 8a2D2 + a4D2 ,
2 2 2a
(2.120)
while for the approximate metric in the Schwarzschild gauge
1 [ ? ]
RLR = 3M + 9M2 ? 8Q2 , (2.121)
2
which is the same as the Reissner-Nordstro?m metric since the potential A(R) is the
same in both cases.
In Fig. 2.11, we plot the binding energy scaled by the total mass, EB/M , versus
the orbital frequency M? for equal masses, ? = 1/4, and for charge-to mass-ratios
q1/m1 = q2/m2 = 0.99, 0.9, 0.5 and q1/m1 = ?q2/m2 = 0.9. The binding energy
diverges at the light ring; to improve readability, we show the plots only up to the
frequency corresponding to R = 1.05RLR or to energy EB/M = 0.015. We plot
the binding energy for four cases: (a) EMd-EOB Hamiltonian in the GHS gauge;
(b) EMd-EOB Hamiltonian in the Schwarzschild gauge; (c) EMd-GHS Hamiltonian
with a = 0, which is EM theory; and (d) EMd-GHS Hamiltonian in the limit where
all charges are zero Q = 0, which is the standard uncharged GR case. [The effective
Hamiltonian for case (c) is that of a charge moving in the Reissner-Nordstro?m
spacetime, and for (d) it is that of a reduced mass in Schwarzschild spacetime.]
The difference between the EM case (a = 0 curve) and the standard astrophysical
scenario of uncharged BHs (Q = 0 curve) quantifies the effect of the electric charges,
while the difference between the EMd Hamiltonian(s) and the EM case quantifies
the effect of the scalar charges.
126
We see from Fig. 2.11 that the electric charges have a larger effect on the
binding energy than the additional scalar charges in EMd theory (except for almost
extreme charges). For small electric charges . 0.5 (lower left panel of Fig. 2.11),
the difference in binding energy between EMd theory and EM theory at the ISCO
is only 9% of the difference between EMd theory and GR with no charges, i.e., the
scalar charge has a very small effect. The difference between the two EMd-EOB
Hamiltonians increases with increasing electric charge and frequency, but they still
agree well. The binding energy of the two Hamiltonians at the ISCO differs by ? 6%
for charge-to-mass ratio 0.99 and by? 0.1% for charge-to-mass ratio 0.5. For charge-
to-mass ratios larger than one, a naked singularity appears in the effective metric
in the Schwarzschild gauge; this is an unphysical feature arising from the choice
of gauge, and thus the EOB Hamiltonian should not be used for small separations
(high frequencies) approaching this singularity. Note that, if one is only interested
in the inspiral, then the comparison of the Hamiltonians via the binding energy can
be stopped already at the ISCO frequency instead of the LR frequency.
The ISCO marks the end of the inspiral phase of the binary coalescence and
the beginning of the plunge. To find the value of the ISCO, we set both the first and
second derivatives of the effective Hamiltonian to zero ?H /?R = 0 = ?2eff Heff/?R
2
and set PR = 0. Then, we solve the two equations numerically for the ISCO radius
and angular momentum. The orbital frequency at ISCO can then be calculated
from Eq. (2.118).
In Fig. 2.11, the location of the ISCO is indicated by the point on each curve.
In Fig. 2.12, we plot the orbital frequency at ISCO, scaled by the mass, i.e., M?ISCO,
127
versus the charge-to-mass ratio q1/m1 with q2/m2 = q1/m1 in the left panel, and
q2/m2 = ?q1/m1 in the right. From the left panel, we see that for high charge-to-
mass ratios, the two EOB Hamiltonians do not agree well at this high frequency.
For same-sign charges, the ISCO orbital frequency is lower than the uncharged case,
which means the ISCO radius is greater than the Schwarzschild value of 6M . This is
because the binding energy of charged BHs is higher (less bound) than the energy of
uncharged BHs, as can be seen from the binding energy in Fig. 2.11. For opposite-
sign charges, the ISCO orbital frequency is higher than the uncharged case because
the binding energy is lower than the energy of uncharged BHs.
2.5 Conclusions
In this work, we analytically modeled the dynamics of binary BHs in EMd
theory. In this theory, electrically charged BHs also carry a scalar charge, whereas
in GR (and many modified theories of gravity) the scalar charge is zero. Thus,
the identification of a BH with scalar charge through GW observations could point
to modifications of gravity in the strong-field regime and violations of the strong
equivalence principle. Observation of a large electric charge on BHs could be a trace
of minicharged dark matter and/or dark photons.
We began by considering the case of a test BH in the background of a more
massive companion in EMd theory, wherein the scalar charge of the test BH de-
creases as it moves radially inwards. Consistent with the results of Ref. [202], we
found that the dimensionless charge ?(?) exhibits a sharp transition [see Figs. 2.1
128
and 2.2]. However, we showed that in a binary system, the scalar charge of the test
BH will change dramatically only very close to the horizon of the background BH
and only if both BHs are nearly-extremally charged. While BHs are not expected
to be able to achieve or maintain enough electric charge within the Standard Model
of particle physics for these features to be observed, but proposed alternatives like
minicharged dark matter and dark photons models may allow such features to be-
come observationally relevant. Our study also showed that binary BHs in EMd
theory will not exhibit non-perturbative phenomena akin to induced or dynamical
scalarization that are found in certain ST theories [see Fig. 2.3].
We then used the PN approximation in EMd theory to study the dynamics of
a two-body system with an arbitrary mass ratio. We derived the two-body 1PN La-
grangian and Hamiltonian, and investigated how the bodies? scalar charges decrease
with their separation at next-to-leading PN order. As in the test-BH case, we expect
that dramatic changes could occur only for nearly-extremal charged BHs on very
compact orbits; this is a regime most easily probed by systems with extreme mass
ratios and/or rapidly spinning BHs. We derived the scalar, vector, and tensor energy
fluxes at next-to-leading PN order. From the energy flux and binding energy, we
calculated the Fourier-domain gravitational waveform for binaries on quasi-circular
orbits using the stationary-phase approximation.
Using our PN result, we discussed the possibility of constraining EMd theory
with GWs. Given current and projected constraints on dipole radiation, we exam-
ined how the degeneracies between electric and scalar charges limit the bounds that
can be set on the EMd parameter a ? constraining this parameter requires one
129
to measure the electric charges of each BH independently, and the strength of this
bound improves for larger total electric charge [see Fig. 2.8]. We also estimated the
observational deviations from GR predicted in EMd theory with two measures: the
dephasing between PN waveforms in the stationary-phase approximation [Fig. 2.9],
and the difference in the number of useful GW cycles [Fig. 2.10]. For ground-based
GW detectors, we found that the presence of electric and scalar charges contributes
. 1 radian to the phase provided the black holes have charge-to-mass ratios of
qi/mi . 0.01 for coupling constant a = 1. We showed that the relative difference in
useful cycles between EMd theory and GR provides an estimate of the fractional cor-
rection to SNR by non-GR corrections; for systems with qi/mi . 0.1, the deviations
from GR affect the total SNR by a few percent.
Finally, we constructed two EOB Hamiltonians for binary BHs in EMd theory:
an EOB Hamiltonian in the GHS gauge, which is based on the exact BH solution,
and an EOB Hamiltonian in the Schwarzschild gauge, which is based on an ap-
proximation to that solution. The EOB Hamiltonian in the GHS gauge is more
physical in the strong-gravity regime, since it exactly reproduces the dynamics of
a test body, and hence will be more accurate for systems with a very asymmetric
mass ratio. The EOB Hamiltonian in Schwarzschild gauge is easier to implement by
taking the existing EOB Hamiltonians in GR as a starting point and adding to it
corrections due to EMd theory. We compared the two Hamiltonians by calculating
the binding energy and the innermost stable circular orbit, and found that they
agree well, except for nearly-extremal charges at high frequencies [see Figs. 2.11 and
2.12]. The binding energy of the two Hamiltonians at the ISCO differs by ? 6% for
130
charge-to-mass ratio 0.99 and by ? 0.1% for charge-to-mass ratio 0.5.
An important goal in future continuations of our work would be the construc-
tion of a full (inspiral-merger-ringdown) EOB waveform model in EMd theory. For
accurate predictions in the late inspiral, one likely needs PN results for the Hamil-
tonian, fluxes, and modes to the same order as they are available in GR, next to a
calibration of the model to NR simulations in EMd theory. Modeling the merger and
ringdown requires predictions for the parameters of the final black hole and its quasi-
normal modes as a function of the EMd coupling constant a (see, e.g., Refs. [227, 228]
for partial results). Since EOB waveform models in existing data-analysis infras-
tructure are formulated in the Schwarzschild gauge, this gauge is probably the best
compromise for the purpose of GW data analysis. This gauge is also better suited
for creating a single EOB waveform model covering various alternative theories; for
example, we demonstrated that our EOB Hamiltonian in the Schwarzschild gauge
can describe both ST and EMd theories. Ultimately, one could aim to construct a
generalized EOB framework that uses a physically motivated parameterization to
encode a range of possible deviations from GR.
Acknowledgments
M.K. thanks the graduate school and the physics department at the University
of Maryland for the International Graduate Research Fellowship. He also thanks the
Max Planck Institute for Gravitational Physics for its hospitality and for co-funding
his stay during the completion of this work.
131
Chapter 3: Gravitational waveforms in scalar-tensor gravity at 2PN
relative order
Authors: Noah Sennett, Sylvain Marsat, and Alessandra Buonanno1
Abstract: We compute the gravitational waveform from a binary system in
scalar-tensor gravity at 2PN relative order. We restrict our calculation to non-
spinning binary systems on quasi-circular orbits and compute the spin-weighted
spherical modes of the radiation. The evolution of the phase of the waveform is
computed in the time and frequency domains. The emission of dipolar radiation
is the lowest-order dissipative process in scalar-tensor gravity. However, stringent
constraints set by current astrophysical observations indicate that this effect is sub-
dominant to quadrupolar radiation for most prospective gravitational-wave sources.
We compute the waveform for systems whose inspiral is driven by: (a) dipolar radia-
tion (e.g., binary pulsars or spontaneously scalarized systems) and (b) quadrupolar
radiation (e.g., typical sources for space-based and ground-based detectors). For
case (a), we provide compete results at 2PN, whereas for case (b), we must in-
troduce unknown terms in the 2PN flux; these unknown terms are suppressed by
constraints on scalar-tensor gravity.
1Originally published as Phys. Rev. D94, 084003 (2016).
132
3.1 Introduction
The observation of gravitational-wave (GW) events GW150914 and GW151226
by Advanced LIGO marks the dawn of GW astronomy [182, 184]. We expect to
observe several such events per year [73, 213, 229, 230] with the upcoming network of
ground-based detectors comprised of Advanced LIGO [231], Advanced VIRGO [232],
KAGRA [233], and LIGO-India [234]. These ground-based detectors can observe
binary systems containing neutron stars and/or stellar-mass black holes (with a
total mass M? 1? 100M); future space-based detectors like the proposed LISA
mission [235] will observe binary systems composed of intermediate-mass and/or
supermassive black holes (M ? 100 ? 107M). Gravitational-wave observations
allow us to not only measure the astrophysical properties of these systems but can
also be used to test general relativity (GR). Because the coalescence of a compact
binary system produces extreme gravitational fields that vary over short time scales,
observations of such events allow us to probe the highly-dynamical, strong-field
regime of gravity for the first time [73, 122].
The detection and analysis of GWs with ground-based detectors require banks
of very accurate template waveforms. The prospects of testing gravity with these
detectors hinge on our ability to model waveforms in both GR and alternative theo-
ries of gravity. Given a GW detection, one can adopt either a theory-independent or
theory-dependent approach to testing GR. A theory-independent test employs wave-
forms that deviate from a GR signal in some generic, parameterized manner (for
examples, see Refs. [117, 118, 213]). One compares an observed GW signal against
133
these template waveforms to constrain the deviations from GR. A theory-dependent
test instead uses waveforms predicted in a particular alternative theory of gravity,
comparing them against the detected GW to estimate the underlying physical pa-
rameters of that theory. Each approach has its advantages: theory-independent
tests can constrain a wide range of alternative theories while theory-dependent tests
can directly constrain the fundamental physics of an alternative theory. Both types
of tests were performed for GW150914 and GW151226 by the LIGO and Virgo
collaborations in Refs. [73, 122]. For a comprehensive review of proposed theory-
independent and theory-dependent tests, see Refs. [79, 88] and references therein.
In this work, we present waveforms in scalar-tensor theories of gravity suit-
able for theory-dependent tests of GR. In particular, we construct ready-to-use
waveforms for the inspiral of non-spinning binary systems accurate up to second
post-Newtonian (2PN) order, i.e., O ((v/c)4) beyond leading order.2 We restrict
our attention to systems on quasi-circular orbits, as binaries formed in the field
are expected to radiate away any initial eccentricity at frequencies too low to be
observable by GW detectors.
Scalar-tensor theories are amongst the most natural alternatives to GR. Specif-
ically, we focus on theories where a single massless scalar ? non-minimally couples
to the metric g?? . Written in the Jordan frame, the action for such theories is given
2We describe post-Newtonian (PN) corrections of order O(c?2n) as ?nPN,? which we also
abbreviate with the notation O(2n).
134
by
? ? [ ]
4 ?g ? ?(?)S = d x ?R g???????? + Sm[g?? ,?], (3.1)
2? ?
where ? = 8?G 3?/c depends on the bare gravitational coupling constant G?. The
action for the matter in the theory Sm is a function of only the metric and mat-
ter degrees of freedom ?; the scalar field does not couple to matter directly, only
indirectly through its interactions with the metric.
The restricted class of scalar-tensor theories described by Eq. (3.1) has been
studied extensively in the literature because it is general enough to manifest many
different deviations from GR yet simple enough that its predictions can be worked
out completely. At 2PN order, the Fokker Lagrangian for a system of point particles
was first computed in Ref. [152] using an effective field-theory approach. The 2PN
metric and equations of motion were computed for bodies composed of perfect fluids
in Ref. [236]. The post-Minkowskian technique of direct integration of the relaxed
Einstein equations (DIRE) was used in a recent series of papers [149?151] to compute
the equations of motion for a system of compact objects at 2.5PN order, as well as
the gravitational waveform and energy flux at 2PN (relative) order for binaries on
generic orbits. For comparison, the entire waveform for non-spinning systems in GR
is known at 3PN order [237], and its quadrupolar and octupolar parts are known at
3.5PN order [238, 239] (see Ref. [30] for a review of existing results in GR).
In this work, we specialize the results of Refs. [149?151] to binary systems on
quasi-circular orbits and present the waveform in a form that can be easily used to
test GR with GWs. This calculation serves as an extension of Ref. [240], in which the
135
leading-order behavior of the GW signal produced by binary systems was computed
in Brans-Dicke theory [178, 179, 241], where the scalar coupling ?(?) = ?BD is
constant. This work extends those earlier findings to higher PN order in a larger
class of scalar-tensor theories.
The paper is organized as follows. In Sec. 3.2, we review some preliminary
information regarding the production and detection of GWs in scalar-tensor theories.
Section 3.3 presents the dynamics for binary systems on quasi-circular orbits. In
Sec. 3.4, we compute the hereditary contributions to the gravitational waveform
from such systems. We present the binding energy and energy flux in Sec. 3.5
and compute the associated orbital phase evolution. In Sec. 3.6, we decompose
the waveform into spin-weighted spherical modes, and in Sec. 3.7, we express these
modes in Fourier space using the stationary phase approximation. We provide some
concluding remarks in Sec. 3.8. Appendix C details the conversion of our notation
to that of Refs. [147, 152], which is commonly found in the literature. Appendix D
contains formulae omitted from the main text for the sake of compactness.
All calculations are done both for systems whose inspiral is driven by dipolar
radiation and for those driven by quadrupolar radiation; this distinction is discussed
in detail in Sec. 3.2.2. Note that the complete 2PN (relative) order results are given
for only the former case (dipolar-radiation driven systems). The results for the latter
case depend on higher-order corrections to the energy flux that have not yet been
computed, but we argue in Sec. 3.5.2 that the impact of these missing terms is very
small.
Henceforth, we work in units where G? = c = 1.
136
3.2 Gravitational waves in scalar-tensor gravity
This section contains information concerning the generation of GWs in scalar-
tensor gravity that will prove useful throughout the rest of the paper despite not
directly contributing to the computation of the waveform. We begin by discussing
the behavior of binary systems, tracing new phenomena not found in GR to viola-
tions of the strong equivalence principle. We then review the current experimental
constraints set on these theories and show that, in most cases, sources for ground-
and space-based GW detectors evolve similarly to as in GR. Finally, we discuss the
response of a detector to a GW in scalar-tensor gravity and delineate the waveform
computed in the subsequent sections.
3.2.1 Binary systems of compact objects
Differences between the dynamics and GW emission of binary systems in
scalar-tensor gravity and those in GR ultimately stem from the non-minimal cou-
pling between the metric and scalar field. As a result, the gravitational ?constant?
experienced by massive bodies depends on the value of the background scalar field
in which they are situated. For test bodies, this dependence can be deduced directly
from Eq. (3.1): the strength of their gravitational interaction scales as ??1.
The gravitational interaction between compact, self-gravitating bodies is more
complex. Because the binding energy of a single self-gravitating body depends on
the interactions between all of its constituents, the body?s mass mA(?) depends on
the local scalar field. This phenomenon is a manifestation of the violation of the
137
Table 3.1: Parameters that govern gravitational wave production in binary systems.
Quantities listed with the subscript 0 are evaluated at the value of the background
scalar field ?0.
Parameter Definition
Weak-field parameters
G G??
?1
0 (4 + 2?0)/(3 + 2?0)
? 1/(4 + 2?0)
?1 (d?/d?) ? ?
2
0 0 /(1? ?)
? 22 (d ?/d?
2) ?20 0?
3/(1? ?)
Strong-field parameters
sA [d lnmA(?)/d ln?]0
s? 2A [d lnmA(?)/d ln?
2]0
s??A [d
3 lnmA(?)/d ln?
3]0
Binary parameters
Newtonian
? 1? ? + ?(1? 2s1)(1? 2s2)
post-Newtonian
? ?2??1?(1? 2s1)(1? 2s2)
?1 ?
?2?(1? 2s2)2 (?1(1? 2s1) + 2?s?1)
? ??22 ?(1? 2s1)2 (?1(1? 2s2) + 2?s?2)
2nd post-Newtonian
?1 ?
?2?(1? ?)(1? 2s )21
? ??22 ?(1? ?)(1? 2s )22
? ?3 31 ? ?(1? 2s2) [(?2 ? 4?21 + ??1)(1? 2s1)? 6?? ?1s1 + 2?2s??1]
? ??32 ?(1? 2s1)3 [(?2 ? 4?21 + ??1)(1? 2s ? 2 ??2)? 6??1s2 + 2? s2]
strong equivalence principle, as the self-interaction of a massive body is dictated
by its composition. As is done in the literature, we adopt an approach proposed
by Eardley [242] to handle the interplay between microphysics and gravity that
determines the connection between the body?s composition and mA(?). We treat
compact objects as point particles whose mass is given by mA(?). Rather than solve
for this function outright, we parameterize it by its expansion about a background
138
field ?0 [
(0) 1 ( ? ) ]mA(?) =mA 1 + sA? + s2 2A + sA ? sA ? + ? ? ? , (3.2)2
where we?ve defined
(0)
mA ?m(A(?0), ) (3.3)
? d lnmAsA ( , (3.4)d ln? )?=?0
2
s?A ?
d lnmA
, (3.5)
d(ln?)2 ?=?0
??? ?0? . (3.6)
?0
The parameter sA is known as the sensitivity of the body. For test bodies, sA = 0,
while for stationary black holes, sA = 1/2 [23].
3
The underlying parameters that govern the orbital dynamics and gravitational
emission of binary systems up to 2PN order are given on the left-hand side of
Table 3.1. These parameters are classified as either weak-field or strong-field : the
former class influence behavior in all gravitational contexts whereas the latter class
only enter in systems with strong gravitational fields, such as those found in self-
gravitating compact objects. The weak- and strong-field parameters appear in only
a small set of combinations, denoted as the binary parameters in Table 3.1. We
3The sensitivity of neutron stars is often estimated to be of the order ? 0.2. While true
in Brans-Dicke theory [242, 243] and some slight variations [146], this result does not hold for
generic choices of ?(?). One of the most popular classes of scalar-tensor theories, those that
allow spontaneous [153] and dynamical scalarization [169, 171], are a striking counterexample. In
these theories, neutron-star sensitivities can be large and negative; the process of spontaneous
scalarization describes stars whose sensitivity diverges, i.e., sA ? ??.
139
have adopted the notation introduced in Refs. [149]; the mapping between these
parameters and the notation used in Refs. [147, 152] is given in Appendix C.
Novel behavior in scalar-tensor gravity stems from violations of the strong
equivalence principle, and thus, is dictated by the strong-field parameters. For
example, dipolar emission, the most prominent new effect not found in GR, is tied
to (s1 ? s 22) . Formally, dipolar radiation is generated at one PN order lower than
quadrupolar radiation (the dominant dissipative channel in GR); in keeping with
the conventions of Refs. [149?151], we demarcate dipolar emission as a ?1PN order
effect.
3.2.2 Generic constraints on scalar-tensor gravity
A hundred years of tests have confirmed that gravity closely resembles GR [79].
Restricting our attention to only those theories that satisfy these constraints, we
must study the regime in which new scalar-tensor effects are small relative to those
also found in GR. In this limit, the structure of the PN expansion is modified; for
example, in the frequency band of interest, the dominant dissipative process is the
emission of Newtonian order quadrupolar radiation rather than the ?1PN dipolar
energy flux. We investigate which systems fall within this regime by first mapping
the current constraints on scalar-tensor theories to the parameters given in Table 3.1.
The best constraints on weak- and strong-field parameters come from a combi-
nation of solar-system experiments and binary-pulsar observations. The weak-field
parameters G, ?, ?1, ?2 are tied to the behavior of the scalar coupling ?(?) near the
140
background value of the scalar field ?0. These quantities can be expressed in terms
of the parametrized post-Newtonian (PPN) parameters ?PPN and ?PPN as well as
the 2PN parameter  introduced in Ref. [152]
2
G = , (3.7)
(1 + ?PPN)?0
1? ?PPN
? = , (3.8)
? 2
2 2?(?PPN ? 1)?0??1 = , (3.9)
1 + ?PPN
((?PPN ? 1) + 24(? 2PPN ? 1) )?2
?? = 02 , (3.10)
1 + ?PPN
?
where we have used the rescaled parameters ??1 ? ?1 ? and ??2 ? ?2? because ?1,
?2 are not well defined in the GR limit.
The current constraints on these parameters are given in Table 3.2. The
constant background field ?0 is undetectable with weak-field measurements ? at
Newtonian order, a redefinition of the field ? ? ?/?0 can be compensated by
the rescaling of the bare gravitational constant G? ? G?/?0 and the redefinition
? ? ?0?. For simplicity, we set ?0 to unity in Table 3.2. Note that the constraint
on  was estimated in Ref. [152] with only binary-pulsar measurements available at
the time; this constraint could be improved by including more recent observations.
The current experimental constraints on the strong-field parameters s , s? , s??A A A
are not as restrictive. The best limits on neutron-star sensitivities come from timing
measurements of pulsar-white-dwarf binaries [245?248]; white dwarfs are expected to
have negligible sensitivity, so the magnitude of dipolar emission is dictated entirely
by the sensitivity of the neutron star. Constraints are typically given in terms of
141
Table 3.2: Constraints on the weak-field parameters in Eqs. (3.7)?(3.10) set by solar-
system and binary-pulsar observations. As discussed in the text, we set ?0 = 1 for
simplicity.
Parameter Constraint Reference
?PPN ? 1 2.3? 10?5 [180]
?PPN ? 1 7.8? 10?5 [79, 244]
 7? 10?2 [152]
G? 1 1.2? 10?5
? 1.2? 10?5
??1 1.6? 10?4
??2 8.8? 10?7
the scalar charge ?A, related to the sensitivity by
?1? 2sA?A = . (3.11)
3 + 2?0
Amongst known pulsar-white-dwarf binaries used to constrain scalar-tensor
theories, PSR J0348+0432 hosts the most massive neutron star [248]. The con-
straints on the scalar dipole reported in Ref. [248] provide an estimate for the maxi-
mum scalar charge that this neutron star can have |? | . 6?10?3A . Extending these
data to an absolute bound on the charge of any neutron star requires the assump-
tion of a particular choice of ?(?) and equation of state. Working within one of
the most popular classes of scalar-tensor theories [153] and selecting certain realistic
equations of state, one can produce a global constraint of |?A| . 10?2 [248, 249].
However, it is conceivable that other theories and/or equations of state allow neu-
tron stars to acquire large scalar charges of ?A ? 1 via the process of spontaneous
scalarization [153] while satisfying all current experimental constraints.
Because the weak-field constraints leave ?0 ? 1/(2?) unbounded, no absolute
bound can be placed on sA. To our knowledge, no constraints have been placed
142
on s? and s??A A either; for neutron stars, these higher derivatives can be orders of
magnitude larger than sA (for example, see Fig. 3 of Ref. [3]).
Excluding the possibility of spontaneous scalarization, the constraints on weak-
field and strong-field parameters ensure that dipole radiation is suppressed in viable
scalar-tensor theories, as can be shown by comparing the relative size of the ?1PN
and Newtonian order flux, given in Refs. [147, 149] and repeated in Eq. (3.48) below.
Despite entering at higher PN order, the next-to-leading order term overpowers the
leading-order term when
( )
24
1 . S (G?M?f)
2/3 , (3.12)
5? 2?
where, for simplicity, we have dropped all terms that are not of order O(??1) and
introduced the scalar dipole
S? ?? ??1/2 (s1 ? s2) . (3.13)
Given the experimental constraints on ? and S?, this threshold is reached at
frequencies f . 100?Hz in binary neutron star or neutron-star stellar-mass-black-
hole systems, and at frequencies f . 5?Hz in neutron-star intermediate-mass-black-
hole systems. Following this argument, ground- and space-based GW detectors
would only observe binary systems whose inspiral is driven by the next-to-leading
order flux.4 On the other hand, the evolution of binary pulsars could be dominated
4Unlike the class of scalar-tensor theories considered here, there are alternative theories in which
binary black holes can emit dipolar radiation (e.g., dilatonic Einstein-Gauss-Bonnet, dynamical
Chern-Simons, etc.). Given the relatively weak constraints on dipolar radiation in vacuum space-
times (compared to those from binary pulsars observations), we note that space-based detectors
143
by dipolar emission. Binary systems that undergo dynamical scalarization may also
be exempt from this verdict, as these systems dynamically generate large scalar
charges that can substantially enhance dipolar emission [169, 171, 250].
Because non-perturbative scalarization phenomena have not been entirely ruled
out, we compute below the gravitational waveform both for systems in which dipo-
lar radiation is dominant and for those in which quadrupolar radiation is dominant.
For conciseness, we refer to the former class of systems as dipole driven (DD) and
the latter class as quadrupole driven (QD).
3.2.3 Detector response
We consider the response of a laser interferometer at spatial coordinates X
generated to an incident GW produced by a distant binary system of size d, where
R ? |X|  d. We assume that far from the binary, the metric and scalar field ap-
proach the Minkowski metric ??? and a constant background value ?0, respectively,
at a rate ? R?1. Let ?? ? ?/?0 be the normalized scalar field. We introduce the
conformally transformed metric
g??? ???g?? , (3.14)
and the gravitational field5
?
h?? ???? ? ?g?g??? . (3.15)
or pulsar timing arrays could, in principle, observe binary black holes driven by dipolar flux. As
discussed below, the GW signal from such systems has a distinct structure from that in GR.
5Note that in Ref. [30] and the references therein, the metric perturbation is defined with an
overall minus sign relative to the definition given here.
144
The metric at the detector takes the form
1 ( )
g?? =??? + h?? ? h??? ??? +O R?2?? , (3.16)
2
where h ? ? h???? is the trace of h?? and h ???? ? ??????h is lowered using the
Minkowski metric. Gravitational-wave detectors use laser interferometry to measure
the separation between mirrors; we treat these mirrors as test masses. Assuming
that the distance between mirrors is smaller than the wavelength of the incident
GWs and that the mirrors move slowly, the separation between the mirrors obeys
??i = ?R0i0j?j, (3.17)
where i, j = 1, 2, 3 are spatial indices. Working at leading order in h?? and ?, the
Riemann tensor is calculated from Eq. (3.16)
1 1 ( )
R0i0j = ? h?ijTT ? ?? N? iN? j ? ?ij , (3.18)2 2
where N? ? X/R and hijTT is the transverse-traceless component of the gravitational
field defined as ( )
ij ip jq ? 1hTT = P P P ijP pq hpq, (3.19)2
where P pq = ?pq ? N?pN? q is the transverse projection operator.
From Eq. (3.18), we see that the GW signal contains a transverse-traceless
mode (as in GR) characterized by the field h?? . In scalar-tensor gravity, there is
an additional transverse breathing mode produced by ?. Extracting this new GW
polarization requires a network of detectors; see Ref. [79] and references therein for
a discussion of the prospects of detecting GW polarizations absent in GR. We focus
exclusively on h?? for the remainder of this work.
145
3.3 Dynamics for quasi-circular orbits
In this section, we specialize the results of Ref. [149] for the 2.5PN dynamics of
binary systems to the case of quasi-circular orbits. Before proceeding, we establish
some notation employed throughout this work. We denote the total mass of the
system by M = m1 + m2 and the symmetric and antisymmetric mass ratio by
? = m1m2/M
2 and ? = (m1 ?m2)/M , respectively. We signify the symmetric and
antisymmetric combinations of parameters given in Table 3.1 by
?1?+ (?1 + ?2) , (3.20a)
2
?? ?
1
(?1 ? ?2) , (3.20b)
2
and in addition to S? above, we also define
S+ ???1/2 (1? s1 ? s2) . (3.21a)
To describe the system?s dynamics, we denote the orbital separation by x = rn,
the relative velocity by v = x?, and the acceleration by a = v?. We construct
an orthonormal moving frame (n,?) and define the orbital frequency ? such that
v = r?n + r??. To avoid confusion, we note that certain variables are used to
denote multiple quantities; for example, ? represents both the frequency and scalar
coupling, while ?,? are used for the phase and scalar field. The usage of each can
be inferred from context.
Our analysis of binary systems on quasi-circular orbits begins with the 2.5PN
146
equation of motion, given in the center-of-mass frame by
? GM? GM?a = n + [n (A1PN + A2PN) + r?v (B1PN +B2 2 2PN)]r r
8? (GM?)2
+ [r?n (A1.5PN + A2.5PN)? v (B3 1.5PN +B2.5PN)] . (3.22)5 r
where the expressions for Ai, Bi can be found in Eqs. (1.4)?(1.5) and (6.12)?(6.13)
of Ref. [149]. It will prove useful also to write the equations of motion in the generic
form
a = (r? ? r?2)n + (r?? + 2r??)? . (3.23)
The restriction of the dynamics to quasi-circular orbits follows the same procedure
as in GR. For such orbits, the only departure from circular motion is induced by
radiation reaction, which enters at 1.5PN order in scalar-tensor theories rather than
the usual 2.5PN order in GR. Expressed symbolically, we have r?, ?? = O(3) [instead
of O(5)], while r? = O(6) [instead of O(10)].
The first term in Eq. (3.23) determines the conservative sector of the dynamics
at 1PN and 2PN order. The scalar product a ?n = ?r?2 +O(6) produces a relation
between the orbital separation and frequency that generalizes Kepler?s law. We
introduce the PN parameters (recall that we work in units where c = 1)
?PN ?
GM?
, (3.24a)
r
x ? (GM??)2/3 , (3.24b)
which differ from their usual definition in GR by an additional factor ?.
At leading order, one obtains r3?2 = Gm? + O(2), or x = ?PN(1 + O(2)).
From there, solving order by order yields
147
[ ( )
2??? 2?+ ? ?
x =?PN 1(+ ?PN ? ? + ? 13 3 3 3
8?22 ?? 16?
2
?? ? 4?
2
? 8???+? ? 2???? 11???? 4???+?PN + + + ?? 9 9 9 9 9 3
8?2? +? ? 4?
2 2? 2 2+ +? 7?+? 4?+ ? ? ? 11? 17?? ???+ + + + + + ? +
? 9 9 9 3 6 36 ) 9 ] 3
2??? ? 2?+? ?
2
+ 2? 4??+ 49? 2?+
+[ ( + + + +) ? + 1 +O(6) , (3.25a)3 3 3 9 3 12 3
?PN =x 1 +(x ?
2??? 2?+ ?
+ + ? ? + 1
3 3 3 3
2 2 2
+x2 ?8??? ? 16??? 4?+ ? ? 8???+? ? 2???? ? ???? ? 4???
? 3 3 3 3 3 3
8?2 ? 4?2 2? ? 5? ? 4? ?2? ?2+ + + ? + + ? ? 7?? ? ???+ + + + + ) ]+ ?? 3 3 3 3 6 12 3 3 3
?2??? 2?+? ?+ 4??++ ? ? ? 65? 2?++ + 1 +O(6) . (3.25b)
3 3 3 3 12 3
Having derived the reduction to quasi-circular orbits for the conservative dy-
namics up to 2PN order, we now turn our attention to the dissipative sector. Al-
though only the leading-order radiation-reaction terms are needed to compute the
2PN relative order dynamics, we provide results up to 2.5PN for the sake of com-
pleteness. Inserting Eq. (3.25) into the relation a ?? = r??+ 2r?? gives the following
expressions for r? and ??
8 8
r? =? ??S2?x2 ? ?? 3RRx +O(7) , (3.26a)3 3
4??S2x9/2 4?? x11/2RR
?? = ? + +O(7) , (3.26b)
G2M2?2 G2M2?2
148
where we have introduced
?24 4?? ?S
2 2
? 8????S 7??S2 4?? 2+S
? + 2? ? ? + ? ? ? + ?RR
5 ? 3 6 ?
? 8??+S
2 2 2
? 2??S? ? ?S? 4???S+S?+ + ? 4??+?S+S? . (3.27)
3 3 2 ? ?
For dipole-driven systems, the second term in Eq. (3.26) is much smaller than
the first. Integrating this equation at leading order gives the evolution of the orbital
separation and frequency
[ ]1/3
rDD(t) = [8??S
2
?(GM?)
2(tc ?]t) (1 +O(2)) , (3.28a)?1/2
?DD(t) = 8??S2?GM?(tc ? t) (1 +O(2)) , (3.28b)
where tc is the time of coalescence. In the quadrupole-driven regime, the first term
in Eq. (3.26) is overpowered by the second. We delay a precise formulation of this
limit until Sec. 3.5, but note that the evolution of the inspiraling orbit will take the
same form as in GR, given[at leading order by ]
256(GM)3
1/4
?
rGR(t) = [ (tc ? t) ] (1 +O(2)) , (3.29a)5
5/3 ?3/8256(GM) ?
?GR(t) = (tc ? t) (1 +O(2)) . (3.29b)
5
The difference in structure between Eqs. (3.28) and (3.29) stems from radiation
reaction entering at a different PN order in the two regimes.
3.4 Radiative coordinates and hereditary contributions
Equipped with the leading-order evolution of the inspiral, we begin our com-
putation of the 2PN order waveform. The waveform was derived for generic orbits
in Ref. [150]; schematically, these results are given by
149
ij 2G(1? ?)M? [h = Qij + P 1/2Qij + PQijTT ]+ P
3/2Qij
R N
+P 3/2Qij + P 2QijC?N N + P
2QijC?N , (3.30)TT
where TT stands for the transverse-traceless projection given in Eq. (3.19), P de-
notes the PN order of each term, and N and C ? N indicate contributions from the
near zone and radiation zone, respectively, as defined in Ref. [150]. The expressions
for P nQij are presented in Eq. (7.2) of Ref. [150]. These terms can be categorized
as either instantaneous or hereditary : instantaneous terms depend only on the state
of the system at the current retarded time, whereas hereditary terms take the form
of integrals extending over the binary?s entire history. In Eq. (3.30), P 3/2QijC?N and
P 2QijC?N are hereditary, while the remaining terms are all instantaneous.
This section details the computation of the hereditary terms for systems on
quasi-circular orbits. First, we re-express the waveform in a radiative coordinate
system, in which the metric perturbation falls off as ? R?1. We then compute
separately the contributions from so-called tail and memory terms.
3.4.1 Radiative coordinates
We begin by transforming the results of Ref. [150] into radiative coordinates.
This reference employed harmonic coordinates X = (t,X), defined by the gauge
condition ??h
?? = 0; however, these coordinates are known to give rise to unwanted
logarithms of R in the far-zone expansion. As shown in Ref. [251], it is possible to
build another set of coordinates X? = (t?, X?), called radiative coordinates, in which
150
these logarithms are eliminated and the metric perturbation hij admits an expansion
in powers of R??1. Here, we will follow the presentation of Ref. [252], in which the
construction of this coordinate system is explicitly written at quadratic order in
the multipolar post-Minkowskian formalism [253]. Note that our definition h?? in
Eq. (3.14) introduces a sign difference with respect to Ref. [252].
Both hereditary pieces, P 3/2QijC?N and P
2QijC?N , contain integrals with a loga-
rithmic kernel, known as tail terms. The logarithmic terms can be expressed as the
second time derivative of the leading-order, linearized metric, as shown by Eq. (2.28)
of Ref. [252]. Written in terms of the retarded time u = t ? R/c, these terms are
given by
( ) ? +? ( )2
P 3/2
d s
QijC?N (u) =2G(1? ?)M ij
( )ln ?
ds Q (u? s) ln , (3.31a)
dt20 2R
+? (+ s )d2 s
P 2Qij 1/2 ijC?N (u) =2G(1? ?)M ds P Q (u? s) ln . (3.31b)ln 2
0 dt 2R + s
Because we are only interested in the R?1 piece of the waveform, we expand
the logarithms according t(o ) ( ) ( )s s
ln = ln +O 1 . (3.32)
2R + s 2R R
We define the radiative coordinates as X?? = X? + ??( ,)with
R
?? = 2G(1? ?)M??0 ln , (3.33)r0
where we have introduced an arbitrary constant length scale r0. The metric pertur-
bation in these new coordinates takes the form
[ ]
h??? = h?? ? ???? ? ???? + ???? ?? + ??? h??? ? , (3.34)X=X?
151
where we have kept only the relevant terms in Eqs. (2.36) and (2.37) of Ref. [252].
The first three terms describe the usual effect of a first order gauge transformation
on the harmonic perturbation; their contribution will be eliminated by the TT
projection. The last term combines with the lower boundary term of the integrals
in Eq. (3.31) to replace R by the new constant r0:
( ) ? +? ( ) ( )d2 s 1
P 3/2QijC?N (u?) =2G(1? ?)M ds Qij(u?? s) ln +O ,
ln 0 dt?
2 2r0 R?2
( ) ? (3.35a)+? ( ) ( )2
ij d s 1P 2QC?N (u?) =2G(1? ?)M ds P 1/2Qij(u?? s) ln +O .
ln 20 dt? 2r0 R?2
(3.35b)
Since the transformation to radiative coordinates affects only the logarithmic terms,
from here on, we drop the notation X?, using instead the ordinary notation X to
signify these new coordinates.
3.4.2 Tail contributions
The tail terms in the waveform arise from back-scattering of the waves on
the curvature of spacetime. In the multipolar post-Minkowskian wave generation
formalism of Refs. [252, 253], they appear as interactions between each multipole
moment and the mass monopole of the system. In the DIRE formalism [254, 255]
used in Refs. [150, 151], tail terms arise from wave-zone contributions to the integrals
over the past light-cone of the observer. Recall that these terms take the form of an
integral with a logarithmic kernel over the past history of the source.
Since we are only interested in the R?1 part of the waveform, we can expand
152
the logarithms as in Eq. (3.32). Using Eq. (3.25) to replace ? with r, the tail terms
then take the generic form ? +? ( )ein?(t?s) s
I = ds
0 r
p(t? ln (3.36)s) 2r0
where n, p are integers and ? is the orbital phase of the binary.
To evaluate Eq. (3.36), we make use of the fact that the radiation-reaction
timescale is much longer than the orbital period. It was shown in Ref. [256] that ig-
noring radiation reaction, i.e., approximating the binary orbit as circular (with con-
stant radius and frequency), introduces an error in these integrals of order O(ln c/c5)
in GR. The same argument holds in scalar-tensor gravity, with the only difference
being that the error is of order O(ln c/c3) for dipole-driven systems due to the dif-
ferent scaling of radiation reaction. Under this assumption that the frequency does
not evolve with s, we write ?(t ? s) ' ?(t) ? s?(t). One can then compute the
resulting integral?s by making use of the formula [256]+? 1 [? ]
dy ei?y ln y = ? sgn(?) + i (?E + ln |?|) , (3.37)
0 ? 2
where ?E is the Euler-Mascheroni constant.
3.4.3 Memory contributions
Memory terms arise in the waveform as integrals of the product of multipoles
without a logarithmic kernel over the history of the source [252, 257]. They can be
separated into so-called DC terms, which are non-oscillatory and accumulate over
the entire lifetime of the system, and AC, oscillatory terms that, by contrast, depend
only on the recent history of the source.
153
The computation of oscillatory memory terms is identical in GR and in scalar-
tensor theories. On quasi-circular orbits, these terms have the structure
? +? ein?(t?s)
J = ds ? , (3.38)0 rp(t s)
with integers n, p. Thanks to the oscillatory factor ein? in the integrand, it can
again be shown (see, e.g., Ref. [258]) that only the recent past contributes in the
integral, so that one can approximate r(t ? s) ' r(t) and ?(t ? s) ' ?(t) ? s?(t)
with a negligible relative error of the same PN order as radiation reaction. For
dipole-driven inspirals, the result is
( )1/2
1 r(t)3 ein?(t)
JDD = +O(3), (3.39)
in GM? rp(t)
whereas for quadrupole-driven inspirals, one obtains
( )1/2
1 r(t)3 ein?(t)
JQD = +O(5). (3.40)
in GM? rp(t)
Note that the only difference between these two cases is the order of the remainders.
The non-oscillatory (DC) memory terms take the form
? +? 1
K = ds ? . (3.41)rp0 (t s)
Their computations in GR and scalar-tensor theory differ. Non-oscillatory terms
are enhanced by the accumulation of the integrand over the long radiation-reaction
timescale, an effect which formally decreases their PN order. The result depends
here on the rate of evolution of the quasi-circular inspiral under radiation reaction.
For dipole-driven systems, these DC memory terms formally appear at the 1.5PN
order in the expression of the multipole moments, but the integration over the
154
radiation-reaction timescale (formally of ?1.5PN order) pushes this contribution
back to Newtonian order. Using the leading-order evolution of the quasi-circular
inspiral given by Eq. (3.28), one obtains
3r3?p(t)
KDD =
8(p? +O(?1), (3.42)3)??S2?(GM?)2
for p > 3.
The contribution from non-oscillatory memory terms in quadrupole-driven sys-
tems is more difficult to compute. Following Eq. (3.12), any system with non-zero
scalar dipole will have been dominated by dipolar radiation at some point during its
lifetime. The transition between the dipole-driven and quadrupole-driven regimes
needs to be accommodated in the integral in Eq. (3.41). Such a calculation goes
beyond the scope of this work.
3.5 Balance equation and phase evolution
Having computed all of the hereditary terms for quasi-circular orbits, one can
use Eqs. (3.25) to express the waveform entirely in terms of the instantaneous orbital
phase ? and frequency ? of the binary. We now need the evolution of these quantities
at 2PN order to finish our calculation of the waveform. This level of accuracy cannot
be achieved using only the dynamics of the binary presented in Sec. 3.3. In place
of the higher-order radiation-reaction force, we use the total energy flux F and the
balance equation
dE
= ?F , (3.43)
dt
155
which can be reformulated using ?? = ? as
d? ? 1 dE/dx= x3/2 . (3.44)
dx GM? F(x)
We calculate the energy for systems restricted to quasi-circular orbits by ap-
plying the results of Sec. 3.3 to those of Ref. [149]. The energy measured in an
arbitrary frame is given by Eq. (6.4) of Ref. [149]. After shifting to the center-of-
mass frame with Eqs. (6.9) and (6.10) of Ref. [149], we reduce this expression to the
case of quasi-circu[lar orb(its using Eqs. (3.25) and (3.26) a)nd obtain
? 1 ?2??? 2?+ ? 2? ? ?Ecirc = M(?x 1 + x + ?
3
2 3 3 3 12 4
2 2
2 ?16??? ? 16??? 4?
2
? ? 8???+? ? 4???? ????+x + + ? ???
? 3 3 3 3 3
16?2 2 2 2
+ +
? 4?+ 4?+? ? 19?+? ? ? ? 19? 11?? 14?+ + + ?+ + + ?
? 3 3 3 3 12 3 )] 3
?? 2? ? 4??? 4?+? ?+ ? ? ? 8??+ 19? 4?+ ? 27+ + + + + . (3.45)
3 3 3 3 24 3 8 3 8
The total emitted energy flux, including both tensor and scalar contributions,
was given for generic orbits in Ref. [151] with the structure
F = F?1 + F0 + F0.5,C + F0.5,C?N + F1 +O(3), (3.46)
where the number in the index indicates the PN order of each term. The ?1PN
term comes from dipolar, scalar radiation and is responsible for the appearance of
radiation-reaction effects at 1.5PN order. Note that while the flux we consider is
given at 1PN (using the order-counting scheme from GR), it corresponds to 2PN
relative order.
The individual terms in Eq. (3.46) are given in the center-of-mass frame in
Eq. (6.8) of Ref. [151]. The term F0.5,C?N includes a logarithmic hereditary term
156
coming from the product of the leading-order term and a tail term (at 1.5PN relative
order) in the scalar waveform. We calculate this tail contribution using the method
detailed in Sec. 3.4.2. Then, as before, we use the results of Sec. 3.3 to compute
the total energy flux for quasi-circular orbits F(x) that will be given in Eqs. (3.48)
and (3.55) below.
Equipped with expressions for the binding energy E(x) and the total energy
flux F(x) both at the 2PN relative order, we proceed to evaluate the orbital phasing
of the binary using Eq. (3.44). Different approaches have been proposed in the
literature to integrate the balance equation, differing by the choice of integration
variables (time or frequency) and by the choice of either numerical integration or
analytical integration of a re-expansion of Eq. (3.44) (see, e.g., Ref. [259] for a
definition and comparison of these so-called Taylor approximants). Our purpose here
is not to compare these different approaches, but to examine the new contributions in
the phasing that arise in scalar-tensor theories. We adopt a method (corresponding
to the TaylorT2 approximant) that provides a result in analytic form: the ratio
?(dE/dx)/F(x) is re-expanded in x, truncated at relative 2PN order, and then
integrated term by term. For this purpose, it will be convenient to introduce the
notation
1 1 dE
?(x) ? ? F . (3.47)GM? (x) dx
Care must be taken before re-expanding this ratio in the PN parameter x: we
distinguish between the dipole-driven case in which the dipolar term F?1 [defined
in Eq. (3.46)] dominates the denominator and the quadrupole-driven case wherein
157
F0 dominates due to the smallness of scalar-tensor parameters.
3.5.1 The dipole-driven regime
We first consider systems whose inspiral is driven by dipolar radiation. As
discussed in Sec. 3.2.2, this regime is reached by binaries with large separations
(binary pulsars) or large scalar dipoles (spontaneously scalarized systems). Dynam-
ically scalarized systems begin in the quadrupole-driven regime but then abruptly
become dipole-driven at some point during their evolution; in principle, one must
account for both stages when modeling their inspiral, but we will not pursue such
a treatment here.6 Factoring out the leading order dipolar flux in Eq. (3.46), we
obtain
2 2 4
FDD 4S(x) = ??? x
[ ]
1 + fDD2 x+ f
DD
3 x
3/2 + fDDx24 +O(5) , (3.48)3G?
where explicit expressions for the coefficients fDDn are given in Eq. (D.2). The leading
order of the flux carries a factor S2? characteristic of dipolar radiation.
In this case, we simply re-expand the ratio ?(x) in x at 2PN relative order and
obtain
3 [ ]
?DD(x) = S 1 + ?
DD
2 x + ?
DD 3/2 DD 2
3 x + ?4 x +O(5) , (3.49)8 2 4???x
where the coefficients ?DDn are given explicitly in Eq. (D.6). By integrating Eq. (3.44)
6Reference [3] (reproduced in Chapter 4) argues that the PN approximation breaks down as
dynamical scalarization occurs, but that a straightforward resummation of PN results can provide
an accurate waveform model valid in this regime.
158
term-by-term, the phasing is then given by
[ ]
? 1 DD 3?(x) = 1 + 3x? 3/2 DD 2 DD
4S2 ??x3/2 2 ? x lnx?3 ? 3x ?4 +O(5) , (3.50)? 2
where we have dropped an arbitrary additive constant that can be fixed by specifying
the value of the phase at a given frequency.
3.5.2 The quadrupole-driven regime
For quadrupole-driven systems, the flux should be expanded about the Newton-
ian-order term F0 in Eq. (3.46) rather than the leading-order ?1PN term. To ac-
complish this reordering of the PN approximation, we expand the flux in the PN
parameter x and an additional parameter that describes the smallness of non-GR
effects. There exists some flexibility in the choice of this second small parameter;
the weak-field parameters listed in Table 3.1 describe the smallness of scalar-tensor
corrections in complementary ways, and these quantities appear in the waveform in
several combinations (e.g., the binary parameters).
We adopt a prescription that generalizes the approach of Ref. [240] to more
generic scalar-tensor theories and to higher PN order. In the present quadrupole-
driven case, we split the flux into pieces independent and dependent on the scalar
dipole
FQD = Fnon-dip + Fdip, (3.51)
159
where we have defined
Fnon-dip ? lim F , (3.52)
S??0
Fdip ?F ?Fnon-dip. (3.53)
We refer to Fdip and Fnon-dip as the ?dipolar part? and ?non-dipolar part? of the
flux, respectively. Note that these labels do not correspond precisely to the mul-
tipolar structure of the source; for example, Fdip contains contributions from time
derivatives of the scalar monopole and quadrupole. Instead, Fdip represents the part
of the flux that vanishes when s1 = s2
We compute the phasing at first order in the small quantity Fdip/Fnon-dip,
employing the approximation
( )
? dE/dx ' ? dE/dx Fdip(x)F F 1? F (3.54)(x) non-dip(x) non-dip(x)
in Eq. (3.47). Evaluating the right-hand side of this equation requires knowledge of
Fdip and Fnon-dip each at 2PN relative order.
We obtain for the dipolar and non-dipolar parts
32?2F ?x
5 [ ]
(x) = 1 + fndnon-dip 2 x +O(3) , (3.55a)5G?
2 2 4
F 4S(x) = ??? x
[ ]
1 + fdx + fd 3/2dip 2 3 x + f
d 2
4 x +O(5) , (3.55b)3G?
where the coefficients fndn and f
d
n can be found in Eqs. (D.10) and (D.14) of Ap-
pendix D. The leading order dipolar part of the flux (3.55b) is the same as in
Eq. (3.48). The leading order non-dipolar part (3.55a) is simply the quadrupolar flux
in GR with an additional factor of ?/?, where we have defined ? ? 1+?/2+ ?S2+/6.
160
Note that because it enters at Newtonian order (rather than ?1PN), the non-
dipolar part of the flux is only known to 1PN relative order. A complete calculation
of the phasing at 2PN relative order requires the 1.5PN and 2PN corrections to the
non-dipolar flux. In place of these unknown terms, we use
F (GR) (ST)non-dip =F2PN + Fnon-dip, (3.56)
with
2 5
F (ST) (ST)1PN 32? x
[ ]
ST 3/2 ST 2
non-dip =Fnon-dip + ? f x + f5G? 3 4 x +O(5) . (3.57)
(GR)
In the above, F2PN is the PN expanded flux in GR up to 2PN order, with the
natural replacement G? ? G?. The first term in Eq. (3.57) denotes the known con-
tributions to the non-dipolar flux that only arise in scalar-tensor theories, which can
be obtained by subtracting the GR terms from (3.55a). We introduce the unknown
coefficients fST3 and f
ST
4 to represent our ignorance of the new scalar-tensor contri-
butions at 1.5PN and 2PN order. In the quadrupole-driven context, experimental
constraints on the weak-field parameters imply that these contributions should be
much smaller than the 2PN GR terms. Moreover, these terms are doubly suppressed
in the second term of Eq. (3.54) because Fdip is already of the first order in the small
scalar-tensor coefficients. We will keep these unknown coefficients throughout our
calculation for completeness.
We repeat the computation of the phasing from Sec. 3.5.1 but using the ap-
proximation (3.54). We write ?(x) = ?non-dip(x) + ?dip(x), where we have defined
161
1 1 dE
?non-dip(x) ?? F , (3.58a)GM? non-dip(x) dx
1 Fdip(x) dE
?dip(x) ? , (3.58b)
GM?F 2non-dip(x) dx
which can be expanded in the form
5 [ ]
? (x) = 1 + ?ndx+ ?ndx3/2 + ?ndx2non-dip 2 3 4 +O(5) , (3.59a)64x5??
25S2? ??
[ ]
?dip(x) = 1 + ?
d
2x+ ?
d 3/2 d 2
3x + ?4x +O(5) . (3.59b)1536x6??2
The expressions for the coefficients ?nd dn , ?n are given in Eqs. (D.16) and (D.20) in
Appendix D. Using the decomposition in Eq. (3.58), we integrate Eq. (3.44) and
obtain the phase evolution
?(x) = ?non-dip(x) + ?dip(x), (3.60)
with
[ ]
1 5 nd 5? nd 3/2 nd 2non-dip(x) =? [1 + ?2 x+ ?3 x + 5?4 x +O(5]) , (3.61a)32x5/2?? 3 2
25S2 ? 7 7
? (x) = ? 1 + ?dx+ ?dx3/2
7 d 2
dip 2 3 + ?4x +O(5) , (3.61b)5376x7/2??2 5 4 3
where we have ignored an arbitrary additive constant phase.
3.6 Spin-weighted spherical modes of the waveform
Combining the results of the previous sections, we present the gravitational
waveform in a convenient form for use with GW detectors. First, we decompose
the waveform hTTij as given in Eq. (3.30) into its plus and cross polarizations. We
162
introduce the spherical coordinates (R,?,?) in the center-of-mass frame and define
the usual orthonormal triad {N?, P?, Q?} where N? = eR, and P? and Q? lie along the
major and minor axes, respectively, of the projection of the orbital plane onto the
plane of the sky. The plus and cross polarizations of the waveform are defined as
the projections
1 ( )
h+ = (P?
TT
iP?j ? Q?iQ?j
2 ) hij , (3.62a)1
h TT? = P?iQ?j + Q?iP?j hij . (3.62b)2
We then decompose the waveform into spin-weighted spherical harmonics according
to [260] ? ??
h+ ? ih? = ?2Y`m(?,?)h`m, (3.63)
`?2 m=?`
where the coefficients h`m are the spin-weighted spherical modes that we wish to
compute.
We introduce, as in GR, a convenient new orbital phase variable that allows
us to formally absorb the logarithms appearing in the polarizations h+, h?:[ ]
? ? ? 2(1? ?)? x3/2 11ln (4r0?) + ?E ? , (3.64)
? 12
where r0 is the length scale associated with the transformation to radiative coordi-
nates introduced in Sec. 3.4.1. This definition differs from its GR counterpart7 [261]
by only a factor of (1? ?)/?. Note that the difference between ? and ? is of at least
3PN relative order because the leading-order term in the phase is formally O(?3)
7Note that in the notation of Ref. [30] and references therein, this redefined phase is denoted
by ?. We instead use ? to avoid confusion with our ? = (m1 ?m2)/M .
163
in dipole-driven systems and O(?5) in quadrupole-driven systems. Given that we
control the phasing of the binary only at 2PN relative order, we can ignore this
correction.
For the mode amplitudes, we adopt th?e notation
? 2GM(1? ?)?x 16?h`m H?`me?im?, (3.65)
R 5
where the appropriate phase factor is scaled out for each mode as well as the leading
order amplitude of the 22 mode, which differs from its value in GR by only a factor
of 1? ?.
Because we consider only non-spinning binaries, and consequently, those on
planar orbits, the modes obey the symmetry relation
h`m = (?1)`h?`,?m . (3.66)
Thus, one needs only the modes with m ? 0 to specify the waveform. Combining
the results of the previous sections, we obtain at 2PN order for the quantities H? 8`m :
( )
4??? 4?+ 2? 55? 107
H?2,2 =1 + x ( ? ? + ?3 3 3 42 42 )
3/2 ?2?? 2? ? 3 S2 ? 1 1+ x ( + i?? ? i?S
2 + i?S2
? ? 2 3 ? 3 +
16?2 2 22 ?? 16??? ? 4?? 8???+? 19???? ? 113?
2
?? ? 16?+ x + + + +?
? 3 3 3 7 63 ?
?16?
2
+? ? 4?
2 23? 2 2+ +? 113?+ ? ? ? 5? ? 74?? ? ? ???+ + + +
? 3 7 63 3 12 21 21 3
8Recall that our definition for h?? (3.14) introduces a sign difference relative to the results
summarized in Ref. [30].
164
)
4?? 4? ? ? 2047?2? ? + + 8??+ ? 1069? ? 4?+ ? 2173+ [ + + + (3.67a)3 3( 3 1512 3 )216 3 1512
1 ? ? ? 5? 17H?2,1 = i? x( 1 + x 2??? 2?+ + + ?3 2 7 28 )]
3/2 ??? i? i? ln(16) ?+x + + + ? i ? i ln(16) ? 4 S2 ? 4i??S?S+i??
? 2? 2? ? 2? 2? 3 ? 3?
? [ (3.67b)
H?3,3 =?
3 15 ?
i ( ? x 1 + x (2??? ?(2?)+ ? ? + 2? ? 4)4 14 ( )
3/2 ?3?? 21i? ? 6i? ln
3 3? ? 21i 6i ln
3
+x + 2 + + 2)] ?
8
i??S2
? 5? ? ? 5? ? 9 ?
? 3 S2 8i??S?S+ 3i? + + i?S2? + (3.67c)10 9? 10
x
H?3,2 = ? [90? 270? + x (?720???? + 240??? + 720?+? ? 240?+
54 35 )]
?365?2[+ 725(? ? 193? ) (3.67d)
i?? x ? ? ? 2? 8H?3,1 = (1 + x 2??? 2?+ ? ?12 14 3 3
?? 7i? 2i? ln(2) ? 7i 2i ln(2) 40
+x3/2 ? + + +)]? ? ? i??S
2
? 5? ? ? 5? ? 3 ?
? 1 i?S2 8i??S?S+ 1 2? + + i?S+ (3.67e)10 3? 10
4x
H?4,4 = ? [990? ? 330 + x (2640???? ? 880??? ? 2640?+? + 880?+
297 35 )]
?1320?? + 440? + 2625?2 ? 6365? + 1779 (3.67f)
9i?(2? ? 1)x3/2
H?4,3 = ? [ ( (3.67g)4 70
1 ?? ? 8??? 8?+ 4?H?4,2 = 5x 3? 1 + x)]8???? ? ? 8?+? + ? 4?? +63 3 3 3
19?2 ? 805? 437+ + (3.67h)
22 66 110
i?(2? ? 1)x3/2
H?4,1 =? ? (3.67i)
84 10
165
? 625i?(2??? 1)x
3/2
H?5,5 = (3.67j)
96 66
? 32 (5?
2 ??5? + 1)x
2
H?5,4 = ? (3.67k)9 165
9 3
H?5,3 = i ?(2? ? 1)x3/2 (3.67l)
32 110
2 (5?2 ??5? + 1)x
2
H?5,2 = (3.67m)
27 55
i?(2? ? 1)x3/2
H?5,1 =? ? (3.67n)
288 385
54 (5?2 ? 2
H?6,6 = ?
5? + 1)x
(3.67o)
5 143
H?6,5 =0 ? ( ) (3.67p)128 2
H? 2 26,4 =? 5? ? 5? + 1 x (3.67q)
495 39
H?6,3 =0 (3.67r)
2 (5?2 ? 5?? + 1)x
2
H?6,2 = (3.67s)
297 65
H?6,1 =0, (3.67t)
where we omit the common remainder O(5) for all modes.
The h`m modes with m = 0 correspond to non-oscillatory memory terms. As
discussed in Sec. 3.4.3, even systems presently driven by quadrupolar radiation will
have undergone a dipole-driven phase in the distant past, which complicates the
calculation of these DC memory terms. Hence, we limit ourselves to the dipole-
driven case and use Eq. (3.42). Working at Newtonian order, the only non-zero
mode is h20, which reads
1
H?DD2,0 = ? +O(2). (3.68)
4 6
166
3.7 Stationary phase approximation
In this section, we compute the Fourier transform of the gravitational waveform
using the stationary phase approximation (SPA). This technique is only applicable
to oscillatory modes; we do not consider the m = 0 modes here.
We adopt the following convention for the Fourier transform of a function g:
? +?
g?(f) ? dt e+2i?ftg(t). (3.69)
??
Note that this convention differs from the standard one, in which the argument of
the exponential has a minus sign. Our convention ensures that modes proportional
to e?im? with positive mode number m and increasing orbital phase ? have power
in positive frequencies in the Fourier domain. Our results can be converted to the
more common convention by taking f ? ?f .
Combining the terms in Eq. (3.65), the h`m modes can be written as
h (t) = A (t)e?im?(t)`m `m , (3.70)
where A`m is the (complex) amplitude. Note that we use ? to describe the phase
rather than ? defined in Eq. (3.64) ? we ignore the 3PN correction ? ? ?, which
can be thought of as a small phase correction to the amplitude at higher order than
we work.
The h`m modes for m =6 0 are rapidly oscillatory, slowly chirping signals. Put
more precisely, during the inspiral, the modes satisfy |A?`m/A`m|  ? and |??|  ?2,
which indicates that the SPA is applicable to the waveform [262]. Applying the
167
Fourier transform (3.69) to Eq. (3.70) gives
hSPA(f) =A (f)e?i?`m(f)?i?/4`m `m , (3.71a)
(m) (m)
?`m(f) =m?(tf )?? 2?ftf , (3.71b)
A (m) 2?`m(f) =A`m(tf ) , (3.71c)(m)
m??(tf )
(m) (m)
where ? = ?? and tf is defined implicitly as the time at which m?(tf ) = 2?f .
Note the m-dependence of this time-to-frequency correspondence; at a given time,
the different harmonics in the signal correspond to gravitational wave emission at
different frequencies.
In keeping with the notation common in the literature, we introduce the new
PN tracking parameter v = x1/2 = (GM??)1/3, tied to the orbital frequency ?. It is
customary to introduce a similar notation for the frequency f as v = (?GM?f)1/3f .
(m)
Since v(tf ) = (2/m)
1/3vf , Eq(. (3.71b) can be rewr)it?ten as
1 ?
?`m(f) = m ?(v)? v3t(v) ?? . (3.72)GM? v=(2/m)1/3vf
We compute the functions ?(v) and t(v) using a similar method to the phasing
as in Eq. (3.44). From the balance equation?(3.44) we deducev
?(v) =?(v0)?
1 dE/dv
? dv v
3 , (3.73a)
GM? v F(v)0
v
? dE/dvt(v) =t(v0) dv F , (3.73b)v (v)0
where v0 is related to the orbital frequency at some reference point in the evolution.
Likewise, the factor entering the Fourier-domain amplitude (3.71c) is computed
using
1 ?GM? 1 dE= F . (3.74)?? 3v2 (v) dv
168
We evaluate Eqs. (3.73) and (3.74) using a prescription akin to that in Sec. 3.5
(corresponding now to the TaylorF2 approximant [259]): the expressions are re-
expanded in v, truncated at relative 2PN order, and then integrated term by term.
For the sake of compactness, we write ?(v), t(v), and 1/?? in terms of the expansion
of the dimensionless ratio ?(x) introduced in Eq. (3.47), using
? 1 1 dEF = 2v?(v
2). (3.75)
GM? (v) dv
3.7.1 The dipole-driven regime
For the dipole-driven regime, we insert Eq. (3.75) into Eq. (3.73) using the
expansion (3.49) and integrate, yielding
?1 [ ]
?DD(v) = 1 + 3?DDv2S [ 2 ? 3?
DD 3
3 v ln v ? 3?DD4 ]v
4 , (3.76a)
4 2 ??v3?
tDD(v) ?1 3
= 1 + ?DDv2S 2 + 2?
DDv33 + 3?
DDv44 , (3.76b)GM? 8 2 ??v6? 2
where we have dropped the integration constants for now.
Combining these two expressions gives the SPA phase (3.72)
[ ]
m 9
?DD(f) =? 1 + ?DD 2`m S v ? 2?
DDv3(1 + 3 ln v)? 9?DDv4 +m?0 ? 2?ft0 ,
8 2???v
3 2 2 3 4
(3.77)
where v is evaluated at v = (2/m)1/3vf , and where we restored constants t0 and ?0
which are the sums of ?(v0), t(v0) with the terms from the lower boundary of the
integrals. The coefficients ?DDn are given in Eq. (D.6).
169
Similarly, the complex amplitud?e is?given by [
ADD 2G
2(1? ?)M2???1/2 4 2 H?`m(v) 1 1
`m (f) = 1 + v
2?DD + v3?DD
R(?1/2|S?| )5] m v5/2 2 2 2 3
1 1
+ v4 ?DD4 ? (?DD2 )2 , (3.78)2 4
where H?`m(v) is given by Eq. (3.67) with the replacement x = v
2 and, as before, v
is evaluated at v = (2/m)1/3vf .
Note that because H?`m are complex they can affect the phase of the waveform.
In particular, they can carry an overall minus sign or factor ?i, which should be
included in the phase of the mode. In addition, higher-order terms can have a factor
?i differing from the one entering at leading order, which induce corrective phases.
However, those phases turn out to be negligible, entering at higher PN order than
the 2PN relative order we consider.
3.7.2 The quadrupole-driven regime
We follow the same treatment for the quadrupole-driven systems as laid out
in Sec. 3.5.2: we split the flux into dipolar and non-dipolar parts, and expand ?(v)
to first order in the ratio Fdip/Fnon-dip according to Eq. (3.54). The first and second
terms in Eq. (3.54) produce a non-dipolar and dipolar contribution, respectively, to
the phase and amplitude of the SPA waveform.
The non-dipolar contribution to the phasing is constructed by inserting
Eq. (3.75) into the integrals in Eq. (3.73) and using the expansion (3.59a). Ignoring
170
the integration constants for now, we find
[ ]
?non-dip
1 5 5
(v) =? [1 + ?
ndv2 + ?ndv3 + 5?ndv4 +O(5) , (3.79a)
32v5?? 3 2 2 3 4 ]
tnon-dip(v) ? 5 4= 1 + ?nd 82 v2 + ?nd 3 nd 43 v + 2?4 v +O(5) , (3.79b)GM? 256v8?? 3 5
and thus, the corresponding contribution to the Fourier-domain phase for the h`m
mode is given by
[ ( )]??
?non-dip
3 20
`m (f) = m ? 1 + ?nd 2 nd2 v + 4?3 v3 + 10?ndv4 ? .256v5?? 9 4 ?v=(2/m)1/3vf
(3.80)
Similarly, we use Eq. (3.59b) to compute the contribution to the phasing from the
dipolar energy flux
[ ]
2
dip 25S?? 7 d 2 7 7? (v) = [1 + ?2v + ?
dv3 + ?d 43 4v +O(5) ,] (3.81a)5376v7??2 5 4 3
tdip(v) 5S2?? 5 10 5= [ 1 +
d 2 d
( ?2v + ?3v
3 + ?dv44 +O)(]5?) , (3.81b)GM? 1536v10??2 4 7 3
2 ?
?dip
5S ? 7 5 35
`m(f) =m
? 1 + ?d 2 d2v + ?3v
3 + ?d 4 ?
3584v7??2 4 2 9 4
v ? . (3.81c)
v=(2/m)1/3vf
Combining these two pieces and restoring the constants ?0 and t0, the Fourier-
domain phase is then simply
?QD(f) = ?non-dip`m `m (f) + ?
dip
`m(f) +m?0 ? 2?ft0. (3.82)
The coefficients ?ndn and ?
d
n are given in Eqs. (D.16) and (D.20). When restricted
to Brans-Dicke theory, Eq. (3.82) reproduces the leading order deviation in the
phase from GR derived in Ref. [240] at order O(1/?BD), which is equivalent to
first order in Fdip/Fnon-dip. For systems containing a very massive black hole, i.e.,
171
s1 = 1/2, s
? ??
1 = s1 = 0 and m1  m2, we recover the phase up to 2PN relative order
derived in Ref. [263].
The computation of the Fou?rier-domain amplitude closely follows that of the
dipole-driven regime. In place of ?(v), one instead uses
? ? [ ]1 ?dip(v)
?(v) ' ?non-dip(v) 1 + . (3.83)
2 ?non-dip(v)
Finally, one re-expands this expression using Eqs. (3.59a) and (3.59b) and inserts
the result in to Eq. (3.71c).
3.8 Conclusions
We have computed the gravitational waveform at 2PN relative order for a
compact binary system on quasi-circular orbits in scalar-tensor theories with a single
massless scalar. The phase and amplitude are presented in ready-to-use form for all
hlm modes. We used the stationary phase approximation to express the waveform in
Fourier space. We performed these calculations for systems whose inspiral is driven
by the emission of dipolar radiation and those driven by quadrupolar flux. Because
of the tight constraints on scalar-tensor gravity, only very low-frequency systems
(e.g., binary pulsars) or those that host non-perturbative scalarization phenomena
(e.g., spontaneous or dynamical scalarization) fall within this first regime ? most
prospective GW sources will be quadrupolar-driven.
We conclude with a brief discussion of the potential utility of our results for
testing GR with GWs. The early inspiral offers the best prospects for detecting
the emission of dipolar radiation by compact binary systems, as radiation reac-
172
tion enters at lower PN order in scalar-tensor theories than in GR. Thus, the best
constraints would come from observation of neutron star-black hole binaries with
space-based detectors.9 Current estimates on the detectability of scalar-tensor ef-
fects have predominantly been made using the leading-order correction to the GW
phase [240, 264?267] (although Ref. [263] used 3.5PN scalar-tensor waveforms in
studying extreme mass ratio inspirals).
Using Eq. (3.82), we estimate the upper bound on the contribution of each
PN correction to the phase. We consider the phase accumulated by a 100? 1.4M
system during an observation period of one year, spanning the frequency range
f ? (0.065Hz, 1Hz), subject to the experimental constraints discussed in Sec. 3.2.2.
Relative to the 7.7 ? 106 cycles produced by the Newtonian-order GR term, the
leading-order scalar-tensor correction decreases the phase by up to ? 600 GW cy-
cles.10 The 1PN relative order correction increases the total phase by another ? 2
cycles, although this piece would be difficult to detect, as it takes the same form as
the leading-order GR term. The 1.5PN relative order correction adds ? 3 GW cycles
to the inspiral, and the 2PN order effect is below the limit of eLISA detectability,
only contributing ? 0.1 cycles over the year. We emphasize that these values are
only an order-of-magnitude estimate of the possible impact of higher-order scalar-
tensor corrections; a more extensive parameter estimation study is needed to truly
9Up to 2PN order, the signal produced by binary black holes is known to be identical to that
in GR up to an undetectable rescaling of the gravitational constant G. [150]
10Note that this bound comes from assuming a neutron-star scalar charge of ?A ? 6 ? 10?3,
whereas Refs. [240, 264?267] considered the maximum charge found in Brans-Dicke gravity ?A ?
2? 10?3.
173
determine the detectability of these effects.
3.9 Acknowledgements
We are grateful to Ryan Lang for providing a Mathematica notebook con-
taining the results of Ref. [151] and to Lijing Shao for useful discussions concern-
ing current binary pulsar constraints. N.S. acknowledges support from NSF Grant
No. PHY-1208881. S.M. acknowledges support from NASA grant 11-ATP-046 and
NASA grant NNX12AN10G at the University of Maryland, College Park. N.S.
thanks the Max Planck Institute for Gravitational Physics for its hospitality during
the completion of this work.
174
Chapter 4: Modeling dynamical scalarization with a resummed post-
Newtonian expansion
Authors: Noah Sennett and Alessandra Buonanno1
Abstract: Despite stringent constraints set by astrophysical observations,
there remain viable scalar-tensor theories that could be distinguished from general
relativity with gravitational-wave detectors. A promising signal predicted in these
alternative theories is dynamical scalarization, which can dramatically affect the
evolution of neutron-star binaries near merger. Motivated by the successful treat-
ment of spontaneous scalarization, we develop a formalism that partially resums
the post-Newtonian expansion to capture dynamical scalarization in a mathemati-
cally consistent manner. We calculate the post-Newtonian order corrections to the
equations of motion and scalar mass of a binary system. Through comparison with
quasi-equilibrium configuration calculations, we verify that this new approximation
scheme can accurately predict the onset and magnitude of dynamical scalarization.
1Originally published as Phys. Rev. D93, 124004 (2016).
175
4.1 Introduction
The detection of gravitational-wave (GW) event GW150914 by Advanced
LIGO heralds a new era of experimental relativity [184]. Every test of the past
hundred years has indicated that gravity behaves as predicted by general rela-
tivity (GR). Until now, the best constraints have come from solar-system exper-
iments [79] and binary-pulsar observations [89, 249]. These measurements probe
the mildly-relativistic, strong-field regime of gravity generated by objects with ve-
locities v/c . 10?3 and gravitational fields ? 2Newt/c . 10?1 (see Table 4 of Ref. [79]
for a summary of model-independent constraints).
For the first time, these constraints can be extended through the direct ob-
servation of strong, dynamical gravitational fields. In particular, GW detectors
can track the coalescence of compact objects in binary systems, a process in which
the objects are highly-relativistic and strongly self-gravitating, with v/c ? 0.5 and
? 2Newt/c ? 0.5. We expect to observe several GWs per year [229, 231] with the
upcoming global network of detectors comprised of Advanced LIGO [230], advanced
Virgo [64], and KAGRA [233].
These ground-based GW detectors will be able to observe binaries of solar-
mass objects for thousands of orbital cycles before merger. Significant effort has
gone into the development of techniques to test GR with these measurements; see
Refs. [88, 268] and references therein. During the first stage of a binary?s coalescence
(the early inspiral), the waveform measured by the detector is well described within
the stationary-phase approximation. The waveform generated in GR for the early
176
inspiral can be approximated by
A(?)
h ?7/6 i?(?;f)GR(?; f) = f e , (4.1)
D
where f is the observed frequency, D is the distance to the binary, and A and ?
are the amplitude and phase of the GW, respectively, dependent on the intrinsic
(e.g. chirp mass, component spins, etc.) and extrinsic (e.g. sky position, time of
coalescence, etc.) parameters of the binary, represented collectively by ?.
Using this signal as a baseline, one can parameterize any non-GR waveform in
the early inspiral as
h(?; f) = hGR(?; f) (1 + ?A(?, ?; f)) ei??(?,?;f) (4.2)
where ? represents the parameters that characterize the alternative theory [117, 118].
Then, given a GW detection, Bayesian inference can be used to estimate ?A and ??
[120, 213, 214, 269?271]. Typically one expands these functions in powers of the fre-
quency f (and its logarithm log f), then performs a hypothesis test to constrain the
corresponding expansion coefficients. This approach can be used either to search
for generic deviations from GR by treating these coefficients independently or to
test a specific alternative theory against GR by relating the coefficients to the un-
derlying physical parameters ?. Using a parameterized waveform that also included
the merger and ringdown signal, both types of tests were done for GW150914 in
Ref. [122]: with the former, the authors constrained the higher-order expansion
coefficients in ??, and with the latter, they placed a lower bound on the Compton
wavelength ?g of the graviton in a hypothetical massive gravity theory [111] (?g is
signified by ? in our notation).
177
However, this type of analysis rests on the assumption that ?A and ?? admit
expansions in powers of f . There exist certain alternative theories of gravity where
this assumption of analyticity breaks down due to phase transitions or resonant
effects [271]. Fortunately, several complementary tests were performed in Ref. [122]
to verify that GW150914 is indeed consistent with GR. Still, our ability to model
non-analytic features in waveforms is essential in case future events do not match
the predictions of GR as closely.
The task of modeling a non-analytic deviation ?? in a generic, theory-independ-
ent way is intractable. Instead, previous work has focused on modeling specific non-
GR phenomena predicted in particular alternative theories of gravity. We continue
this effort here, focusing on dynamical scalarization (DS), an effect that can arise
in neutron-star binaries in certain scalar-tensor (ST) theories of gravity [169, 171].
Previous efforts to model this effect have simply grafted models of DS onto inde-
pendently developed analytic approximations of the inspiral [170, 175, 271]. In this
work, we propose a new perturbative formalism that incorporates DS from first
principles. Our aim is to lay the groundwork for a model whose accuracy can be
improved iteratively in a way that is more straightforward and self-consistent than
previous methods.
The paper is organized as follows. In Sec. 4.2, we examine the relationship
between DS and the better understood phenomenon of spontaneous scalarization.
From this discussion, we motivate a resummation of the post-Newtonian formalism
to incorporate DS, which is then developed in Sec. 4.3. We derive the equations
of motion for a neutron-star binary up to next-to-leading order in Secs. 4.4 and
178
4.5. In Secs. 4.6 and 4.7, we calculate its scalar mass (a measure of the system?s
scalarization) at the same order. As a test of its validity, in Sec. 4.8, we compare
our model with numerical quasi-equilibrium configurations of neutron stars [172] and
previous analytical models [170]. We provide a summary in Sec. 4.9 and outline the
future work needed to produce waveforms with our model.
4.2 Non-perturbative phenomena in scalar-tensor gravity
Scalar-tensor theories of gravity are amongst the most natural and well-motivated
alternatives to GR. We consider the class of theories detailed in Ref. [147], in which
a massless scalar field couples non-minimally to the metric, effectively allowing a
spin-0 polarization of the graviton. These theories are described by the action
? ? [ ]
4 c
3 ?g ? ?(?)S = d x ?R g???????? + Sm[g?? ,?], (4.3)
16?G ?
where ? represents all of the matter degrees of freedom in the theory. Note that in
the limit that ? ? ?, the scalar field relaxes to a constant value, and the theory
reduces to GR with the modified gravitational constant Geff = G/?; we refer to this
extreme as the GR limit.
The form of the action in Eq. (4.3) is known as the ?Jordan frame? action.
Alternatively, the action can be cast into the ?Einstein frame? by performing a
conformal transformation g??? ? ?g?? as
? ?
c3 ?g? [ ] [ ? ? ]
S = d4x R?? 2g???? ??? ?? + S e? 2d??/ 3+2?(??)? ? m g??? ,? , (4.4)
16?G
179
where we have introduced the scala?r field?
? 3 + 2?(?)?? d? . (4.5)
2?
From Eq. (4.4), we see that the coupling of the scalar field to matter (through the
metric g???) is characterized by
a = (3 + 2?)?1/2. (4.6)
Measurable phenomena absent in GR arise in theories whose coupling is linear
in ??
B??
a = . (4.7)
2
This coupling can be expressed in terms of Jordan frame variables as
1
=B log ?, (4.8)
?(?) + 3/2
and imposes the relation between ? and ??
? = exp(B??2/2). (4.9)
Damour and Esposito-Fare?se discovered an instability in the scalar field trig-
gered by the presence of relativistic matter in theories with B > 0 [153].2 For
2Based off the work of Refs. [152, 272, 273], the authors of Ref. [175] recently discussed a
related instability in this theory that would cause the scalar field to grow rapidly over cosmological
timescales throughout the Universe. Consequently, the scalar field today would be so large that
its presence would have already been detected by solar-system experiments. The addition of a
potential V (?) or slight modification of ?(?) could ameliorate this issue while preserving the
neutron star phenomena discussed in this work (for example, see Ref. [161]). As is done in the
literature, we ignore here this cosmological problem.
180
sufficiently large B, compact neutron stars were found to undergo a phase transi-
tion now known as spontaneous scalarization. Spontaneously scalarized stars are
expected to behave differently than their (un-scalarized) GR counterparts (see Refs.
[274?278] for examples of such deviations).
Observation of a scalarized star would be a smoking gun for modifying gravity;
in turn, our lack of evidence for such stars places constraints on this class of ST
theories [247]. Because scalarization arises from the non-linear interaction between
strong gravitational fields and matter, it is unconstrained by weak-field experiments
and GW150914. However, pulsar timing measurements have ruled out nearly all
theories that can sustain spontaneous scalarization [247].
Dynamical scalarization is a similar phenomenon revealed by recent numerical-
relativity simulations that is not ruled out by binary-pulsar observations [169, 171].
In a binary system, neutron stars too diffuse to spontaneously scalarize in isolation
were found to scalarize collectively. Despite the name, DS has also been found
in recent quasi-equilibrium calculations [172]; the phenomenon is caused by the
proximity of the neutron stars rather than their dynamical evolution. The onset of
DS produces an abrupt change in the stars? motion, generating sharp features in the
GW signal produced by the binary.
Gravitational-wave detectors may be able to extend the current constraints
on ST theories by searching for DS [175]. This endeavor hinges on our ability to
accurately and efficiently model GWs from binaries that undergo DS. Such effects
have been modeled by using a Heaviside function for ?? in Eq. (4.2) [175, 271] or
augmenting the post-Newtonian (PN) evolution of the binary with a semi-analytic
181
feedback model [170]. This work follows a general strategy similar to that of Ref.
[170]. However, we adopt a top-down approach to incorporate DS into the PN
formalism in hopes of creating a model that is more consistent and streamlined
conceptually (see Appendix G for a detailed analysis of the results of Ref. [170]).
The PN expansion is an effective tool for analytically approximating the evolu-
tion of binary systems of interest to ground-based GW detectors. In this approach,
one expands solutions to the Einstein equations about flat space in the small pa-
rameter  ? GM/rc2 ? (v/c)2 < 1, where M, r, v represent the characteristic mass,
distance, and velocity scales in the problem, respectively. In ST theories, this ex-
pansion is done about the Minkowski metric ??? and background field ?0 (assumed
to be constant and homogeneous over the time and distance scales of the evolution
of a binary system). We refer to the n+1 corrections to these quantities as the
?nPN? fields. We define non-perturbative phenomena as behavior found in the full
gravitational theory that cannot be recovered at any finite PN order.
In the remainder of this section, we argue that DS is a non-perturbative phe-
nomenon. First, we review the analytic treatment of spontaneous scalarization, de-
scribing the way in which the phenomenon has been identified as non-perturbative
and then incorporated into the PN expansion in a rigorous manner. We then perform
a similar analysis for DS and present a quantitative argument that the phenomenon
is non-perturbative. Finally, we describe how the analytic treatment of spontaneous
scalarization could be adapted to incorporate DS into the PN formalism.
182
4.2.1 Spontaneous scalarization: single neutron star
In ST theories, static, spherically symmetric spacetimes are characterized by
three parameters: the asymptotic field ?0, the Arnowitt-Deser-Misner (ADM) mass
m, and the scalar charge ? [147, 279]. These parameters can be extracted from the
asymptotic behavior of the metric and scalar field
( )
2Gm ( )
lim gij = 1 + | | ?ij +O |x|
?2 , (4.10)
|x|?? x c2
2G?0m? ( )
lim ? =?0 + | | +O |x|
?2 , (4.11)
|x|?? x c2
where we have defined
?0 ? ? ?1 B log ?0= . (4.12)
3 + 2?(?0) 2
It was shown in Ref. [147] that the scalar charge of an isolated star can be
written in the PN expansi[on as ( ) ( ) ]2
Gm Gm
? = ?0 1 + A1 + A2 + ? ? ? , (4.13)
Rc2 Rc2
where R is the radius of the body, and the coefficients Ai are of order unity. Because
?0 vanishes in the GR limit, one finds that the right hand side of Eq. (4.13),
truncated at any finite order, must vanish as well. However, as first discovered in Ref.
[153], exactly solving the geometry numerically shows that a sufficiently compact
body can sustain an appreciable scalar charge even when ?0 = 0 (corresponding
to the GR limit ?0 = 1).
3 Thus, we would describe this scalarization as non-
perturbative (in the sense defined above). Figure 4.1 depicts the sharp growth in
3Because of the additional prefactor of ? , the |x|?10 term in Eq. (4.11) vanishes even for
183
0.8 ??0? =
=10-
=10-
3
4
0
0.6
0 10-5
0.4
0.2
0.0
0.00 0.05 0.10 0.15 0.20 0.25
Gm/Rc2
Figure 4.1: The scalar charge of an isolated non-spinning neutron star
as a function of its compactness in the limit that ?0 approaches zero
(the GR limit). We use the theory parameter B = 9 with a piecewise
polytropic fit to the APR4 equation of state detailed in Ref. [206].
scalar charge in the limit ?0 ? 0 as one increases the compactness of a neutron
star. For this figure and all that follow, we use a piecewise polytropic fit [206] to
the APR4 equation of state given in Ref. [280].
The tension between the analytic and numerical results suggests that the PN
expansion must break down beyond some compactness Gm/Rc2 for this class of ST
theories. The scalar charge is non-analytic at this critical compactness, at which
point the isolated body undergoes a phase transition. Analogous to ferromagnetism,
the derivative of the charge diverges when ?0 approaches zero, indicating that this
transition is of second order. Beyond the critical point, the vanishing of ?0 in the GR
limit is compensated by the divergence of the bracketed sum in Eq. (4.13). The only
spontaneously scalarized stars in the GR limit. The dramatic effect of spontaneous scalarization
is m( ore e)asily seen through ?? [given in Eq. (4.9)], which can be approximated as ?? = ??
Gm?
0 + |x|c2 +
O |x|?2 , where ?0 = ?(??0).
184
Scalar Charge
astrophysical objects that could reach this critical compactness are neutron stars and
black holes. However, no-hair theorems protect isolated black holes from developing
a scalar charge [23]. We focus exclusively on neutron stars for the remainder of this
work.
In anticipation of our discussion of dynamical scalarization, we briefly review
how spontaneous scalarization is incorporated into analytic models of binary pulsars.
A binary system of non-spinning stars is characterized by two length scales: the
characteristic size of the bodies R and their separation r. As in the case of an
isolated body, the individual stars can spontaneously scalarize if they exceed some
critical compactness, at which point the PN expansion no longer accurately predicts
the evolution of the binary. Damour and Esposito-Fare?se developed the ?post-
Keplerian? (PK) expansion to accommodate such systems [147, 208] (not to be
confused with the ?parameterized post-Keplerian? formalism for modeling binary
pulsars in generic alternative theories [281]). In the PK approach, one expands only
in Gm/rc2, leaving quantities dependent on R unexpanded (e.g the scalar charge
?). Equivalently, one can recombine the sum in powers of Gm/Rc2 in the PN
expansion to produce the PK expansion. The relationship between the PN and PK
expansions is summarized in the bottom two panels of Fig. 4.2 (the remaining panels
are discussed in Sec. 4.2.2). Spontaneous scalarization is captured by explicitly
including all of the terms in Eq. (4.13) at each order in the PK expansion.
185
Full Theory (Unexpanded)
Post-Dickean (PD)
Small Parameters: Gm/rc2, v/c
Partially resum Gm/rc2, v/c
Post-Keplerian (PK)
Small Parameters: Gm/rc2, v/c
Resum Gm/Rc2, vinternal/c
Post-Newtonian (PN)
Small Parameters: Gm/Rc2, v 2internal/c, Gm/rc , v/c
Figure 4.2: Analytic approximations of ST theories. Starting from the
PN expansion about the Minkowski metric ??? and the background field
?0, one resums all expansions in the compactness Gm/Rc
2 and asso-
ciated internal velocity vInternal/c to capture spontaneous scalarization.
Recombining these expansions produces the PK approximation. To cap-
ture dynamical scalarization, one resums the PK expansion in Gm/rc2
and v/c. Fully recombining these expansions reproduces the original
(unexpanded) theory. Instead, one partially resums the PK expansions
to generate the PD approximation.
4.2.2 Dynamical scalarization: neutron-star binaries
Despite its successful application to binary pulsars, the PK approximation
does not predict dynamical scalarization. The asymptotic scalar field for a binary
system has been computed recently to 1.5PK order in Ref. [151].4 For a system
4In the literature, the distinction between the PN and PK expansions is often overlooked; the
PK expansion (i.e., the approximation in which power series in Gm/Rc2 have been resummed) is
often referred to as the ?PN expansion,? (for example Refs. [79, 149, 151]). To avoid confusion,
we have taken care to distinguish the two in Sec. 4.2 when discussing spontaneous scalarization.
Because both the PN and PK expansions fail to capture dynamical scalarization, starting from
Sec. 4.3, we continue the popular conflation of these two approximation schemes, referring to the
expansions collectively as ?PN.?
186
containing neutron stars too diffuse to spontaneously scalarize individually, the PK
prediction of the total scalar charge remains small as the binary coalesces. However,
numerical-relativity calculations indicate that the scalar charge can greatly increase
beyond this estimate as the two neutron stars draw close [169, 171, 172]. We post-
pone a quantitative comparison between these analytic and numerical predictions
until Sec. 4.8 (see Fig. 4.6); we must first formulate a precise measure of the scalar-
ization of a binary system. Akin to spontaneous scalarization, we suspect that the
mismatch between analytic and numerical results stems from a breakdown of the
PK expansion. We posit that DS is a non-perturbative phenomenon, and hence the
PK expansion needs to be suitably modified to capture it.
To support this intuition, we carefully examine how the mass and scalar charge
of a star depend on the nearby scalar field. For an isolated body, these are the
relations m(?0) and ?(?0) where ?0, m, and ? are defined in Eqs. (4.10) and (4.11).
As shown in Appendix A of Ref. [147], the scalar charge is related to the mass by
( )
? d logmA?A(?0) =?0 1 2 . (4.14)
d log ?0
The dependence of the mass on ?0 can only be found by numerically solving the
Tolman-Oppenheimer-Volkoff (TOV) equations modified for ST gravity with a given
equation of state [153].
The mass and scalar charge of each neutron star in a binary system can be
similarly determined provided that the system is well-separated (R/r  1). Working
at leading order in R/r, each star can be treated as an isolated body immersed in
?
the scalar field produced by its partner [147]. At a distance |x| = d ? Rr from
187
each star (?far? from the star relative to R), the metric and scalar field will behave
as in Eqs. (4.10) and (4.11) with ?0 replaced by the background field produced
by the other star. As above, we numerically solve the modified TOV equations to
relate the mass m and sc?alar charge ? to this background scalar field. Because we
work in the limit d/r = R/r ? 0, this matching occurs effectively at each star
relative to r, the smallest distance scale relevant to GW generation.5 In this limit,
the TOV equations provide us with the dependence on the mass and charge on the
local scalar field for each body in a binary system, i.e. the functions m(?) and ?(?)
where ? is evaluated at the star.
To analytically model these relations using the PK approximation, one must
expand m and ? about the background field ?0, where now ?0 is the value taken very
far from the binary system at |x|  r. Because the analytic form of the function
m(?) is unknown, Eardley [242] proposed the agnostic expansion
[ ( ) ](0) 1
mA(?) = mA 1 + sA? + s
2
A ? sA + s?A ?2 + ? ? ? , (4.15)2
where
(0)
mA ? m(A(?0), ) (4.16)
? d logmAsA ( ) , (4.17)d log ? ?=?0
? d
2 logmA
sA ? , (4.18)d(log ?)2 ?=?0
? ?? ?0 ? Gm? . (4.19)
?0 rc2
5In this work, we ignore all effects that arise from the finite size of the neutron stars. Such
effects could influence the dynamics of a binary system of scalarized stars at 1PK order [282].
188
We plot the magnitude of the coefficients in Eq. (4.15) in Fig. 4.3 across a
range of scalar field values reached during the coalescence of a binary neutron star
system [169, 171]. Using the model of Ref. [170], we estimate that DS occurs when
the field at each body reaches a value of
? ? 10?4, (4.20)
depicted as the pink region in the figure.
For neutron stars with realistic, piecewise polytropic equations of states (e.g.
fits to APR4 and H4 defined in Ref. [206]), we find that for ? near this maximal
value, ???? ?Cn+1 ??? ? 103 ? 105, (4.21)Cn
for n = 1, 2, where Ci is the coefficient of the i-th PN correction in Eq. (4.15).
Comparing Eqs. (4.20) and (4.21), we see that the rapid growth of the ex-
pansion coefficients in Eq. (4.15) can overpower the ?smallness? of our expansion
parameter ?. In particular, the relative contribution of each term on the right hand
side of Eq. (4.15) does not diminish as one moves to increasingly higher order.
These symptoms indicate that m(?) may not be analytic in this regime, and thus,
the PK expansion would break down at this point in the binary?s evolution. Inspired
by the treatment of spontaneous scalarization, we posit that the best way to work
around this restriction is to resum the expansion in Eq. (4.15). The hierarchy of
these expansions is outlined in Fig. 4.2.
Unfortunately, such a prescription is not as straightforward as the case for
spontaneous scalarization. To capture spontaneous scalarization, one simply ?un-
189
1010
Dynamical
109 ||C1|||C2| Scalarization108 C3
107 ?104
106 
105 
104 
103 ?104
102 
101 
100 
10-7 10-6 10-?5 10-4 10-3Log 0
Figure 4.3: Magnitude of the coefficients of the expansion m(?) =
m(0) (1 + C 21? + C2? + ? ? ? ) across the typical scalar field values
achieved during the evolution of a compact binary. Values shown here
are for an isolated body with m(?0) = 1.35M and APR4 equation of
state with B = 9. Interpolation errors dominate the computation of C3
for small values of log ?0; we omit these regions of the curve.
expands? all expansions in Gm/Rc2 in the PN approximation [i.e. those of the
form as in Eq. (4.13)], leaving only expansions in Gm/rc2 and the corresponding
orbital velocity v/c. Completely resumming these expansions would reproduce the
full ST theory. Instead, we need to choose certain quantities dependent on Gm/rc2
to resum, and leave the rest expanded. Based on the discussion above, we suspect
that the best quantities to keep unexpanded are the mass m(?) and its derivatives
(including the scalar charge ?(?)). However, a priori, there is no clear indication of
precisely ?what to resum.? We need to incorporate the flexibility of this choice into
our model.
190
4.3 The post-Dickean expansion
4.3.1 Action and field equations
We refer to our method of resumming the PN expansion as the ?post-Dickean?
(PD) approach ? named after Robert Dicke, one of several pioneers of ST gravity
[177?179, 283, 284] who made many important contributions to experimental rela-
tivity throughout his career. For notational convenience, we introduce an auxiliary
field ? that is related to ? in a small neighborhood of each particle?s worldline.6 This
new field is used to demarcate the resummed variables (m and its derivatives). We
explicitly constrain ? in the matter action via the Lagrange multipliers ?A
?? ?
S 2 4m ? c d x d?A?(4) (x? ?A(?A))? (mA(?, ?) + ?A(?A) (F (?)? ?)) ,
A
(4.22)
where the arbitrary functions m(?, ?) and F (?) encode our choice of how to resum
the mass and scalar charge, respectively.
6Formally, the matching of ? and ? is done at the boundary of the body zone, defined at a
?
distance d ? Rr from each body. As justified in Appendix A of Ref. [147], in the limit that
d/r ? 0, we can represent each body as a point particle; in this limit, the matching of the two
field variables is done exactly on each body?s worldline.
191
With this expression, the action in Eq. (4.3) gives rise to the field equations
F (?(?A(?A))) =?(?A(?A)), (4.23)
u?A?
Dm
? (m(?, ?)u
? ) =? ??A ( ? ) (4.24)D?
1 ?(?) 1
R ? Rg = ? ?? ?? g g???? ?? ? ? ?? ??????
2 ?2 2
(4.25)
1 8?G
+ (???(??? g???) + T?? ,? ?c4 )
1 8?G ? 16?G DT d?? = T ? ? g???????? , (4.26)
3 + 2?(?) c4 c4 D? d?
where we have defined
D ? ? dF ?+ , (4.27)
D? ?? d? ??
?? ??2c ?SmT ?g ?g??
? ?? (4.28)
=c3 ?g d?AmA(?, ?)u? ? (4)AuA? (x? ?A(?A)),
A
d??
where ?A, ?
? A
A, and uA = are the worldline, proper time, and four velocity ofd?A
particle A, respectively.
In this work, we focus on only a few, natural choices form and F , given in Table
4.1. Physically, the choice of m(RJ) corresponds to resumming the mass measured
in the Jordan frame, while the choice and m(RE) corresponds to resumming the
Einstein-frame mass
m(?)
m(E)(?) = ? . (4.29)
?
The choices of F (?) and F (??) respectively equate the auxiliary field ? to ? and ??,
defined in Eq. (4.9). We refer to the joint selection of m and F as the resumma-
tion scheme. The PD parameterization also encompasses the (non-resummed) PN
192
Table 4.1: Resummation schemes discussed in this work. We abbreviate m(?, ?)
with m and F (?) with F .
F (?) F (??)
(RJ) m = m(?) m?= m(?)m
F = ? F = 2 log ?/B
m = (?/?)1/2m(?) m = ?1?/2e?B?2/4(RE) m(?)m
F = ? F = 2 log ?/B
m(PN) m = m(?)
expansion; this limit is reached with the choice of m(PN) given in the table (recall
that here ?PN? is used to refer collectively to the post-Newtonian and -Keplerian
approximations).
4.3.2 Relaxed field equations
To solve Eqs. (4.25) and (4.26), we employ a technique known as direction
integration of the relaxed Einstein equations, originally developed in GR in Refs.
[285?289] and then extended to ST gravity in Refs. [149?151]. The remainder of
this section closely follows the framework presented in Sec. II.B of Ref. [149]. We
define
?
g?? ? ?gg?? , (4.30)
H???? ?g??g?? ? g??g??. (4.31)
As in general relativity, the following identity holds:
H????
16?G
,?? =(?g)(2R ?Rg ???? ?? + tLL), (4.32)c4
193
where t??LL is the Landau-Lifshitz pseudotensor.
We assume that far from any sources, the metric reduces to the Minkowski
metric ??? and that the scalar field approaches a constant value ?0. Let ? ? ?/?0
be the normalized scalar field. We introduce the conformally transformed metric
g??? ? ?g?? , (4.33)
the gravitational field
?
h??? ? ??? ? ?g?g??? , (4.34)
and the ?conformal gothic metric?
?
g??? ? ?g?g??? . (4.35)
We impose the Lorentz gauge condition
? h???? =0. (4.36)
Substituting Eqs. (4.30)?(4.34) into the gauge condition (4.36), the field equa-
tion (4.25) is rewritten as
16?G
?h?
?? =? ??? , (4.37)
c2
where ? is the Minkowski space d?Alembertian and
2
??? ? ? ? c( g) T ?? + (??? + ???) , (4.38)
?0c2 16?G
S
??? ?16?G [(?g)t?? ]((g???) + ? h???? h???)? h???LL ? ? ? ? ??4 ? ?h? , (4.39)c
??? ?3 + 2?S ?????? g???g??? ?
1
g??? g??? , (4.40)
?2 2
194
where the notation [(?g)t??LL](g???) indicates that the Landau-Lifshitz pseudotensor
should be calculated using g? rather than the physical metric g. Similarly, the scalar
field equation (4.26) can be recast into the form
8?G
?? =? ?s, (4.41)
c2
with ( )
?? 1 ?? ? ? DT c
2
?s [g ( T )2? ? h?
???????
3 + 2? ? c20 ]D? 8?G (4.42)
c2 d 3 + 2?
+ log ???? ?g?
??
? .
16?G d? ?2
The differential equations (4.37) and (4.41) can be solved formally using the standard
flat-space Green?s function; we only consider retarded solutions, i.e. those with no
incoming radiation
?
?? 4G 3 ? ?
??(t? |x? x?|,x?)
h? (t,x) = d?x | ? ?| , (4.43)c2 x x
2G ? ?
?(t,x) =1 + d3 ?
?s(t? |x? x |,x )
x
c2 |x? ?| , (4.44)x
where the integration constant is explicitly added to enforce the asymptotic bound-
ary condition on the scalar field. By construction, the constraint equation Eq. (4.23)
acts as an additional boundary condition on the scalar field along the worldline of
each body; this constraint distinguishes our work from the PN solutions found in
Refs. [149?151].
We approximate the formal solutions given in Eqs. (4.43) and (4.44) with an
expansion in terms of  ? (v/c)2 ? Gm/rc2. However, to capture the strong-field
effects behind dynamical scalarization, we expand only the metric g?? and scalar
field ?, leaving ? unexpanded. Note that ? appears only in the function mA(?, ?) in
195
Eqs. (4.37) and (4.41). Thus, by not expanding ?, we effectively resum the variable
mass found in the PN treatment. This treatment also resums the scalar charge,
which is governed by the derivative of m [see Eq. (4.14) or (4.76)]. The constraint
equation Eq. (4.23) is used to solve ? exactly on each worldline at a given order in
.
4.4 Structure of the near-zone fields
The resummation detailed above only enters through the sources, i.e. the
stress-energy tensor T ?? and its derivatives. As such, we adopt the same techniques
used for the PN calculation of the metric and scalar field in Refs. [149?151]. We
summarize this approach below, leaving our results in terms of T ?? and its deriva-
tives. For more detail, see Secs. III and IV of Ref. [149].
The integration in Eqs. (4.43) and (4.44) is done over the flat space past null
cone C emanating from the point (t,x). We divide this three-dimensional hypersur-
face into two regions. For matter sources of characteristic size S, we define the near
zone as the worldtube with |x| < R where R ? S/v is the characteristic wavelength
of the emitted gravitational radiation. The radiation zone is the region outside of
the near zone, that is, |x| > R. We demarcate the intersection of C with the near
zone as N and the intersection of C with the radiation zone as C ? N .
We focus first on finding the metric and scalar field in the near zone, as these
determine the equations of motion of the binary system through Eq. (4.24). Fol-
196
lowing Refs. [149, 288], we establish the following notation
N ? h?00, Ki ? h?0i,
(4.45)
Bij ? h?ij, B ? h?ii.
To post-Newtonian order, we express the metric in terms of these fields using Eqs.
(4.33) and (4.34)
( ) ( ) ( )
? 1 1 ? 3 2 ? 1 1g00 = 1 + N(+)? + B N N???
2 +O , (4.46)
2 2 8 2 c6
1
g i0i =? [K +(O ,)] ( ) (4.47)c5
1 1
gij =?ij 1 + N ?? +O , (4.48)
2 c4
where ? was defined in Eq. (4.19).
At the point (t,x) in the near zone, the near-zone contribution to the integrals
in Eqs. (4.43) and (4.44) can be expanded in powers of |x? x?|
? 00 ? ? ( )4G ? (t,x ) 2G 1
NN (t,x) =
3
| ? ?| d x
? + ?2t ?
00(t,x?)|x? x?|d3x? +N?M +O ,
c2 4 6M x x c M c
? ( ) (4.49)
4G ? 0i(t,x?) 1
KiN (t,x) =
3 ?
? | ? ?| d x +K
i
?M +O( ) , (4.50)c2 M x x c5
4G ? ij(t,x?) 1
BijN (t,x) =
3 ? ij
c2 ? |x? x?| d x +B?M?+O , (4.51)M c4
2G ?s(t,x
?)
? (t,x) = d3x? ? 2G? ? (t,x?)d3 ?N
c2 M |?x? ?| 3 t sx c M ( )x (4.52)
G
+ ?2 ? (t,x?)|x? x?| 3 ? O 1s d x + ,
c4 t 6M c
where M is a constant-time hypersurface which covers the near zone and we have
used Eq. (4.36) to eliminate the first order correction in Eq. (4.49). There will
197
also be a contribution to the fields at (t,x) from the radiation zone, but these only
enter at higher order [149]. The boundary terms N?M, K
i , Bij?M ?M depend on the
value of R. Because the left hand side of Eqs. (4.49)?(4.52) should not depend on
the arbitrarily chosen boundary between the near and radiation zones, we argue (as
in Ref. [149]) that these terms are exactly cancelled by the contributions from the
radiation zone. This cancellation was shown explicitly in GR in Refs. [287, 288].
All that remains is to expand the sources ??? and ?s. We first define the
densities
? ? (T 00 + T ii)c?2, (4.53)
?i ? T 0ic?2, (4.54)
?ij ? T ijc?2, (4.55)
?s ? ?
T 2?DT
+ . (4.56)
c2 c2 D?
We expand Eqs. (4.38) and (4.42) to post-Newtonian order
[ ( ) ( )]
00 1 G 7 G?
2
0 1
? = ? ? ?ii + 4?U ? (?U)2 ? 6?Us ? (?Us)2 ,
?0 ?0c2 8? ? c20 8?
(4.57)
?i
? 0i = ,[ ] (4.58)?0
ii 1 ii ? 1 G ? 2 ? 1 G?
2
0
? = [? ( U) (?U )
2
s ] (4.59)?0 8? ? 20c 8? ? 20c
? 2 2 20 G G(B ? 2?0 ) 1 G(B + 4?0 )
?s = ?s + 2 ?
2
2 s
U + ?
2 s
Us ? (?Us) , (4.60)
?0 ?0c ?0c 8? ?0c2
198
where we have introduced the potentials
?
?(t,x?? )U 3 ?? ? d x , (4.61)M |x? x |
?s(t,x
?)
U 3 ?s ? | ? ?| d x . (4.62)M x x
Plugging these expressions back into Eqs. (4.46)?(4.52), the 1PD metric and scalar
field are given by
2G 2G? 2G 2G2 2 20
g =? 1 + U + U ? M? ? U2 G ?0(B ? 4?0) 200 + U ?
G ?0
UU
? c2 2
s s s
0 ?0c c3 ?2c4 2?2c4
s 2 4
0 0 ?0c ( )
4G2?0 s ? 12G
2?0 G
2?0(B ? 8?0) s G G?0 O 1+ ? ?2s + ? + X? + X?s + ,
?20c
4 2 ?2 4 2 4 2s 4 4 60c ?0c ?0c ?0c c
( ) (4.63)
? 4G i 1g0i = [ V +O , ] ( ) (4.64)? c2 50 c
2G 2G?0 1
gij =?ij 1 + U ? Us +O , (4.65)
? c2 20 ?0c c4
2G?0Us ? 2G? =?0 + [ M?2 sc c3 ] ( )
G?0 G(B + 4?0)
+ U2
G s G(B ? 8?0) s 1
c2 2? c2 s
+ 4 ? + ? + X?s +O ,
0 ?0c2
2 ?0c2
2s c6
(4.66)
199
with the additional potentials
?
M ? ? (t,x?)d3x?s ? s , (4.67)
i ? ?
i(t,x?)
V 3 ?? | d x , (4.68)x? x?|
? (t,x?s ? s )U(t,x
?)
? d3x?2 ? | ? ?| , (4.69)x x
? ?(t,x
?)Us(t,x
?)
? 3 ?2s ? | ? ?| d x , (4.70)x x
s ?s(t,x
?)U ?s(t,x )
?2s ? ? | ? ?| d
3x?, (4.71)
x x
X ? ? ?(t,x
?)|x? x?|d3x?, (4.72)
Xs ? ?s(t,x?)|x? x?|d3x?, (4.73)
4.5 Two-body Equations of motion
4.5.1 Newtonian order
We now apply these calculations to a binary system whose stress-energy ten-
sor is given by Eq. (4.28). To highlight the novel aspects of the PD approach, we
explicitly work out the leading-order equations of motion here before calculating
their higher-order corrections in the following section. In keeping with PN conven-
tions, we describe the leading order as Newtonian and the next-to-leading order as
post-Dickean or ?1PD.?
At Newtonian order, the densities defined in Eqs. (4.53) and (4.54) are given
200
by
? ( )1
? = mA(?, ?)?
(3)(x? xA) +O , (4.74)2
? cA ( )?A(?, ?)
?s = mA(?, ?) ?
(3)(x? xA) +O
1
. (4.75)
?0 c2
A
where we have introduced the(scalar ch)arge1/2(
of each body )
B log ? D logmA
?A(?, ?) ? 1? 2? . (4.76)
2 D?
Our definition of the scalar charge is the natural generalization of the expression
used in Ref. [147]; with no resummation, i.e. m(?, ?) = m(?), one recovers the
definition
(E)
?d logm? = AA , (4.77)
d??
where ?? is defined in Eq. (4.9).
Evaluating Eq. (4.66) at Newtonian order, the scalar field for a 2-body system
is given by
G? 20 Us
? =?0 + 2 ,
c2 ( ) (4.78)
2Gm1?0?1 1
=?0 + +O + (1
 2) ,
c2r 31 c
where we have adopted the shorthand
mA ?mA(?(xA), ?(xA)), ?A ? ?A(?(xA), ?(xA)),
(4.79)
rA ? |x? xA|, nA ? (x? xA)/rA.
Because mA and ?A depend on ?, these quantities must be expanded around the
background field ?0. We suppress these expansions (given in Appendix E) through-
out the remainder of this work for notational convenience, denoting with the short-
201
hand in Eq. (4.79) that the mass and charge should be expanded and truncated at
the appropriate PD order.
On each worldline, we exactly solve (i.e. not perturbatively) Eq. (4.23), ig-
noring the divergent terms ?that arise from self-interactions of each body????? + 2Gm2?0?2? 0 2 , if F (?) = ??(x1) =? c r?? ? (4.80)?? Gm2?20 + 2 , if F (?) = 2 log ?c r B
?(x2) = (1 2) , (4.81)
where r ? |x1 ? x2| is the orbital separatio?n of the binary and
? 2?0 2 log ?0??0 = . (4.82)
B B
Note that this system of equations cannot be solved analytically, as mA and ?A
depend on ? along each worldline. This final step is analogous to the feedback
model proposed in Ref. [170]; with the choice of F (??) given in Table 4.1, we exactly
reproduce this model.
Plugging in the expressions for the metric and scalar field into Eq. (4.24), we
find the Newtonian equations of motion
i ? Gm2 (1 + ?1?2)a1 = ni, (4.83)? r20
ai2 = (1
 2) , (4.84)
where n ? (x1 ? x2)/r. The mass mA and scalar charge ?A depend on the choice
of resummation scheme; their leading order piece is given in Appendix E.
202
4.5.2 Post-Dickean order
To find the equations of motion of the binary to next order in c?2, we expand
the stress-energy tensor and evaluate the potentials introduced in Sec. 4.4 (see
Appendix F).
On each worldline, we plug the above potentials into Eq. (4.66) and numeri-
cally solve Eq. (4.23)
?
??????? 2G?0(m2?? 2
Gm2?2 [
??0 + + ) ??0(v2 ? n)
2
??? ?0rc
2 ? 40rc
???? B 2 Gm2? +? if F (?) = ?(2 + 2?02 ?0r ) ]
??? ? 2 Gm?(x 11) =?? 2?0 ?1 + ?0 (3 +[ ?1?2) , (4.85)?? ?0r???? Gm2?2 Gm2?2 1??? 2?? 0 + ( + ? )(v2 ? n) ?2 4?? ?0rc ?0rc 2? ] if F (?) =
2 log ?
? 3 1 Gm
B
1
+ ?0?1 + ?1?2 ,
2 2 ?0r
?(x2) = (1
 2) . (4.86)
Substituting Eqs. (4.63)-(4.66) into Eq. (4.24), we find the following equation
of motions for each particle [
i ? Gm2 (1 + ?1?2) Gm2a(1) = ni + ni ? (1? ? ? 2 22 2 2 1 2) v ? 2(v ? 2v1 ? v2)?0r ?0r c 1 2
3 ? 2 Gm1+ (1 + ?1?2) (v2 ]n) + (5 + ?0?1) (1 + ?1?2)2 ?0r
Gm2 Gm2
+4 (1 + ?1?2) + (v1 ? v )i2 [4 (v1 ? n)? (3? ?2 1?2) (v2 ? n)] ,?0r ?0r c2
(4.87)
ai(2) = (1
 2) , (4.88)
203
where m and ? themselves receive post-Dickean corrections dependent on the re-
summation scheme used (see Appendix E).
For reference later, the 1PN equations of motion (with no resummation of the
mass) are recovered with the choice m(PN) [
Gm?2 (1 + ??1??2) Gm?2 ( )
ai i i1 (PN) =? n + n ? (1? ?? 2 21??2) v1 ? 2 v2 ? 2v1 ? v2? 20r ?0
3 (r
2c2
+ (1 + ?? ?? ) (v ? n)2 + 4 + 4?? ?? ? ???
)
2 Gm?2
1 2 2 1 2 ??
2( ) ] 1 2 ?0rGm?1
+ (5 + ??1??2) (1 + ??1??2)? ??? 22??1 ?0r
Gm?2
+ (v1 ? v2)i [4 (v1 ? n)? (3? ??2 2 1??2) (v2 ? n)] ,?0r c
(4.89)
ai2 (PN) = (1
 2) , (4.90)
where we have introduced the shorthand
m?i ? mi((?0), ) (4.91)
? ? d logmi??i ?0 1 2 ( , (4.92)d log ? ?=?)0
? ? B??
2
i ? d logmi??i 4?22? 0 . (4.93)0 d(log ?)2 ?=?0
The apparent differences between Eqs. (4.87)?(4.88) and Eqs. (4.89)?(4.90)
are simply artifacts of the different notations. The disparities stem from the presence
in Eq. (4.89) of higher-order terms from expansions like Eq. (4.15). These terms are
absorbed into the definitions of mA and ?A in the PD expansion [see Eq. (4.79)]. We
emphasize the differences between these two notations because the analytic model
proposed in Ref. [170] directly adapted the equations of motion written as in Eqs.
204
(4.89) and (4.90). Beyond post-Newtonian order, we expect a greater proportion of
the corresponding terms in each notation to differ.
For a generic resummation scheme, the 1PD Eqs. (4.87) and (4.88) are not
solutions to the Euler-Lagrange equations for any Fokker Lagrangian (a Lagrangian
dependent solely on the the positions and velocities of the two bodies). A simple
calculation reveals that the equations of motion can be integrated back to such a
Lagrangian only when no resummation is performed, i.e. when m(PN) is used.7 The
absence of a PD Fokker Lagrangian suggests that our model of DS requires the
two-body phase space to be augmented with additional degrees of freedom besides
the bodies? positions and velocities, such as the scalar field ?. We conjecture that
any other extension to the PN formalism to incorporate DS will also require new,
dynamical degrees of freedom.
4.6 Structure of the far-zone fields
Having solved the dynamics of the binary, we now shift our attention to ob-
servables that can be extracted from the asymptotic geometry of the system. Our
interest in this type of quantity is twofold. First, such objects encode all information
needed to estimate GW signals (e.g. the waveform and its phase evolution estimated
from the Bondi mass and flux). Second, there are several gauge-invariant quantities
defined asymptotically that are easily computed in numerical relativity (e.g. the
ADM mass and angular momentum) and thus can be used to directly check the
7This result contradicts the assertion of Ref. [170] that such a Lagrangian can be constructed
by resumming (or not expanding) the scalar charge ? in the PK Lagrangian of Ref. [208].
205
validity of our model. For simplicity, in this work we restrict our attention to the
scalar mass, a coordinate independent measure of a spacetime?s scalarization. We
define the scalar mass at retarded time ? as
2 ?
MS(?) ??
c
?ij?i? dSj, (4.94)
8?G ? |x|??t?|x|=?
? ?0c
2
= ?ij?i? dSj, (4.95)
8?G |x|??
t?|x|=?
where Sj is the surface-area element in flat space. We leave the other useful quan-
tities described above for future work.
Calculating the scalar mass requires knowledge of the scalar field at a distance
|x| = R R (recall that R is the boundary of the near zone). As in the near zone,
we will recycle the tools used to determine the scalar field in the radiation zone from
previous PN calculations. We summarize this calculation for a generic stress-energy
tensor below; for more detail, see Refs. [150, 151]
At the order at which we work, the scalar field at null infinity receives contri-
butions from both the near and radiation zones, which we denote as ?N and ?C?N ,
respectively. We compute each piece separately, dropping any terms dependent on
R, which we assume will cancel when the pieces are combined (as was done in Sec.
4.4).
4.6.1 Near-zone contribution to the scalar field
The contribution to the scalar field at the point (t,x) in the radiation zone
from points (t?,x?) in the near zone is found by expanding the integral expression
206
given in Eq. (4.44) in powers of |x?|/R
?? ?2G 1 ?m (N? ? x?)
?N = ?s(?,x
?) d3x?, (4.96)
c2+m? m! ?t
m
m=0 M?
? ( R )
2G (?1)m
= ? ? ? ? 1? Ik1???km(?) , (4.97)
c2
k k
m! 1 m R s
m=0
where N? ? x/R, M? is the intersection of the near zone with a hypersurface of
constant retarded time ? = t ? R, and we have introduced the scalar multipole
moments
?
Ik1???kms (?) ? ?s(?,x)xk1 ? ? ?xkmd3x. (4.98)
M?
We note that the terms that fall off faster that R?1 in Eq. (4.97) will not contribute
to the scalar mass; dropping these terms, the remaining piece of the scalar field is
given by
?? 2G 1 m
? = N?k1 ? ? ? dN?km Ik1???kmN (?), (4.99)
Rc2+mm! dtm s
m=0
We also note that only terms with even parity (with respect to inversions of x) will
contribute to the scalar mass. These are the terms in Eq. (4.99) with even m.
The source ?s in the near zone is needed at higher order than what was given
207
in Eq. (4[.60) to calculate the 1PD scalar mass ]
? 2 2 20 G G(B ? 2?0 ) 1 G(B + 4?0 )
?s = ?s + 2 ?s{U + ?sUs ? (?Us)
2
?0 ? c2 ? c20 0 8? ?0c2
G?20(1 + ?
2) G [ ]
+ 0 ? 2U2 ? (2B ? 4?2s 0)(UUs + ?s 22)? 12?0?2s ? (4?}1 ? X?)?2c40 ?0
G(B2 ? 10B?2 + 8?2) G(B ? 8?2 2 2
+ 0 0 U2 + 0
)(B ? 2?0) s B ? 2?? 0{ s 2s + X?s4?0 2?0 2G?2? 0 8UU?s + 16V j?jU?s + 8?ij1 ?i?jUs ? (B + 4?20)(U?2s ??Us ??X?s)8??2c40
G?2(6B + 8?20 0) ? 2 G(B ? 8?
2
0)(B + 4?
2
0)+ Us ( Us) + ?U ss ???
[ ?0 ? 2s0 ]}
?G ?4(B + 4?2)?U ???s ? 8P ij0 s 2 2 ? 2 iji?jUs ? 8?0P2s?i?jUs ,?0
(4.100)
where, in addition to the potentials introduced in Sec. 4.4, we define
? ii
? ? (t,x
?)
? 3 ?1 ? | d x , (4.101)x? x?|ij ?
?ij1 ?
? (t,x ) 3 ?
?| ? ?| d x , (4.102)x x
ij ? 1 ?
?
iU(t,x )?jU(t,x
?)
P 3 ?2 ? | ? ?| d x , (4.103)4? x x
ij ? 1 ?
?
iUs(t, x )?jUs(t, x
?)
P 3 ?2s 4? |x? x?| d x . (4.104)
4.6.2 Radiation-zone contribution to the scalar field
We rewrite the integral in Eq. (4.44) in a more useful way when working far
from the system
?
2G ? (R? + ? ?,x?)?(t? ? t+ |x? x?| ?R?s )
? = 4 ?
c2 |x? x?| d x , (4.105)
208
where R? = |x?| and ? ? = t??R?. Thus, the contribution to the scalar field from the
radiation zone (i.e.?R? > R) ?is given?by
2G ? 2? 1 ? ? ?
? = d? ?
?s(? +R ,x ) ? 2 ?
C?N ? ? d? (R ) d(cos ? )c2 ???2R 0 1?? t? ? ? N?? ? x (4.106)
2G ??2R ?s(?
? +R?,x?)
+ (R?)2d2??
c2 ?? t? ? ? ? N?? ? x
where ? = (? ? ? ?)(2R? 2R+ ? ? ? ?)/(2RR) and N?? = x?/R?. The source ?s takes
a different form in the radiation zone than in Eq. (4.100). To the order at which we
work, the source in the radiation zo
B + 4?2 [
ne is given by ]
? =? 0 c2(??)2 2 1s ? ?? ? N??. (4.107)
32?G?20 8?G
The stress-energy tensor does not appear in this expression (under the guise of ?
or ?s) because the radiation zone does not contain any matter. In computing the
source ?s, we can ignore the radiation-zone contribution to the scalar field, as the
corresponding contributions to the source will enter at beyond the order that we
work. Thus, we use the scalar field as given in Eq. (4.97); the metric field N can
be expanded in a similar way. At this order, only the monopole and dipole pieces
of these fields appear in ?s.
4G I
N = + ? ? ? , ( ) (4.108)c2 R
2G Is ? 2G I
i
? = ? si + ? ? ? , (4.109)
c2 R c2 R
where the mass monopole moment I is defined as in Eq. (4.98) with ? 00. Plugging
these expressions into(Eq. (4.107), w)e find ( ... )
? G(B + 4?
2) I 20 sI?s (Is) ? G II?s ? G I I
j
s II?j? s js = + + N?
2??2c2 ( R3c R4 ?c4 R2 ?c40 R2c) R3 (4.110)
? G(B + 4?
2 j j j
0) IsI?s 2IsI?s 2IsIs I?sI?js I? jsI+ + + + s N? j.
?c2 R3c2 R4c R5 R3c2 R4c
209
where we?ve used the fact that the moments are functions of retarded time, so that
?jIj j js = ?I?sN? /c. The first line of Eq. (4.110) contains the lowest order terms,
which enter at c?3 order relative to the leading contribution to ?N from the near
zone, while the second line contains terms that are suppressed by one additional
factor of c.
We note that all of the terms in ?s in the radiation zone take the form
1 f(?)
? (l, n) = N?k1 ? ? ? N?kls . (4.111)
4? Rn
With this information, each corresponding term in Eq. (4.106) can be rewritten as
[?
2G R
? ? ]
?C?N (l, n) = N?
k1 ? ? ? N?kl f(? ? 2s)A(s, R)ds+ f(? ? 2s)B(s, R)ds ,
Rc2 0 R
(4.112)
with
? R+s
A(s, R) ? Pl(?)? ? dp, (4.113)pn 1RR+s Pl(?)
B(s, R) ? ? dp, (4.114)pn 1s
?R + 2s ? 2s(R + s)? , (4.115)
R Rp
and where Pl(?) are Legendre polynomials.
Given Eq. (4.110), we see that l = 0, 1 and n = 2 ? 5 integrals contribute
to the scalar field at this order. However, by inspection, the l = 1 terms have odd
parity, and thus will not contribute to the scalar mass. The l = 0 contributions [in
210
the notation of Eq. (4?.112)] are given by
4G2 ? ( ( ? u) [ ]
?C?N (0, 2) =? du log R + ? II?s
Rc6 (?? 2 2 ) u ? (4.116)
? ? u
) [ ] ? [ ]
log R+( ? II?s ? logR du II?s ,2 2 u?2R ??2R u
2G2 B + 4?2 I2(?) I2(?)
? 0 s sC?N (0, 3) = ?
Rc?5 ?20 ( 2R 2R )) (4.117)?
? [I
2] [I2s ]
du s u ? u?2R ,
?? (2(R?+ ? ?(u)2 (2R+ ? ? u)2
4G2 B + 4?2 ? [I2] [I2 )s ]
?C?N (0, 4) =
0 du s u ? u?2R
Rc4 ?20? ??[ ] )(2R + ? ? u)2 (2R+ ? ? u)2 (4.118)1 ?? du I2
(2R)2 s ,u??2R
where we have used the shorthand [fg]x = f(x)g(x). Nearly all of these terms are
hereditary, i.e. depend on the full history of the system up to the retarded time ? .
The one exception is the first term in Eq. (4.117), but this term falls off too quickly
with R to contribute to the scalar mass.
4.7 Two-body Scalar mass
Having expressed the scalar field in the radiation zone entirely in terms of
the (even) scalar multipole moments, we now specialize to an inspiraling binary
system. Plugging the potentials for a two-body system (Appendix F) into Eq.
(4.100), we integrate to find the scalar moments. Integrals containing ?s can be
evaluated directly as they contain delta functions at the worldlines of the bodies.
The remaining terms are integrated by parts, using techniques analogous to those
outlined in Sec. III of Ref. [150]. The multipoles needed to compute the scalar mass
211
at 1PD order are{given by [
I ?0m
2 4 2
1?1 ? v1 ? Gm2s = ( 1 )(1 + ?0(?2)?
v1 Gm2 ?3 (v2 ? n)+
? 2c2 ? rc2 8c40 0 ?0rc2 )2 c2
2?0(1? ? ? 22 0)?B?2 v1 B?2 ? 8?0 + 6? 22?0 (v1 ? n)2+( 4?0) +c2 ( 4?0 ) c2 ]
?2(B + 4?
2
0) (v1 ? v2) ? B?2 ? 16?
2
0 + 4?2?0 (v1 ? n) (v2 ? n)+
4?[ c2 4? c20 0
G2m m 2 21 2 ?1 (B ? 6?0)?2 (B + 6? 6?0)?1?2+ +
2 2 4 ]?0?1 ??0r c 2 4?0 [ ? 4? ] }?(B + 2?2)? ?20 1 2 G2m22 ? Gm2 B?2 4?0(1? ?2?0)+ (a1 ? n)
4? 2?2r2 40 0 c ? c
4
0 2?0
+ (1
 2) ,
[ ] (4.119)
i j 2 2
I ij ?0m1?1x1x1 ? v1 ? Gm2 Gm1m2?1?2(B + 4? )rs = 1 (1 + ? 0 ij2 2 0?2) + ??0 2c ?0rc 4?2 20?0c
+ (1
 2) ,
(4.120)
? m ? xixj k lI ijkl 0 1 1= 1 1x1x1s + (1
 2) . (4.121)?0
We evaluate Eq. (4.99) with these moments to compute the near zone contribution
to the scalar field. Before proceeding, we briefly detail how time derivatives of the
masses mi and scalar charges ?i are handled. Recall that the dependence of each
body?s mass (and scalar charge) on the local scalar field is decomposed into a re-
summed and expanded piece, represented by its dependence on ? and ?, respectively.
Thus, the derivative of the mass would be given by
dmA ?mA ? ?mA= v ?
dt ?? A
???+ vA???, (4.122)??
212
where v? ? 0A = uA/uA. To reinforce that the fields ? and ? really represent the same
physical scalar, we relate the two through Eq. (4.23). Thus (assuming differentia-
bility), their gradients along each worldline are related as
? dFuA??? = u
?
A???. (4.123)d?
In truth, because we expand only ? and not ?, Eqs. (4.23) and (4.123) only hold in
an approximate sense [e.g. up to 1PD order when using Eq. (4.85)]. Nevertheless,
one finds that ( )
dmA DmA ? O 1= v ???+ . (4.124)
dt D? c4
Because the time dependence of the mass enters only through the scalar field (whose
leading order term is constant), its derivative is suppressed by an additional factor
of c?2 more than dimensional analysis would suggest, i.e. m?/m ? c?2. This sup-
pression greatly simplifies our calculation of the scalar field.
Equipped with the scalar moments and a prescription for differentiating with
respect to time, we calculate the near-zone contribution to the scalar field of a binary
system
(?1) (0) (1)
?N =?N + ?N + ?N , (4.125)
with
(?1) 2G?0m1?1
?N = + (1
 2) (4.126)? Rc20 {
(0) 2G?
2 2
0m1?1
?N = ?
v1 (N? ? v1)+ ? Gm2 (1)+}?2 2 2 2 0?2?0Rc 2c c ?0rc (4.127)
(N? ? x1)2 ? (N? ? x1)(N? ? x2)
+(1 + ?1?2) + (1
 2)
r2
213
{ [( ) ( )
(1) 2G?
4 2
0m1?1 ? v1 Gm2 1 + ?0?2 v1 ? 4? ?0?2 (v1 ? n)
2
?N = +?0Rc2 8c4 ?0rc2 ] 2 c2 [ 2 c2
?3(v2 ? n)
2 4(v1 ? n)(v2 ? n) ? G
2m1m2 1 ? 5?0?2+ ?
2 2 ] 2 4 0?1 ?2c c ?0r c 2 2
(6 +B ? 6?20)? ? 2 2 21 2+ + 2?2 ?0?1?2 3G m2( 4 )2 ? ?2 2?20r2c4
? v
2 Gm (1 + ? ? ) (v ? N?)21 2 0 2 1+
2c2 ?0rc2 c2
Gm2
+ [(?
(x1 ? N?)(v2 ? N?)
(4(v1(? n) + (3? ?1?2)(v2 ? n))?0rc2 )rc )
Gm2 ? B D(logm1?1) (v1 ? n)+ ( 8 ( 2?0 + + 4?0?0 ) ?2 + 2?1)?2 2?0rc ?0 D? c]
? ? B D(logm1?1) (v2 ? n) (x1 ? N?)(v1 ? N?)[5 2?0 + + 4?0?0?0 ] ?2 ? ?1?2D? c rc
Gm 22 4?0(v1 ? n) ? 4?
2
0(v2 ? n) (x2 ? N?)(v2 ? N?)+
?0rc2 [ c c rc
Gm 2 22 ?(1? 3?1?2)v1 ? 2v2 4(v1 ? v2) 3(1 + ?1?2)(v
2
2 ? n)
+ + +
? 2 2 2 20rc 2c c c 2c2 ]
Gm1(1 + ?1?2)(5 + ?0?1) Gm2(1 + ?1?2)(5 + ?0?2) (x1 ? N?)(n ? N?)
+ [ ( ( +? rc20 ?0r)c2 ) r
Gm2 1 B D(logm ? ) v
2
1 1
+ 2 1
?0rc2 (2 (? ?0 + + 2?0?0 ) )?2 + ?1?22?0 D? c2
1 B D(logm1?
2
1) v
+ (1? (?0 + + 2?0?0 ?
2
2
2 2? D? ) c20 )
?1 ? B D(logm1?1) (v1 ? v2)(3 (2?0 + + 4?0?0 )?2 + ?1?22 ? D? ) c20
?3 ? B D(logm1?1) (v
2
1 ? n)
2 ?0 + + 2?0?0 ?2 + ?1?2
2 ( ( 2? 20 D? ) ) c
?3 ? B D(logm1?1) (v2 ? n)
2
(1 ?0 + + 2?0?0 ?22 ( 2? D? ) c20 )
3
+ (3?
B D(logm1?1) (v1 ? n)(v2 ? n)
2?0 + + 4?0?0 ?2 + ?1?2
2 ? D? c20
?1 2(+ 3?1?2 + 2?
2?20 2 + ?
2
1?
2
2 2 ) )
? B D(logm1?1) Gm2?0 + + 2?0?0 (1 + ?1?2)?2
2? D? ? 20 0rc
214
(
?1 1(+ ?
2
1?2 + 2?0?
2
2 2 ) ) ]
? B D(logm1?1) Gm1 (x1 ? N?)
2
[ ?0 + + 2?0?0 (1 + ?1?2)?22?0 D? ? 20rc r2
Gm 22 ?0v
2
1 ?
2v2 2?2(v1 ? v 22) 3? (v1 ? n)2 3?2(v2 ? n)2
+ + 0 2 ? 0 ? 0 ? 0
?0rc2 c2 c2 c2 c2] c2
6?2(v ? n)(v ? n)) G?2m G?20 1 2 ? 0 2 ? 0m1 (x
2 4
2 ? N?) (v1 ? N?)
+ +
c2 ? 2 20rc ?0rc r2 c4
?2Gm2(1 + ?1?2) (x
2 2
1 ? N?) (v1 ? N?) ? 6Gm2(1 + ?1?2) (x1 ? N?)(v
2
1 ? N?) (n ? N?)
?0rc2 r2c2 ? 2 20rc rc
2Gm2(1 + ?1?2) (x1 ? N?)2(v1 ? N?)(v2 ? N?)
+
?0rc2 [ r2c2? ? ]6Gm2(1 + ?1?2) (v1 n) (v2 n) (x1 ? N?)2(v1 ? N?)(n ? N?)
+ ?
?0rc2 [ c c ] r2c
Gm2(1 + ?1?2) (v1 ? n) ? (v2 ? n) (x
3
1 ? N?) (v1 ? N?)
+
? rc20 [ c c ] r3c
?Gm2(1 + ?1?2) (v1 ? n) ? (v2 ? n) (x1 ? N?)
3(v2 ? N?)
?0rc2 c c r3c
3G2m22(1 + ?
2 2 2
1?2) (x1 ? N?) (n ? N?)
+
2?20r
2c4 [ r2
Gm2(1 + ?1?2) v
2
1 v
2
2 ? (v1 ? v2) ? 5(v1 ? n)
2 5(v1 ? n)(v2 ? n)
+ + +
? rc20 2c2 2c2 c2 2c2
? ] c
2 }
5(v n)2 Gm (1 + ? ? ) Gm (1 + ? ? ) (x ? N?)3? 2 ? 1 1 2 ? 2 1 2 1 (n ? N?)
2c2 3?0rc2 3? rc2 r30
+ (1
 2) , (4.128)
where we have dropped the pieces that do not contribute to the scalar mass and
have used Eqs. (4.83) and (4.84) to eliminate the bodies? accelerations.
To the order at which we work, the radiation-zone contribution to scalar mass
is zero. The scalar monopole Is is the only multipole that enters in Eqs. (4.116)?
(4.118); as discussed above, at leading order, the monopole is constant in time.
This insight allows one to trivially evaluate these hereditary integrals. The non-zero
215
terms either depend on the arbitrarily chosen boundary R (and thus are canceled by
near-zone contributions to the scalar field) or fall off too quickly with R to contribute
to the scalar mass.
Computing the scalar mass from the scalar field given in Eqs. (4.125)?(4.128)
is most easily done in the center of mass frame. However, we cannot compute the
exact transformation to this frame in the PD formalism without first calculating
the total momentum of the system.8 Instead, we consider frames in which the two
bodies? positions are related by x1 ? ?x2. Without dissipative effects, we expect
the center of mass frame to satisfy this criterion.
Furthermore, we restrict our attention to binary systems undergoing circular
motion. Neutron-star binaries are expected to radiate away any eccentricity rela-
tively early in their evolution, long before they would be detectable by ground-based
experiments like LIGO, thereby justifying this approximation.
We plug the expression for the scalar field in Eq. (4.125) into Eq. (4.95)
to obtain the scalar mass. This surface integral can be computed easily using the
standard angular coordinates (?, ?) on the coordinate sphere of radius R. The
scalar mass takes the exact same form as the scalar field with the N?-dependent
8In the PN formalism, the transformation to the center of mass frame is derived by forcing the
total momentum of the binary system to vanish. The momentum is difficult to calculate within
the PD approach because the equations of motion cannot be derived from a Lagrangian dependent
solely on the particles? positions and velocities. Thus, the exact transformation to the center of
mass frame remains unknown.
216
terms replaced by the geometric quantities derived below
? ( ) ?
? f(?, ?)?i dSi = f(?, ?)d(cos ?)d?, (4.129)
R?? R
and
?
4?
? (N? ? xA)(N? ? xB)d(cos ?)d? = ??ABxAxB, (4.130)3
? (N? ? vA)(N? ?
4?
vB)d(cos ?)d? = ??ABvAvB, (4.131)
3
4?
? (N? ? xA)(N? ? xB)(N? ? xC)(N? ? xD)d(cos ?)d? = ??AB??CDxAxBxCxD, (4.132)5
? (N? ?
4?
vA)(N? ? vB)(N? ? vC)(N? ? vD)d(cos ?)d? = ??AB??CDvAvBvCvD, (4.133)
5
(N? ? 4?xA)(N? ? xB)(N? ? vC)(N? ? vD)d(cos ?)d? = ??AB??CDxAxBvCvD, (4.134)
15
where we have defined
?????1, if A = B
??AB ? ???? . (4.135)?1, if A =6 B
The scalar mass is gi[ven by ( ( ) )]
m 21?1?0 v Gm2 1 + ?1?2 r1
MS = [ ]1?
1 ? 1 + ?0?2 +
?0 6c2 ? 20rc 3 r (4.136)
+ 1PD + (1
 2) ,
where the 1PD terms are represented only schematically for the sake of compactness.
4.8 Validity of the Post-Dickean expansion
The PD expansion was motivated through analogy: spontaneous and dynami-
cal scalarization are suspected to arise from similar mechanisms, and so the analytic
217
fGW [Hz] fGW [Hz]
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200
- ? ? ? ? ? ? ? ? ? ? ? ? ?10 5 10-5
???? ( ( ) (?)) ?? ? ? ?
? ? ? ??
m RJ(RJ), F(?? )
10-6 ((m( ), F (?? ))) 10-6? ? m RE , F
- ? QE10 7 ?? 10
-7
? ??
10-
? ? ?
8 Newtonian 1PD 10-8 ???? ??? ?
100 100 
10-1 10-1
10-2 10-2
0.00 0.01 0.02 ?/ 0.03 0.04 0.05 0.00 0.01 0.02GM c3 GM?/ 0.03 0.04 0.05c3
Figure 4.4: Scalar mass of a (1.35+1.35)M neutron-star binary system
on a circular orbit as a function of the orbital angular frequency and
gravitational wave frequency (fGW = ?/?). The scalar mass is computed
at Newtonian (dashed) and 1PD (solid) order for resummation schemes
listed in Table 4.1. We also plot the quasi-equilibrium configuration
calculations (QE) reported in Ref. [172] (dotted). The bottom panels
depict the magnitude of the fractional error between the PD and quasi-
equilibrium results. We use the APR4 equation of state with (left) B =
9, ??0 = 3.33? 10?11 and (right) B = 8.4, ?? = 3.45? 10?110 .
218
Fractional Error MS [M?]
Fractional Error MS [M?]
techniques applied to the former (resummation of expansions in Gm/Rc2) should
also be used with latter (partial resummation of expansions in Gm/rc2). While
such reasoning seems plausible, ultimately, the validity of our model can only be
checked via comparison with high-precision numerical calculations. In absence of
long numerical-relativity simulations of DS, we compare the PD approximation to
recent quasi-equilibrium configuration calculations. We also closely examine the
differences between the PD approximation and the analytic model proposed in Ref.
[170] for completeness
4.8.1 Quasi-equilibrium configurations
The scalar mass of an equal-mass binary system was calculated along sequences
of quasi-equilibrium configurations in Ref. [172]. Inherent to these calculations is the
assumption of a conformally flat and stationary spacetime; physically, each configu-
ration represents a binary system following a circular orbit. Despite neglecting the
loss of energy and angular momentum through the emission of gravitational radia-
tion, this setup is believed to closely resemble the adiabatic inspiral of a neutron-star
binary system. Systematic errors enter these quasi-equilibrium calculations through
the physical assumptions made above and imperfect numerical convergence, par-
ticularly at higher frequencies. At present, the magnitude of these errors is not
well-understood.
We compare the PD predictions of the scalar mass with these numerical results
to validate the accuracy of the model. Figure 4.4 depicts the scalar mass as a
219
function of orbital frequency ? for a (1.35+1.35) M binary system, where the
PD corrections to Kepler?s third law for an equal mass system (derived from the
equations of motion)
GM(1 + ?22 ) ? G
2M2(1 + ?2)(11 + 2?0? + ?
2)
? = , (4.137)
r3?0 4r4?2 20c
are used to replace the r-dependence in Eq. (4.136), and where M = m1 +m2 and
? = ?1 = ?2. The scalar mass is computed at Newtonian (dashed) and 1PD (solid)
order; note that the former calculation is done consistently at Newtonian order
[e.g. only the first term in Eq. (4.137) is used]. We employ the APR4 equation of
state, for which the allowed range of theory parameters in which DS can occur is
spanned by B ? [8, 9] (see Ref. [171] for more detail). From this range, we focus
on the cases B = 9 and B = 8.4, corresponding to the choices ??0 = 3.33 ? 10?11
and ??0 = 3.45 ? 10?11 considered in Ref. [172]. For all PD calculations, we use a
Newton-Raphson method to numerically solve Eqs. (4.85) and (4.86) to within a
fractional error of 10?7.
Recall that the PD expansion encodes a flexibility in ?what to resum? in the
choice of m(?, ?) and F (?). We compare each combination of the choices in Table
4.1 in Fig. 4.4, denoting each resummation scheme by the pair (m,F ). The scalar
mass estimated with the (m(RJ), F (?)) and (m(RE), F (?)) resummation schemes differ
by only ? 0.01%; to improve legibility, we only plot the former (in red).
The two most important features depicted in Fig. 4.4 that we hope to recover
with our model are the frequency at which DS occurs ?DS and the magnitude of the
scalar mass after scalarization. We extract the onset of DS from the figure using
220
the fitting procedure detailed in Ref. [172]; these values are given in Table 4.2. One
finds that the scalar mass MS can be wel?l approximated by( ( ) ) ??2 10/3 ?
M ?1, if ? < ?DSS
1 + =
M?0 ???? (4.138)a0 + a1x, if ? > ?DS
2/3
where x ? (GM?/c3) . We determine the coefficients a0 and a1 by fitting the high
frequency part of the curves in Fig. 4.4 and then find ?DS from the intersection of
this linear function with 1.
The 1PD predictions for both the location and magnitude of scalarization
match the results of Ref. [172] at the . 10% level for the choice F (??). (Note that
the peaks in the relative error seen in the bottom panels of Fig. 4.4 stem from
the slight misalignment of the scalar mass predictions at the sharp onset of DS.)
Interestingly, for systems that scalarize later in the inspiral (i.e. smaller values of B),
the Newtonian order prediction in the (m(RE), F (??)) scheme agrees more closely with
the numerical results. Without a more comprehensive study of various resummation
schemes or the PD expansion at higher order, it is difficult to say whether this
agreement is coincidental.
The choice of m(?, ?) seems to have little effect on the scalar mass predictions
of the PD model. The two resummation schemes with F (?) are essentially indistin-
guishable, while the schemes with F (??) appear to converge to within a few percent
at 1PD order.
On the other hand, the choice of F (?) drastically alters the growth of the scalar
mass. Of the two options presented in Table 4.1, only F (??) reproduces the sharp
221
Table 4.2: Orbital angular frequency and gravitational wave frequency at which
dynamical scalarization occurs (fGW = ?/?) for the systems considered in Fig. 4.4.
Only resummation schemes with the choice F (??) produce DS. For comparison, we
list the results of the quasi-equilibrium configuration calculations (QE) of Ref. [172].
B Model Order GM? 3 GWDS/c fDS [Hz]
9.0 (m(RJ), F (??)) Newtonian 0.0044 106
9.0 (m(RJ), F (??)) 1PD 0.0047 112
9.0 (m(RE), F (??)) Newtonian 0.0052 124
9.0 (m(RE), F (??)) 1PD 0.0051 122
9.0 QE ?? 0.0051 123
8.4 (m(RJ), F (??)) Newtonian 0.0282 674
8.4 (m(RJ), F (??)) 1PD 0.0212 508
8.4 (m(RE), F (??)) Newtonian 0.0217 520
8.4 (m(RE), F (??)) 1PD 0.0212 508
8.4 QE ?? 0.0223 534
transition consistent with dynamical scalarization. The significance of the choice of
F can be seen by studying the behavior of the scalar charge ?(?, ?). Because the
definition of ? relies on the choice of resummation scheme [see Eq. (4.23)], we invert
this definition and instead consider the dependence of the charge on an auxiliary
field ? that is the same in all resumma?tio?n schemes, defined as? ??? 2 log ?
? 2 log(F
?1(?)) ?? , if F (?) = ?? B? = ?? ? (4.139)B ?, if F (?) = 2 log ?
B
Figure 4.5 shows the leading order piece of the scalar charge ? in the (m(RJ), F (?))
and (m(RE), F (??)) resummation schemes given in Eqs. (E.2) and (E.4). The re-
summed scalar charges in each scheme agree at ? = ??0, but they scale as
(RJ,?) ? d logm ? d logm 1 2 d logm 1? e?B? /2 ? , (4.140)
d? d? B? d? ?
(E)
(RE,??) ? d logm ? d logm ? B? ? d logm? , (4.141)
d? d? 2 d?
where we have used the fact that ? 1.
222
100 ?(RJ, ?)
10-1 ??RE,?? ?
10-2
10-3
10-4
10-5
10-5 10-4 10?-3 10-2 10-1
Figure 4.5: Newtonian order contribution to the scalar charge ? of each
neutron star in a (1.35+1.35)M binary system as a function of the aux-
iliary field ? in the (m(RJ), F (?)) and (m(RE), F (??)) resummation schemes.
We use the APR4 equation of state with B = 9, ?? = 3.33? 10?110 .
Without the additional factor of ??1, the scalar charge in the (m(RE), F (??))
scheme grows with the local scalar field (the red curve in Fig. 4.5). This trend
enables a positive feedback loop that ultimately emulates DS [170]. Intuitively, an
increase in the field ? at one body increases its charge ?, which, in turn, increases
the field ? at the other body (and so on). No such feedback is possible within the
(m(RJ), F (?)) resummation scheme because ? does not increase with greater ?.
4.8.2 Earlier analytic models
The first analytic model of DS was proposed in Ref. [170]. This model used
the 2.5PN equations of motion computed in Ref. [149], but altered the coefficients
using a feedback mechanism designed to mimic DS. To 1PN order, these modified
equations of motion are given in Eqs. (4.89)?(4.92) but with the important difference
that m?i and ??i are evaluated at an enhanced field value ?B instead of at ?0. To
223
Newtonian order ?
fGW [Hz] fGW [Hz]
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200
- ? ? ? ? ? ? ? ( ? ? ? ? ?
?
10 5 ?? ( ) 10-5? ? ? ?
? ? ??
? 1PD (m RJ( ), F(?? ))- 1PD m RE , F(?? )) ?
? ?
10 6 ? Palenzue+la et al 10-6? 1PN MS 2PN EOM- QE10 7 -7
????
? 10
- - ? ? ?
??
10 8 10 8 ???? ???
101 101 
100 100 
10-1 10-1
10-2 10-2
0.00 0.01 0.02 ?/ 0.03 0.04 0.05 0.00 0.01 0.02 ?/ 0.03 0.04 0.05GM c3 GM c3
Figure 4.6: Scalar mass as a function of orbital frequency ? and gravi-
tational wave frequency fGW of the binary system depicted in Fig. 4.4.
The post-Dickean curves are calculated at 1PD order with the resum-
mation schemes using F (??). The model proposed in Ref. [170] is plotted
in black alongside the variations that we develop in Appendix G, which
are collectively depicted by the pink region. For comparison, we plot the
1PN scalar mass (red) computed using the results of Ref. [151] (we use
the 2PN equations of motion of Ref. [149] to restrict to circular orbits).
The bottom panels depict the magnitude of the fractional error between
the models and the quasi-equilibrium configurations (QE) of Ref. [172].
determine ?B the authors numerically solved the Newtonian order relations
(2) (2)
(1) Gm?2(? )??2(? )
?B =?? +
B B
0 , (4.142)
?0rc2
(2)
?B = (1
 2) . (4.143)
Additionally, the authors explicitly set the derivatives of the scalar charge ???, ???? to
zero.
As reported in Ref. [170], this model captures dynamical scalarization and pro-
duces results (qualitatively) consistent with numerical-relativity simulations. The
224
Fractional Error MS [M?]
Fractional Error MS [M?]
model is easily implemented because it directly augments the PN results of Ref.
[149] with Eqs. (4.142) and (4.143). However, mixing the Newtonian order feedback
mechanism with higher-order equations of motion produces technical ambiguities in
the model; we address these uncertainties in greater detail in Appendix G.
Comparing Eq. (4.80) with Eq. (4.142), we immediately see that our PD
approach recovers the feedback mechanism of Ref. [170] at Newtonian order with
the resummation schemes that use F (??). Similarly, comparing Eq. (4.83) with Eq.
(4.89), we see that the equations of motion for the binary system agree at Newtonian
order with those of Ref. [170] provided we also use m(RJ).
Disparities arise between the two formalisms beyond Newtonian order. For the
same resummation scheme adopted above, the auxiliary field given in Eq. (4.85) is
the natural extension of the feedback model of Ref. [170] to higher order. Beyond
the difference between ? and ?B, the equations of motion of each approach [Eq.
(4.87) and Eq. (4.89)] differ only in the terms proportional to r?2 (recall that mi
and ?i receive PD corrections as discussed in Appendix E). However, as discussed
at the end of Sec. 4.5, we expect a greater proportion of terms in the model of Ref.
[170] to disagree with the PD equations of motion beyond post-Newtonian order.
To provide some context of the PD expansion?s place relative to previous mod-
els, the scalar mass predicted by each of the analytic approximations discussed above
is plotted in Fig. 4.6. As discussed in Sec 4.2.2, the unaltered PN approximation
(denoted in red) does not reproduce DS, giving a scalar mass orders of magnitude
smaller than numerical predictions. In contrast, the PD approximation (blue and
green) agrees with the quasi-equilibrium calculations (dotted black) reported in Ref.
225
[172] at the level of . 10% when equipped with the proper resummation scheme.
This level of accuracy is comparable to that achieved by the analytic model pro-
posed in Ref. [170] (solid black). In addition, the technical ambiguities found in
this earlier model (see Appendix G) generate some systematic uncertainty in its
predictions. As a rough estimate of this uncertainty, we denote with the pink region
the range of values spanned by all of the alternatives considered in Figs. G.1 and
G.2. The PD formalism alleviates this issue by resumming the PN approximation
in a mathematically consistent way, albeit with a freedom in the exact choice of
quantities to resum.
4.9 Conclusions
In this work, we proposed the post-Dickean expansion, a new model of dynam-
ical scalarization constructed by resumming the post-Newtonian expansion. The
motivation for this approach stems from the success of previous analytic treatments
of spontaneous scalarization, a phenomenon suspected to be closely related to DS.
By appropriating tools from recent PN calculations [149?151], we derived the equa-
tions of motion and the scalar mass (a measure of scalarization) of a binary system
at post-Newtonian order. Comparisons with recent numerical results [172] indicate
that our new formalism captures DS accurately. The PD model exactly coincides
with the analytic model introduced in Ref. [170] at leading order, but the ambigui-
ties that arise at higher order in that earlier work are avoided with the PD approach
because of its more rigorous and self-consistent formulation.
226
While this work establishes a framework for modeling DS, several further steps
remain before it can be used to generate waveforms needed to test GR with GW
detectors. Fortunately, most of these remaining calculations are straightforward,
albeit lengthy. The waveform was recently computed to 2PN order in Ref. [2, 150].9
Similarly to what was done in Secs. 4.4 and 4.6, the PD waveform can be calculated
in precisely the same way as the PN result with a slightly modified stress-energy
tensor. To reach the 2PD accuracy, one would also need to derive the equations of
motion at that order. Again, all of the necessary steps have been completed for the
PN calculation [149], so one can simply recycle that work with a new stress-energy
tensor to produce the corresponding PD result.
The evolution of a binary system directly impacts the GW signal it produces.
Thus, in conjunction with the waveform calculation sketched above, one would need
to estimate the phase evolution of a binary in the PD formalism. One approach,
analogous to what was done in Refs. [170, 175], would be to directly integrate
the equations of motion. However, earlier surveys of PN models in GR indicate
that such a procedure can produce unreliable waveforms [290]. Instead, a better
approximation can be found by balancing the change in the (conservative) binding
energy and the radiated flux far from the system. The flux was computed to 1PN
order in Ref. [151]; this calculation could be redone in the PD expansion with a
9Advanced LIGO is most sensitive to a GW?s transverse-traceless polarizations, for which the
2PN calculation was done. An additional transverse ?breathing? mode would accompany the
signal; this third polarization is determined by ? and has only been computed to 1.5PN order
[151].
227
modified stress-energy tensor.
Unfortunately, the PD binding energy cannot be easily recomputed with ex-
isting PN work. To date, this energy has been calculated in the PN approach by
integrating the (conservative) equations of motion to produce a Lagrangian and per-
forming a Legendre transformation. However, as discussed at the end of Sec. 4.5,
no such Lagrangian exists for the PD equations of motion because of the presence
of the auxiliary field ?. Without this shortcut, one would need to instead calcu-
late the ADM energy at spatial infinity. To our knowledge, the full asymptotic
metric has not been computed at spatial infinity to any PN order for the class of
ST theories we consider. In principle, the 1PD energy could be estimated at null
infinity because the system is fully conservative up to that order, but the results of
Ref. [150] would have to be considerably extended, as the author computed only
the traceless piece of the asymptotic metric. A more systematic approach should
mimic the PN calculation of the ADM Hamiltonian in GR [291], in which all of the
gravitational degrees are integrated out, leaving an energy dependent only on each
body?s position, momentum, and local scalar field ?.
Besides the litany of PN results that need to be recomputed in the PD formal-
ism to produce waveforms, the model could offer a better physical understanding of
DS. Surprisingly, we found that the PD predictions were largely independent of the
choice m(?, ?) in the resummation scheme. While this result needs to be confirmed
with a more comprehensive survey of possible schemes, the dependence of our for-
malism on the sole function F (?) suggests that DS could be modeled with a single
effective potential for the scalar charge at the level of the action. An analogous
228
method was employed in Ref. [292], in which the quadrupole modes of a neutron
star were promoted to dynamical field variables governed by an effective potential
to model their response to the tidal fields produced by a companion black hole. This
procedure could be adopted for dynamical scalarization, where each body?s scalar
monopole (i.e. scalar charge) dynamically responds to the monopolar scalar field
sourced by the companion star [293]. This investigation could offer a more intuitive
view of DS as a non-linear phenomenon.
4.10 Acknowledgements
We are grateful to Gilles Esposito-Fare?se, Jan Steinhoff, Enrico Barausse, and
Nicola?s Yunes for useful discussions and comments. N.S. and partially A.B. acknowl-
edge support from NSF Grant No. PHY-1208881. A.B. also acknowledges partial
support from NASA Grant NNX12AN10G. N.S. thanks the Max Planck Institute
for Gravitational Physics for its hospitality during the completion of this work.
229
Chapter 5: Effective action model of dynamically scalarizing binary
neutron stars
Authors: Noah Sennett, Lijing Shao, and Jan Steinhoff 1
Abstract: Gravitational waves can be used to test general relativity (GR)
in the highly dynamical strong-field regime. Scalar-tensor theories of gravity are
natural alternatives to GR that can manifest nonperturbative phenomena in neutron
stars (NSs). One such phenomenon, known as dynamical scalarization, occurs in
coalescing binary NS systems. Ground-based gravitational-wave detectors may be
sensitive to this effect, and thus could potentially further constrain scalar-tensor
theories. This type of analysis requires waveform models of dynamically scalarizing
systems; in this work we devise an analytic model of dynamical scalarization using
an effective action approach. For the first time, we compute the Newtonian-order
Hamiltonian describing the dynamics of a dynamically scalarizing binary in a self-
consistent manner. Despite only working to leading order, the model accurately
predicts the frequency at which dynamical scalarization occurs. In conjunction with
Landau theory, our model allows one to definitively establish dynamical scalarization
as a second-order phase transition. We also connect dynamical scalarization to the
1Originally published as Phys. Rev. D96, 084019 (2017).
230
related phenomena of spontaneous scalarization and induced scalarization; these
phenomena are naturally encompassed into our effective action approach.
5.1 Introduction
Over a century of experiments have shown that general relativity (GR) very
accurately describes the behavior of gravity. The bulk of these tests have come from
measurements of gravitationally bound systems, either with electromagnetic obser-
vations of our Solar System [79] and binary pulsars [249, 294] or with gravitational-
wave (GW) observations of coalescing binary black holes [73, 122, 123, 184]. Com-
bined, these systems probe GR over a large phase space, with gravitational fields
whose relative strength and dynamism span many orders of magnitude [89, 187, 295,
296]. However, one corner of parameter space that has not yet been directly tested is
the highly-dynamical, strong-field regime of gravity coupled to matter, which would
be reached in the merger of a neutron star (NS) binary system.
GWs from coalescing binary neutron stars (BNSs) are expected to be detected
by Advanced LIGO in the near future [297]. Tests of GR are done using Bayesian
inference [122], comparing the relative probability that the measured data are consis-
tent with a GR waveform over a non-GR waveform to search for possible deviations
from GR. Waveforms in alternative theories of gravity can be written schematically
in the Fourier domain as
h(?; f) = h (?; f) [1 + ?A(?; f)] ei??(?;f)GR , (5.1)
where f is the observed GW frequency, ? represents the intrinsic (e.g., component
231
masses, spins, etc.) and extrinsic (e.g., distance, sky position, etc.) parameters
of the binary. We have used hGR(?; f) to represent the expected waveform in GR
while ?A and ?? are the deviations in the amplitude and phase, respectively, from
GR [117, 118, 120, 176]. One makes an ansatz for ?A as parameterized by a set of
coefficients {?i} and for ?? as parameterized by another set of coefficients {?j}. A
common choice?the so-called restricted waveforms?is for ?A to be identically zero
while, for frequencies corresponding to the inspiral, ?? is expanded in powers of the
frequency f and its logarithm log f [118, 120, 176]. For this choice, the parameters
{?j} are simply the coefficients of the power series in f and log f?they measure the
deviations from GR that appear at each order in a post-Newtonian (PN) expansion
of the phase. Because this approach makes no reference to a particular alternative
theory of gravity, constraining the parameters {?j} can simultaneously constrain
many alternative theories using appropriate mappings.
However, this theory-agnostic approach does not capture all possible devia-
tions from GR because it relies on the assumption that ??(?; f) admits a series
expansion in f and log f during the early inspiral. In this paper, we study a par-
ticular class of scalar-tensor theories of gravity in which BNSs can undergo a phase
transition known as dynamical scalarization [169]; the GW signals from such sys-
tems cannot be expanded in a simple power series. Through this phenomenon,
BNSs abruptly transition from a configuration that closely resembles a BNS in GR
to a drastically different state. Previous efforts to model dynamically scalarizing
systems have relied on phenomonological waveform models or analytic approxima-
tions of the equations of motion [3, 170, 175, 271]. We continue these efforts in this
232
work by reformulating the PN dynamics of BNSs with dynamical scalar charges in
a manner analogous to the treatment of dynamical tides in GR [298, 299]. Using
this approach, we explicitly construct a two-body Hamiltonian that incorporates dy-
namical scalarization; in contrast, in Refs. [3, 170], only the PN equations of motion
were calculated. Our results comprise an important step towards fully-consistent
waveform models of dynamical scalarization and offer a clear interpretation of the
phenomenon as a phase transition.
The paper is organized as follows. In Sec. 5.2, we provide an overview to
scalar-tensor theories and certain nonperturbative phenomena for NSs. In Sec. 5.3
we construct an effective action to model the dynamical scalarization of BNSs. In
Sec. 5.4 we compare results obtained from our model to previous analytic approaches
and numerical quasi-equilibrium (QE) configuration calculations. In Sec. 5.5, we
use our model to solidify the interpretation of dynamical scalarization as a phase
transition and then discuss possible extensions to the model. Finally, we present
some concluding remarks in Sec. 5.6.
Throughout the paper we use the conventions of Misner, Thorne, and Wheeler [15]
for the metric signature and Riemann tensor. We work in units in which the speed
of light and the bare gravitational constant in the Einstein frame are unity.
5.2 Nonperturbative phenomena in scalar-tensor gravity
Scalar-tensor theories of gravity are amongst the most natural and well-motiva-
ted alternatives to GR [79, 89]. We consider the class of theories detailed in
233
Ref. [147], in which a massless scalar field couples nonminimally to the metric.
These theories are described in the Jordan frame by the action,
? ?? [ ]g? ?(?)
S = d4x ?R?? g????????? + Sm[g??? , ?], (5.2)
16?G? ?
where ? represents all of the matter degrees of freedom in the theory and G? is the
bare gravitational coupling constant in the Jordan frame. Alternatively, the action
can be written in the Einstein frame by performing a conformal transformation,
g?? ? ?g??? , as
? ? [ ]
S = d4
?g
x [R? 2g????????] + S 2m A (?)g?? , ? , (5.3)
16?
where we have introduced the scalar field,
? ?
? 3 + 2?(?)? d? , (5.4)
2?
and defined ( ? )
? ? ? d?A(?) exp . (5.5)
3 + 2?(?)
Varying the Einstein-frame action yields the field equations
R?? ?
1
Rg ???? = 8?T?? + 2??????? g??g ??????, (5.6)
2
? = 4??(?)T, (5.7)
where T ?? ? 2(?g)?1/2?Sm/?g?? is the stress-energy tensor of matter, T ? g??T ??
is its trace, and we have introduced the coupling,
? ?d logA?(?) = (3 + 2?)?1/2. (5.8)
d?
234
Much of the seminal research in scalar-tensor alternatives to GR considered the
simple choice of a constant coupling ?, corresponding to Jordan-Fierz-Brans-Dicke
theory [178, 179, 300]. This theory is currently well-constrained by measurements
from the Cassini probe [180] and of binary pulsars [249, 294]; future observations
by Advanced LIGO are not expected to improve these constraints [301]. Instead, in
this work we consider theories whose coupling is linear in ?,
?(?) = ???. (5.9)
Such theories can give rise to phenomena that are potentially detectable by Ad-
vanced LIGO while evading the bounds set by the Cassini probe [153, 169].2 In
2Cosmological considerations can further constrain the class of theories with the coupling given
by Eq. (5.9). In particular, when ? is negative, the theory evolves rapidly away from GR over
cosmological timescales [152, 175, 273]; this evolution cannot be reconciled with current Solar
System observations without fine-tuning the theory at some point in the distant past. One can
solve this cosmological issue by generalizing the coupling (5.9) to a higher-order polynomial in
?, which causes the scalar field to evolve to a local minimum of A(?) rather than diverge [302].
However, when expanded around this local minimum, the leading order term of the modified
coupling ?(?) will have the opposite sign as in Eq. (5.9), and thus such theories no longer manifest
the nonperturbative scalarization phenomena that we study here [302]. Alternatively, one can
add a mass term for the scalar field to Eq. (5.3) to evade the cosmological constraints on these
theories [152, 303]. Neutron stars can undergo nonperturbative phenomena analogous to those we
consider here when immersed in a constant background massive scalar field [95, 161]. However,
recent work has revealed that this background field should in fact oscillate over relatively short
timescales in massive scalar-tensor theories [303]. It remains to be seen whether NSs embedded in
an oscillatory background scalar field can also exhibit nonperturbative phenomena. As is commonly
done in the literature [3, 147, 152, 153, 169?172, 175], we ignore these cosmological concerns here.
235
particular, for sufficiently negative ?, such theories can manifest spontaneous scalar-
ization, dynamical scalarization, and induced scalarization.3
Before discussing these phenomena in detail, we briefly examine the structure
of NS solutions to Eqs. (5.6) and (5.7) to establish some useful notation. For simplic-
ity, we consider a static matter source. Working far from all matter, one can expand
the metric about a Minkowskian background in powers of  ? mE/r  1 where mE
is the total mass (measured in the Einstein frame) using the post-Minkowskian for-
malism (see Ref. [30] and references within). To leading order in , Eq. (5.7) reduces
to the Poisson equation on a flat background, whose solution in this region takes
the generic form,
( )
Q 1
?(r) = ?0 + +O , (5.10)
r r2
where we have introduced a constant background field ?0 and defined the scalar
charge Q as the scalar monopole moment of the source.
5.2.1 Spontaneous scalarization
Damour and Esposito-Fare?se discovered that the presence of relativistic mat-
ter in theories with negative ? can trigger an instability in the scalar field [153]. In
such theories, a sufficiently compact NS can undergo a phase transition known as
spontaneous scalarization corresponding to the spontaneous breaking of the sym-
metry ? ? ?? in Eq. (5.3). Given current constraints from binary pulsars (see
below) [6, 248, 249, 294, 305], numerical solutions to Eqs. (5.6) and (5.7) reveal that
3See Refs. [156, 157, 304] for a discussion of similar phenomena in theories with positive ?.
236
an isolated NS can develop a scalar charge of order
Q
. 10?1, (5.11)
mE
through spontaneous scalarization. This figure should be contrasted with a PN
prediction for this quantity,
Q ( )
= ???0 1 + a1C + a2C2 + ? ? ?
mE
? ( ) (5.12). 10 5 1 + a1C + a C22 + ? ? ? ,
where the coefficients ai are of order unity and C ? mE/R is the compactness of
the NS [147]. The drastic difference in magnitude between Eqs. (5.11) and (5.12)
indicates that the PN expansion does not predict spontaneous scalarization. In
this sense, we describe spontaneous scalarization as nonperturbative; loosely speak-
ing, one must include every term in the infinite sum in Eq. (5.12) to recover the
phenomenon.
The best constraints on spontaneous scalarization come from timing measure-
ments of white dwarf-NS binaries (see, e.g., Refs. [6, 152, 247?249, 305]). Unlike
NSs, white dwarfs (WDs) are too diffuse to develop any significant scalar charge
through spontaneous scalarization. Consequently, WD-NS binaries can emit sub-
stantial scalar dipole flux Fdip, which scales as( )2 ( )2
F ? QNS ? QWD QNSdip ? , (5.13)
mE E ENS mWD mNS
where mEWD and m
E
NS are the masses, and QWD and QNS are the scalar charges
of the WD and NS, respectively. Pulsar timing experiments are sensitive to any
anomalous decrease in the orbital period of the binary, and thus can constrain Fdip
237
and consequently Q ENS/mNS; we refer readers to Ref. [6] for the current best limits
on spontaneous scalarization from pulsar timing.
5.2.2 Dynamical and induced scalarization
More recently, a similar phenomenon, known as dynamical scalarization, was
uncovered in numerical-relativity (NR) simulations of BNSs in the same class of
scalar-tensor theories with negative ? [169, 171, 172]. These simulations considered
binary systems composed of NSs too diffuse to undergo spontaneous scalarization
in isolation. As the binaries coalesced, it was found that the presence of a com-
panion allowed the NSs to scalarize abruptly, developing scalar charges of the same
order of magnitude as might occur through spontaneous scalarization. A related
phenomenon, known as induced scalarization, was also discovered [169], in which a
spontaneously scalarized star generates a scalar charge on a companion too diffuse
to scalarize in isolation. For simplicity, we primarily focus on dynamical scalariza-
tion in this work; however, the model we develop can be applied to systems that
undergo induced scalarization as well.
Numerical relativity simulations show that dynamical and induced scalar-
ization hasten the plunge and merger of BNSs relative to the same systems in
GR [169, 171]. Two factors dictate the difference in merger time for scalarized
versus unscalarized systems: (i) an enhancement in energy flux, and (ii) a modifica-
tion to the binding energy. A scalarized BNS system will emit energy more rapidly
than an unscalarized system; the dissipative channels available in GR (e.g., tensor
238
quadrupole radiation) are enhanced for bodies with scalar charge and new chan-
nels become available (e.g., scalar dipole radiation). Modifications to the binding
energy of scalarized systems are not well understood. In Ref. [172], the binding
energy was argued to decrease (in magnitude) in scalarized systems, prompting an
earlier merger, whereas in this paper, we argue that it should instead increase (see
Sec. 5.4.2 for more detail).
Advanced LIGO will be able to distinguish between the coalescence of scalar-
ized and unscalarized NSs provided that their scalar charges: (i) are sufficiently
large and (ii) develop early enough in the inspiral (in the case of dynamical scalar-
ization) [6, 175, 271]. Observation of such scalarization would provide direct evidence
for modifications of GR in the strong-field regime; conversely, lack of evidence of
scalarization can further constrain the space of viable scalar-tensor theories. De-
pending on the NS masses and equation of state (EOS) observed in coalescing BNS
systems, Advanced LIGO could provide constraints competitive with current binary-
pulsar limits [6].
Searches for deviations from GR with GWs rely on accurate and faithful wave-
form models in modified gravity. Several models of dynamical scalarization have
been proposed in the literature, but none at the level of sophistication of waveforms
in GR. The simplest of these approaches phenomenologically model ??(?; f) to re-
produce features expected to arise in dynamically scalarized systems. For example,
one can model ??(?; f) by a polynomial in f to capture effects such as scalar dipole
radiation and/or use a Heaviside step function to mimic the abrupt growth of scalar
charge and hastened merger triggered by dynamical scalarization. Detectability
239
studies reveal that such models may be sufficient to identify dynamical scalarization
with Advanced LIGO [175, 271]. However, the accuracy of phenomenological wave-
form models cannot be established a priori. Ultimately, one must validate and/or
calibrate these models using independent waveforms. In GR, this comparison is
made with both analytic and NR waveforms (e.g., the IMRPhenom waveform fam-
ily [306]). Because very few NR simulations of dynamical scalarization have been
produced to date, one must rely solely on more sophisticated analytic models of this
phenomenon to verify the accuracy of phenomenological models.
A more sophisticated approach towards waveform modeling, and one we shall
pursue in the present work, is to solve the field equations (5.6) and (5.7) in some
perturbative fashion (see Sec. 5.3). The PN approximation is an example of such
an approach; PN waveforms are useful inspiral models in their own right and also
serve as the foundation for more refined waveform models, such as the effective-
one-body (EOB) formalism [38, 39]. Dynamical scalarization can be modeled by
augmenting [170] or resumming [3] the PN dynamics in scalar-tensor gravity; such
modifications are necessary because dynamical scalarization is a nonperturbative
phenomenon in the same sense as spontaneous scalarization [3]. Both of these ana-
lytic approaches suffer from two shortcomings. First, simulating the dynamics with
these models requires one to solve a system of algebraic equations at each moment in
time involving the function mE(?), which measures the complete (nonperturbative)
dependence of the NS mass on the scalar field in which it is immersed. Second,
these approaches only model the dynamics at the level of the equations of motion;
no rigorous formulation of the two-body Hamiltonian has been constructed.
240
In the next section, we develop a new analytic model of dynamical scalariza-
tion that addresses these shortcomings using an effective-action approach. First,
the scalar charges Q are given by roots of a system of polynomial equations; for sys-
tems with no background scalar field ?0, the algebraic system reduces to a pair of
cubic equations that have a closed-form solution. These algebraic equations depend
on only two new parameters per NS [as opposed to the complete functions mE(?)]
that can be directly interpreted as the separation at which dynamical scalarization
begins and the magnitude of scalar charge that develops. Second, the new model
allows one to construct a simple two-body Hamiltonian and thus also compute the
binding energy of a binary system. The Hamiltonian is a fundamental building block
in the construction of perturbative waveform models. For example, the binding en-
ergy, in conjunction with the energy flux, allows one to compute the phase evolution
through the balance equation [30], and the Hamiltonian is the natural starting point
in constructing an EOB description of the dynamics. Additionally, our new formula-
tion allows for a more nuanced interpretation of dynamical scalarization as a phase
transition than exists in the literature and more intimately connects dynamical and
spontaneous scalarization.
5.3 Effective action with a dynamical scalar charge
We construct a model for dynamical scalarization by explicitly re-parameterizing
the standard point-particle action for a BNS in terms of the scalar charges of its
components. This approach closely resembles the treatment of extended bodies in
241
GR in terms of their multipolar structure; in fact, as can be seen from Eq. (5.10),
the scalar charge is simply the scalar monopole moment of an extended body. The
gravitational fields (tensor and scalar) produced by a system of compact bodies
can be represented completely in terms of the bodies? multipoles through matched
asymptotic expansions [30, 147]. In turn, these external fields affect the multipolar
structure of the compact bodies. This response must be included into the point-
particle model in some way. For example, a constant external tidal field Gi1...i will`
induce a multipole Qi1...i as determined by the tidal deformability ?` `
Qi1...i = ??`Gi1...i . (5.14)` `
(See Ref. [307] for more detail.) A more sophisticated model is needed to capture
dynamical tides, i.e., tidal fields that vary on periods comparable to the relaxation
timescale of the compact body (see Refs. [292, 299, 308, 309] and references therein).
As can be seen from the arguments of the matter action Sm in Eq. (5.3),
compact objects in scalar-tensor gravity interact with the scalar field in conjunction
with the Einstein frame metric. For non-self-gravitating objects (i.e., test particles),
this interaction is characterized simply by A(?). However, the internal gravitational
interactions in self-gravitating objects can dramatically change the couplings to the
metric and scalar field; these differences represent violations of the strong equiva-
lence principle. As first proposed by Eardley [310], the response of a body?s mass
monopole mE to an external scalar field can be encoded into a generic function
mE(?). As shown in Appendix A of Ref. [147], the scalar monopole Q induced by
242
an external scalar field is given by
?dmEQ = . (5.15)
d?
For bodies immersed in weak scalar fields, Eq. (5.15) reduces to a linear relation
analogous to Eq. (5.14). However dynamical scalarization occurs outside of this
linear regime: the complete expression mE(?) is needed to accurately model this
phenomenon.
In this section, we develop a model inspired by the treatment of non-adiabatic
tides in GR [292, 299, 308, 309]. We rewrite the point-particle action using Q in
place of ? and promote the scalar charge Q to a dynamical degree of freedom. We
find that this action can be expressed as a simple effective action for a dynamical
scalar charge linearly coupled to an external scalar field. The complete function
mE(?) is condensed into the coupling coefficients (or ?form factors?) in the effective
action. Thus, the predictions of the model are parameterized by a small set of
coefficients and are easy to study without reference to any particular BNS system;
in contrast previous analytic models [3, 170] required the full form of m(?) to be
predictive.
In Sec. 5.3.1, we develop the framework for our new effective point-particle
action for a single NS and discuss possible extensions for future work. Using this
approach, we compute the dynamics for a binary system of two point particles in
Sec. 5.3.2.
243
5.3.1 The effective point-particle action
We begin with the standard model of the orbital dynamics of compact objects
in scalar-tensor gravity. If the orbital separation is much larger than the size of the
bodies, one can represent each star as a point particle governed by an action of the
form [147, 208, 276], ? ?
Sm = ? d? ?u?u?mE(?), (5.16)
where z?(?) is the object?s worldline parametrized by a generic parameter ?, u? ? dz?/d?
is its four-velocity, and mE(?) is its Einstein-frame mass as a function of the scalar
field along the worldline ?(z?). Inserting the source (5.16) into Eq. (5.7), one finds
that the compact object generates a scalar field given by
? ??u?u? dmE
2? =4? d? ? ?(4)? (x
? ? z?), (5.17)
g d?
where the derivative of the mass is evaluated at ?(z?). Similarly, the influence of
the object on the metric can be found by inserting Eq. (5.16) into Eq. (5.6).
Next, we convert the action (5.16) from a function of the external field ?
imposed on the body to one of the scalar charge Q. These two quantities offer
complementary descriptions of the local geometry of the compact body; one can
convert between the two using Eq. (5.15). To rewrite the action as a function of Q,
we adopt a method first introduced in Ref. [152]; we define a new potential m(Q)
given by the Legendre transformation of the mass mE(?),
m(Q) ? mE(?) +Q?. (5.18)
244
Inserting this definition into Eq?. (5.16), the action reads?
Sm = ? d? ?u?u? [m(Q)?Q?] . (5.19)
Now we promote Q to an independent degree of freedom in the model; variation of
the action with respect to this variable gives an additional equation of motion in
the dynamics.
The notation in Eq. (5.18) is intentionally suggestive; as we will show in
Sec. 5.3.2, m(Q) assumes the role of the particle?s mass in the orbital dynamics
rather than mE(?). A natural analogy can be drawn to thermodynamics: consider,
for example, an ideal gas composed of a fixed number of particles held at a constant
temperature. The state of the system can be described by either its pressure?an
intrinsic quantity, analogous to ??or its volume?an extrinsic quantity, analogous
to Q. While the internal energy?analogous to mE(?)?has a natural interpretation
as the thermal energy of the gas, it is often more convenient to use the free energy?
analogous to m(Q)?to describe certain physical processes. As was discussed in
Ref. [152] (and will be revisited in Sec. 5.5), the equilibrium state for an isolated NS
will minimize the function m(Q); again, this quantity plays the role of an effective
free energy of each NS in a binary system.
We expand the potential m(Q) in a power series to quartic order,
c(2) c(3) c(4) ( )
m(Q) = c(0) + c(1)Q+ Q2 + Q3 + Q4 +O Q5 . (5.20)
2! 3! 4!
Recall that the action (5.3) equipped with the coupling (5.9) is invariant under the
symmetry ? ? ??. Thus, we expect the mass of an isolated NS mE(?) to also
respect this symmetry, even in the presence of spontaneous scalarization. From
245
Eq. (5.15), we see that this parity transformation will also send Q? ?Q. Perform-
ing both of these transformations leaves the right hand side of Eq. (5.18) unchanged,
and thus we can conclude that m(Q) must be an even function of Q.
Some of the coefficients c(n) have an immediate interpretation. The leading
c(0) describes the body?s mass in absence of any scalar charge, i.e., the ADM mass
in GR, and so we also denote it as c(0) = m(0). Furthermore, a background scalar
field ?0 can be handled by working instead with the field,
?? ? ?? ?0, (5.21)
leading to an additional coupling ?Q?0 in the Lagrangian. This term can be ab-
sorbed into m(Q) by setting c(1) = ??0, and thus we can interpret c(1) as a cos-
mologically imposed background scalar field. Note that the addition of a nonzero
scalar background weakly breaks the symmetry ? ? ?? in the point-particle ac-
tion, prompting us to relax the conclusion that m(Q) is a strictly even function.
However, all other odd powers of Q will still vanish, i.e., c(3) = 0.
Given the discussion above, our model for m(Q) reduces to
c(2) c(4)
m(Q) = m(0) ? ?0Q+ Q2 + Q4 +O(Q6). (5.22)
2 24
Potentials of this form are widely used to describe systems that exhibit spontaneous
symmetry breaking (see also Sec. 5.5); the Higgs mechanism is one notable exam-
ple [311]. Reference [152] employed a similar potential to model isolated NSs near
the critical point for spontaneous scalarization. In the present work, we show that
the ansatz (5.22) remains valid for NSs far from this critical point; we describe the
246
procedure by which we numerically compute the various coefficients for a particular
NS in Sec. 5.4.1.
One ingredient conspicuously absent from our effective action (5.19) is the
dynamical response of the scalar charges to changes in the scalar field. In truth,
our model is only valid in the adiabatic limit, wherein the external fields evolve
over timescales much longer than the relaxation time of NSs. Given the abrupt
nature of dynamical scalarization, the validity of our assumption of adiabaticity
should be studied in greater detail; we reserve this analysis for future work. If one
rapidly changes the external scalar field, the NS?s scalar charge cannot respond in-
stantaneously. In general, physical systems undergo (harmonic) oscillations around
equilibrium configurations under small perturbations. Thus, one expects the scalar
charge to behave approximately like a harmonic oscillator driven by the external
fields, characterized by an action of the form (5(.19) with )
c(2) Q?2
m(Q, Q?) = m(0) ? ?0Q+ +Q2 , (5.23)
2 ?2 ?0u u?
where Q? = dQ/d? and ?0 is the resonant frequency of this scalar mode. This
action is analogous to the dynamical tidal model in Ref. [299]: Q corresponds to the
dynamical quadrupole, ? to the tidal field, 1/c(2) to the tidal deformability, and ?0
to the oscillation mode frequency. In general, one should add separate dynamical
degrees of freedom for every oscillation mode of the NS. Identifying all dynamical
degrees of freedom relevant for the scale of interest is very important in constructing
an effective action (see, e.g., Ref. [312]). Note that when the dynamics of the system
occur much more slowly than the resonant frequency, i.e. Q? ?0, and we restore the
247
Q4 interaction, Eq. (5.23) reduces to the adiabatic model (5.22) considered earlier.
Viewed from an effective field theory perspective, our effective action model
of dynamical scalarization may appear too simplistic. In general, one should add to
the action all possible combinations of Q, u?, the scalar field ?, and the curvature
(and derivatives of these variables) allowed by the symmetries of the theory, up to
terms negligible for the desired accuracy of the model. Not all of these interactions
are independent, since some might be related by redefinitions of the other dynamical
variables; the redundant terms should be dropped. In the present model, we consider
only couplings of the scalar charge to itself, as well as a linear coupling of the
charge to the scalar field. A broader class of interactions would allow our model
to reproduce other interesting phenomena. For example, the induction of scalar
charges on black holes from time-varying external fields can be modeled with an
effective action [313, 314]. We delay such an investigation for future work; for the
present work, the effective action model given by Eqs. (5.19) and (5.22) is sufficient
to reproduce dynamical scalarization.
5.3.2 Dynamics of a binary system
We now turn to the task of translating the action [which will contain a copy
of Eq. (5.19) for each NS] into a Hamiltonian describing the orbital dynamics of a
BNS. Using the PN approximation, we expand the metric and scalar field in powers
of v/c and solve the field equations (5.6) and (5.7) at each order. An efficient
method for solving the two-body dynamics is through a Fokker action4 together with
4This means to insert the perturbative solution to the field equations into the full action.
248
a diagrammatic method to represent the perturbative expansion [208]. Similarly,
one can integrate out the fields perturbatively using techniques from quantum field
theory [315], i.e., Feynman integrals and diagrams.
We consider only the leading-order (Newtonian) approximation of the orbital
dynamics in the present work. Thus, the accuracy of our model will degrade towards
the end of the inspiral. However, because Advanced LIGO is only sensitive to
dynamical scalarization that occurs in the very early inspiral [175, 271], our model
can still be applied to the systems of scientific interest; we pursue extensions of our
model to higher PN order in future work.
The PN expansions of the metric g?? and the scalar field ?? are given by
( )
g?? =??? + h +O c?4(?? ) , (5.24)
?? =? +O c?4 , (5.25)
where ??? is the Minkowski metric and ?? vanishes at infinity by construction. The
leading-order PN corrections enter with the following powers of c:
( ) ( )
h00 ? O (c
?2
) , h
?3
0i ? O( c ) ,
h ? O c?4 , ? ? O c?2ij . (5.26)
Inserting the expansions (5.24) and (5.25) into the field equations (5.6) and (5.7)
with the source (5.19), one finds the Newtonian-order solution to the metric and
249
scalar field,
mA(QA) mB(QB) ( )
h00(x, t) =
?4
| ? | + | ? +O c , (5.27a)x zA(t) x zB(t)|
QA QB ( ? )?(x, t) = | ? | + 4| ? | +O c , (5.27b)x( zA)(t) x zB(t)
h0i(x, t) ? O ?3(c ) , (5.27c)
hij(x, t) ? O c?4 , (5.27d)
where the labels A and B distinguish the two NSs. Henceforth, we suppress the
explicit dependence of each body?s mass m on its corresponding scalar charge for
notational convenience.
Inserting these solutions into the action and dropping singular self-interactions,
we find the?leadi[ng-order two-body action, ]
? ? ? mA 2 mB mAmB QAQBS dt mA mB + vA + v2B + + , (5.28)2 2 r r
where vi ? dzi/dt is the Newtonian velocity and r ? |zA ? zB| and we have cor-
rected for any double counting. Legendre transforming the Lagrangian yields the
Hamiltonian,
p2 p2 mAmB QAQB
H = mA +m
A B
B + + ? ? , (5.29)
2mA 2mB r r
where the canonical momenta are pA,B = mA,BvA,B. The equation of motion for
QA reads ( )
(4)
?H ? (2) c0 = = z ? + c A QBA 0 A QA + Q3A ? , (5.30)?QA 6 r
with the redshift given by
?H p2 mB
zA = = 1? A ? , (5.31)
?mA 2m2A r
250
and the equation of motion for QB takes the same form but with the body labels
exchanged A ? B. The scalar charges are given by the roots of these two cubic
equations.5 Closed form solutions can be found using computer algebra for ?0 6= 0,
but the result is rather lengthy and not very illuminating; we do not provide them
here for space considerations.
While Eq. (5.30) may seem daunting, simple analytic solutions for the scalar
charge can be easily found in special, but very relevant cases. We restrict our
attention to the theories that exactly preserve the symmetry ? ? ??, i.e., we set
the background scalar field ?0 = 0. Next, for simplicity, we will neglect the O(c?2)
corrections to the redshift zA in Eq. (5.30); including these terms does not change
the qualitative behavior of the solutions discussed below. Finally, we specialize to
the case of equal-mass binaries and assume that the NSs have identical properties,
(0) (0) (i) (i)
i.e. mA = mB and cA = cB . Under these assumptions, Eq. (5.30) reduces to[ ]
?H ? 1
(4)
0 = = 2Q ? c(2) ? c Q2 , (5.32)
?Q r 6
where we have dropped the body labels. As expected, the trivial solution Q = 0
satisfies this equation. However, this is not necessarily the only solution; if the
trivial solution is unstable, the BNS system will transition to a state with nonzero
scalar charge. The requirement for stability,
2
? ? H (2) ? 20 = 2c + c(4)Q2, (5.33)
?Q2 r
5For consistency, we truncate Eq. (5.30) at cubic order in the scalar charges, e.g. dropping the
term proportional to Q3AQB that would arise from the product of mB and Q
3
A.
251
is violated for Q = 0 when 1/r > c(2). The stable solutions therefore read,
???? 0 ? ? for 1/r ? c(2)Q = ??? , (5.34)? 6 1 ? c(2) for 1/r ? c(2)
c(4) r
which contains a phase transition at rDS = 1/c
(2).
Equation (5.34) provides some intuition into the physical interpretation of the
coefficients c(2) and c(4). The parameter c(2) determines the orbital scale of the phase
transition to the scalarized regime, where the scalar-parity symmetry is broken and
the solution bifurcates. The parameter c(4) determines the size of the scalar charge
in this regime. Notice that for negative c(2) the NS is scalarized for all r. In fact,
this situation corresponds to spontaneous scalarization; we discuss the connection
between spontaneous and dynamical scalarization in greater detail in Sec. 5.5.
Finally, we compute the Newtonian-order equations of motion for each NS.
Working from the Hamiltonian (5.29), the equations of motion are given by,
?mB (1 + ?A?B)z?A = n, (5.35)
r2
where ?A,B ? QA,B/mA,B and n ? (zA ? zB)/r. Note that ?A differs from the
quantity found in Eqs. (5.11)? (5.13) because it uses m(Q) in place of mE(?). We
also derive Kepler?s third law for circular orbits
2 (mA +mB) (1 + ?A?B)? = , (5.36)
r3
where ? is the orbital frequency.
252
5.4 Results
The previous sections aimed to motivate and develop a novel analytic model
of dynamical scalarization; in this section, we test the accuracy of this approach
by comparing against previous models [3] and numerical QE configuration calcula-
tions [172]. The dynamics are determined entirely by the coefficients c(i), as can be
seen by inserting Eq. (5.22) and the solution of the cubic equations (5.30) for QA,B
into the Hamiltonian (5.29). These coefficients characterize the behavior of each
compact body in isolation, and thus can be computed straightforwardly.
To facilitate comparison with previous work, we restrict our attention to the
binary systems considered in Refs. [3, 172]. We consider (1.35 + 1.35)M nonspin-
ning BNS systems with a piecewise polytropic fit to the APR4 EOS; see Ref. [206]
for more details on this EOS and its polytropic fit. We examine configurations with
? = ?4.2 and ? = ?4.5, where ? characterizes the strength of the scalar cou-
?
pling (5.9). Finally, we add the background scalar field ?0 = 10
?5/ ?2?, which
satisfies binary-pulsar constraints for this EOS [171].
5.4.1 Computing c(i)
The coefficients c(i) describe how the energy of an isolated NS varies with its
scalar charge Q. Thus, to extract these coefficients, we study the behavior of the NS
under infinitesimal changes in Q. In practice, we compute sequences of NS solutions
with equal baryonic mass with incremental changes to the mass mE, scalar charge
Q, and asymptotic field ?. Spherically symmetric solutions for perfect fluid stars are
253
10-2 ?????????????- ?????????
??
4 ???? ???? ?
?????????
10 ????
- ??? ????
?????
10 6 ?????????
? ??????????
?????
??????????????
?
10-8 ??????????
- ??????????
?? ? ?????
???????
10 10 ????? ??
???
- ??
???????????
10 1
10-2
10-3
10-4 10-3 10-2 10-1
Figure 5.1: Potential m as a function of scalar charge Q for a 1.35M
NS with the APR4 EOS. Top: The numerical values and polynomial
fit are plotted with points and lines, respectively, for ? = ?4.2 (red)
and ? = ?4.5 (dashed black). We have subtracted the leading-order
term m(0) = 1.35M from m to improve readability. Bottom: We plot
the fractional error in m(Q) ? m(0) between the numerical data and
polynomial fits.
found by solving the Tolman-Oppenheimer-Volkoff (TOV) equations; the extensions
of these equations to scalar-tensor gravity were derived in Refs. [152, 153]. We solve
these equations using fourth order Runge-Kutta methods and use standard shooting
techniques to construct solutions with the same baryonic mass. The quantities mE,
Q, ? parameterize the asymptotic behavior of each numerical solution; we extract
mE, Q, ? using the relations detailed in Refs. [152, 153]. Equipped with these
quantities, we then compute m(Q) using Eq. (5.18).
We compute the coefficients c(i) by fitting the numerically computed m(Q)
with a polynomial of the form (5.22). The numerical values and polynomial fit
254
of m(Q) are plotted with dots and solid lines, respectively, in the top panel of
Fig. 5.1 for the NS parameters discussed above. To improve readability, we have
subtracted the leading-order coefficient m(0) = 1.35M from m. The values for c
(i)
computed through the polynomial fit are also given in Fig. 5.1; the i-th coefficient
has dimension of [mass]i?1. The bottom panel of the figure shows the fractional error
between numerical values and polynomial fits of m ?m(0). We see that deviations
are generally of the order . 0.01%, slightly worsening as the charge increases. The
range in Q plotted here covers the typical range achievable by this NS over an
entire inspiral in which dynamical scalarization occurs. As a check of our initial
ansatz (5.22), we also fit the data to polynomials including Q3, Q5 and Q6 terms;
we find that these additional powers of Q shift our estimates for c(i) by less than
? 0.1% and only marginally improve the overall agreement to data.
5.4.2 Comparison against previous models
As a first test of our model, we compute the scalar charge Q as a function of
frequency. Because we only consider equal-mass systems, this relation can be found
by solving the cubic equation (5.30) for Q = QA = QB as a function of separation
r. Then, by inserting this result into Eq. (5.36), we determine an exact relation
between r and the orbital frequency ?. Finally, we invert this relation and insert it
into the solution to Eq. (5.32) to find an implicit expression for Q(?). We plot Q(?)
in Fig. 5.2 computed with our model in red. The lower axis gives the dimensionless
255
0 200 400 600 800 1000 1200
100
? ? ? ? ? ?
?
? ? ?
?
? ? ? ?
?
? ? ? ??
? ?
-1 ? ?10 ? ?
?
10-2
? ?
?
? ?
10-3 ? ?? ?
?
?
?
??
???
?
10-4
0.00 0.01 0.02 0.03 0.04 0.05
Figure 5.2: Scalar charge of each star as a function of frequency for a
(1.35 + 1.35)M BNS with the APR4 EOS. The lower axis indicates
the orbital frequency ?; the upper axis shows the dominant GW fre-
quency fGW = ?/?. The model developed here using an effective action
is shown in red. The analytic post-Dickean (PD) model of Ref. [3] is
shown in blue. The numerical calculations of quasi-equilibrium (QE)
configurations performed in Ref. [172] are shown in black. The curves
depicting earlier scalarization were computed with ? = ?4.5; the other
set of curves correspond to ? = ?4.2.
orbital frequency, normalized by the total rest mass M , which we define as,
? (0) (0)M mA +mB , (5.37)
i.e. the sum of the component ADM masses in GR. The upper axis gives the
dominant frequency fGW = ?/? of the GWs produced by the binary in hertz.
We plot in blue the predictions of the post-Dickean (PD) model constructed
in Ref. [3]. The PD approach resums the PN dynamics to reproduce dynamical
scalarization. To accomplish this resummation, one promotes the mass mE and its
derivatives to functions of two scalar fields mE(?, ?), Then, one field (?) is integrated
256
out of the point-particle action (5.16) through a standard PN expansion, while the
other (?) is treated as a new dynamical degree of freedom in the theory. In this way,
the PD approximation resembles the model presented here. Both methods introduce
new degrees of freedom at the level of the action, and extremizing the action with
respect to these quantities yields algebraic equations that relate the quantities to
the bodies? positions and momenta. However, in the PD approach, these equations
involve the potentially complicated function mE(?, ?) and its derivatives, whereas in
the formalism presented here, one needs only the coefficients c(i). In the notation of
?
Ref. [3], we define the natural analog of the scalar charge as Q ? m(RE,?)?(RE,?)/ ?0
and plot this quantity in the figure; see Eqs. (A3) and (A4) in Ref. [3] for the explicit
definitions of these quantities. The blue curve shown in Fig. 5.2 corresponds to the
next-to-leading-order dynamics in an expansion in c?2.
Finally, we plot the results of the numerical QE configuration calculations
performed in Ref. [172] with black dots. These calculations were made under the
assumption of conformal flatness and stationarity; physically, each configuration
represents a binary on an exactly circular orbit emitting no GWs. This setup is
used to approximate a BNS during its adiabatic inspiral. The scalar mass MS of the
total system, defined in the Jordan frame, was computed in Ref. [172]. To convert
this quantity to the scalar charge of the full system, we use Qtot = MS/ (???0);
this conversion is discussed in detail in footnote 2 of Ref. [172]. For simplicity, we
assume that the component scalar charges are simply half of the total scalar charge,
Q = Qtot/2.
As evidenced by Fig. 5.2, we find very close agreement to previous predictions
257
of the evolution of the scalar charge with our effective-action model. A key feature
is the frequency ?DS at which dynamical scalarization occurs. As discussed above,
our model predicts the onset of dynamical scalarization when the binary separation
r = 1/c(2). Converting the separation into an orbital frequency using Eq. (5.36),
we find agreement to within . 10% compared to the values presented in Table
II of Ref. [3] for both the PD model and the QE configuration calculations.6 We
emphasize that our effective action model is in no way calibrated to fit the QE
results; the only numerical input to the model comes from isolated NS solutions of
the TOV equations.
Having computed Q(?), we now compute the energy of the binary system as
a function of frequency. We define the binding energy EB of the binary as,
EB ? H ?M, (5.38)
and use Eq. (5.29) to evaluate the Hamiltonian. Using Eq. (5.36) to convert r to
? we plot the binding energy (normalized by the total mass M) as a function of
orbital frequency in Fig. 5.3. We also plot the binding energy computed from QE
configurations in Ref. [172] as dashed lines and the 4PN prediction for nonspinning
point particles in GR [316] as a green dashed-dotted line.7 To improve comparison,
we have added to the predictions of our effective-actio(n m)odel (computed at New-
6The agreement can be slightly improved by neglecting the O c?2 contributions to the redshift
variables (5.31) that enter into Eq. (5.30)
7We use the 4PN binding energy in GR as our benchmark rather than more sophisticated
estimates for simplicity. For the frequency range we consider, the 4PN energy is visually indistin-
guishable from the predictions of the EOB formalism [45].
258
0.000
??
?? ????? ?
?
-0.005 ?
?
??
?
? ?
?
??? ?
??
-0.010 ??
?
? ?
? ?
?
?? ?? ???
-0.015
0.00 0.01 0.02 0.03 0.04 0.05
Figure 5.3: Binding energy EB normalized by the total mass M = 2.7M
as a function of orbital frequency for the same BNS system as in Fig. 5.2.
The predictions of the effective action model introduced here are shown
in solid lines; we add to our Newtonian-order result the 1PN, 2PN, 3PN,
and 4PN contributions found in GR. The QE configuration calculations
performed in Ref. [172] are shown with dashed lines. Red (black) curves
correspond to ? = ?4.2 (? = ?4.5). For comparison, we have also
plotted in green the 4PN prediction for the point-particle binding energy
in GR.
259
tonian order), the 1PN, 2PN, 3PN, and 4PN corrections to the binding energy in
GR. These corrections raise the binding energy closer to the other curves in Fig. 5.3,
but do not influence the ordering of the various curves, and thus do not affect our
conclusions.
As expected, prior to the onset of dynamical scalarization, the binding en-
ergy closely resembles that of the corresponding system in GR. After dynamical
scalarization occurs, we find significant differences between our analytic model and
the QE results of Ref. [172]: the present model predicts an increase in the mag-
nitude of the binding energy |EB| relative to GR whereas the QE computations
indicate that the magnitude should decrease. Given the interpretation of dynamical
scalarization as a phase transition detailed in Sec. 5.5, one expects the scalarized
binary to be more tightly bound than the corresponding unscalarized binary, i.e.,
the GR prediction. If this were not the case, dynamical scalarization would be an
endothermic process (requiring energy input) and the ? ? ?? symmetry would
not spontaneously break. Based on this intuition, the predictions of our model in
Fig. 5.3 appear qualitatively correct. The cause of the disagreement between our
model and Ref. [172] remains unclear. The discrepancy could stem from the as-
sumption of conformal flatness and/or the presence of tidal interactions absent in
our point-particle model of the dynamics. However, to explain the disagreement in
Fig. 5.3, these factors would need to play a more significant role in the presence of
scalar charges; analogous calculations done in GR agree with analytic point-particle
predictions of the binding energy much more closely than the deviations shown in
Fig. 5.3 (see, e.g., Ref. [317]).
260
5.5 Dynamical scalarization as a phase transition
Having validated its accuracy in Sec. 5.4, in this section we explore an impor-
tant conceptual implication of our effective action model: we definitively establish
dynamical scalarization as a second-order phase transition. Using the Landau theory
of phase transitions [318], we discuss the scalarization of an isolated NS (spontaneous
scalarization), an equal-mass BNS (dynamical scalarization), and an unequal-mass
BNS (spontaneous, induced, and dynamical scalarization).
The approach by Landau [318] allows one to relate certain types of phase tran-
sitions to broken symmetries. We begin with a schematic review, closely following
Ref. [318]. Consider a system described by a set of state variables ? and thermody-
namic potential ?(?) that undergoes a second-order transition between two phases
at some critical point ??. The degree of symmetry in each phase can be described by
an order parameter ?. We choose the order parameter such that it vanishes for the
phase with greater symmetry, but in the other phase, the breaking of some of these
symmetries causes ? to be nonzero. To exhibit a second-order phase transition, the
thermodynamic potential must admit an expansion near the critical point of the
form
( )
?(?, ?) = ? (?) + ? (?)?2 + ? (?)?40 2 4 +O ?6 , (5.39)
where the coefficients obey the following conditions:
?4(?) >0, (5.40)
?2(?
?) =0. (5.41)
261
The first condition guarantees that the system has an equilibrium solution (found
at the minimum of ?). We discuss the second condition below.
For states ?above? ??, i.e., those for which ?2(?) > 0, the potential (5.39)
is positive definite, and so the system reaches equilibrium in the more symmetric
state (the one in which ? vanishes). However, as one passes through the point
??, the coefficient ?2(?) changes sign; now the potential (5.39) is minimized for
configurations with nonzero values of ?.
In anticipation of later discussion, we generalize the treatment above to sys-
tems described by a vector order parameter ? ? Rn, where Euclidean coordinates
are denoted with unitalicized Latin indices. In this generalization, the functions
?m(?) become rank-m tensors of dimension n such that Eq. (5.39) becomes
( )
?(?,?) =?0(?) + [?2(?)]ab ?
a?b + [?4(?)] ?
a
abcd ?
b?c?d +O ?6 . (5.42)
The conditions (5.40) and (5.41) must be appropriately extended, as well. To
ensure that the system has an equilibrium solution, we require that ?4 be positive
definite, in the sense that
[?4(?)]
a b c d n
abcd ? ? ? ? > 0, ?? ? R . (5.43)
The n-dimensional generalization of Eq. (5.41) is
det ([?2(?
?)]ab) = 0. (5.44)
Note that in the phase with greater symmetry, our assumption that ? is minimized
when ? vanishes ensures that all eigenvalues of the matrix [? ?2(? )]ab must be posi-
tive. In the less symmetric phase, at least one of the eigenvalues must be negative;
262
however, the determinant of the matrix remains positive if an even number of eigen-
vectors have negative eigenvalues.
5.5.1 Spontaneous scalarization of an isolated body
The classical illustration of a second-order phase transition is spontaneous
magnetization in a ferromagnet at the Curie temperature TC . In this example, ?
is the energy E of the system and ? represents the temperature and external mag-
netic field B. The order parameter ? is the total magnetization M ? ??E/?B,
which is thermodynamically conjugate to B. Inspired by this example, Damour and
Esposito-Fare?se [152] considered a phenomenological model of spontaneous scalariza-
tion following the Landau ansatz (5.39). Starting from the total energy of an isolated
NS mE(?), the authors selected the potential m(Q), defined as in Eq. (5.18), to play
the role of ?. The bulk properties of the NS are its baryonic mass m? and external
scalar field ?. Analogous to spontaneous magnetization, the authors identified the
order parameter Q as the conjugate variable to the scalar field [c.f. Eq. (5.15)].8
The behavior of the potential m around the critical baryonic mass m?cr was modeled
by [152]
1 1
m(Q) = a (m?cr ? m?)Q2 + bQ4, (5.45)
2 4
where a and b are constant (positive) coefficients. Above the critical baryonic mass,
NSs equilibrate in configurations with nonzero scalar charge.
8The notation of Ref. [152] differs from that used here. The original notation can be recovered
with the following substitutions: ?? ?0, Q? ?A, mE(?)? mA(?A, ?0), m(Q)? ?(?A).
263
By design, our point-particle model (5.22) takes the same form as Eq. (5.45),
and thus can model spontaneous scalarization as well. Unlike Eq. (5.45), we do
not factor out any mass-dependence of the coefficients c(i). As demonstrated in
Section 5.4.1, our model remains valid for stars with m? 6? m?cr?these stars were not
considered in Ref. [152]. The coefficient c(2) plays the role of ?2 in the Landau
ansatz (5.39); note that this coefficient depends on the properties of the NS (e.g.,
the mass and EOS) and on the scalar-tensor coupling (characterized by ?). The
critical point at which a NS transitions from an unscalarized state (Q = 0) to a
spontaneously scalarized state (Q 6= 0) occurs when c(2) is zero. Neutron stars
with negative values of c(2) must spontaneously scalarize; the unscalarized state is
unstable.
5.5.2 Dynamical scalarization of equal-mass binaries
With our effective action model, we can now apply this analysis to a binary
system of NSs. For simplicity, we begin by studying equal-mass systems with zero
background scalar field ?0. We assume that NSs have the same properties as well,
(i) (i)
i.e., cA = cB . For illustrative purposes, we drop the p
2 and m/r terms in the
Hamiltonian (5.29); restoring these terms does not affect the qualitative behavior
we describe below.
Under these assumptions, the Hamiltonian is given by
( )
(4)
H = 2m(0) + c(2) ? 1 cQ2 + Q4, (5.46)
r 12
where we have dropped the body labels. This expression takes the same form
264
Figure 5.4: Illustration of the Hamiltonian of an equal-mass BNS as
(2)
function of scalar charge and effective coefficient c = c(2)eff ? 1/r. Solu-
tions to the equations of motion are highlighted with solid lines. When
(2)
ceff becomes negative, the trivial solutions Q = 0 become unstable. The
bottom lower plane shows the projection of the solutions.
as Eq. (5.39). Using the same analysis as in the previous subsection, we show
that dynamical scalarization is a second-order phase transition that occurs at a
separation r = 1/c(2)DS ; this conclusion agrees with our prediction in Eq. (5.34). By
comparing Eqs. (5.22) and (5.46) we see that an equal-mass dynamically scalarizing
(2)
system behaves like an isolated NS with an effective coefficient c ? c(2)eff ? 1/r that
decreases as the binary coalesces.
In Fig. 5.4, we plot the simplified Hamiltonian (5.46) as a function of charge
and effective coefficient c(2) ? 1/r. For positive values of this effective coefficient,
the energy is minimized in the trivial configuration Q = 0. Below the critical
265
Figure 5.5: Illustration of the binding energy as a function of scalar
charge for BNSs that undergo: (left) spontaneous scalarization, (middle)
induced scalarization, and (right) dynamical scalarization. Equilibrium
solutions are highlighted with dots. The solutions are projected onto the
(QA, QB) plane below; colored arrows indicate the flow of these solutions
as the binary coalesces.
(2)
point ceff = 0, the unscalarized state becomes unstable; instead, the binary system
transitions into a scalarized state. The bottom plane shows the projection of the
equilibrium solutions in black. As predicted by Eq. (5.34), the stable solutions
bifurcate at the critical point, spontaneously breaking the scalar-parity symmetry
of the theory. Note that this entire discussion can be applied directly to isolated
NSs that undergo spontaneous scalarization by taking r ??.
5.5.3 Scalarization of unequal-mass binaries
Finally, we turn our attention to the critical phenomena that can occur in
unequal-mass binaries. The (vector) order parameter ? ? R2 is given by
(?1, ?2) = (QA, QB) . (5.47)
266
Again, we assume that the background scalar field ?0 vanishes and drop the p
2
and m/r terms in the Hamiltonian (5.29); these simplifications do not affect the
qualitative behavior described below. Under these assumptions, the Hamiltonian
takes the same form as Eq. (5.42) with
(0) (0)
?0 =m?A +mB , ? (5.48)
1
[? ?? (2)2]ab =2 ? c ?r
?1
A ??? , (5.49)
?r?1 (2)
1 (
cB )
(4) (4)
[?4]abcd = c ?
1?1?1?1 + c ?2?2?2A a b c d B a b c?
2
d . (5.50)24
We examine the Hamiltonian (5.29) for systems that undergo:
(2) (2)
1. Spontaneous scalarization: Both stars are initially scalarized (cA < 0, cB <
0),
(2) (2)
2. Induced scalarization: Only one star is initially scalarized (cA > 0, cB < 0),
(2) (2)
3. Dynamical scalarization: Neither star is initially scalarized (cA > 0, cB > 0).
For all three cases, we restrict our attention to binaries following circular orbits. The
binding energy is shown in Fig. 5.5 as a function of the NS charges. We show only
the slices of the full graph H(QA, QB) that pass through equilibrium solutions; for
comparison, these curves correspond to the thin black lines on the surface in Fig. 5.4.
Moving from left to right, the plots correspond to spontaneous, induced, and dy-
namical scalarization, respectively. Moving downwards in each plot, the green, red,
and blue curves depict the binding energy at progressively smaller separations. The
equilibrium solutions are denoted with dots on the curves and are projected onto the
267
(QA, QB)-plane in the color corresponding to their separation. The colored arrows
depict the flow of equilibrium solutions as the separation decreases.
At large separations (green), there exist four stable configurations for sponta-
neously scalarized binaries (left panel): each NS can exhibit a positive or negative
scalar charge, and the choices for each are uncorrelated. However, as the separation
decreases (red and blue), configurations in which the two stars have opposite-parity
charges become energetically unfavorable. As indicated by the pink arrows, these so-
lutions flow towards the origin and transform into a saddle point, i.e., this branch of
solutions becomes unstable. Thus, at this critical separation (red) there exists a new
phase transition distinct from those discussed above. From Eqs. (5.44) and (5.49),
(2) (2)
we find that this critical point occurs at a separation of r? = (c ?1/2A cB ) . Unlike
with dynamical scalarization, the more symmetric state phase occurs at separations
smaller than r?. The equilibrium solutions with charges of the same sign flow away
from the origin as the binary coalesces. The charge of each spontaneously scalarized
star will continue to grow during the inspiral due to feedback from its companion.
Binaries that undergo induced scalarization (middle panel) begin with an un-
scalarized star QA = 0 and a scalarized star QB 6= 0 (green). As the stars are
brought closer together (red), the unscalarized star rapidly develops scalar charge,
whereas the initially scalarized star remains (approximately) unchanged. However,
as the separation decreases further (blue), the two charges become of the same or-
der of magnitude and continue to increase at roughly the same rate through the
remainder of the coalescence. Unlike for spontaneous and dynamical scalarization,
the branches of equilibrium solutions are disjoint throughout the entire coalescence,
268
(2) (2)
i.e. the colored arrows in Fig. 5.5 never meet. Because cA and cB have opposite
signs, the determinant of [?2]ab [given in Eq. (5.49)] is negative for all separations.
Induced scalarization fails to meet condition (5.44) and therefore cannot be classified
as a phase transition.
Finally, initially unscalarized unequal-mass binaries (right panel) evolve simi-
larly as in Fig. 5.4. As seen in Fig. 5.5, the binary system begins in an unscalarized
state (green). At the critical transition point (red), the effective c(2) coefficient
vanishes; beyond that point (blue), scalarization becomes energetically favorable.
Again, Eqs. (5.44) and (5.49) reveal that dynamical scalarization occurs at a sepa-
(2) (2) ?1/2 (2) (2)ration of rDS = (cA cB ) , which reduces to the result in Sec. 5.5.2 when cA = cB .
As before, the scalar charges continue to grow after the onset of dynamical scalar-
ization.
5.6 Conclusions
In the present paper, we developed a new point-particle model for NSs in
scalar-tensor gravity that can reproduce spontaneous, induced, and dynamical scalar-
ization. The model parametrizes the various scalarization phenomena by just two
coefficients c(2), c(4) for each NS. This approach should be contrasted with previous
analytic models of dynamical scalarization [3, 170], which relied upon numerically
solving equations containing the generic function mE(?). For the first time, we have
computed a two-body Hamiltonian that incorporates dynamical scalarization in a
self-consistent manner (see Ref. [3] for a discussion of previous attempts). Observ-
269
ables derived from the model at leading order in the PN expansion were shown to be
in good agreement with earlier analytic models and numerical QE calculations. The
identification of the relevant dynamical variables in the effective action is crucial to
our model.
Analogous to the analysis done in Ref. [152] concerning spontaneous scalariza-
tion, our model rigorously establishes dynamical scalarization as a phase transition
as per Landau theory [318]. Additionally, it demonstrates the intimate connec-
tion between spontaneous and dynamical scalarization. The mapping between an
equal-mass BNS undergoing dynamical scalarization and an effective spontaneously
scalarized NS is detailed in Sec. 5.5.2.
Our effective action stands as an important first step towards accurate ana-
lytic waveforms of dynamically scalarizing BNSs. The model benefits from its close
analogy to the effective action model of dynamical tides detailed in Refs. [292, 299]?
the dynamical scalar monopole Q here corresponds to the dynamical gravitational
quadrupole therein. References [292, 299] derived an accurate EOB [38, 39] wave-
form model incorporating dynamical tidal interactions. Using this model as a tem-
plate, one could construct an analogous model for dynamical scalar-tensor effects.
This construction will require calculations of dissipative effects and higher PN order
results for the conservative dynamics.
Another avenue for future work is the addition of kinetic-energy terms to the
effective action as in Eq. (5.23). Resonant effects play an important role in the
dynamical tides model of Refs. [292, 299]; it remains to be seen whether analogous
effects could be important with dynamical scalar charges. Formulating the effective
270
action in this manner offers a conceptual advantage over the current model, as it
guarantees that all of the equations of motion are ordinary differential equations
(rather than a mix of nonlinear algebraic and differential equations).
Finally, an intriguing extension of this work is to theories with a massive
scalar field. Pulsar timing cannot constrain sufficiently short-range scalar fields, so a
much wider range of parameter space of massive scalar-tensor theories remains to be
constrained by GW observations than that of theories with a massless scalar [161].
The PN dynamics of a simple massive scalar-tensor theory were investigated in
Ref. [94], and spontaneous scalarization of isolated NSs was studied in Refs. [95,
161]; the framework we have presented above could synthesize these results with
appropriate modifications to the field equations (5.6) and (5.7).
5.7 Acknowledgments
We are grateful to Andrea Taracchini for discussions and Alessandra Buonanno
for helpful comments.
271
Chapter 6: Theory-agnostic modeling of dynamical scalarization in
compact binaries
Authors: Mohammed Khalil, Noah Sennett, Jan Steinhoff, and Alessandra
Buonanno1
Abstract: Gravitational wave observations can provide unprecedented insight
into the fundamental nature of gravity and allow for novel tests of modifications to
General Relativity. One proposed modification suggests that gravity may undergo a
phase transition in the strong-field regime; the detection of such a new phase would
comprise a smoking-gun for corrections to General Relativity at the classical level.
Several classes of modified gravity predict the existence of such a transition?known
as spontaneous scalarization?associated with the spontaneous symmetry breaking
of a scalar field near a compact object. Using a strong-field-agnostic effective-field-
theory approach, we show that all theories that exhibit spontaneous scalarization
can also manifest dynamical scalarization, a phase transition associated with sym-
metry breaking in a binary system. We derive an effective point-particle action that
provides a simple parametrization describing both phenomena, which establishes a
foundation for theory-agnostic searches for scalarization in gravitational-wave obser-
1Originally published as Phys. Rev. D100, 124013 (2019).
272
vations. This parametrization can be mapped onto any theory in which scalarization
occurs; we demonstrate this point explicitly for binary black holes with a toy model
of modified electrodynamics.
6.1 Introduction
Classical gravity described by General Relativity (GR) has passed many exper-
imental tests, from the scale of the Solar System [79] and binary pulsars [249, 294] to
the coalescence of binary black holes (BHs) [10, 73, 122, 123, 184] and neutron stars
(NSs) [9]. Despite its observational success, certain theoretical aspects of GR (e.g.,
its nonrenormalizability and its prediction of singularities [319]) impede progress
toward a complete theory of quantum gravity; yet, strong-field modifications of the
theory may alleviate these issues [83].
Gravitational wave (GW) observations probe the nonlinear, strong-field be-
havior of gravity and thus can be used to search for (or constrain) deviations from
GR in this regime. Because detectors are typically dominated by experimental noise,
sophisticated methods are required to extract GW signals. The most sensitive of
these techniques rely on modeled predictions of signals (gravitational waveforms),
which are matched against the data. This same approach can be adopted to test
gravity with GWs; to do so requires accurate signal models that faithfully incorpo-
rate the effects from the strong-field deviations one hopes to constrain [79]. Ideally,
these models would be agnostic about details of the strong-field modifications to
GR, so that a single test could constrain a variety of alternative theories of gravity.
273
This work establishes a framework for such tests given the hypothetical sce-
nario in which the gravitational sector manifests phase transitions, with only one
phase corresponding to classical GR. This proposal comprises an attractive target
for binary pulsar and GW tests of gravity; if the transition between phases arises
only in the strong-gravity regime (e.g., in the presence of large curvature, relativistic
matter, etc.), then such a theory could generate deviations from GR in compact bi-
nary systems while simultaneously evading stringent constraints set by weak-gravity
tests. We consider the case wherein the ?new? phases arise via spontaneous sym-
metry breaking in the gravitational sector. Similar phase transitions occur in many
areas of contemporary physics?perhaps the most famous example is the electroweak
symmetry breaking through the Higgs field [320?322]?so it is sensible to consider
their appearance in gravity as well. As a first step, we focus on a simple set of
such gravitational theories, in which the transition from GR to a new phase most
closely resembles the spontaneous magnetization of a ferromagnet; however, these
theories can also be extended to instead replicate the standard Higgs mechanism in
the gravitational sector [323, 324].
Specifically, we investigate the nonlinear scalarization of nonrotating com-
pact objects (BHs and NSs), which arises from spontaneous symmetry breaking of
an additional scalar component of gravity [4, 152]. Spontaneous scalarization?the
scalarization of a single, isolated object?has been found in several scalar extensions
of GR, including massless [153?159] and massive [160?162] scalar-tensor (ST) theo-
ries and extended scalar-tensor-Gauss-Bonnet (ESTGB) theories [163?168]. Similar
phenomena can also occur for vector [325, 326], gauge [324], and spinor [327] fields.
274
In contrast, dynamical scalarization?scalarization that occurs during the coales-
cence of a binary system?has been demonstrated and modeled only for NS binaries
in ST theories [3, 4, 169?172, 175, 328].
A scalarized compact object emits scalar radiation when accelerated, analo-
gous to an accelerated electric charge. In a binary system, the emission of scalar
waves augments the energy dissipation through the (tensor) GWs found in GR,
hastening the orbital decay. Radio observations of binary pulsars [6, 249, 294, 329]
and GW observations of coalescing BHs and NSs [9, 10] are sensitive to anomalous
energy fluxes, and thus can be used to constrain the presence of scalarization in
such binaries. In addition to the tensor radiation mentioned above, binaries con-
taining spontaneously scalarized components also emit scalar radiation throughout
their entire evolution. In contrast, dynamically scalarizing binaries transition from
an unscalarized (GR) state to a scalarized (non-GR) state at some critical orbital
separation, emitting scalar waves only after this point. Because this is a second-
order phase transition [4], the emitted GWs contain a sharp feature corresponding
to the onset of dynamical scalarization. This feature cannot be replicated within
typical theory-agnostic frameworks used to test gravity [117, 118, 120, 176], as these
only consider smooth deviations from GR predictions, e.g., modifications to the
coefficients of a power-series expansion of the phase evolution.
While one could attempt to model dynamical scalarization phenomenologically
by adding nonanalytic functions to such frameworks [175, 271], in this work, we
propose a complementary theory-agnostic approach. We focus on a specific non-
GR effect, here scalarization, but remain agnostic toward the particular alternative
275
theory of gravity in which it occurs. The basis for our framework is effective field
theory. Scalarization arises from strong-field, nonlinear scalar interactions in the
vicinity of compact objects; the details of this short-distance physics depends on
the specific alternative to GR that one considers. By integrating out these short-
distance scales, we construct an effective point-particle action for scalarizing bodies
in which the relevant details of the modification to GR are encapsulated in a small
set of form factors. The coefficients of these couplings offer a concise parametrization
ideal for searches for scalarization with GWs. The essential step in constructing this
effective theory is identifying the fields and symmetries relevant to this phenomenon.
Starting from the perspective that scalarization coincides with the appearance of a
tachyonic scalar mode of the compact object, we derive the unique leading-order
effective action valid near the critical point of the phase transition. Though this
effective action matches that of Ref. [4]?which describes the scalarization of NSs in
ST theories2?the approach described here is valid for a broader range of non-GR
theories.
Our proposed parametrization of scalarization is directly analogous to the
standard treatment of tidal interactions in compact binary systems. Tidal effects
enter GW observables through a set of parameters that characterize the response of
each compact object to external tidal fields [308]. These parameters are determined
by the structure of the compact bodies?for example, the short-distance nuclear
interactions occurring in the interior of a NS. This description of tidal effects is
applicable to a broad range of nuclear models (i.e., NS equations of state) and offers
2Dynamical scalarization was also modeled at the level of equations of motion in Refs. [3, 170].
276
a more convenient parametrization of unknown nuclear physics for GW measure-
ments [186, 330] than directly incorporating nuclear physics into GW models. From
the perspective of modeling compact binaries, the primary difference between tidal
effects and scalarization is that the latter is an inherently nonlinear phenomenon,
necessitating higher-order interactions in an effective action.
Beyond offering a convenient parametrization for GW tests of gravity, our ef-
fective action also elucidates certain generic properties of scalarization phenomena.
Using a simple analysis of energetics based on the effective theory, we argue that
any theory that admits spontaneous scalarization must also admit dynamical scalar-
ization. Additionally, this type of analysis can provide further insights regarding the
(nonperturbative) stability of scalarized configurations and the critical phenomena
close to the scalarization phase transition. We illustrate these points by applying our
energetics analysis to a simple Einstein-Maxwell-scalar (EMS) theory in which elec-
trically charged BHs can spontaneously scalarize, complementing previous results
for NSs in ST theories [4].
The paper is organized as follows. In Sec. 6.2, we first review the mechanism
of scalarization as the spontaneous breaking of the Z2 symmetry of a scalar field
driven by a linear scalar-mode instability. Then, we construct an effective worldline
action for a compact object interacting with a scalar field valid near the onset of
scalarization. In Sec. 6.3, we discuss how the relevant coefficients in the action can
be matched to the energetics of an isolated static compact object in an external
scalar field, demonstrating the procedure explicitly with BHs in the EMS theory of
Ref. [158]. In Sec. 6.4, we employ the effective action to further investigate scalar-
277
ization in this EMS theory: we examine the stability of scalarized configurations,
compute the critical exponents of the scalarization phase transition, and predict the
frequency at which dynamical scalarization occurs for binary BHs. We also argue
that dynamical scalarization is as ubiquitous as spontaneous scalarization in modi-
fied theories of gravity. Finally, in Sec. 6.5, we summarize the main implications of
our findings, and discuss future applications of our framework. The appendixes pro-
vide a derivation of a more general effective action and details on the construction
of numerical solutions for isolated BHs in EMS theory.3
6.2 Linear mode instability and effective action close to critical point
In this section, we review the connection between the appearance of an unsta-
ble scalar mode in an unscalarized compact object and the existence of a scalarized
state for the same body. We then derive an effective action close to this critical
point at which this mode becomes unstable.
As an illustrative toy model for this discussion, we consider the modified theory
of electrodynamics introduced in Ref. [158] (hereafter referred to as EMS theory for
brevity), whose action is given by
? ??g
S 4 ? ??field = d x [R? 2???? ?? f(?)F F?? ] , (6.1)
16?
where R is the Ricci scalar, g is the determinant of the metric g?? , and F?? =
3Throughout this work, we use the conventions of Misner, Thorne, and Wheeler [15] for the
metric signature and Riemann tensor and work in units in which the speed of light and bare
gravitational constant are unity.
278
??A? ? ??A? is the electromagnetic field tensor. In this work, we consider two
choices of scalar couplings:
???2f1(?) =e( , ) (6.2)?1
1
f2(?) = 1 + ??
2 ? ?2?4 , (6.3)
2
where ? is a dimensionless coupling constant. While the two couplings have the
same behavior near ? = 0, their behavior for large field value differs drastically.
The former choice was used in Ref. [158] to construct stable scalarized BH solutions,
whereas we introduce the latter in this work to demonstrate a theory in which no
stable scalarized BH configurations exist (see Sec. 6.4).
The absence of any linear coupling of ? to the Maxwell term implies that any
solution in Einstein-Maxwell (EM) theory, i.e., with ? = 0, also solves the field
equations of Eq. (6.1); however, stable solutions in EM theory may be unstable in
EMS theory. To see this, we write the scalar-field equation schematically as
f ?(?)
2? = m2 2 ??eff ?, meff = F F?? . (6.4)4?
We consider an electrically charged BH, for which F ??F?? < 0 and thus the effective-
mass squared m2eff is negative for ? < 0. One can decompose ? into Fourier modes
with frequency ? and wave vector k, which satisfy the dispersion relation ?2 ?
k2 +m2eff(k), where curvature corrections have been dropped for simplicity. We see
that if m2eff(k) is sufficiently negative, then ?
2 is also negative, leading to a tachyonic
instability. The critical point at which this tachyonic instability first appears can
be determined by identifying linearly unstable quasinormal scalar modes of the EM
279
full theory matching asymptotics effective
M M m(q)? q?IR(y) theory
E E cA
Q Q q
? IR0 ?0 ? (y)
worldline
y?(?)
+
IR projection
Figure 6.1: An illustration of the approach used in this work (read from
left to right). (Left) The fields describing an isolated compact object in
equilibrium are decomposed into short- (UV) and long- (IR) wavelength
modes, depicted in blue and red, respectively. (Center) We integrate out
the UV modes via an IR projection. (Right) By matching asymptotics,
we identify the coarse-grained compact object with an effective point-
particle model that describes the IR sector of the full theory. Section 6.3
contains a detailed description of this procedure and the definitions of
all quantities shown above.
solution [164, 331] or by constructing sequences of fully nonlinear, static scalarized
solutions (as we do here) [158, 331].
The tachyonic instability drives the body away from the unscalarized solution,
thereby breaking the symmetry ? ? ?? in Eq. (6.1). For a stable scalarized equi-
librium configuration to exist, this instability must saturate in the nonlinear regime
[332]. These two conditions?the existence of a tachyonic instability and its eventual
saturation?are satisfied in all of the theories discussed previously [153?168, 324?
327]. The only difference between these theories is the form of m2eff; for example, in
ST theories m2eff depends on the stress-energy tensor, while in ESTGB it depends
on the Gauss-Bonnet invariant. Indeed, theories which meet these two criteria can
be straightforwardly constructed, which is the reason why scalarization is such a
ubiquitous phenomenon.
280
An even simpler perspective on scalarization arises from a coarse-grained, or
effective, theory. Let us derive it explicitly. We start by splitting the fields into the
short- (or ultraviolet, UV) and long- (or infrared, IR) wavelength regimes separated
by the object?s size ? R, i.e., ? = ?IR + ?UV, and spatially average over (integrate
out) the UV parts. This effectively shrinks the compact object to a point and its
effective action is given by an integral over a worldline y?(?), where ? is the proper
time (see Fig. 6.1 for a schematic illustration). Dynamical short-length-scale pro-
cesses like oscillations of the object are represented by dynamical variables on the
worldline. For simplicity, we assume that we can also average over fast oscillation
modes and only retain the monopolar mode associated to a linear tachyonic insta-
bility, denoted by q(?). This mode q(?) can indeed be excited by IR fields, since its
frequency (or effective mass) vanishes at the critical point.
Effective actions are usually constructed by making an ansatz respecting cer-
tain symmetries and including only terms up to a given power in the cutoff between
IR and UV scales. The relevant symmetries here are diffeomorphism, U(1)-gauge,
worldline-reparametrization, time-reversal4, and scalar-inversion invariance. The
last reads ? ? ?? in the full theory, so in the effective action it decomposes into
simultaneous IR ?IR ? ??IR and UV q ? ?q transformations. The IR fields are
of order ?IR ? O(R/r) on the worldline, where r is the typical IR scale (e.g., the
separation of a binary). Now, the oscillator equation for the mode q(?) driven by
4Time reversal is an approximate symmetry of compact objects in an adiabatic setup, like
the inspiral of a binary system. In this case, a compact object?s entropy remains approximately
constant.
281
the IR field ?IR can be schematically written as
c q? + V ?
c(2) c(4)
q?2 (q) = ?
IR(y), V (q) = q2 + q4 + . . . , (6.5)
2 4!
where ? = d/d? and the c... are constant coefficients determined by the UV physics.
(The singular self-field contribution to ?IR(y) must be removed using some regu-
larization prescription.) The normalization of q is chosen to fix the coefficient of
?IR(y); for all that follows, we simply assume that cq?2 > 0. Close to the critical
point, the quadratic term in V is negligible, and thus from Eq. (6.5), one finds that
for equilibrium configurations (q? = 0), q scales as q3 ? ?IR. More generally, the
mode q oscillates around this equilibrium point provided that the IR field evolves
slowly relative to the frequency of the mode, i.e., ??IR = y?????
IR  q?/q; this con-
dition is satisfied for binary systems on quasicircular orbits (which we restrict our
attention to in this work), but could be violated for highly e?ccentric orbits. For
small perturbations around equilibrium, one finds that q? ? ? ?IR and q? ? ??IR
where ? ? (q? q0)/q0  1 is the fractional deviation from the equilibrium point q0.
Using the scaling relations derived above, we construct the most generic effec-
tive action for a nonrotating compact object with a dynamical mode q(?) described
by Eq. (6.5) close to th?e crit[ical point, up to order O(R
2/r2)
crit cq?2S = d? q?2 + ?IR ? ? c(2) 2 ? c(4)(y)q c 4CO (0)2 ( )] q q2 4!
R2
? + c A
IR ?
[ A ? (y)y? +O , (6.6)r2
cq?2
= d? q?2 + ?IR(y)q ?(m(q)2 )]
2
+ c AIRA ? (y)y?
? +O R , (6.7)
r2
282
where CO stands for compact object and for later convenience we define
m(q) ? c(0) + V (q). (6.8)
Terms containing time derivatives of ?IR all enter at higher order in R/r than we
work, e.g., ??IRq ? ??IRq? ? ?IRq? ? O(R2/r2), and thus are absent in Eq. (6.7). A
reparametrization-invariant action is?obtained by replacing d? with
dy? dy?
d? = d? ?gIR??(y) , (6.9)d? d?
where ? is an arbitrary affine parameter, and replacing derivatives d/d? accordingly.
The complete effective action reads
Seff = S
IR + Scritfield CO , (6.10)
where more copies of ScritCO can be added depending on the number of objects in the
system and SIRfield is given by Eq. (6.1) with IR labels on the fields. The equations of
motion and field equations are obtained by independent variations of y?(?), q(?),
and ?IR(x), gIR(x), AIR?? ? (x).
The simplicity of ScritCO is striking, but we recall that it is only valid close to the
critical point of a monopolar, tachyonic, linear instability of a scalar mode. (A more
generic effective action valid away from the critical point is discussed in Appendix
H.) Despite its simplicity, the effective action (6.10) is theory agnostic, in the sense
that it is constructed assuming only the scalar-inversion symmetry and that the
nonrotating compact object hosts such a mode; in particular, it should hold for the
cases studied in Refs. [153?168, 324?327] and similar work to come. We emphasize
that strong-field UV physics at the body scale is parametrized through the numerical
283
coefficients c..., which can be matched to a specific theory and compact object, or
be constrained directly from observations (analogous to tidal parameters [9, 330]).
6.3 Matching strong-field physics into black-hole solutions
As an illustrative example of the effective-action framework derived above, we
now compute the sought-after coefficients c... for BHs in EMS theory.
For this purpose, we match a BH solution in the full theory (6.1) to a generic
solution of the coarse-grained effective theory (6.10) for an isolated body. Schemat-
ically, the former represents the full solution at all scales, while the latter only
represents its projection onto IR scales. We focus first on BH solutions of the full
theory (6.1), restricting our attention to equilibrium/static, electrically charged,
spherically symmetric solutions.
In EMS theory, this family of solutions is characterized by three independent
parameters, which we take to be the electric charge E , the BH entropy S, and the
asymptotic scalar field ?0, assuming a vanishing asymptotic electromagnetic field
and an asymptotically flat metric. The electric charge is globally conserved by
the U(1) symmetry of the theory and the entropy remains constant under reversible
processes, which we have implicitly restricted ourselves to by assuming time-reversal
symmetry in the effective action. Thus, we use a sequence of solutions with fixed E
and S to represent the response of a BH to a varying scalar background ?0. Since
EMS theory modifies electrodynamics, but not gravity or the coupling to gravity,
the entropy of a charged BH is the same as in GR, i.e., it is proportional to the
284
0.4
2.0
1.5 0.2
1.0
0.0
0.5
0.0 -0.2
-0.5
-0.4
-1.0
-0.2 -0.1 0.0 0.1 0.2 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15
Figure 6.2: The potential V (q) for ? = ?8 and different electric charges
E . The left and right panels correspond to BHs in EMS theory with cou-
pling f1(?) [Eq. (6.2)] and f2(?) [Eq. (6.3)], respectively. The numerical
solutions and polynomial fits are indicated with points and lines, respec-
tively.
horizon area. See also Ref. [158] for the first law of BH thermodynamics in EMS
theory.
The asym?ptotic?beha?vior of?the?fields take the form
???
?
??
? ????? ?
? ? Q(? )
??
?
A ?? = ???
0 ??
? ?0 ????+?
??? 0 ??
??2
0 ??? ?Ee 0 ???
? 1
? | | +O(|X|
?2), (6.11)
X
g00 ?1 2M(?0)
where Q(?0) is the scalar charge of the BH andM(?0) is its gravitational mass.5 We
construct these solutions numerically (see Appendix I for details), and then compute
?0,M, and Q directly from their asymptotic behavior.
Next, we turn our attention to the description of these BH solutions in the
5The quantities Q(?0) and M(?0) describing the asymptotic behavior of the solution also
depend on the parameters E and S, but we suppress the dependence in our notation for brevity.
Derivatives of Q and M are taken holding E and S constant.
285
effective theory (6.10). We set up the solution under the same boundary condi-
tions as the numerical sequence described above; in particular, we look at isolated
equilibrium configurations, y?? IR ? IR ? IR?g?? ? y ??A0 ? y ??? ? q? ? 0. We construct
a coordinate system x in which the worldline has spatial components y = 0, so
that all fields are independent of time. Furthermore, relying on the fact that both
solutions are asymptotically flat, we choose the coordinates x such that they match
the numerical coordinates X in the asymptotic region, i.e., x = X + O(|X|?1).
Then workin?g to line?ar or?der in?the fi? ?
elds, we find ?
?? ?IR? ???? ???
?0 ??? ??? q ?? ? ??? 1IR? AIR ?? = ???? 0 ????+???? ?c e?[? (y)]
2 ???? | | + . . . . (6.12)0 A x
gIR IR00 ?1 2[m(q)? ? (y)q]
These fields are singular when evaluated on the worldline, x = y = 0. This can
be cured by appropriately regularizing the solution; here, we simply keep the finite
part and drop the singular self-field part, e.g., ?IR(y) = ?0. The situation is anal-
ogous to the singular fields that arise in electrostatics when an extended source is
approximated by a point charge.
In addition to a solution for the fields, a variation of q in the effective action
leads to ( )
dm dV c(4)q
3 R2
?0 = = = c(2)q + +O . (6.13)
dq dq 3! r2
The matching now consists of identifying the IR-scale fields in the solution of
the full theory (6.11) with the fields predicted in the IR effective theory (6.12). We
extract the IR-scale fields from the former solution using an appropriate IR projector
P IR[?], such that the matching conditions are given explicitly as P IR[?] = ?IR (and
286
likewise for the other fields). Such a projector is most easily formulated in the Fourier
domain, so we first compute the (spatial) Fourier transform of the fields (6.11),
denoted by a tilde
4?Q(?0)
??(K) = ?0?(K) + +O(|K|?1). (6.14)
K2
We employ the simple projector P IR[??] ? ??(K)?(KIR ? |K|), where ? is the
Heaviside function and KIR is the cutoff scale. Applying this projection to Eq. (6.14)
and taking the inverse Fourier transform, one finds that P IR[?] takes the same form
as Eq. (6.11) on scales longer than the cutoff, i.e., for |X|  1/KIR. Then, our
matching conditions P IR[?] = ?IR, P IR[A ] = AIR, P IR0 0 [g00] = g
IR
00 reduce to
Q(?0) = q, E = cA, M(?0) = m(q)? ?0q, (6.15)
where we have used ?IR(y) = ?0 as discussed above. Note that the last equation
and m?(q) = ?0 (6.13) reveal that the two measures of energyM and m are related
by a Legendre transformation of the conjugate variables (q, ?0). Hence, we find
that Q = q = ?M?(?0), in agreement with the first law of BH thermodynamics
[333, 334].
While M(?0) is the gravitational mass of the system, m(q) represents the
?gravitational free energy? of the body (see also Sec. III.A of Ref. [4]). That is, m
is the mass/energy with the potential energy ??0q (due to the external scalar field)
subtracted fromM. We find below that m?notM?serves as better representation
of ?point-particle mass? found in the Lagrangian or Hamiltonian description of a
binary system; of course, both quantities reduce to the standard ADM mass in GR
in the absence of scalarization. Furthermore, away from the critical point, it is not
287
necessary to treat the mode q as a dynamical variable. This means that we can set
q? = 0 and remove q from the action (6.7). The latter is achieved by virtue of the
Legendre transformation between m(q) in Eq. (6.7) and M(?IR(y)),
? [ ]
SCO = d? ?M(?IR(y)) + EAIR(y)y??? + . . . . (6.16)
We see thatM plays the role of the Eardley mass [204] in the action now. We note
that since the Eardley mass and m(q) are related by a Legendre transformation,
they contain the same information.
To compute the values of the various c..., we numerically construct a sequence
of BH solutions as described above and extract the functions M(?0) and Q(?0).
From there we obtain m(q) and V (q) numerically from Eq. (6.15), as illustrated in
Fig. 6.2. Each curve indicates a BH sequence with a different electric charge-to-mass
ratio E/m(0), where m(0) ? c(0) is the mass of the isolated BH with no scalar charge.
The points indicate the numerically computed solutions, which are calculated by
solving the field equations with different boundary conditions for the scalar field
(see Appendix I). The solid lines are polynomial fits of the form (6.5), from which
we extract the values of the coefficients c(2) and c(4).
It is remarkable that from equilibrium solutions one can fix the potential of a
dynamical (nonequilibrium) mode to order q4. This connection is nontrivial, and it
breaks down when one relaxes the assumption of being close to the critical point. For
instance, if terms like (?IR)2 are included in Eq. (6.7), then Q 6= q; or consider the
case of a minimally coupled scalar field (? = 0), wherein no-hair theorems [23, 148]
guarantee that M(?0) = const?the energy of an equilibrium BH obviously does
288
not encode any information about dynamical modes.
Having described how to compute the coefficients c... in the effective action (6.7),
the following section illustrates how this action can be used to study spontaneous
and dynamical scalarization.
6.4 Modeling strong-gravity effects within the theory-agnostic frame-
work
In this section, we show how spontaneous and dynamical scalarization can be
understood from the effective theory (6.7), based on a simple analysis of energetics.
We use this effective action to investigate the properties of these critical phenomena,
namely their critical exponents. Though we use EMS theory to make quantitative
predictions throughout this section, we emphasize again that the qualitative behav-
ior we find should hold generically for all theories in which spontaneous scalarization
occurs. More specifically, theories in which scalarized configurations are stable are
analogous to EMS theory with scalar coupling f1(?), whereas those where such
configurations are unstable correspond to the coupling f2(?) (see the following sub-
section for details). Extending the predictions made below to other theories only
requires computation of the effective mode potential V (q), as described in the pre-
vious section, and the inclusion of any new long-range fields not present in EMS
theory that impact the motion of binary systems.
289
6.4.1 Spontaneous scalarization
Recall that a spontaneously scalarized object is one that hosts a nonzero scalar
charge even in the absence of an external scalar field ?0 = 0. From Eq. (6.13) we
see that ?0 = 0 corresponds to extrema of V (q) for equilibrium configurations. Fur-
thermore, since V (q) is the oscillation-mode potential, q dynamically evolves into
a minimum of V (q) [see Eq. (6.5)]. Thus, the existence of spontaneously scalar-
ized configurations is indicated by nontrivial extrema of V (q), and the stability of
these configurations depends on whether such points are local minima (stable) or
maxima (unstable). For example, the left panel Fig. 6.2 depicts the appearance of
spontaneously scalarized BH solutions as one increases the charge-to-mass ratio in
EMS theory with coupling f1(?). Without enough electric charge (e.g., the red and
orange curves, with c(2) > 0), the EM (unscalarized) solutions are the only stable
BH solutions, but by increasing the charge beyond a critical value (e.g., the green
and blue curves, with c(2) < 0), the EM solution becomes unstable and the stable
solutions instead occur at nonvanishing values of q.
Our approach allows one to determine the values of the coupling ? and the
electric charge E at which spontaneous scalarization first occurs (c(2) = 0) using
only sequences of equilibrium BH solutions. A more direct approach employed in
the past was to search for instabilities of linear, dynamical scalar perturbations on a
(GR) Reissner-Nordstro?m background [331]. We find that the two methods provide
the same predictions. For the choice of coupling f1(?), we compute the critical
coupling as a function of the electric charge ?crit(E) where c(2) = 0 and find that our
290
results agree with the predictions of Ref. [331] at the onset of the linear instability
of the ` = 0 scalar mode to within 1%. For theories in which scalarized solutions
are easy to construct, like the EMS theory considered here, our approach can more
efficiently compute this critical point than a perturbative stability analysis. We see
that the effective potential V (q) provides strong indications for a linear scalar-mode
instability and its nonlinear saturation.
The same energetics argument reveals drastically different behavior in the
EMS theory with coupling f2(?), depicted in the right panel of Fig. 6.2. Recall that
f1(?) ? f2(?) for small field values, but the two choices differ in the nonlinear regime,
which will dramatically impact the stability of scalarized solutions. This distinction
is reflected in our effective action by the sign of c(4); this coefficient is positive for the
choice of coupling f1(?) and negative for f2(?). Our simple energetics arguments
reveal that above some critical electric charge (e.g., the green and blue curves with
c(2) < 0), no spontaneously scalarized solutions exist, whereas below this value (e.g.,
the red and orange curves with c(2) > 0) spontaneously scalarized solutions may
exist, but are unstable to scalar perturbations. In the former case (the green and
blue curves), there is no sign of a nonlinear saturation of the tachyonic instability
of the EM solution; no stable equilibrium solutions seem to exist. However, it is
impossible to infer how the unstable EM solutions would evolve using our effective
theory, since the assumption of time-reversal symmetry (or constant BH entropy)
will likely break down. Numerical-relativity simulations are needed to answer this
question (or the construction of a more generic effective theory).
The importance of nonlinear interactions in stabilizing spontaneously scalar-
291
ized solutions has been studied extensively in the context of ESTGB theories [335?
337]. For those theories, exponential couplings (equivalent to our f1) or quartic cou-
plings (f ? ??2 + ?4) provide stable scalarized solutions, whereas with quadratic
couplings (f ? ??2), all scalarized solutions are unstable. Interestingly, quadratic
couplings predict stable scalarized solutions in EMS theories [338], but our analysis
suggests that stability is not guaranteed for generic couplings. The stability analy-
ses in these references involve studying linearized perturbations on scalarized back-
grounds. Though technically only valid near the critical point of the spontaneous
scalarization phase transition and for small q, our approach offers a much easier
alternative for estimating stability. We find that our approach correctly reproduces
the findings of these stability analyses for scalarized BHs in EMS theories [338, 339].
6.4.2 Critical exponents of phase transition in gravity
The point-particle action (6.7) also offers some insight into the critical behav-
ior that arises near the onset of spontaneous scalarization. For this discussion, we
restrict our attention to the scalar coupling f1(?), for which spontaneously scalar-
ized configurations are stable. Considering the various coefficients c... as functions
of the electric charge E and entropy S of a BH and the overall coupling constant ?,
the effective potential V (q) corresponds precisely to the standard Landau model of
second-order phase transitions [318, 340]. Compared to the archetypal example of
ferromagnetism, the role of temperature T is played by either E or ?. This connec-
tion reveals that (i) spontaneous scalarization is a second-order phase transition and
292
(ii) the critical exponents characterizing this phase transition match the universal
values predicted by the Landau model. Point (i) was already demonstrated for NSs
in ST theories in Ref. [4], but point (ii) is new to this work; we elaborate on (ii)
below.
Critical exponents dictate how a system behaves close to a critical point (e.g.,
the location of a second-order phase transition). Such phenomena have first been dis-
covered in GR in the context of critical collapse [341?344], but also appear in pertur-
bations of extremal BHs [345, 346]. Applied to the current example of spontaneous
scalarization, we study how the structure of the BH solutions varies as we approach
the critical point at which spontaneous scalarization first occurs, parametrized by
? ? 0 where ? could be either ? = (? ? ?c)/?c at fixed E (identifying temperature
as T ? ?1/?) or ? = (E ?Ec)/Ec at fixed ? (identifying T ? 1/E). For example, the
critical exponent ? of a Landau model is given by the scaling of the order parameter
q ? ?? as ? ? 0+.
The effective potential V (q) in Eq. (6.5) depends on the properties of the
BH solution; this dependence is suppressed in the notation used in the previous
section, but here we explicitly restore it. In particular, close to the critical point,
the potential takes the form
c(2)(?) c(4)(?)
V (q; ?) = q2 + q4. (6.17)
2 4!
If c(2)(?) and c(4)(?) are analytic functions, they must take the following form near
? = 0,
( )
c(2)(?) = a ? +O ?2 , c(4)(?) = b+O (?) , (6.18)
293
0.150
0.125
0.100
0.075
0.050
0.025
0.01 0.02 0.05 0.10
Figure 6.3: Scalar charge q as a function of ? ? (???c)/?c for different
electric charges with coupling f1(?) [Eq. (6.2)]. The numerical solutions
are indicated with points, while the solid lines are best fits with slope
1/2. We see that the solutions agree well with the expected scaling
q ? ?1/2.
where a and b are positive const?ants. Then, th?e minima of V (q; ?) occur at
? ?6c(2) ? ?6aq = = ?1/2, (6.19)
c(4) b
thus, q ? ?1/2 as ? ? 0+. For this system, q represents an order parameter, and
thus the critical exponent ? is 1/2.
We numerically confirm this claim by computing the scalar charge q of elec-
trically charged BHs as a function of ? near the critical point. Fixing the electric
charge E , we first determine the critical coupling ?crit at which c(2) vanishes. We
then compute q for couplings just below this value, i.e., for ? = (???crit)/?crit & 0.
The dependence of q on ? is depicted in Fig. 6.3; we find that q ? ?1/2 agrees well
with our numerical results.
Similarly, we compute the other standard critical exponents describing the
294
phase transition. In particular, both the analytic model (6.17) and our numerical
solutions indicate that ? = 1 where ? = dq/d? ? |?|??0 for ? ? 0?, and ? = 3
1/?
where q ? ?0 at ? = 0. These findings are consistent with the Landau model for
phase transitions. It would be interesting to find a correspondence to a correlation
length in the future, so that all standard critical exponents can be studied. The
introduction of a correlation length (becoming infinite as c(2) ? 0) as another scale
next to the size of the compact object could also allow for a more formalized power
counting for the construction of the effective action close to the critical point.
6.4.3 Dynamical scalarization
We now employ our effective action to study the dynamical scalarization of
binary systems in EMS theory, only considering the scalar coupling f1(?) except
where noted. For this purpose, we integrate out the remaining IR fields from the
complete action (i.e., the field part, suitable gauge-fixing parts, and a copy of ScritCO
for each body).6 We employ a weak-field and slow-motion (i.e., post-Newtonian,
PN) approximation. These approximations are not independent here, since a wide
separation of the binary (weak field) implies slow motion due to the third Kepler law
for bound binaries. The leading order in this approximation is just the Newtonian
limit of the relativistic theory we are considering. Therefore, the Lagrangian of the
6Strictly speaking, the IR fields are split again into body-scale and radiation-scale parts, and
we integrate out the body-scale fields [315]. This would be necessary for a treatment of radiation
from the binary using effective-field-theory methods [347].
295
binary to leading order (LO) r[eads7 ] [ ]
LO ? y?
2
A y?
2
B cq?2,AL = m 2A 1? ?mB 1? + q?
2 2 2 A
(6.20)
cq?2,B 2 mAmB EAEB qAqB+ q?B + + + ,2 r r r
where A and B label the bodies, r = |yA?yB| is their separation, and in this section
the dot ? indicates a derivative with respect to coordinate time. We have suppressed
the dependence of mA on qA for brevity, but recall that the ?free energy? of each
body takes the form
mA(qA) = m(0),A + V (qA)
c(2),A 2 c(4),A= c 4(0),A + qA + qA. (6.21)2 4!
Notice that mA(qA) plays the role of the body?s mass in the binary Lagrangian
because (i) it couples to gravity like a mass, see Eq. (6.7), and (ii) it is independent
of the fields, so that for the purpose of integrating out the fields it can be treated
as a constant. The Hamiltonian for the binary can be obtained via a Legendre
transformation
p2 p2 p2 2LO A B q,A pq,BH = mA +mB + + + +
2mA 2mB 2cq?2,A 2cq?2,B (6.22)
? mAmB EAEB ? qAqB+ ,
r r r
with the pairs of canonical variables (yA/B,pA/B) and (qA/B, pq,A/B).
Following Ref. [4], let us now consider a special case that allows simple analytic
solutions for the scalar charges of the bodies. We henceforth assume that the scalar
charges evolve adiabatically pq,A/B ? 0 and that the two bodies are identical, i.e.,
7We have also gauge-fixed the worldline parameters to the coordinate time ? = t as usual in
the PN approximation.
296
q ? qA = qB, c(2) ? c(2),A = c(2),B, etc. The Hamiltonian in the center-of-mass
system p ? pA = ?pB now reads
2 2 2 2
HLO,adiab.
p ? m E ? q= 2m+ + . (6.23)
m r r r
Under these assumptions and recalling again that m = m(q), the equation of motion
for the scalar charge q is given by
?HLO,adiab. ( )? c(4) 2q0 p?q = = 2z c 3(2)q + q ? , (6.24)
?q 6 r
with the redshift
p2 m
z ? 1? ? . (6.25)
2m2 r
For simplicity, we neglect relativistic corrections to the redshift from here onward,
i.e., z ? 1; restoring these corrections does not affect the qualitative behavior that
we describe. Equation (6.24) has three solutions: an unscalarized solution with
q = 0 and a two scalarized solutions with nonzero q of opposite signs. The condition
for stability of these solutions is that they are located at a minimum of the energy
of the binary,
?2HLO,adiab.
0 ? ? 22c(2) ? + c(4)q2, (6.26)
?q2 r
which is violated for q?= 0 when 1/r > c(2). Hence, the stable solutions are given by???? 0 for 1/r ? c
q = ????
? ? (2) , (6.27)
? 6 1 ? c(2) for 1/r ? c(2)
c(4) r
which contain a phase transition at r = 1/c(2) corresponding to the spontaneous
breaking of the q ? ?q symmetry of the effective action. Recall that c(2) < 0
297
0 100 200 300
0.15
0.10
0.05
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Figure 6.4: Scalar charge Qtot of an equal mass binary as a func-
tion of orbital frequency ? or GW frequency fGW = ?/? for coupling
f1(?) [Eq. (6.2)] with ? = ?8.
corresponds to the case where each object is spontaneously scalarized, for which the
bottom condition in Eq. (6.27) always holds, and thus q 6= 0 over all separations.
Restricting our attention to circular orbits, we plot in Fig. 6.4 the total scalar
charge of the binary Qtot ? qA + qB as a function of orbital frequency, given by
Kepler?s law as ( )
2 1 qAqB EAEB? = (mA +mB) 1 + ? . (6.28)
r3 mAmB mAmB
For simplicity, we only show the positive scalar charge branch of solutions. The
frequency is shown both as the dimensionless combination M? with M ? (0) (0)mA +mB
and as the equivalent GW frequency fGW = ?/? for a (30M + 30M) binary
system. The plotted curves correspond to solutions with coupling constant ? = ?8,
and the colors correspond to different values of the electric charge. The scalar charge
vanishes below the onset of dynamical scalarization; the scalar charge grows abruptly
298
at some critical frequency ?scal (as evidenced by kinks in the plotted curves) because
dynamical scalarization is a second-order phase transition?see Ref. [4] for a more
detailed argument that dynamical scalarization is a phase transition.
In Fig. 6.5, we depict the scalarization of binary systems for various charge-to-
mass ratios and couplings ?. The solid lines indicate the critical frequency ?scal at
which dynamical scalarization begins; the heavily shaded regions above these lines
correspond to dynamically scalarized binaries after the onset of this transition. The
critical point c(2) = 0, corresponding to ?scal ? 0, represents the division between
binaries that dynamically scalarize and those whose component BHs (individually)
spontaneously scalarize; we depict all spontaneously scalarized configurations with a
lighter shading. Thus, we see that our effective action, which was matched to isolated
objects and models spontaneous scalarization, predicts dynamical scalarization as
well. Refined predictions can be obtained by perturbatively calculating the binary
Lagrangian to higher PN orders and also the emitted radiation; both are possible
using effective-field-theory techniques [315, 347, 348] or more traditional methods
where extended bodies are represented by point particles, e.g., Refs. [30, 210].
The same calculation can be repeated for binary systems with the choice
f(?) = f2(?). As before, the unscalarized q = 0 solution is stable for above
r = 1/c(2). However, because c(4) < 0, no stable dynamically scalarized branch
exists below that separation; instead, the system becomes ?dynamically? unstable
after this critical point. The phase diagram for this choice of coupling takes the
same form as Fig. 6.5, but here the shaded regions correspond to scenarios in which
no stable configuration exists. At the onset of instability, the scalar radiation will
299
0.3 300
0.2 200
0.1 100
0.0 0
0.3 300
0.2 200
0.1 100
0.0 0
0.3 300
0.2 200
0.1 100
0.0 0
0.2 0.4 0.6 0.8 1.0
Figure 6.5: Scalarization of binary systems with various electric charges
for scalar coupling f1(?) [Eq. (6.2)] with different coupling strengths ?.
Lightly and heavily shaded regions indicate spontaneously and dynam-
ically scalarized configurations, respectively. The solid lines depict the
onset of dynamical scalarization ?scal as a function of electric charge.
likely grow rapidly and the GW frequency will decrease more rapidly compared to
the EM case. However, long-term predictions are not possible with our effective
theory since the assumption of time-reversal symmetry likely breaks down, as for
the unstable isolated BHs.
6.5 Conclusions
In this work, we developed a simple energetic analysis of spontaneous and
dynamical scalarization, based on a strong-field-agnostic effective-field-theory ap-
proach (extendable beyond the scalar-field case [324, 325, 327] in the future). We
demonstrated our analysis for BHs with modified electrodynamics here, comple-
menting the study of NS in ST gravity from Ref. [4]. The theory-agnostic nature
300
of our approach allowed us to draw general conclusions about scalarization As an
example, we found that dynamical scalarization generically occurs in theories that
admit spontaneous scalarization. Specific examples of theories for which our findings
apply include those discussed in Refs. [154, 155, 161, 163?167, 324?327].
The recent discovery of spontaneous scalarization in ESTGB theories [163?
167] has sparked significant interest in this topic. Our work predicts that dynam-
ical scalarization can occur in binary systems in these theories and allows one to
straightforwardly estimate the orbital frequency at which it occurs using only in-
formation derivable from isolated BH solutions. Such information is valuable for
guiding numerical-relativity simulations in these theories, for which there has been
recent progress [349]. Eventually higher PN orders in specific theories could be
added to our model to derive more accurate predictions, and the framework could
be extended to massive scalars or other types of new fields.
We demonstrated how scalarization, as an exemplary strong-gravity modifica-
tion of GR, is parametrized by just a few constants in the effective action (which
vanish in GR). Hence, our effective action provides an ideal foundation for a strong-
field-agnostic framework for testing dynamical scalarization. Ultimately, one needs
to incorporate self-consistently such effects into gravitational waveform models. This
undertaking will require the computation of dissipative effects (analogous to the
standard PN treatment in GR) and the mode dynamics during scalarization, char-
acterized (in part) by cq?2 . The effective action can also, in principle, be extended to
include other strong-field effects influencing the inspiral of a binary, e.g., phenomena
like floating orbits [350, 351] or induced hair growth [314, 352].
301
Independent of GW tests of GR, further study of dynamical scalarization could
offer some insight into the nonlinear behavior of merging binary NSs in GR. As dis-
cussed in the Introduction, scalarization and tidal interactions enter models of the
inspiral dynamics in a similar manner; in fact, the effective action treatment of dy-
namical tides (in GR) [299] is completely analogous to the approach adopted here
for scalarization. Unlike the case with tides, our model of scalarization includes non-
linear interactions via the q4 term. Nonlinear tidal effects could be relevant for GW
observations of binary NSs [353, 354], but are difficult to handle in GR. Furthermore,
mode instabilities also occur for NSs in GR [355, 356]. Dynamical scalarization can
be used as a toy model for these types of effects, and further exploration of this
non-GR phenomenon could improve gravitational waveform modeling in GR.
Our effective action approach allowed us to study the critical phenomena at the
onset of scalarization; further study could also provide insight into critical phenom-
ena in GR. The critical exponents we obtained numerically agree with the analytic
predictions from Landau?s mean-field treatment of ferromagnetism [318, 340]. Only
missing here is a proper definition of correlation length, which we leave for future
work. In GR, the BH limit of compact objects has been suggested to play the role of
a critical point and lead to the quasiuniversal relations for NS properties [357, 358].
These quasiuniversal relations are invaluable for GW science because they reduce
the number of independent parameters needed to describe binary NSs, improving
the statistical uncertainty of measurements. In the BH limit (for nonrotating con-
figurations), the leading tidal parameter vanishes [307, 359, 360], like c(2) at the
critical point here. Better theoretical understanding of the origin of these universal
302
relations could help improve their accuracy; utilizing information from the critical
phenomena at the BH limit is a compelling idea, and scalarization could serve as a
toy model in that regard.
6.6 Acknowledgements
We thank Emanuele Berti, Caio Macedo, Ne?stor Ortiz, and Fethi Ramazanog?lu
for useful discussions.
303
Chapter 7: Constraining nonperturbative strong-field effects in scalar-
tensor gravity by combining pulsar timing and laser-interferometer
gravitational-wave detectors
Authors: Lijing Shao, Noah Sennett, Alessandra Buonanno, Michael Kramer,
and Norbert Wex 1
Abstract: Pulsar timing and laser-interferometer gravitational-wave (GW)
detectors are superb laboratories to study gravity theories in the strong-field regime.
Here we combine those tools to test the mono-scalar-tensor theory of Damour and
Esposito-Fare?se (DEF), which predicts nonperturbative scalarization phenomena for
neutron stars (NSs). First, applying Markov-chain Monte Carlo techniques, we use
the absence of dipolar radiation in the pulsar-timing observations of five binary
systems composed of a NS and a white dwarf, and eleven equations of state (EOSs)
for NSs, to derive the most stringent constraints on the two free parameters of the
DEF scalar-tensor theory. Since the binary-pulsar bounds depend on the NS mass
and the EOS, we find that current pulsar-timing observations leave scalarization
windows, i.e., regions of parameter space where scalarization can still be prominent.
Then, we investigate if these scalarization windows could be closed and if pulsar-
1Originally published as Phys. Rev. X7, 041025 (2017).
304
timing constraints could be improved by laser-interferometer GW detectors, when
spontaneous (or dynamical) scalarization sets in during the early (or late) stages
of a binary NS (BNS) evolution. For the early inspiral of a BNS carrying constant
scalar charge, we employ a Fisher matrix analysis to show that Advanced LIGO can
improve pulsar-timing constraints for some EOSs, and next-generation detectors,
such as the Cosmic Explorer and Einstein Telescope, will be able to improve those
bounds for all eleven EOSs. Using the late inspiral of a BNS, we estimate that
for some of the EOSs under consideration the onset of dynamical scalarization can
happen early enough to improve the constraints on the DEF parameters obtained
by combining the five binary pulsars. Thus, in the near future the complementarity
of pulsar timing and direct observations of GWs on the ground will be extremely
valuable in probing gravity theories in the strong-field regime.
7.1 Introduction
In general relativity (GR), gravity is mediated solely by a rank-2 tensor,
namely the spacetime metric g?? . Scalar-tensor theories of gravity, which add a
scalar component to the gravitational interaction, are popular alternatives to GR.
Though first proposed in 1921 [361], contemporary interest in these theories has been
spurred by their potential connection to inflation and dark energy, as well as possi-
ble unified theories of quantum gravity [87]. A modern framework for the class of
scalar-tensor theories we consider was developed in Refs. [97, 147, 178, 179, 300, 362]
(see also more generic Horndeski scalar-tensor theories in Ref. [90]).
305
Ultimately, the existence (or absence) of scalar degrees of freedom in grav-
ity will be decided by experiments. Most scalar-tensor theories are designed to
be metric theories of gravity, that is, they respect the Einstein equivalence princi-
ple [79, 97]. Therefore, precision tests of the weak-equivalence principle, the local
Lorentz invariance, and the local position invariance in flat spacetime are unable to
constrain them [79, 89, 97, 305]. However, such theories generally violate the strong-
equivalence principle. Tests of the strong-equivalence principle with self-gravitating
bodies provide an ideal window to experimentally search for (or rule out) the scalar
sector of gravity [79, 97, 363].
Particularly prominent violations of the strong-equivalence principle are known
to arise in the class of massless mono-scalar-tensor theories, studied by Damour
and Esposito-Fare?se in the form of nonperturbative strong-field effects in neutron
stars (NSs) [152, 153, 276]. In this work, we investigate the extent to which pulsar
timing and ground-based gravitational-wave (GW) observations can constrain these
phenomena (space-based GW experiments [266, 364, 365] are beyond the scope of
this work). Our results demonstrate that, depending on the parameters of binary
systems and NS equations of state (EOSs), these two types of experiments can
provide complementary bounds on scalar-tensor theories [79, 89, 171, 187, 305].
These results are especially timely as new instruments come online in the upcoming
years in both fields [366, 367].
The paper is organized as follows. In the next section, we briefly review
two nonperturbative phenomena, notably spontaneous scalarization [152, 153] and
dynamical scalarization [3, 169?172], that arise in certain scalar-tensor theories of
306
gravity. Then, in Sec. 7.3, we derive stringent constraints on these theories by com-
bining state-of-the-art pulsar observations of five NS-white dwarf (WD) systems. In
Sec. 7.4, we employ these constraints and investigate the potential detectability of
nonperturbative effects in binary NS (BNS) systems using the Advanced Laser Inter-
ferometer Gravitational-wave Observatory (LIGO) [63] and next-generation ground-
based detectors. Finally, in Sec. 7.5, we discuss the main results and implications
of our finding, and give perspectives for future observations.
7.2 Nonperturbative strong-field phenomena in scalar-tensor gravity
In this work, we focus on the class of mono-scalar-tensor theories that are
defined by the following action in the Einstein-frame [152, 153, 178, 362],
c4
?
d4x? [ ]
S = ?g [R ? 2g??? ?? ?? V (?)] + S ? ;A2(?)g?? ? ? ? m m
16?G c ? ??
, (7.1)
?
where G? is the bare gravitational coupling constant, g
?
?? is the Einstein metric
with its determinant g?, R? ? g??? R??? is the Ricci scalar, ?m collectively denotes
the matter content, and A(?) is the (conformal) coupling function that depends on
the scalar field, ?. Henceforth, for simplicity, we assume that the potential, V (?),
is a slowly varying function that changes on scales much larger than typical length
scales of the system that we consider, thus, we set V (?) = 0 in our calculation.
The field equations are derived with the least-action principle [97, 147] for g???
307
and ?,
( )
? 8?G? ? 1R ? ??? =2??????+ T?? ? T g?? , (7.2)c4 2
4?G
 ?g?? =? ?(?)T? , (7.3)
c4
with the energy-momentum tensor of matter fields, T ?? ? 2c (?g )?1/2 ?S /?g?? ? m ?? ,
and the field-dependent coupling strength between the scalar field and the trace of
the energy-momentum tensor of matter fields, ?(?) ? ? lnA(?)/??.
Following Damour and Esposito-Fare?se [147, 152], we consider a polynomial
form for lnA(?) up to quadratic order, that is A(?) = exp (? 20? /2), and denote
?0 ? ?(?0) = ?0?0 with ?0 the asymptotic value of ? at infinity. This particular
scalar-tensor theory (henceforth, DEF theory) is completely characterized by two
parameters (?0, ?0) and for systems dominated by strong-field gravity, such as NSs,
can give rise to potentially observable, nonperturbative physical phenomena [153,
169]. Weak-field Solar-system experiments, generally, only probe the ?0-dimension
or the combination ?0?
2
0 in the (?0, ?0) parameter space (see Refs. [79, 368] and
references therein). In GR, ?0 = ?0 = 0.
Using a perfect-fluid description of the energy-momentum tensor for NSs in the
Jordan frame, in 1993 Damour and Esposito-Fare?se derived the Tolman-Oppenheimer-
Volkoff (TOV) equations [153] for a NS in their scalar-tensor gravity theory. Inter-
estingly, they discovered a phase-transition phenomenon when ?0 . ?4, largely ir-
respective of the ?0 value (a nonzero ?0 tends to smooth the phase transition [152]).
The phenomenon was named spontaneous scalarization. With a suitable (?0, ?0),
the ?effective scalar coupling? that a NS develops, ?A ? ? lnmA/??0 (the baryonic
308
100
10-1
10-2
10-3
10-4
10-5
1.0 1.2 1.4 1.6 1.8 2.0 2.2
mA / M?
Figure 7.1: Illustration of spontaneous scalarization in the DEF gravity,
in comparison to individual binary-pulsar limits, for a NS with EOS SLy4
and |?0| = 10?5. The blue curves correspond to (from top to bottom)
?0 = ?4.5,?4.4,?4.3, and ?4.2; the grey curves in between differ in ?0
in steps of 0.01. We indicate with triangles the 90% CL upper limits
on the effective scalar coupling |?A| from the individual pulsars listed in
Table 7.1. We can clearly see a scalarization window at mA ? 1.7M.
mass of NS is fixed while taking the derivative), could be O(1) when the NS mass,
mA, is within a certain EOS-dependent range
2. For masses below and above this
range, the effective scalar coupling is much smaller 3. In Fig. 7.1 we show an exam-
ple of spontaneous scalarization for a NS with the realistic EOS SLy4, and compare
it to existing individual binary-pulsar constraints.
In general, if two compact bodies in a binary have effective scalar couplings, ?A
and ?B, they produce gravitational dipolar radiation ? (??)2, with ?? ? ?A??B,
2For black holes, the effective scalar coupling equals to zero. Therefore, the tests performed
with binary black holes [122] do not directly apply to the DEF theory.
3For sufficiently negative ?0 (. ?4.6), NSs do not de-scalarize before reaching their maximum
mass, i.e. spontaneous scalarization is found for all NSs above a certain critical mass, which depends
on the actual value of ?0 and the EOS [152, 153].
309
|?A |
which is at a lower post-Newtonian (PN) order than the canonical quadrupolar ra-
diation in GR [152] 4. In Ref. [276], Damour and Esposito-Fare?se for the first time
compared limits on the DEF gravity arising from Solar system and binary pulsar
experiments with expected limits from ground-based GW detectors like LIGO and
Virgo. The analysis in Ref. [276] is based on soft (by now excluded [248, 369]),
medium and stiff EOSs, and for the LIGO/Virgo experiment it assumes a BNS
merger with PSR B1913+16 like masses (1.44M and 1.39M), as well as a 1.4M-
10M NS-BH merger. Damour and Esposito-Fare?se come to the conclusion that
binary-pulsar experiments would generally be expected to put more stringent con-
straints on the parameters (?0, ?0) than ground-based detectors, such as LIGO and
Virgo. Since then, several analyses have followed [247?249, 370, 371], but typically
those studies did not probe a large set of NS masses and EOSs. Considering ad-
vances in our knowledge of NSs and more sensitive current and future ground-based
detectors, we revisit this study here. Quite interestingly, as pointed out in a first
study in Ref. [171], the constraints on the parameters (?0, ?0) from binary pulsars
depend quite crucially on the EOSs and the masses of the NSs, in particular in the
parameter space that allows for spontaneous scalarization. By taking into account
this dependence when setting bounds from pulsar timing, we shall find that current
4In this work, generally we denote with nPN the O(v2n/c2n) corrections to the leading Newto-
nian dynamics (equations of motion). Therefore, the gravitational dipolar radiation reaction is at
1.5 PN, and the quadrupolar radiation is at 2.5 PN. In the GW phasing, when there is no potential
confusion we sometimes refer to the quadrupolar (dipolar) radiation as 0 PN (?1 PN), as typically
done in the literature.
310
and future GW detectors on the Earth might still be able to exclude certain specific
regions of the parameter space (?0, ?0) that are not probed by binary pulsars yet.
Twenty years after the discovery of spontaneous scalarization, Barausse et
al. [169] found another interesting nonperturbative phenomenon in a certain pa-
rameter space of the DEF theory. This time the scalarization does not take place
for a NS in isolation, but for NSs in a merging binary. Indeed, modeling the BNS
evolution in numerical relativity, Barausse et al. found that the two NSs can scalar-
ize even if initially, at large separation, they are not scalarized. This phenomenon
is called dynamical scalarization, and its onset is determined by the binary com-
pactness instead of the NS compactness. Reference [169] also demonstrated that a
spontaneously scalarized NS can generate scalar hair on its initially unscalarized NS
companion in a binary system through a process known as induced scalarization.
Dynamical and induced scalarization cause BNSs to merge earlier [169, 171, 172]
than in GR, resulting in a significant modification to the GW phasing that is po-
tentially detectable by ground-based GW interferometers [3, 169, 170, 175].
Finally, it is important to note that cosmological solutions in the strictly mass-
less limit of the DEF theory are known to evolve away from GR when ?0 is neg-
ative [175, 272, 273, 302]; to be consistent with current Solar-system observations,
such cosmologies require significant fine tuning of initial conditions.5 Various mod-
ifications to the theory have been considered to cure this fine-tuning problem, for
example, by adding higher order polynomial terms to lnA(?) [302] or including a
5For cosmologies in the scalar-tensor theories with a positive ?0, we refer readers to Refs. [272,
273, 372].
311
mass term V (?) = 2m ?2? in the action [95, 152, 161, 303]. To date, none of these
proposals have produced scalar-tensor theories that: (i) satisfy cosmological and
weak-field gravity constraints, (ii) generate an asymptotic field ?0 that is stable
over timescales relevant to binary pulsars and GW sources, and (iii) give rise to the
nonperturbative phenomena present in DEF theory. As is commonly done in the
literature [3, 147, 152, 153, 169?172, 175], we will ignore these cosmological concerns
in this work and focus only on (massless) DEF theory.
7.3 Constraints from binary pulsars
Until now, binary pulsars have provided the most stringent limits to the DEF
theory [152, 247?249, 375]. These limits were usually obtained with individual pul-
sar systems and with representative EOSs [249] 6. Here, by contrast, we combine
observational results from multiple pulsar systems employing Markov-chain Monte
Carlo (MCMC) simulations [380]. In particular, we pick the five NS-WD bina-
ries that are the most constraining systems in testing spontaneous scalarization:
PSRs J0348+0432 [248], J1012+5307 [373], J1738+0333 [247], J1909?3744 [374],
and J2222?0137 [375]. We choose these five binaries basing on the binary nature
(namely, NS-WD binaries), the timing precision that has been achieved, and the NS
masses. These aspects are important to the study here, and see Refs. [89, 294, 305]
for more discussion. For convenience, we list the parameters of these binaries in
Table 7.1, and notice that it is the combination of their P? intb and the NS mass that
6An exception is Ref. [294], where, for individual binary pulsars, the most conservative limits
in the (?0, ?0) parameter space across different EOSs are presented.
312
Table 7.1: Binary parameters of the five NS-WD systems that we use to con-
strain the DEF theory [247, 248, 373?375]. The observed time derivatives of the
orbit period Pb are corrected using the latest Galactic potential of Ref. [376]. For
PSRs J0348+0432, J1012+5307 and J1738+0333, the mass ratios were obtained
combining radio timing and optical high-resolution spectroscopy, while the compan-
ion masses are determined from the Balmer lines of the WD spectra based on WD
models [248, 377?379]. For PSRs J1909?3744 and J2222?0137, the masses were
calculated from the Shapiro delay, where the range of the Shapiro delay gives di-
rectly the companion mass, and the pulsar mass is then being derived from the mass
function, using the shape of the observed Shapiro delay to determine the orbital in-
clination [374, 375]. The masses below are based on GR as the underlying gravity
theory. However, since the companion WD is a weakly self-gravitating body, they
are practically the same in the DEF theory (with a difference . 10?4). We give in
parentheses the standard 1-? errors in units of the least significant digit(s).
Pulsar J0348+0432 [248] J1012+5307 [373] J1738+0333 [247]
Orbital period, Pb (d) 0.102424062722(7) 0.60467271355(3) 0.3547907398724(13)
Eccentricity, e 2.6(9)? 10?6 1.2(3)? 10?6 3.4(11)? 10?7
Observed P?b, P?
obs ?1
b (fs s ) ?273(45) ?50(14) ?17.0(31)
Intrinsic P? , P? intb b (fs s
?1) ?274(45) ?5(9) ?27.72(64)
Mass ratio, q ? mp/mc 11.70(13) 10.5(5) 8.1(2)
Pulsar mass, mobsp (M) . . . . . . . . .
WD mass, mobs (M ) 0.1715+0.0045 +0.0073c  ?0.0030 0.174(7) 0.1817?0.0054
Pulsar J1909?3744 [374] J2222?0137 [375]
Orbital period, Pb (d) 1.533449474406(13) 2.44576454(18)
Eccentricity, e 1.14(10)? 10?7 0.00038096(4)
Observed P? , P? obsb b (fs s
?1) ?503(6) 200(90)
Intrinsic P? , P? int (fs s?1b b ) ?6(15) ?60(90)
Mass ratio, q ? mp/mc . . . . . .
Pulsar mass, mobsp (M) 1.540(27) 1.76(6)
WD mass, mobsc (M) 0.2130(24) 1.293(25)
313
AP3 H4 MS2 WFF1
AP4 MPA1 PAL1 WFF2
ENG MS0 SLy4
2.5
2.0
1.5
1.0
0.5
0
10 11 12 13 14 15
R [km]
Figure 7.2: The mass-radius relation of NSs (in GR) for the 11 EOSs that
are adopted in the study. The mass constraint (with 1-? uncertainty)
from PSR J0348+0432 [248] is depicted in grey. The color coding for
different EOSs is kept consistent for all figures in this work.
makes them particularly suitable for the test of spontaneous scalarization. We ob-
tain the limits using 11 different EOSs that have the maximum NS mass above
2M [381]. The names of these EOSs are AP3, AP4, ENG, H4, MPA1, MS0, MS2,
PAL1, SLy4, WFF1, and WFF2 (see Refs. [381, 382] for reviews). Figure 7.2 shows
the mass-radius relation of NSs in GR for these EOSs. As evidenced by the spread
of the curves in the figure, we believe that these EOSs are sufficient to cover the
different EOS-dependent properties of spontaneous scalarization, and at the same
time satisfy the two-solar-mass limit from pulsar-timing observations [248, 369].
Markov-chain Monte Carlo techniques allow us to preform parameter esti-
mation within the Bayesian framework. These methods provide the posterior dis-
tributions of the underlying parameters that are consistent with observations. In
Bayesian analysis, given data D and a hypothesis H (here, the DEF theory), the
314
M [M]
marginalized posterior distribution of (?0, ?0) is given by [380],?
|D H I P (D|?0, ?0,?,H, I)P (?0, ?0,?|H, I)P (?0, ?0 , , ) = D|H I d? , (7.4)P ( , )
where I is all other relevant prior background knowledge and ? collectively denotes
all other unknown parameters besides (?0, ?0), which are marginalized over to obtain
the marginalized posterior distributions for just (?0, ?0) [see below for more details].
In the above equation, given H and I, P (?0, ?0,?|H, I) is the prior on (?0, ?0,?),
P (D|?0, ?0,?,H, I) ? L is the likelihood, and P (D|H, I) is the model evidence.
As said, we use MCMC techniques to explore the posterior in Eq. (7.4). We discuss
below our choices for the priors and the likelihood function [see Eq. (7.9)]. We
assume that observations with different binary pulsars are independent.
We now explain how we employ the MCMC technique to get the posterior by
combining binary pulsar systems. Let us assume that N pulsars (N = 1, 2, 5, see
below) are used to constrain the (?0, ?0) parameter space. To obtain a complete
description of the gravitational dipolar radiation of these systems in the{DEF theor}y,
(i)
we need N + 2 free parameters in the MCMC runs, which are ? = ?0, ?0, ??c ,
(i)
where ??c (i = 1, ? ? ? , N) is the Jordan-frame central matter density of pulsar i 7.
As an initial value to the TOV solver, we also need the value of the scalar field in the
(i)
center of a NS, ?c , but the latter is fixed iteratively by requiring that all pulsars
(i) (i)
have a common asymptotic value of ?, ?0 ? ?0/?0. Given ??c and ?c for pulsar
7Actually, in the full calculation we need some other quantities, as well, for example, the orbital
period, Pb, and the orbital eccentricity, e. Those quantities are observationally very well determined
(see Table 7.1), thus we use their central values and find that our constraints on (?0, ?0) do not
change on a relevant scale when we take into account the errors on those quantities.
315
i, we integrate the modified TOV equations (see Eq. (7) in Ref. [153] or Eq. (3.6)
in Ref. [152]) with initial conditions given by Eq. (3.14) in Ref. [152]. During the
integration, we use tabulated data of EOSs, and linearly interpolate them in the
logarithmic space of the matter density, ??, the pressure, p?, and the number density,
n? [381]. Note that only one quantity among {??, p?, n?} is free, while the others are
determined by the EOS. The end products of the integration provide us, for each
(i) (i)
pulsar, the gravitational mass, mA , the baryonic mass, m?A , the NS radius, R
(i),
(i)
and the effective scalar coupling, ?A [152].
For the MCMC runs we use a uniform prior on log10 |?0| for |?0| ? [5 ?
10?6, 3.4 ? 10?3], where 3.4 ? 10?3 is the limit obtained from the Cassini space-
craft [180, 368]. We pick the parameter ?0 uniformly in the range [?5,?4], which
corresponds to a sufficiently large parameter space where the scalarization phenom-
ena can take place [153, 169]. During the exploration of the parameter space, we
restrict the values of (?0, ?0) to this rectangle region, as well, in order to avoid
overusing {comp}utational time in uninteresting regions. The initial central matter
(i)
densities, ??c , are picked around their GR values, but they are allowed to explore
a very large range in the simulations. As we discuss below, we perform convergence
tests and verify that when evolving the chains all parameters in ? quickly lose mem-
ory of their initial values.
During the MCMC runs, we evolve the N + 2 free parameters according to
an affine-invariant ensemble sampler, which was implemented in the emcee package
[383, 384] 8. At every step, we solve the N sets of modified TOV equations on
8http://dan.iel.fm/emcee
316
the fly, using for the companion masses of the binary pulsars the values listed in
Table 7.1 9.
Then, for the decay of the binary?s orbital period, which enters the likelihood
function [see Eq. (7.9)], we use the dipolar contribution from the scalar field and
the quadrupolar contribution from the tensor field as given by the following, well
known, formulae [147, 385],
( )
dipole ?2?G? 2? mpmcP?b = g(e) (?A ? ? 20) , (7.5)c3 Pb
5/3 ( m)p +mc5/3
P? quadb = ?
192?G? 2? mpmc
f(e) , (7.6)
5c5 Pb (mp +m
1/3
c)
with
( )
? e
2 ( )?5/2
g(e) (1 + 1? e
2
) , (7.7)2
? 73 2 37
( )
f(e) 1 + e + e4 1? e2 ?7/2 . (7.8)
24 96
We find that higher order terms, as well as the subdominant scalar quadrupolar
radiation, give negligible contributions to this study. Notice that in Eq. (7.5), we
have replaced the effective scalar coupling of the WD companion with the linear
9The masses in PSRs J0348+0432, J1012+5307, and J1738+0333 are based on a combination of
radio timing of the pulsars and optical spectroscopic observation of the WDs. The derivation of the
masses only depends on the well-understood WD atmosphere model in combination with gravity
at Newtonian order, and the mass ratio q, which is free of any explicit strong-field effects [368].
Therefore, even within the DEF theory, these masses are valid [247, 248]. For PSRs J1909?3744
and J2222?0137, the masses are derived from the range and shape of the Shapiro delay [152, 249].
Since for the weakly self-gravitating WD companion |?B | ? |?0|  1, these masses are practically
identical to the GR masses in Table 7.1.
317
matter-scalar coupling constant, since for a weakly self-gravitating WD ?A ' ?0 in
the ?0 range of interest.
10 Furthermore, we can approximate the bare gravitational
constant G? in the above equations with the Newtonian gravitational constant GN =
G 2?(1 + ?0), since |?0|  1 (e.g., from the Cassini spacecraft [180, 368]).
We construct the logarit?hm(ic likelihoo)d for (the MCMC runs as,?N
L ? ?1 ? 2 )
?
P? int th
2
b ? P?b mp/mc ? qln + ? , (7.9)
2 ?obs ?obs
i=1 P? qb
where for PSRs J1909[?( 3744 and )J2222?] 0137 we replace the second term in the2
squared brackets with m obs obsp ?mp /?m . In Eq. (7.9), the predicted orbital decayp
from {the theory i}s P? th ? P? dipole +P? quad, and ?obsb b b X is the observational uncertainty for
X ? P? intb , q,mp , as given in Table 7.1. Note that P? thb and mp implicitly depend
on (?0, ?0,?), through direct integration of TOV equations in the DEF theory.
For each EOS, we perform four separate MCMC runs:
(i) 1 pulsar: PSR J0348+0432 (J0348);
(ii) 1 pulsar: PSR J1738+0333 (J1738);
(iii) combining 2 pulsars: PSRs J0348+0432 and J1738+0333 (2PSRs);
(iv) combining 5 pulsars: PSRs J0348+0432, J1012+5307, J1738+0333, J1909?3744
and J2222?0137 (5PSRs).
We pick J0348 and J1738 due to their mass difference (2.01M and 1.46M respec-
tively), and their high timing precision (see Table 7.1), which leads to interesting
differences in the constraints on the DEF parameters, especially on ?0. For each run,
10In this context, see footnote ?d? in Ref. [370], concerning WDs and very large (positive) ?0.
318
68% CL
90% CL
5.0
EOS = SLy4 68% CL
90% CL
4.8
4.6
4.4
4.2
4.0
5.0 4.5 4.0 3.5 3.0 2.5
log10|?0|
Figure 7.3: The marginalized 2-d distribution of (log10 |?0|,??0) from
MCMC runs on the five pulsars listed in Table 7.1, for the EOS SLy4.
The marginalized 1-d distributions and the extraction of upper limits
are illustrated in upper and right panels.
we accumulate sufficient MCMC samples to guarantee the convergence of MCMC
runs. By using the Gelman-Rubin statistic [386], we find that, for each EOS, 200,000
samples for cases J0348 and J1738, and 400,000 samples for cases 2PSRs and 5PSRs,
are enough, respectively. We discard the first half chain points of these 44 runs
(4 cases ? 11 EOSs) as the burn-in phase [384, 387], while we use the remaining
samples to do inference on the parameters (?0, ?0).
As an example, we show in Fig. 7.3 the marginalized 2-d distribution in the
parameter space of (log10 |?0|,??0) for the case 5PSRs and the EOS SLy4. As
mentioned above, we distribute the initial values of log10 |?0| and ??0 uniformly in
the rectangle region of Fig. 7.3. We see that after MCMC simulations, the region
319
??0
Table 7.2: Limits on the parameters of the massless mono-scalar-tensor DEF the-
ory for different EOSs when applying the MCMC analysis to the five pulsars
J0348+0432, J1012+5307, J1738+0333, J1909?3744, and J2222?0137. Results at
68% and 90% CLs are listed. |? |maxA is the maximum effective scalar coupling that
a NS could still possess without violating the limits, and mmaxA is the corresponding
(gravitational) mass at this maximum effective scalar coupling (see Figure 7.5).
68% confidence level
EOS |? | ?? mmax/M |? |max0 0 A  A
AP3 6.5? 10?5 4.21 1.83 1.1? 10?3
AP4 5.5? 10?5 4.24 1.71 1.2? 10?3
ENG 6.0? 10?5 4.21 1.80 1.0? 10?3
H4 5.7? 10?5 4.24 1.91 1.3? 10?3
MPA1 5.7? 10?5 4.22 1.92 1.1? 10?3
MS0 7.7? 10?5 4.28 2.26 2.7? 10?3
MS2 7.9? 10?5 4.26 2.24 2.1? 10?3
PAL1 7.3? 10?5 4.21 1.99 1.2? 10?3
SLy4 5.2? 10?5 4.23 1.71 1.1? 10?3
WFF1 5.3? 10?5 4.21 1.58 9.1? 10?4
WFF2 5.5? 10?5 4.24 1.68 1.2? 10?3
90% confidence level
EOS |? | ?? mmax/M |? |max0 0 A  A
AP3 1.5? 10?4 4.29 1.85 6.9? 10?3
AP4 1.4? 10?4 4.31 1.73 1.0? 10?2
ENG 1.6? 10?4 4.30 1.81 8.2? 10?3
H4 1.7? 10?4 4.33 1.92 2.8? 10?2
MPA1 1.6? 10?4 4.30 1.93 8.4? 10?3
MS0 2.0? 10?4 4.38 2.26 1.0? 10?1
MS2 2.4? 10?4 4.36 2.26 8.0? 10?2
PAL1 2.0? 10?4 4.29 2.00 8.2? 10?3
SLy4 1.4? 10?4 4.33 1.72 2.2? 10?2
WFF1 1.3? 10?4 4.30 1.60 6.9? 10?3
WFF2 1.4? 10?4 4.32 1.70 1.4? 10?2
320
10-2
AP3
10-3 AP4
ENG
10-4 H4
MPA1
10-5 MS0
5.0 MS2
4.8 PAL1
4.6 SLy4
4.4 WFF1
4.2 WFF2
4.0
J0348 J1738 2PSRs 5PSRs
Figure 7.4: Marginalized upper limits on |?0| (upper) and ??0 (lower)
at 90% CL. These limits are obtained from PSRs J0348+0432 (J0348),
J1738+0333 (J1738), a combination of them (2PSRs), and a combina-
tion of PSRs J0348+0432, J1012+5307, J1738+0333, J1909?3744 and
J2222?0137 (5PSRs).
with large |?0| or large (negative) ?0 is no longer populated, and only a small corner
is consistent with the observational results of the five NS-WD binary pulsars.
Furthermore, we extract the upper limits of log10 |?0| and ??0 from their
marginalized 1-d distributions at 68% and 90% CLs. We summarize the upper limits
at 90% CL from all 44 runs in Fig. 7.4. It is interesting to observe the following
facts. First, for all EOSs, J1738 gives a more constraining limit on ?0 than J0348.
This result might be due to the fact that the ?obs of J1738 is about two orders of
P?b
magnitude smaller than that of J0348, thus giving a tighter limit on ?20 by roughly
the same order of magnitude. Second, the constraints on ?0 from J0348 and J1738
are extremely EOS-dependent. This should be a consequence of the masses of the
NSs, which are (in GR) 1.46M for J1738, and 2.01M for J0348. For EOSs that
favour spontaneous scalarization at around 1.4?1.5M, J1738 gives a better limit,
321
??0 |?0|
AP3 H4 MS2 WFF1
AP4 MPA1 PAL1 WFF2
ENG MS0 SLy4
10-1
10-2
10-3
10-4
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
mA/M?
Figure 7.5: The effective scalar coupling |?A| that an isolated NS could
still develop after taking into account the 95% CL constraints from the
five pulsars (see Table 7.2). The point of the maximum |?A| is marked
with a dot, and the values (and the corresponding masses) are listed in
Table 7.2.
while for EOSs that favour spontaneous scalarization at around 2M, J0348 gives
a better limit. This trend is also consistent with Fig. 7.5 (to be introduced below).
Third, by combining two pulsars (2PSRs), NSs are limited to scalarize at neither
1.4?1.5M nor ? 2M. Therefore, almost for all EOSs, ?0 is well constrained.
This result demonstrates the power of properly using multiple pulsars with different
NS masses to constrain the DEF parameter space for any EOS. Fourth, we obtain
the most stringent constraints with five pulsars (5PSRs). This is especially true for
?0, which is constrained at the level of ? ?4.2 (68% CL) and ? ?4.3 (90% CL)
for all EOSs. Finally, we list in Table 7.2 the marginalized 1-d limits for 5PSRs.
We shall use them in the next section when combining binary pulsars with laser-
interferometer GW observations.
322
|?A|
Considering the results that we have obtained when combining the five pulsars
(5PSRs), one could wonder whether isolated NSs can still be strongly scalarized. To
address this question, we use the limits on (?0, ?0) and calculate the effective scalar
coupling that a NS can still develop as a function of the NS mass, for the 11 EOSs
used in this work. The results at 90% CL are summarized in Fig. 7.5, while in
Table 7.2 we list the maximally allowed effective scalar couplings at 68% and 90%
CLs, and their corresponding (gravitational) NS masses (marked as dots in Fig. 7.5).
Figure 7.5 clearly shows the nonperturbative nature of the scalarization phe-
nomenon. The (absolute values of the) maximally allowed effective scalar coupling
for NSs can be as large as O(10?2) and even 0.1 if the limits at 90% CL are used,
while those values are . 10?3 if one uses the limits at 68% CL (not shown in Fig. 7.5,
but listed in Table 7.2). Furthermore, quite remarkably Fig. 7.5 shows that there
are scalarization windows (this feature could also be seen in Fig. 7.1 for the EOS
SLy4). What we mean is the following. The NS masses for the five most constrain-
ing pulsars are 1.46M (PSR J1738+0333) 1.54M (PSR J1909?3744), 1.76M
(PSR J2222?0137), 1.83M (PSR 1012+5307), and 2.01M (PSR J0348+0432).
For these specific masses, using the 11 EOSs that can give rise to spontaneous scalar-
ization, we have constrained stringently the DEF parameters. However, some EOSs
can still allow NS to scalarize strongly (i.e., acquire large effective scalar couplings)
for other values of the masses. As Fig. 7.5 shows, using limits at 90% CL, NSs with
EOSs AP4, SLy4, and WFF2 can still have |?A| & O(10?2) if the NS masses are
in the range mA = 1.70?1.73M, and NSs with EOS H4 can still be scalarized to
|?A| ? 0.03 with mA ' 1.92M, and NSs with EOSs MS0 and MS2 can still be
323
strongly scalarized to |?A| ' 0.1 with mA ' 2.26M. Those scalarization windows
could be closed in the future if binary pulsars with these masses are discovered and
their gravitational dipolar radiation is constrained by pulsar timing. As we shall
discuss in Sec. 7.4, the presence of scalarization windows also opens the interesting
possibility to close these gaps with future GW observations from BNSs, if the NS?s
masses lie in the scalarization window.
7.4 Projected sensitivities for laser-interferometer gravitational-wave
detectors
Having determined constraints on the DEF?s parameter space from binary
pulsars (see Table 7.2) and found scalarization windows, we now address the question
of whether present and future laser-interferometer GW observations on the ground
can still improve these limits and close the gaps. Two scenarios are considered:
(i) asymmetric BNS systems, equipped with separation-independent effective scalar
couplings, whose gravitational dipolar radiation during the inspiral can modify the
GW phasing [240, 276], and (ii) BNS systems that dynamically develop scalarization
during the late stage of the inspiral, leading to significant, nonperturbative changes
in the GW signal [3, 169, 171, 172]. A complete description of BNSs in the DEF
theory should include both effects. However, complete waveform models from the
theory are still not available, so here we investigate the two scenarios separately to
obtain some conservative understanding of the whole picture.
324
7.4.1 Dipole radiation for binary neutron-star inspirals
The presence of a scalar field can significantly modify the inspiral of an asym-
metric BNS system, due to the additional energy radiated off by the scalar degree
of freedom. The most prominent effect is a modification of the phase evolution in
GW signals. For two NSs with effective scalar couplings ?A and ?B respectively, one
finds for the evolution of the orbita[l frequency, ?, up to 2.5 PN]order [2, 28, 240],
?? ? 96
= (??)2V3 + ?V5 +O(V6) , (7.10)
?2 1 + ?A?B 5
where ?? ? ?A ? ?B, ? ? mAmB/M2, and the (dimensionless) ?characteristic?
velocity,
V ? [G? (1 + ? ? )M?]1/3A B /c . (7.11)
The quantity ? is given in Refs. [147, 248]. In GR one has ?A = ?B = 0 and ? = 1.
Note that there is also a subdominant contribution from the scalar quadrupolar
waves at 2.5 PN order, which however can be absorbed by a . 1% change in the
mass parameters. Here we assume that ?A and ?B are constant during the inspiral,
and their values are obtained from isolated NSs. This assumption is valid as long
as the induced or dynamical scalarization mechanisms are not triggered.
For an asymmetric compact binary where ?A 6= ?B, the most prominent devi-
ation from the GR phase evolution is determined by the dipole term in Eq. (7.10),
i.e., the contribution ? V3. To leading order, the offset in the number of GW cycles
in band until merger due to the dipole term is given by
25
?N ?1 ?7 2dipole ' ? ? Vin (??) , (7.12)21504?
325
Table 7.3: The number of GW cycles in GR, NGR, for a BNS merger with masses
(1.25M, 1.7M) for frequencies f > fin, and its change due to the dipole radiation
in the DEF?s theory, ?Ndipole, assuming |??| ? |?A ? ?B| ' 0.0199, which comes
from the maximally allowed effective scalar couplings for the EOS SLy4 at 90% CL
(see Figure 7.5). The limits on the contributions from leading-order spin-orbit and
spin-spin terms, |?N?| and |?N?|, are listed where the (dimensionless) spins of the
double pulsar (when it merges in 86 Myr) are used. For |?N?| and |?N?|, we also
give in parentheses when both NSs are spinning at the maximal spin that we have
ever observed (in an eclipsing binary pulsar J1748?2446ad).
Detector fin (Hz) NGR ?Ndipole |?N?| |?N?|
aLIGO 10 1.5? 104 ?3.7? 101 < 0.76 (< 3.5? 101) < 1.8? 10?6 (< 0.43)
CE 5 4.8? 104 ?1.9? 102 < 1.2 (< 5.6? 101) < 2.3? 10?6 (< 0.55)
ET 1 7.0? 105 ?8.1? 103 < 3.5 (< 1.6? 102) < 3.9? 10?6 (< 0.93)
where Vin corresponds to V in Eq. (7.11) when the merging system enters the
band of the GW detector, i.e., when ? = ?fin (see Refs. [240, 276] for details).
In the above equation we have used the approximation ? ' 1 and the fact that
Vin is much smaller than V just before merger. Within the approximation of
Eq. (7.12) one can use Vin ' (GNM?f 1/3in) /c, i.e., replacing the effective gravi-
tational constant G? (1 + ?A?B) in Eq. (7.11) by the Newtonian gravitational con-
stant GN ? G 2? (1 + ?0) [147, 152]. Again, we stress that Eq. (7.12) is based on the
assumption that the effective scalar couplings of the two NSs, ?A and ?B, remain
unchanged during the inspiral in the detector?s sensitive frequency band. It there-
fore neglects the phenomenon of induced scalarization, which can occur in a BNS
system, when the unscalarized NS is sufficiently exposed to the scalar field of the
scalarized companion [170]. This can reduce the dipolar radiation considerably on
short ranges if ?A approaches ?B, and lead to a characteristic change in the late
phase evolution of the merging BNSs. Studies on the dynamically changing effective
scalar couplings are performed in the next subsection.
326
To obtain a rough understanding of the effects of dipolar radiation, let us
calculate the dephasing from GR by an asymmetric BNS inspiral withmA = 1.25M
and mB = 1.7M. According to Fig. 7.5, at present, binary-pulsar experiments
cannot exclude |?A| as large as 10?2?10?1 for NSs of a certain mass range, which
depends on the EOS. For the EOS SLy4 we find from the corresponding (dark green)
curve in Fig. 7.5 |?A| ' 0.0007 and |?B| ' 0.0206, hence |??| ? |?A??B| ' 0.0199.
In our study we consider the Advanced LIGO (aLIGO) detectors at design
sensitivity [63], and future ground-based detectors, such as the Cosmic Explorer
(CE), and the Einstein Telescope (ET). We use the starting frequencies, fin = 10 Hz
for aLIGO, fin = 5 Hz for CE, and fin = 1 Hz for ET [367, 388, 389]. In Table 7.3
we list the number of GW cycles as predicted by GR, NGR, and the change in
the number of cycles caused by the existence of a dipole radiation for a BNS signal,
?Ndipole. From Table 7.3 we can already see that, given the BNS parameters, current
bounds by pulsars still leave room for significant time-domain phasing modifications
in BNS mergers, in particular if one of the NSs falls into the scalarization window of
? 1.7M (for EOSs AP4, SLy4, and WFF2) to ? 1.9M (for the EOS H4), or if one
NS?s mass significantly exceeds 2M (for EOSs MS0 and MS2). As reference points,
we list in Table 7.3 also the changes in the number of GW cycles from spin-orbit and
spin-spin effects. Indeed, from the leading-order spin-orbit (1.5 PN) and spin-spin
(2 PN) contributions to the GW phasing [259, 364], one has
?N ' 5 ??1 ?2? Vin ? , (7.13)64?
N '? 5? ??1? V?1
32? in
? , (7.14)
327
where
? ( )1 m2
? = 113 i + 75? L? ? ?i , (7.15)
12 [ M
2
i=A,B
? ( )( )]
? = ?247?A ? ?B + 721 L? ? ?A L? ? ?B . (7.16)
48
The (dimensionless) spins of a BNS system, ?A and ?B, are likely to be small in
magnitude. The parameters ? and ? are maximized when two spins are aligned
with the direction of the orbital angular momentum, L?. The limits on |?N?| and
|?N?| are listed in Table 7.3 where we have used the (dimensionless) spins of the
double pulsar system that is the only double NS system where two spins are precisely
measured. When the double pulsar evolves to the time of its merger in 86 Myr from
now, one has |?A| ' 0.014 and |?B| ' 0.00002 [390], assuming a canonical moment
of inertia 1038 kg m2 for NSs. As we can see from Table 7.3, if the spins of BNSs
to be discovered by GW detectors are comparable to that of the double pulsar, the
inclusion of spins only affects the number of GW cycles at percentage level at most.
In addition, because ground-based detectors could observe BNSs from a population
different from the one observed with pulsar timing, we also give |?N?| and |?N?| in
Table 7.3 when the (dimensionless) spin of the fastest rotating pulsar ever observed,
PSR J1748?2446ad (P = 1.4 ms) [391], is used for both NSs 11. Even in this extreme
case with |?A| = |?B| ' 0.26 (assuming a canonical mass 1.4M), |?Ndipole| is still
larger (or comparable in the case of the Advanced LIGO) than the upper limits of
|?N?| and |?N?|.
11Notice that PSR J1748?2446ad is not in a double NS binary. Its companion is probably a
bloated main-sequence star that recycles the pulsar to a large spin [391].
328
The dephasing quantity, ?Ndipole, is nevertheless a crude indicator for realis-
tic detectability. In reality, one has to consider various degeneracy between binary
parameters, the waveform templates that are used for detection and parameter es-
timation, the power spectral density (PSD) of noises in GW detectors, Sn(f), the
signal-to-noise ratio (SNR) of an event ?, and so on. In order to obtain more
quantitative estimates of the constraints on dipolar radiation that can be expected
from GW detectors, one would need to compute Bayes factors between two alterna-
tives [392] or apply cutting-edge parameter-estimation techniques, for example the
MCMC or nested sampling [78]. However, given the limited scope of our analysis, for
simplicity, here, we adopt the Fisher-matrix approach [240, 266, 393, 394], although
we are aware of the fact that for events with mild SNR (? ? 10), the Fisher matrix
can have pitfalls [395]. In Appendix J we review the main Fisher-matrix tools that
we use. ?
We summarize in Fig. 7.6 their dimensionless noise spectral density fSn(f)
[367, 388, 389], and show in the figure also an hypothetical BNS signal. For all the
studies, we fix the luminosity distance to DL = 200 Mpc for aLIGO, CE, and ET.
Indeed, within such a distance, aLIGO alone is supposed to observe 0.2?200 BNS
events annually at design sensitivity [229]. With the four-site network incorporating
LIGO-India at design sensitivity, the number of detectable BNS events will dou-
ble [229]. Therefore, it is a realistic setting to discuss BNS events for aLIGO; to
be conservative, we only consider a two-detector network for aLIGO in our study.
CE and ET have better sensitivities, thus will have larger SNRs for these events;
besides, they will be able to detect a larger number of BNSs, including those with
329
10-17 aLIGO CE ET
10-19
10-21
10-23
10-25
100 101 102 103 104
Frequency [Hz]
?
Figure 7.6: Dimensionless noise spectral density fSn(f) of aLIGO,
CE and ET GW detectors. The quantity Sn(f?) is the one-side de-sign PSD [367, 388, 389]. The dashed line describ?es the?? (dimensionless)
pattern-averaged characteristic strain hc(f) ? 2f ?h?(f)? for a BNS with
rest-frame masses (1.25M, 1.63M) at 200 Mpc (z ' 0.0438), up to
the innermost-stable circular orbit given by Eq. (J.3).
unfavourable orientations. Using the standard cosmological model [396], the red-
shift associated to DL = 200 Mpc is z ' 0.0438, and we take it into account in our
Fisher-matrix calculation, even if it generates a small effect. Moreover, we always
report masses in the rest frame of a BNS system.
The Fisher matrix is constructed as usual from the Fourier-domain waveform
h?(f) [240, 393, 394], ( ? )
?ab ? ??ah?(f) ??bh?(f) , (7.17)
with ?ah?(f) being the partial derivative to the parameter labeled ?a? (see Ap-
{pendix J for definitions and n}otations). We use the waveform parameters
lnA, ln ?, lnM, tc,?c, (??)2 to construct the 6 ? 6 Fisher matrix, ?ab. The in-
verse of the Fisher matrix is the correlation matrix for these parameters, from where
330
?
hc(f) or fSn(f)
AP3 H4 MS2 WFF1
AP4 MPA1 PAL1 WFF2
ENG MS0 SLy4
10-1
aLIGO : 10Hz
10-2
CE : 5Hz
10-3
ET : 1Hz
10-4
1.3 1.6 1.9 2.2 2.5 2.8
mB/M?
Figure 7.7: The sensitivities of aLIGO, CE, and ET to |??| (namely
the uncertainty, ? (|??|), obtained from the inverse Fisher matrix) are
depicted with dashed lines, as a function of mB, for a pattern-averaged
BNS inspiral signal with rest-frame component masses (mA = 1.25M,
mB). The starting frequencies of GW detectors are labeled. Luminosity
distance DL = 200 Mpc is assumed. The sensitivity to |??| from GW
detectors scales with SNR as ??1/2. The maximum available values of
|??| for 11 EOSs, saturating the limits from binary pulsars at 90% CL,
are shown in solid lines. If a sensitivity curve (dashed) is below a solid
curve, the corresponding GW detector has the potential to improve the
limit from binary pulsars for this particular EOS, with BNSs of suitable
masses.
331
max|??|  or  ?(|??|)
Advanced LIGO Cosmic Explorer Einstein Telescope 2.0
lnM 1.6
1.2
ln? 0.8
A 0.4ln
0.0
tc 0.4
0.8
?c
1.2
|??|2 1.6
2.0
Figure 7.8: The correlations between six parameters, obtained
from the inverse Fisher matrix in the matched-filter analysis for
a BNS with rest-frame masses (1.25M, 1.63M). F (?cor) ?
log10 [(1 + ?cor) / (1? ?cor)]??cor log10 2 is a function defined in Ref. [363]
such that it counts 9?s in the limit of large correlations [e.g., F (0.99) '
+2, F (?0.9) ' ?1, and F (0) = 0]. On diagonal, F (?cor = 1) diverges
and is plotted in black.
we can read their uncertainties and correlations [240, 266, 393, 394].
In Fig. 7.7 we plot in dashed lines the uncertainties in |??| obtained with
three GW detectors (aLIGO, CE, and ET) for an asymmetric BNS with rest-frame
masses mA = 1.25M and mB > 1.25M, located at DL = 200 Mpc. For a BNS
of masses, for example, (1.25M, 1.63M) which are the most probable masses for
the newly discovered asymmetric double-NS binary pulsar PSR J1913+1102 [397],
we find that aLIGO, CE, and ET can detect its merger at 200 Mpc with ? = 10.6,
450, and 153, respectively, after averaging over pattern functions and assuming two
detectors in each case. The characteristic strain of such a BNS is illustrated in the
frequency domain in Fig. 7.6. In the large SNR limit, the uncertainties in |??| scale
with the SNR as ??1/2. In Fig. 7.8, we give the correlations between parameters
obtained from the matched-filter analysis. We find that due to its low-frequency
332
lnM
ln?
lnA
tc
?c
|??|2
lnM
ln?
lnA
tc
?c
|??|2
lnM
ln?
lnA
tc
?c
|??|2
F(?cor)
sensitivity ET can break some degeneracy between parameters better than aLIGO
and CE do.
In Fig. 7.7, we show with solid lines the maximum values of |??| at 90% CL
from pulsars for 11 EOSs (calculated from Fig. 7.5). If for some NS?s mass range a
solid line (which is associated to a certain EOS) is above a dashed line (associated
to a certain detector), then for NSs described by that EOS, the corresponding GW
detector has potential to further improve the DEF?s parameters with the observation
of a BNS within that mass range. From the figure we can see that, with the expected
design sensitivity curves of aLIGO, CE, and ET [367, 388, 389],
? aLIGO has potential to further improve the current limits from binary pulsars
with a discovery of a BNS of suitable masses, if the EOS of NSs is one of (or
similar to) H4, MS0, MS2, SLy4, and WFF2;
? CE and ET, due to their low-frequency sensitivity and better PSD curves, are
able to significantly improve current limits from binary pulsars on the DEF?s
parameters, no matter what the real EOS of NSs is.
We stress that those conclusions are obtained with a Fisher-matrix analysis, and
should be made more robust in the future using more sophisticated tools, notably
Bayesian analysis.
Constraints outside the spontaneous-scalarization regime
With the results above it is fairly straightforward to calculate the limits from
aLIGO, CE and ET on |?0| when ?0 is outside the spontaneous scalarization regime,
333
i.e., ?0 & ?4, and compare them to existing limits from the Solar system and
pulsars [276]. For completeness, we present the relevant results here. Shibata et al.
[171] have shown that for small ?0 there exists a simple relation between ?A, ?0 and
mA as long as spontaneous scalarization does not set in (see Eq. (44) in Ref. [171]),
which in our notation reads 12
?A ' A(A)? (mA, ?0; EOS)?0 . (7.18)0
With this equation at hand one can directly convert the limits from ground-based
GW detectors of Fig. 7.7, for any give?n ?0 & ?4, i?nto limits for |?0| via?? ?? ??|?0| = ?? ? . (7.19)A(A) ?A(B) ??0 ?0
Figure 7.9 gives the results for two different mass configurations and the EOS AP4.
A more stiff EOS would generally lead to less constraining limits for ground-based
GW detectors and binary pulsars. As one can see, in the range ?0 & ?4 current Solar
system and pulsar tests are already clearly more constraining than what aLIGO is
expected to obtain. For CE and ET, only inspirals with a very massive component
will provide constraints that are better than present limits, for a limited range of ?0
(see also aLIGO [398] and ET [398, 399] limits from a NS-BH inspiral for the special
case of ?0 = 0, i.e. the Jordan-Fierz-Brans-Dicke gravity). By the time CE or ET is
operational, however, the expected limits from GAIA [400] and SKA [89] will have
left little room for ground-based GW observatories in the regime. The space-based
12 (A)In principle, there is still a weak dependence on ?0 in A? . However, this dependence becomes0
very small for |?0| . 10?2, as it scales with ?20. Therefore the ?0-dependence is absolutely negligible
for the parameter space explored here.
334
?1.5
PSR J1738+0333
?2.0
Cassini
?2.5
GAIA
?3.0
aLIGO: (1.25M, 1.6M) aLIGO: (1.25M, 2.0M)
CE: (1.25M, 1.6M) CE: (1.25M, 2.0M)
ET: (1.25M, 1.6M) ET: (1.25M, 2.0M)
?3.5?4 ?2 0 2 4 6 8 10
?0
Figure 7.9: Upper limits on |?0| as a function of ?0 for aLIGO (green),
CE (blue), and ET (orange) [276]. The dashed lines correspond to
a 1.25M/1.6M BNS merger, and the dotted lines correspond to a
1.25M/2.0M BNS merger. The chosen EOS is AP4. For compar-
ison we have plotted Solar system limits (grey) and the limits from
PSR J1738+0333 (magenta), which currently gives the best limit for
?0 & 3. The limit from Cassini [180] and the limit expected from
GAIA [400] are also shown.
GW observatories LISA [266, 364] and DECIGO/BBO [365] could in principle also
provide limits on the DEF theory from an inspiral of a NS into an intermediate mass
black hole, provided such BHs exist. However, the resulting limits on |?0| are not
expected to be better than limits from future ground-based GW observatories [266].
It is worthy to mention that for very large (positive) ?0, say, ?
2 3
0 & 10 ?10 , massive
NSs might develop instabilities [156, 304], which is beyond the scope of Fig. 7.9.
335
log10 |?0|
7.4.2 Dynamical scalarization
In addition to the nonlinear gravitational self-interaction testable with binary
pulsars, GW detectors probe the nonlinear interactions between coalescing NSs. Dy-
namical scalarization stems from the interplay between these two regimes of strong
gravity and thus offers a promising means of complementing pulsar timing con-
straints on scalar-tensor theories.
Numerical relativity simulations have demonstrated that dynamical scalariza-
tion can significantly alter the late-time behavior of a BNS system. If this transition
occurs before merger, the sudden growth of effective scalar couplings impacts the
system?s gravitational binding energy and energy flux so as to shorten the time to
merger [169, 171, 172].
The prospective detectability of this effect was investigated in Refs. [175, 271]
using Bayesian model selection. The authors sought to recover injected inspiral
waveforms containing dynamical scalarization with template banks constructed from
similar waveforms. The injected signals and template banks used PN waveforms
augmented with various non-analytic models of dynamical scalarization. To mimic
the abrupt activation of the dipole emission at the onset of dynamical scalarization,
Ref. [271] added a ?1 PN correction modulated by a Heaviside function to a GR
waveform, i.e., signals of the form
h?(f) = h? (f)ei??1PN(f)?(f?f?)GR , (7.20)
where ??1 PN(f) = bf
?7/3, and b and f? are parameters of the model. Injected
336
Table 7.4: Frequency fDS at which dynamical scalarization occurs for various equal-
mass binaries, given in Hz. Results are given for theories that saturate the con-
straints given in Table 7.2 at 68% and 90% CLs. Binary systems are specified
by their component NS masses, given in units of M. We highlight systems that
scalarize at frequencies below 50 Hz with boldface.
68% confidence level
EOS 1.3?1.3 1.5?1.5 1.7?1.7 1.9?1.9
AP3 838 354 123 84
AP4 577 183 57 199
ENG 858 358 118 102
H4 1301 650 235 51
MPA1 955 436 162 67
MS0 1503 854 422 165
MS2 1471 843 426 177
PAL1 1350 693 287 95
SLy4 674 217 66 356
WFF1 386 118 128 841
WFF2 519 154 57 302
90% confidence level
EOS 1.3?1.3 1.5?1.5 1.7?1.7 1.9?1.9
AP3 694 246 50 20
AP4 461 105 8 109
ENG 694 236 39 24
H4 1131 513 131 <1
MPA1 809 325 84 11
MS0 1320 700 302 81
MS2 1290 687 306 88
PAL1 1190 570 193 32
SLy4 508 106 <1 197
WFF1 251 33 35 608
WFF2 391 72 <1 181
337
signals were recovered with a template bank of waveforms of the same form. In
Ref. [175], the authors injected waveforms constructed in Ref. [170] by integrating
the 2.5 PN equations of motion combined with a semi-analytic model of scalarization,
then performed parameter estimation using both templates that included?1 PN and
0 PN scalar-tensor effects throughout the entire inspiral and those that modeled their
sudden activation as in Ref. [271]
Combined, these analyses provide a loose criterion for whether a dynamically
scalarizing BNS system could be distinguished from the corresponding system in
GR by aLIGO. The key characteristic of such systems is the frequency fDS at which
dynamical scalarization occurs. To be distinguishable from a GR waveform, a sig-
nificant portion of the dynamically scalarized signal?s SNR must occur after fDS,
or equivalently, fDS must be sufficiently lower than the merger frequency. Using
waveforms of the form of Eq. (7.20), the authors found in Ref. [271] that dynamical
scalarization can only be observed with aLIGO if fDS . 50?100 Hz. In only one
injection considered in Ref. [175] was dynamical scalarization detectable, occurring
at fDS ? 80 Hz. Understandably, these analyses rely on some initial assumptions
that may bias these estimates away from the real detectability criteria, such as the
limited range of masses and EOS considered and ignoring any degeneracies intro-
duced by the merger and ringdown portions of the waveform or by the inclusion of
spins. Ignoring these subtleties for the moment, we investigate whether the pulsar-
timing constraints described in Sec. 7.3 can exclude the possibility of observing
dynamical scalarization with aLIGO using the conservative detectability criterion
from Refs. [175, 271] that scalarization must occur by fDS . 50 Hz.
338
We consider binary systems composed of NSs with masses ranging from 1.3M
to 1.9M. We compute fDS within the ?post-Dickean? (PD) framework, a resum-
mation of the PN expansion formulated in Ref. [3]. This model introduces new
dynamical degrees of freedom that capture the nonperturbative growth of the scalar
field using a semi-analytic feedback loop. This approach provides a mathematically
consistent backing to previous models of dynamical scalarization [170]. The model
incor(porates a c)ertain flexibility in the choice of resummed quantities; we adopt
the m(RE), F (??) scheme outlined in Table I of Ref. [3] because it was found to
give the best agreement with numerical computations of quasi-equilibrium config-
urations [172]. For clarity, we dress quantities defined in the PD framework with
tildes and leave quantities defined in the PN framework unadorned; in the limit that
no resummation is performed, the PD quantities reduce to their PN analogs.
Within the PD framework, the effective scalar coupling of each NS is promoted
to a function of both the asymptotic scalar field ?0 and the local scalar field in
which the body is immersed, i.e. ??A = ??A(?0, ?A). Unlike in the PN treatment,
this coupling evolves as the BNS coalesces. Similarly, the inertial mass of each body
m?A(?0, ?A) evolves in the PD framework. However this mass varies by no more
than 0.01%, so in practice, one can simply use the PN mass mA in place of m?A.
We define the mass-averaged scalar coupling of the system as
? m?A??A + m?B??B?? , (7.21)
m?A + m?B
where precise definitions of m?A and ??A are given in Eqs. (A3) and (A4) in Ref. [3].
Note that for equal-mass binaries, we have ??A = ??B = ??.
339
fGW [Hz] fGW [Hz]
0 300 600 900 1200 1500 0 300 600 900 1200
100
10?1
10?2
AP3
?3 (1.3M, 1.3M) (1.5M, 1.5M) AP410
ENG
0 0.02 0.04 0.06 0 0.02 0.04 0.06 H4
GM?/c3 GM?/c3 MPA1
MS0
f [Hz] f MS2GW GW [Hz]
PAL1
0 300 600 900 1200 0 200 400 600 800 1000 SLy4
100 WFF1
WFF2
10?1
10?2
10?3 (1.7M, 1.7M) (1.9M, 1.9M)
0 0.02 0.04 0.06 0 0.02 0.04 0.06
GM?/c3 GM?/c3
Figure 7.10: Mass-averaged scalar coupling as a function of orbital angu-
lar frequency for equal-mass BNS systems with masses (1.3M, 1.3M),
(1.5M, 1.5M), (1.7M, 1.7M), and (1.9M, 1.9M). We use the
limits on (?0, ?0) at 90% CLs, given in Table 7.2, for each EOS. The cor-
responding GW frequency is given along the top axis, with fGW = ?/?.
Dashed vertical lines highlight the conservative detectability criterion for
aLIGO that fDS . 50 Hz, derived in Refs. [175, 271].
340
|??| |??|
Table 7.5: Frequency fDS (in Hz) at which dynamical scalarization occurs for various
unequal-mass binaries with the EOS MPA1. Results are given for theories that
saturate the constraints given in Table 7.2 at 68% and 90% CLs. Binary systems
are specified by their component NS masses, given in units of M. We highlight
systems that scalarize at frequencies below 50 Hz with boldface.
68% confidence level
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
1.2 1340 1130 942 767 611 471 364 304
1.3 . . . 955 795 649 515 399 306 258
1.4 . . . . . . 661 538 427 329 253 212
1.5 . . . . . . . . . 436 347 268 206 172
1.6 . . . . . . . . . . . . 274 212 163 136
1.7 . . . . . . . . . . . . . . . 162 126 105
1.8 . . . . . . . . . . . . . . . . . . 96 80
1.9 . . . . . . . . . . . . . . . . . . . . . 67
90% confidence level
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
1.2 1160 973 788 618 458 315 194 119
1.3 . . . 809 653 511 380 262 158 99
1.4 . . . . . . 529 415 305 210 129 79
1.5 . . . . . . . . . 325 240 165 101 62
1.6 . . . . . . . . . . . . 178 120 73 46
1.7 . . . . . . . . . . . . . . . 84 51 31
1.8 . . . . . . . . . . . . . . . . . . 32 19
1.9 . . . . . . . . . . . . . . . . . . . . . 11
341
Following the work of Ref. [3], we compute the mass-averaged scalar coupling
as a function of frequency for binaries on quasi-circular orbits to 1 PD order. The av-
erage scalar coupling is plotted in Fig. 7.10 for equal-mass binaries for theories that
saturate the pulsar-timing constraints at 90% CLs, as given in Table 7.2. Scalariza-
tion occurs earlier for larger mass systems, with an ordering (by EOS) determined
by the magnitude of ( )
d?A
?A = . (7.22)
d? ?=?0
To compute this quantity, one takes the difference in effective scalar couplings of
NSs (of equal baryonic mass) with infinitesimally different asymptotic scalar fields
?0; however, for non-spontaneously scalarized stars, ?A is given approximately by
?0|?A|
?A ? | | , (7.23)?0
provided that |?A| is sufficiently small. Binaries with spontaneously scalarized stars
begin with an appreciable effective scalar coupling at large separations that continues
to grow as they coalesce. In light of this remark, we note that there is no obser-
vational distinction between spontaneous (or induced) scalarization and dynamical
scalarization that occurs at sufficiently low frequencies; for example, compare the
scalarization of (1.7M, 1.7M) systems composed of NSs with the EOSs SLy4,
AP4, and WFF1 (the dark green, red, and beige curves in the lower left panel of
Fig. 7.10, respectively).
The sharp feature for the WFF1 EOS in the (1.9M, 1.9M) system occurs
because of the relatively low mass at which spontaneous scalarization occurs for
this particular EOS. We provide a more detailed analysis of this phenomenon in
342
Appendix K. Similarly abrupt transitions occur for other EOSs in more massive
binary systems with individual masses & 2M.
We adopt the method introduced in Ref. [172] to extract fDS. The average
effective scalar coupling can be closely fit by the piecewise function
( )
2 10/31 + ?? = 1 + a1 (x? xDS) ? (x? xDS) (7.24)
where ? is the Heaviside function, a1 and xDS are fitting parameters, and x ?
(G?M?/c
3)2/3. In practice, we identify xDS with the peak in the second derivative
of the lefthand side of Eq. (7.24) with respect to x. The gravitational wave frequency
at which dynamical scalarization occurs is then given by fDS = ?DS/?. In Ref. [3],
the PD prediction was found to reproduce numerical-relativity results to within an
error of . 10% with this fitting procedure.
The dynamical scalarization frequencies for the configurations considered in
Fig. 7.10 are given in Table 7.4 for theories constrained at the 68% and 90% CLs.
Systems containing spontaneously scalarized stars (i.e., those with appreciable ef-
fective scalar coupling even in isolation) are demarcated as scalarizing below 1 Hz;
as noted above, these systems would be indistinguishable to GW detectors from
those that dynamically scalarize below 1 Hz. For clarity, we highlight the systems in
Table 7.4 that scalarize (dynamically or spontaneously) below 50 Hz. Recall that,
under our definition, induced scalarization occurs in binaries comprised of one ini-
tially scalarized star and one initially unscalarized star; this asymmetry cannot be
achieved in equal-mass systems like those discussed above.
We next consider the onset of dynamical scalarization in unequal-mass sys-
343
tems. For the sake of compactness, we show in Table 7.5 the dynamical scalarization
frequencies for binaries with NS masses of 1.2M to 1.9M with just the MPA1
EOS. We find that the total mass plays a more important role in determining the
onset of dynamical scalarization than the mass ratio. Fixing the total mass, we find
that scalarization occurs earlier in more asymmetric binaries of lower mass (e.g.,
M . 3.2M for the MPA1 EOS). None of the systems listed in Table 7.5 undergo
induced scalarization. As before, we highlight the systems in Table 7.5 that scalarize
below 50 Hz.
To summarize, Tables 7.4 and 7.5 demonstrate that binary-pulsar constraints
cannot entirely rule out the possibility of dynamical scalarization occurring at fre-
quencies fDS . 50 Hz at 90% CL. Initial detectability studies ? Refs. [175, 271]
discussed above ? suggest that this early scalarization should be observable with
aLIGO (although these conclusions should be confirmed with future work in light
of the limitations of these works; see above). Thus, failure to detect dynamical
scalarization in future GW observations could provide tighter constraints on the
parameters (?0, ?0) in DEF theory than pulsar timing. However, as can be seen
from Table 7.4, the prospects of producing such complementary constraints depend
critically on the observed NS masses and the EOS of NSs.
7.5 Conclusions
In this work, we have studied the scalarization phenomena [153, 169] in the
massless mono-scalar-tensor theory of gravity of DEF with pulsar timing and laser-
344
interferometer GW detectors on the Earth. We now summarize the key conclusions
of our analysis.
1. The spontaneous scalarization phenomenon [153] occurs at different NS mass
ranges for different EOSs [171]. Therefore, in a well-timed relativistic binary-
pulsar system with a specific NS mass, the scalar-tensor gravity might be
stringently constrained for some EOSs whose spontaneous-scalarization phe-
nomenon occurs near that specific NS mass. However, in general, strong scalar-
ization could still take place if NSs are described by an EOS whose scalarization
occurs at a mass different from the one observed.
2. Combining two well-timed binary-pulsar systems with quite different NS masses,
one could in principle constrain the scalar-tensor theory with whatever EOS
Nature provides us. Using MCMC simulations, we showed in Sec. 7.3 that,
combining five binary pulsars [247, 248, 373?375] that best constrain gravi-
tational dipolar radiation, we can already bound the scalarization parameter,
?0, to be & ?4.28 at 68% CL and & ?4.38 at 90% CL, for any of the eleven
EOSs that we have considered.
3. Nevertheless, because of the limited distribution of masses of the five chosen
binary pulsars, we found that if the EOS of NSs were similar to one of AP4,
SLy4, or WFF2, NSs with masses of mA ' 1.70?1.73M could still develop an
effective scalar coupling & O(10?2). This is also true for the EOS H4 with
mA ' 1.92M, and for the EOSs MS0 and MS2 with mA ' 2.26M (see
Fig. 7.5).
345
4. Using the upper limits on the effective scalar coupling of NSs from binary
pulsars, we found that for BNSs in the frequency bands of aLIGO, CE, and ET,
we could still have a large time-domain dephasing in the number of GW cycles,
on the orders of O(101), O(102), and O(103), respectively (see Table 7.3).
5. We performed a Fisher-matrix study of BNS inspiral signals using aLIGO,
CE, and ET. We found that for BNSs at a luminosity distance DL = 200 Mpc,
where we expect to observe those sources, aLIGO can still improve the limits
from binary pulsars for a couple of EOSs with BNSs of suitable masses. CE
(whose bandwith starts at 5 Hz) can improve the current limits for all EOSs,
while ET (whose bandwith starts at 1 Hz) will provide us with even more
significant improvements over current constraints for all EOSs. This is mainly
due its better low-frequency sensitivity. Our conclusions for aLIGO differ
from the one obtained in Refs. [276, 370, 371], where the authors concluded
that pulsar timing would do better than aLIGO in constraining scalar-tensor
theories. The main reason of this difference comes from a better understanding
and larger span of the NS masses and EOSs during the past two decades [248,
369], and the different PSD for aLIGO used in Refs. [240, 276]. If we restricted
the analysis to the same NS masses and the same EOS used in Ref. [276], we
would recover the same conclusions as in Ref. [276] (see Fig. 7.7).
6. We investigated dynamical scalarization in equal-mass and unequal-mass BNS
systems. With the criterion that the dynamical scalarization transition fre-
quency must fall below ? 50 Hz [175, 271] to be detectable, we found aLIGO
346
could be able to observe this phenomenon given the constraints obtained from
binary-pulsar timing, even away from the scalarization windows. We found
that the prospects for observing dynamical scalarization with GW detectors
depends critically on the NS EOS?for example, dynamical scalarization of
NSs with the MS0 EOS could not be detected with aLIGO. Producing new
constraints on scalar-tensor theories from GW searches for dynamical scalar-
ization requires waveform models that can faithfully reproduce this nonper-
turbative phenomenon; ultimately, these conclusions should be revisited once
such models are developed.
Our comparisons between binary pulsars and GWs made use of the current
limits of the former and the expected limits of the latter. It shows that advanced and
next-generation ground-based GW detectors have potential to further improve the
current limits set by pulsar timing. Nevertheless, the binary-pulsar limits will also
improve over time, especially if suitable systems filling the scalarization windows
are discovered in future pulsar surveys. Better mass measurements of currently
known pulsars will also help in narrowing down the constraints, especially with
PSRs J1012+5307 [373] and J1913+1102 [397], whose observational uncertainties in
masses are still large, and they might have the right masses to close the windows
below 2M. To reach this goal, the next generation of radio telescopes, such as
FAST and SKA will play a particularly important role [366, 401].
347
7.6 Acknowledgments
We thank Paulo Freire, Ian Harry, Jan Steinhoff, Thomas Tauris, and Nicola?s
Yunes for helpful discussions. We are grateful to Jim Lattimer for providing us
with tabulated data for neutron-star equations of state. The Markov-chain Monte
Carlo runs were performed on the Vulcan cluster at the Max Planck Institute for
Gravitational Physics in Potsdam.
348
Chapter 8: Distinguishing boson stars from black holes and neutron
stars from tidal interactions in inspiraling binary systems
Authors: Noah Sennett, Tanja Hinderer, Jan Steinhoff, Alessandra Buo-
nanno, and Serguei Ossokine1
Abstract: Binary systems containing boson stars?self-gravitating configura-
tions of a complex scalar field? can potentially mimic black holes or neutron stars
as gravitational-wave sources. We investigate the extent to which tidal effects in the
gravitational-wave signal can be used to discriminate between these standard sources
and boson stars. We consider spherically symmetric boson stars within two classes
of scalar self-interactions: an effective-field-theoretically motivated quartic potential
and a solitonic potential constructed to produce very compact stars. We compute
the tidal deformability parameter characterizing the dominant tidal imprint in the
gravitational-wave signals for a large span of the parameter space of each boson star
model, covering the entire space in the quartic case, and an extensive portion of
interest in the solitonic case. We find that the tidal deformability for boson stars
with a quartic self-interaction is bounded below by ?min ? 280 and for those with
a solitonic interaction by ?min ? 1.3. We summarize our results as ready-to-use
1Originally published as Phys. Rev. D96, 024002 (2017).
349
fits for practical applications. Employing a Fisher matrix analysis, we estimate the
precision with which Advanced LIGO and third-generation detectors can measure
these tidal parameters using the inspiral portion of the signal. We discuss a novel
strategy to improve the distinguishability between black holes/neutrons stars and
boson stars by combining tidal deformability measurements of each compact ob-
ject in a binary system, thereby eliminating the scaling ambiguities in each boson
star model. Our analysis shows that current-generation detectors can potentially
distinguish boson stars with quartic potentials from black holes, as well as from
neutron-star binaries if they have either a large total mass or a large (asymmetric)
mass ratio. Discriminating solitonic boson stars from black holes using only tidal ef-
fects during the inspiral will be difficult with Advanced LIGO, but third-generation
detectors should be able to distinguish between binary black holes and these binary
boson stars.
8.1 Introduction
Observations of gravitational waves (GWs) by Advanced LIGO [63], soon to be
joined by Advanced Virgo [64], KAGRA [233], and LIGO-India [402], open a new
window to the strong-field regime of general relativity (GR). A major target for
these detectors are the GW signals produced by the coalescences of binary systems
of compact bodies. Within the standard astrophysical catalog, only black holes
(BHs) and neutron stars (NSs) are sufficiently compact to generate GWs detectable
by current-generation ground-based instruments. To test the dynamical, non-linear
350
regime of gravity with GWs, one compares the relative likelihood that an observed
signal was produced by the coalescence of BHs or NSs as predicted by GR against
the possibility that it was produced by the merger of either: (a) BHs or NSs in
alternative theories of gravity or (b) exotic compact objects in GR. Here, we pursue
tests within the second class. Several possible exotic objects have been proposed
that could mimic BHs or NSs, such as: quark stars [403]?stars whose interiors
attain such high temperature and pressure that baryonic matter transitions to a
phase of ultra-dense quark matter; boson stars (BSs) [404, 405]?self-gravitating
configurations of a complex scalar field or fields; gravastars [406, 407]?localized
vacuum regions of non-zero cosmological constant contained within a finite radius;
and axion stars [408, 409]?self-gravitating configurations of a real scalar field whose
self-interaction takes the same form as the QCD axion.
The coalescence of a binary system can be classified into three phases? the
inspiral, merger, and ringdown? each of which can be modeled with different tools.
The inspiral describes the early evolution of the binary and can be studied within the
post-Newtonian (PN) approximation, a series expansion in powers of the relative ve-
locity v/c (see Ref. [30] and references within). As the binary shrinks and eventually
merges, strong, highly-dynamical gravitational fields are generated; the merger is
only directly computable using numerical relativity (NR). Finally, during ringdown,
the resultant object relaxes to an equilibrium state through the emission of GWs
whose (complex) frequencies are given by the object?s quasinormal modes (QNMs),
calculable through perturbation theory (see Ref. [410] and references within). Com-
plete GW signals are built by synthesizing results from these three regimes from
351
first principles with the effective-one-body (EOB) formalism [38, 45] or phenomeno-
logically, through frequency-domain fits [411, 412] of inspiral, merger and ringdown
waveforms.
An understanding of how exotic objects behave during each of these phases is
necessary to determine whether GW detectors can distinguish them from conven-
tional sources (i.e., BHs or NSs). Significant work in this direction has already been
completed. The structure of spherically-symmetric compact objects is imprinted in
the PN inspiral through tidal interactions that arise at 5PN order (i.e., as a (v/c)10
order correction to the Newtonian dynamics). Tidal interactions are characterized
by the object?s tidal deformability, which has recently been computed for gravastars
[413, 414] and ?mini? BSs [415]. During the completion of this work, an indepen-
dent investigation on the tidal deformability of several classes of exotic compact
objects, including examples of the BS models considered here, was performed in
Ref. [416]; details of the similarities and differences to this work are discussed in
Sec. 8.7 below. Additional signatures of exotic objects include the magnitude of the
spin-induced quadrupole moment and the absence of tidal heating. The possibility
of discriminating BHs from exotic objects with these two effects was discussed in
Refs. [417] and [418], respectively?we will not consider these effects in this work.
The merger of BSs has been studied using NR in head-on collisions [419, 420] and
following circular orbits [421]. The QNMs have been computed for BSs [422?424]
and gravastars [425?427].
In this work, we compute the tidal deformability of two models of BSs: ?mas-
sive? BSs [428] characterized by a quartic self-interaction and non-topological soli-
352
tonic BSs [429]. The self-interactions investigated here allow for the formation of
compact BSs, in contrast to the ?mini? BSs considered in Ref. [415]. We perform
an extensive analysis of the BS parameter space within these models, thereby go-
ing beyond previous work in Ref. [416], which was limited to a specific choice of
parameter characterizing the self-interaction for each model. Special consideration
must be given to the choice of the numerical method because BSs are constructed
by solving stiff differential equations?we employ relaxation methods to overcome
this problem [430]. Our new findings show that for massive BSs, the tidal deforma-
bility ? (defined below) is bounded below by ?min ? 280 for stable configurations,
while for solitonic BSs the deformability can reach ?min ? 1.3. For comparison, the
deformability of NSs is ?NS & O(10) and for BHs ?BH = 0. We compactly summa-
rize our results as fits for convenient use in future gravitational wave data analysis
studies. In addition, we employ the Fisher matrix formalism to study the prospects
for distinguishing BSs from NSs or BHs with current and future gravitational-wave
detectors based on tidal effects during the inspiral. Prospective constraints on the
combined tidal deformability parameters of both objects in a binary were also shown
for two fiducial cases in Ref. [416]. Our findings are consistent with the conclusions
drawn in Ref. [416]; we discuss a new type of analysis that can strengthen the claims
made therein on the distinguishability of BSs from BHs and NSs by combining in-
formation on each body in a binary system.
The paper is organized as follows. Section 8.2 introduces the BS models in-
vestigated herein. We provide the necessary formalism for computing the tidal de-
formability in Sec. 8.3, and describe the numerical methods we employ in Sec. 8.4.
353
In Sec. 8.5, we compute the tidal deformability, providing results that range from
the weak-coupling limit to the strong-coupling limit as well as numerical fits for the
tidal deformability. Finally, in Sec. 8.6 we discuss the prospects of testing the exis-
tence of stellar-mass BSs using GW detectors and provide some concluding remarks
in Sec. 8.7.
We use the signature (?,+,+,+) for the metric and natural units?~ = G =
c = 1, but explicitly restore factors of the Planck mass mPlanck = ~c/G in
places to improve clarity. The convention for the curvature tensor is such that
? ? a ? ? ? a = R?? ? ? ? ? ? ???a? , where ?? is the covariant derivative and a? is a
generic covector.
8.2 Boson star basics
Boson stars?self-gravitating configurations of a (classical) complex scalar
field?have been studied extensively in the literature, both as potential dark matter
candidates and as tractable toy models for testing generic properties of compact
objects in GR. Boson stars are described by the Einstein-Klein-Gordon action
? ? [ ]
S = d4x ? Rg ???????? ? V (|?|2) , (8.1)
16?
where ? denotes complex conjugation. The only experimentally confirmed elemen-
tary scalar field is the Higgs boson [431, 432], which is an unlikely candidate to
form a BS because it readily decays to W and Z bosons. However, other massive
scalar fields have been postulated in many theories beyond the Standard Model,
e.g., bosonic superpartners predicted by supersymmetric extensions [433].
354
The Einstein equations derived from the action (8.1) are given by
? 1R?? g??R = 8?T?
2 ??
, (8.2)
with
T? =? ??? ? +? ??? ?? g (????? ? + V (|?|2)). (8.3)?? ? ? ? ? ?? ?
The accompanying Klein-Gordon equation is
??? dV?? = | | ?, (8.4)d ? 2
along with its complex conjugate.
The earliest proposals for a BS contained a single non-interacting scalar field [434?
436], that is
( )
V |?|2 = ?2|?|2, (8.5)
where ? is the mass of the boson. The free Einstein-Klein-Gordon action also de-
scribes the second-quantized theory of a real scalar field; thus, this class of BS can
also be interpreted as a gravitationally bound Bose-Einstein condensate [436]. The
maximum mass for BSs with the potential given in Eq. (8.5) isMmax ? 0.633m2Planck/?,
or in units of solar mass, Mmax/M ? 85peV/?. The corresponding compactness
for this BS is Cmax ? 0.08 [434], where the compactness C of a body is given by
the ratio between its mass M and radius R.2 Because this maximum mass scales
2Formally, BSs have no surface, so the notion of a radius (and hence compactness) is inherently
ambiguous. One common convention is to define the radius as that of a shell containing a fixed
fraction of the total mass of the star (e.g., R99 where m(r = R99) = 0.99m(r =?)). To avoid
355
more slowly with ? than the Chandrasekhar limit for a degenerate fermionic star
MCH ? m3 2Planck/mFermion, this class of BSs is referred to as mini BSs. The tidal
deformability was computed in this model in Ref. [415]
Since the seminal work of the 1960s [434?436], BSs with various scalar self-
interactions have been studied. We consider two such models in this work, both
which reduce to mini BSs in the weak-coupling limit. The first BS model we consider
is massive BSs [428], with a potential given by
?
Vmassive(|?|2) = ?2|?|2 + |?|4, (8.6)
2
which is repulsive for ? ? 0. In the strong-coupling limit ? ?2/m2Planck, spherically
?
symmetric BSs obtain a maximum mass of Mmax ? 0.044 ?m3 2Planck/? [428]. In
?
units of the solar mass M this reads Mmax/M ? ?(0.3GeV/?)2. Such configura-
tions are roughly as compact as NSs, with an effective compactness of Cmax ? 0.158
[437, 438]. This BS model is a natural candidate from an effective-field-theoretical
perspective because the potential in Eq. (8.6) contains all renormalizable self-interactions
for a scalar field, i.e., other interactions that scale as higher powers of |?| are ex-
pected to be suppressed far from the Planck scale. The ?natural? values of ? ? 1
and ?  mPlanck yield the strong-coupling limit of the potential (8.6). Because
it is the most theoretically plausible BS model, we investigate the strong-coupling
regime of this interaction in detail in Section 8.5.1.
The second class of BS that we consider is the solitonic BS model [429], char-
this ambiguity, our results are given in terms of quantities that can be extracted directly from the
asymptotic geometry of the BS: the total mass M and dimensionless tidal deformability ? (defined
below).
356
acterized by the potential
( )
2 2
V (|?|2solitonic ) = ?2|
2|?|
?|2 1? . (8.7)
?20
?
This potential admits a false vacuum solution at |?| = ?0/ 2. One can construct
spherically symmetric BSs whose interior closely resembles this false vacuum state
and whose exterior is nearly vacuum |?| ? 0; the transition between the false vac-
uum and true vacuum occurs over a surface of width ?r ? ??1. In the strong-
coupling limit ?0  mPlanck, the maximum mass of non-rotating BSs is Mmax ?
0.0198m4 2Planck/(??0), or Mmax/M ? (?/?0)2(0.7PeV/?)3 [429]. The corresponding
compactness Cmax ? 0.349 approaches that of a BH CBH = 1/2 [429].3 The main
motivation for considering the potential (8.7) is as a model of very compact objects
that could even possess a light-ring when C > 1/3. In this work, we will only con-
sider solitonic BSs as potential BH mimickers, as NSs could be mimicked by the
more natural massive BS model.
In this work, we restrict our attention to only non-rotating BSs. Axisymmetric
(rotating) BSs have been constructed for the models we consider [440?443], but these
solutions are significantly more complex than those that are spherically symmetric
(non-rotating). The energy density of a rotating BS forms a toroidal topology, van-
ishing at the star?s center. Because its angular momentum is quantized, a rotating
BS cannot be constructed in the slow-rotation limit, i.e. by adding infinitesimal
rotation to a spherically symmetric solution [444].
3This compactness is still lower than the theoretical Buchdahl limit of C ? 4/9 for isotropic
perfect fluid stars that respect the strong energy condition [439].
357
8.3 Tidal perturbations of spherically-symmetric boson stars
We consider linear tidal perturbations of a non-rotating BSs. We work within
the adiabatic limit, that is we assume that the external tidal field varies on timescales
much longer than any oscillation period of the star or relaxation timescale to reach a
microphysical equilibrium. These conditions are typically satisfied during the inspi-
ral of compact binaries. Close to merger, the assumptions concerning the separation
of timescales can break down and the tides can become dynamical [292, 299, 445,
446]; we ignore these complications here. The computation of the tidal deformability
of NSs in general relativity was first addressed in Refs. [308, 447] and was extended
in Refs. [307, 359].
8.3.1 Background configuration
Here we review the equations describing a spherically symmetric BS [404,
428, 434], which is the background configuration that we use to compute the tidal
perturbations in the following subsection. We follow the presentation in Ref. [424].
The metric written in polar-areal coordinates reads
ds2 = ?ev(r)dt2 + eu(r)dr2 + r2(d?2 + sin2 ?d?20 ). (8.8)
As an ansatz for the background scalar field, we use the decomposition
?0(t, r) = ?0(r)e
?i?t. (8.9)
358
Inserting Eqs. (8.8) and (8.9) into Eqs. (8.2)?(8.4) gives
( ? )
e?u (?
u 1 1
+ ? = ?8??, (8.10a)
r r2 r2
v?
)
?u 1 ? 1( e + = 8?prad, (8.10b)r r? ?)2 r2
???
2 v ? u ? ( )
0 + + ?0 = e
u U0 ? ?2e?v ?0, (8.10c)
r 2
where a prime denotes differentiation with respect to r, U0 = U(?0), U(?) =
dV/d|?|2. Because the coefficients in Eq. (8.10c) are real numbers, we can restrict
?0(r) to be a real function without loss of generality. We have also defined the
effective density and pressures
t
? ?? T? = ?2e?v?2 + e?u(??t 0 0)2 + V0, (8.11)
p ?T?r = ?2e?v 2rad r ?0 + e?u(??0)2 ? V0, (8.12)
ptan ? ??T 2? = ? e?v?20 ? e?u(??0)2 ? V0, (8.13)
where V0 = V (?0). Note that BSs behave as anisotropic fluid stars with pressure
anisotropy given by
? = prad ? ptan = 2e?u(??0)2. (8.14)
An additional relation derived from Eqs. (8.2)?(8.9) that will be used to simplify
the perturbation equations discussed in the next subsection is
p? ?(prad + ?)
[ ( ) ] 2?
rad = e
u 1 + 8?r2prad ? 1 ? . (8.15)
2r r
We restrict our attention to ground-state configurations of the BS, in which
?0(r) has no nodes. The background fields exhibit the following asymptotic behavior
359
limm(r) ? r3, lim m(r) ?M, (8.16a)
r?0 r??
lim v(r) ? v(c), lim v(r) ? 0, (8.16b)
r?0 r??
?
(c) 1 2 2
lim ?0(r) ? ? , lim ? (r) ? e?r ? ??0 0 , (8.16c)
r?0 r?? r
(c)
where M is the BS mass, v(c) and ?0 are constants, and m(r) is defined such that( )
?u(r) ? 2m(r)e = 1 . (8.17)
r
8.3.2 Tidal perturbations
We now consider small perturbations to the metric and scalar field defined
such that
(0)
g?? = g?? + h??, (8.18)
? = ?0 + ??. (8.19)
We restrict our attention to static perturbations in the polar sector, which describe
the effect of an external electric-type tidal field. Working in the Regge-Wheeler
gauge [448], the perturbations take the form
? [ ]
h??dx
?dx? = Y v 2 ulm(?, ?) e h0(r)dt +e h2(r)dr
2 + r2k(r)(d?2 + r sin2 ?d?2) ,
l?|m|
(8.20a)
and
? ?1(r)
?? = Ylm(?, ?)e
?i?t, (8.20b)
r
l?|m|
360
where Ylm are scalar spherical harmonics.
We insert the perturbed metric and scalar field from Eqs. (8.18)?(8.20) into
the Einstein and Klein-Gordon equations, Eqs. (8.2) and (8.4), and expand to first
order in the perturbations. For the metric functions, the (?, ?)-component of the
Einstein equations gives h2 = h0, and the (r, r)- and (r, ?)-components can be used
to algebraically eliminate k and k? in favor of h0 and its derivatives. Finally, the
(t, t)-component leads to the following second-order differential equation:
u ?
h??
e h ( )
0 +
0 1 + e?u ? 8?r2V0
r
? 32?e
u?1 [
??
( ) ( )]
?1 + e?u 2[ 0 ? 8?r prad + r?0 U0 ? 2?
2e?v
r2
h0e
u ]
+ ?16?r2V0 ? l(l + 1)? eu(1? e?u + 8?r2p 2rad) + 64?r2?2?2 ?v
r2 0
e = 0,
(8.21)
where we have also used the background equations (8.10a), (8.10b), and (8.15). From
the linear perturbations to the Klein-Gordon equation, together with the results for
the metric perturbations and the background equations, we obtain
eu?? ?
? ( )
? + 11 [ (1? e
?u ? 8?r2V0
r ) ( )]
? euh ?? ?1 + e?u0 [ 0 ? 8?r
2prad + r?0 U
2
0 ? 2? e?v
(8.22)
eu?1 ( )
+ 8?r2V0 ? 1 + e?u ? l(l + 1)? r2 U + 2W ?22 ] 0 0r 0
2 2+r e?v?2 ? 32?e?ur2 (??0) = 0,
where W0 = W (?0) with W (?) = dU/d|?|2. These perturbation equations were
also independently derived in Ref. [416] and are a special case of generic linear
perturbations considered in the context of QNMs (see, e.g., Refs. [422?424]). As
a check, we combined the three first-order and one algebraic constraint for the
361
spacetime perturbations from Ref. [424] into one second-order equation for h0, which
agrees with Eq. (8.21) in the limit of static perturbations. For the special case of
mini BSs, the tidal perturbation equations were also obtained in Ref. [415].
The perturbations exhibit the following asymptotic behavior [424]
lim h0(r) ? rl,( ) ( ) (8.23a)r?0 r ?(l+1)? r llim h0(r) c1 + c?? 2 , (8.23b)r M M
lim ?1(r) ? rl+1, (8.23c)
r?0
? ?
2 2 2 2 2
lim ? (r) ? rM? / ? ?? e?r ? ?? . (8.23d)
r?? 1
8.3.3 Extracting the tidal deformability
The BS tidal deformability can be obtained in a similar manner as with NSs
[307, 359, 447]. Working in the (nearly) vacuum region far from the center of the
BS, the formalism developed for NSs remains (approximately) valid. For simplic-
ity, we consider only l = 2 perturbations for the remainder of this section. The
generalization of these results to arbitrary l is detailed in Ref. [307].
As shown in Eqs. (8.16) and (8.23), very far from the center of the BS, the
system approaches vacuum exponentially. Neglecting the vanishingly small contri-
butions from the scalar field, the metric perturbation reduces to the general form
[ ]
hvac0 = c
1 1
1Q?22(x) + c2P?22(x) +O (?0) , (?1) , (8.24)
where we have defined x ? r/M ? 1, P?22 and Q?22 are the associated Legendre func-
tions of the first and second kind, respectively, normalized as in Ref. [307] such that
362
P?22 ? x2 and Q?22 ? 1/x3 when x ??. The coefficients c1 and c2 are the same as
in Eq. (8.23b).
In the BS?s local asymptotic rest frame, the metric far from the star?s center
takes the form [449]
Q ( ) ( )? 2M 3 ijg? = 1 + + ninj 1 ij 100 ? ? +O
r (r3 ) [ 3 ] r4 (8.25)
? E xixj +O r3ij +O (?0)1 , (?1)1 ,
where ni = xi/r, Eij is the external tidal field, and Qij is the induced quadrupole
moment. Working to linear order in Eij, the tidal deformability ?Tidal is defined such
that
Qij = ??TidalEij. (8.26)
For our purposes, it will be convenient to instead work with the dimensionless quan-
tity
? ?Tidal? . (8.27)
M5
Comparing Eqs. (8.24) and (8.25), one finds that the tidal deformability can be
extracted from the asymptotic behavior of h0 using
c1
? = . (8.28)
3c2
From Eq. (8.24), the logarithmic derivative
d log h ?0 rh
y ? = 0 , (8.29)
d log r h0
takes the form
3?Q??22(x) + P?
?
y(x) = (1 + x) 22
(x)
, (8.30)
3?Q?22(x) + P22(x)
363
or equivalently ( )
?
?1 (1 + x)P?22(x)? y(x)P?22(x)? = . (8.31)
3 (1 + x)Q??22(x)? y(x)Q?22(x)
Starting from a numerical solution to the perturbation equations (8.21) and (8.22),
one obtains the deformability ? by first computing y from Eq. (8.29) and then
evaluating Eq. (8.31) at a particular extraction radius xExtract far from the center of
the BS. Details concerning the numerical extraction are described in Sec. 8.4 below.
8.4 Solving the background and perturbation equations
The background equations (8.10a)?(8.10c) and perturbation equations (8.21)?
(8.22) form systems of coupled ordinary differential equations. These equations can
be simplified by rescaling the coordinates and fields by ? (the mass of the boson
field). To ease the comparison with previous work, we extend the definitions given
in Ref. [424]: for massive BSs, we use
m2 2
r ? Planckr? m m?(r?), m(r)? Planck ,
? ?
8??2? ?? ? ???? , ? , (8.32)
m2 2Planck mPlanck
m 2Planck??0(r?)
?0(r)? , ?1(r)?
mPlanck??1(r?) ,
(8?)1/2 ?(8?)1/2
while for solitonic BSs, we use
m2 2
r ? Planckr? m, m(r)? Planckm?(r?) ,
??0? ??0?
?mPlanck??0?0 , ? ?
??0???
, (8.33)
(8?)1/2 m2Planck
?0??0(r?) m
2
? (r)? , ? (r)? Planck??1(r?)0 1 ,
(2)1/2 (16?)1/2?
364
where factors of the Planck mass have been restored for clarity.
Finding solutions with the proper asymptotic behavior [Eqs. (8.16) and (8.23)]
requires one to specify boundary conditions at both r? = 0 and r? = ?. To impose
these boundary conditions precisely, we integrate over a compactified radial coordi-
nate
r?
? = , (8.34)
N + r?
as is done in Ref. [450], where N is a parameter tuned so that exponential tails
in the variables ??0 and ??1 [see Eqs. (8.16) and (8.23)] begin near the center of the
domain ? ? [0, 1]. For massive BSs, we use N ranging from 20 to 60 depending on
the body?s compactness; for solitonic BSs we use N between 1 and 10.
Ground-state solutions to the background equations (8.10a)?(8.10c) can be
(c)
completely parameterized by the central scalar field ??0 and frequency ??. To deter-
mine the ground state frequency, we formally promote ?? to an unknown constant
function of r? and simultaneously solve both the background equations and
???(r?) = 0. (8.35)
We impose the following boundary conditions on this combined system:
(c)
u(0) =0, ??0(0) = ??0 , ??
?
0(0) = 0,
(8.36)
v(?) =0, ??0(?) = 0.
Here, the inner boundary conditions ensure regularity at the origin, and the outer
conditions guarantee asymptotic flatness.
The background and pertrubation equations are stiff, and therefore the shoot-
ing techniques usually used to solve two-point boundary value problems require
365
signficant fine-tuning to converge to a solution [424]. To avoid these difficulties,
we use a standard relaxation algorithm that more easily finds a solution given a
reasonable initial guess [430]. Once a solution is found for a particular choice of the
(c)
central scalar field ??0 and scalar coupling (i.e., ? for massive BSs or ?0 for solitonic
BSs), this solution can be used as an initial guess to obtain nearby solutions. By
iterating this process, one can efficiently generate many BS configurations.
After finding a background solution, we solve the perturbation equations (8.21)
and (8.22). To improve numerical behavior of the perturbation equations near the
boundaries, we factor out the dominant r? dependence and instead solve for
h?0(r?) ? h0r??2, (8.37)
??1(r?) ? ??1r??3. (8.38)
We employ the boundary conditions
(c)
h?0(0) = h?0 , h?
?
0(0) = 0,
(8.39)
???1(0) = 0, ??1(?) = 0,
(c)
where the normalization h?0 is an arbitrary non-zero constant.
Finally, we compute the tidal deformability using Eq. (8.31) in the nearly
vacuum region x  1. At very large distances, the exponential falloff of ?0 and
?1 is difficult to resolve numerically. This numerical error propagates through the
computation of the tidal deformability in Eq. (8.31) for very large values of x. We
find that extracting ? at smaller radii provides more numerically stable results, with
a typical variation of ? 0.1% for different choices of extraction radius xExtract. For
consistency, we extract ? at the radius at which y attains its maximum.
366
0 1 5 10 50 100 ?
2.0
1.5
1.0
520
517
514
0.0 0.2 0.4 0.6 0.8 1.0
Figure 8.1: Perturbations of a massive BS as a function of rescaled
coordinate r? and compactified coordinate ? for a star of mass
M = 3.78m2Planck/? with coupling ?? = 300. Top panel: The background
density ?0 (dashed) and its first-order perturbation ?? (solid), rescaled to
fit on the same plot. Middle panel: Logarithmic derivative y of the metric
perturbation. The tidal deformability ? is calculated using the numer-
ically computed solution (black) at the peak of y (dot-dashed vertical
line). Using this value for ?, we plot corresponding expected behavior in
vacuum (red) as given by Eq. (8.30). Bottom panel: Tidal deformability
computed from Eq. (8.31) as a function of extraction radius xExtract.
367
Figure 8.1 demonstrates our procedure for computing the tidal deformability.
The background and perturbation equations are solved for a massive BS with a
coupling of ?? = 300 using a compactified coordinate with N = 20. The profile
of the effective density ?, decomposed into its background value ?0 and first order
correction ??, is shown in the top panel for a star of mass 3.78m2Planck/?. Note that
the magnitude of the perturbation is proportional to the strength of the external
tidal field; to improve readability, we have scaled ?? to match the size of ?0.
The middle panel of Fig. 8.1 shows the computed logarithmic derivative y
across the entire spacetime (black). We calculate the deformability with Eq. (8.31)
using the peak value of y, located at the dot-dashed line. Comparing with the top
panel, one sees that the scalar field is negligible in this region, justifying our use
of formulae valid in vacuum. The bottom panel depicts the typical variation of
? computed at different locations xExtract?our procedure yields consistent results
provided one works reasonably close to the edge of the BS. As a check, we insert
the computed value of ? back into the vacuum solution for y given in Eq. (8.30),
plotted in red in the middle panel. As expected, this curve closely matches the
numerically computed solution at large radii, but deviates upon entering a region
with non-negligible scalar field.
368
104 104
103 103
102 102
0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Figure 8.2: Dimensionless tidal deformability of a massive BS as a func-
tion of mass in units of (left) m2 2 1/2Planck/? and (right) mPlanck?? /?. For
each value of ??, the most compact stable configuration is highlighted
with a colored dot. The arrows indicate the direction towards the strong-
coupling regime, i.e. of increasing ??.
8.5 Results
8.5.1 Massive Boson Stars
The dimensionless tidal deformability of massive BSs is given as a function of
the rescaled total mass M? [defined as in Eq. (8.32)] in the left panel of Fig. 8.2. The
deformability in the weak-coupling limit ?? = 0 is given by the dotted black curve;
this limit corresponds to the mini BS model considered in Ref. [415].4 One finds
that the tidal deformability of the most massive stable star (colored dots) decreases
from ? ? 900 in the weak-coupling limit towards ? ? 280 as ?? is increased. For
large values of ??, the deformability exhibits a universal relation when written in
terms of the rescaled mass M?/??1/2 in the sense that the results for large ?? rapidly
4In Ref. [415], the authors computed the quantity kBS, related to the quantity ? presented here
by kBS = ?M
10. The quantity kE2 , computed in Ref. [416] for mini, massive, and solitonic BSs, is
related to ? by kE = (4?/5)1/22 ?.
369
approach a fixed curve as the coupling strength increases. This convergence towards
the ?? = ? relation is illustrated in the right panel of Fig. 8.2, in which the x-axis
is rescaled by an additional factor of ??1/2 relative to the left panel; in both panels,
we have added black arrows to indicate the direction of increasing ??. Employing
this rescaling of the mass, we compute the relation ?(M?, ??) in the strong-coupling
limit ???? below. The tidal deformability in this limit is plotted in Fig. 8.2 with
a dashed black curve.
The gap in tidal deformability between BSs, for which the lowest values are
? & 280, and NSs, where for soft equations of state and large masses ? & 10, can
be understood by comparing the relative size or compactness C = M/R of each
object. From the definitions (8.26) and (8.27), one expects the tidal deformability
to scale as ? ? 1/C5. In the strong-coupling limit, stable massive BSs can attain a
compactness of Cmax ? 0.158; note that in the exact strong-coupling limit ?? = ?,
BSs develop a surface, and thus their compactness can be defined unambiguously. A
NS of comparable compactness has a tidal deformability that is only ? 0?25% larger
than that of BSs. However, NS models predict stable stars with approximately twice
the compactness that can be attained by massive BSs, and thus, their minimum tidal
deformability is correspondingly much lower.
As argued in Sec. 8.2, the strong-coupling limit of massive BSs is the most
plausible model investigated in this paper from an effective field theory perspective.
We analyze the tidal deformability in this limit in greater detail. To study the
strong-coupling limit of ????, we employ a different set of rescalings introduced,
370
first in Ref. [428]:
m2 1/2 2 1/2? Planck?? r? ? mPlanck?? m?(r?)r , m(r) ,
? ?
? 8??
2?? ? ???? , ? , (8.40)
m2 2Planck mPlanck
?mPlanck??0(r?) ? m
2
? (r) , ? (r) Planck
??1(r?)
0 1 ,
(8???)1/2 ?(8?)1/2
where we have kept the previous notation for ?? to emphasize that it is the same
quantity as defined in Eq. (8.32).
Keeping terms only a(t leading o)rder in ??
?1  1, Eqs. (8.10a)?(8.10c) become
u? 1 1 3??4
e?u ? (+ ?) = ?2??
2
0 ? 0 , (8.41)r? r?2 r?2 2
?
?u v 1 ? 1 ??
4
e + = 0 , (8.42)
r? ( r?2 r?2) 2
2 ?v ? 1/2??0 = ?? e 1 , (8.43)
where a prime denotes differentiation with respect to r?. Note that in particular,
Eq. (8.10c) becomes an algebraic equation, reducing the system to a pair of first
order differential equations.
Turning now to the perturbation equations, we use these rescalings and find
that to lead[ing order in ???1, Eqs. (8.21]) and (8.22) become
eu?? h
? r?2 ( ) [u
h + 0 1? e?2v 4 e h0?? + e?u + 1 ? +eu(1? e?u)20 + l(l + 1)r? 2 r??2 ] (8.44)
r?4eu ( )4 (
+ 1(? e?v??)2 + r?2 eu(1? e?v??2)2 + 10e?v??2(1? e?
)
v??2)? 2 = 0,
4
h0r? 1 + ??
2
0
??1 = . (8.45)
2??0
As with the background fields, the equation for the scalar field ??1 becomes algebraic
in this limit. Note that the scalar perturbation diverges as one approaches the
371
102 102
106
1 10
3
10 101
100
0.3 0.6 0.9
100 100
0 2 4 6 8 10 12 14 0.1 0.2 0.3 0.4 0.5
Figure 8.3: Dimensionless tidal deformability as a function BS mass in
units of (left) m2Planck/? and (right) m
2
Planck/(???
2
0). For each value of
??0, the most compact stable BS is highlighted with a colored dot. The
inset plot in the left panel shows both stable and unstable configurations
over a larger range of ? to illustrate the weak-coupling limit ??0 ? ?
(dotted black). The arrows indicate the direction towards the strong-
coupling regime, i.e. of decreasing ??0; while not plotted explicitly, the
strong-coupling limit ??0 ? 0 corresponds the accumulation of curves in
the right panel in the direction of the arrow.
surface of the BS, defined as the shell on which ??0 vanishes. Nevertheless, the
metric perturbation h0 remains smooth over this surface.
We integrate the simplified background equations (8.41) and (8.42) and then
the perturbation equation (8.44) using Runge-Kutta methods. We compute the
tidal deformability using Eq. (8.31) evaluated at the surface of the BS, and plot the
results in the right panel of Fig. 8.2 (dashed black).
8.5.2 Solitonic boson stars
The dimensionless tidal deformability of solitonic BSs is given as a function
of the mass in Fig. 8.3. As in Fig. 8.2, the colored dots highlight the most massive
stable configuration for different choices of the scalar coupling ??0. To aid comparison
372
with the massive BS model, in the left panel we rescale the mass by an additional
factor of ??0 relative to the definition of M? in Eq. (8.33).
When the coupling ??0 is strong, solitonic BSs can manifest two stable phases
that can be smoothly connected through a sequence of unstable configurations [451].
The large plot in the left panel only shows stable configurations on the more compact
branch of configurations. In the weak-coupling limit ??0 ??, solitonic BSs reduce
to the free field model considered in Ref. [415]. To illustrate this limit, we show in the
smaller inset the tidal deformability for both phases of BSs as well as the unstable
configurations that bridge the two branches of solutions. The weak-coupling limit
is depicted with a dotted black curve. We find that the tidal deformability of the
less compact phase of BSs smoothly transitions from ??? in the strong-coupling
limit (?? ? 0)50 to ? ? 900 in the weak-coupling limit (??0 ? ?). Because their
tidal deformability is so large, diffuse solitonic BSs of this kind would not serve as
effective BH mimickers, and we will not discuss them for the remainder of this work.
However, it should be noted that only this phase of stable configurations exists when
?0 & 0.23mPlanck.
Focusing now on the more compact phase of solitonic BSs, one finds that the
tidal deformability of the most massive stable star (colored dots) decreases towards
? ? 1.3 as ??0 is decreased. As before, the relation between a rescaled mass and ?
approaches a finite limit in the strong coupling limit. We illustrate this in the right
5In the exact strong-coupling limit ??0 = 0, this diffuse phase of solitonic BSs vanishes [429].
However, the tidal deformability of this branch of BS configurations can be made arbitrarily large
by choosing ??0 to be sufficiently small.
373
panel of Fig. 8.3 by rescaling the mass by an additional factor of ???10 relative to the
definition in Eq. (8.33). While we do not examine the exact strong-coupling limit
??0 ? 0 here, we find that the minimum deformability has converged to within a few
percent of ? = 1.3 for 0.03mPlanck ? ?0 ? 0.05mPlanck.
8.5.3 Fits for the relation between M and ?
In this section we provide fits to our results for practical use in data analysis
studies, focusing on the regime that is the most relevant region of the parameter
space for BH and NS mimickers.
For massive BS, it is convenient to express the fit in terms of the variable
1
w = , (8.46)
1 + ??/8
which provides an estimate of the maximum mass in the weak-coupling limit M?max ?
?
2/(? w) [452] and has a compact range 0 ? w ? 1. A fit for massive BSs that is
accurate6 to ? 1%[ for ? ? 10
5 and up to the maximum mass is given by
? ]? 22.39 ? 143.5 305.6wM? = [0.529 + + wlog ? (log ?)2 (log ?)3 ] (8.47)
? 20.99 99.1 149.7+ 0.828 + ? + (1? w).
log ? (log ?)2 (log ?)3
The maximum mass where the BSs become unstable can be obtained from the
extremum of this fit, which also determines the lower bound for ?.
6The accuracy quoted here corresponds to the prediction for the mass at fixed ? and coupling
constant. The error in ? at a fixed mass can be much larger, because ? has a large gradient when
varying the mass, which even diverges at the maximum mass. The applicability of our fits must
be judged by the accuracy with which the masses can be measured from a GW signal.
374
In the solitonic case, a global fit for the tidal deformability for all possible
values of ?0 is difficult to obtain due to qualitative differences between the weak-
and strong-coupling regimes. However, small values of ?0 are most interesting, since
they allow for the widest range for the tidal deformability and compactness. A fit
for ?0 = 0.05mPlanck accurate to better than 1% and valid for ? ? 104 (and again
up to the maximum mass) reads
1079.8 10240
log(?0M?) = ?30.834 + ? . (8.48)
log ? + 19 (log ? + 19)2
This fit is expected to be accurate for 0 ? ?0 . 0.05mPlanck, i.e., including the strong
coupling limit ?0 = 0, within a few percent. Notice that this fit remains valid through
tidal deformabilities of the same magnitude as that of NSs.
8.6 Prospective constraints
8.6.1 Estimating the precision of tidal deformability measurements
Gravitational-wave detectors will be able to probe the structure of compact
objects through their tidal interactions in binary systems, in addition to effects
seen in the merger and ringdown phases. In this section, we discuss the possibility
of distinguishing BSs from NSs and BHs using only tidal effects. We emphasize
that our results in this section are based on several approximations and should be
viewed only as estimates that provide lower bounds on the errors and can be used
to identify promising scenarios for future studies with Bayesian data analysis and
improved waveform models.
375
The parameter estimation method based on the Fisher information matrix is
discussed in detail in Ref. [394]. This approximation yields only a lower bound
on the errors that would be obtained from a Bayesian analysis. We assume that a
detection criterion for a GW signal h(t;?) has been met, where ? are the parameters
characterizing the signal: the distance D to the source, time of merger tc, five
positional angles on the sky, plane of the orbit, orbital phase at some given time ?c,
as well as a set of intrinsic parameters such as orbital eccentricity, masses, spins, and
tidal parameters of the bodies. Given the detector output s = h(t) + n, where n is
the noise, the probability p(?|s) that the signal is characterized by the parameters
? is
p(?|s) ? p(0) 1e? (h(?)?s|h(?)?s)2 , (8.49)
where p(0) represents a priori knowledge. Here, the inner product (?|?) is determined
by the statistical properties of t?he noise and is given by? h??| 1(f)h?2(f) + h??2(f)h?1(f)(h1 h2) = 2 df, (8.50)
0 Sn(f)
where Sn(f) is the spectral density describing the Gaussian part of the detector
noise. For a measurement, one determines the set of best-fit parameters ?? that
maximize the probabili?ty distribution function (8.49). In the regime of large signal-
to-noise ratio SNR = (h|h), for a given incident GW in different realizations of
the noise, the probability distribution p(?|s) is approximately given by
p(?|s) ? p(0) ? 1e ?ij??i??j2 , (8.51)
where (
?h ???? )?h?ij = , (8.52)??i ??j
376
is the so-called Fisher information matrix. For a uniform prior p(0), the distribu-
tion (8.51) is a multivariate Gaussian with covariance matrix ?ij = (??1)ij and the
root-mean-square measurement errors in ?i are given by
? ?
?(??i)2? = (??1)ii, (8.53)
where angular brackets denote an average over the probability distribution func-
tion (8.51).
We next discuss the model h?(f,?) for the signal. For a binary inspiral, the
Fourier transform of the dominant mode of the signal has the form
h?(f,?) = A(f,?)ei?(f,?). (8.54)
Using a PN expansion and the stationary-phase approximation (SPA), the phase ?
is computed from the energy balance argument by solving
d2? 2 (dE/d?)
= = 2 , (8.55)
d?2 d?/dt E?GW
where E is the energy of the binary system, E?GW is the energy flux in GWs, and
? = ?f is the orbital[frequency. The result is of the form( ) ]3
? = 1 + ? (?)x+ . . .+ ?NewtM 1PN tidal + ?
5 6
5PN(?) x +O(x ) , (8.56)
128(? f)5/2
with x = (?Mf)2/3, M = m +m , ? = m m /M2,M = ?3/51 2 1 2 M , and the dominant
tidal contribution is
?Newttidal = ?
39
??. (8.57)
2
Here, ?? is the weighted average of th[e(individual)tidal deformabilitie]s, given by
16 m2 m
5
??(m ,m ,? ,? ) = 1 + 12 11 2 1 2 ?1 + (1? 2) . (8.58)
13 m 51 M
377
The phasing in Eq. (8.56) is known as the ?TaylorF2 approximant.? Specifically,
we use here the 3.5PN point-particle terms [30] and the 1PN tidal terms [453]. At
1PN order, a second combination of tidal deformability parameters enters into the
phasing in addition to ??. This additional parameter vanishes for equal-mass binaries
and will be difficult to measure with Advanced LIGO [454, 455]. For simplicity, we
omit this term from our analysis.
The tidal correction terms in Eq. (8.56) enter with a high power of the fre-
quency, indicating that most of the information on these effects comes from the late
inspiral. This is also the regime where the PN approximation for the point-mass
dynamics becomes inaccurate. To estimate the size of the systematic errors intro-
duced by using the TaylorF2 waveform model in our analysis, we compare the model
against predictions from a tidal EOB (TEOB) model. The accuracy of the TEOB
waveform model has been verified for comparable-mass binaries through comparison
with NR simulations; see, for example, Ref. [292]. For our comparison, we use the
same TEOB model as in Ref. [292]. The point-mass part of this model?known
as ?SEOBNRv2??has been calibrated with binary black hole (BBH) results from
NR simulations. The added tidal effects are adiabatic quadrupolar tides including
tidal terms at relative 2PN order in the EOB Hamiltonian and 1PN order in the
fluxes and waveform amplitudes. The SPA phase for the TEOB model is computed
by solving the EOB evolution equations to obtain ?(t), numerically inverting this
result for t(?), and solving Eq. (8.55) to arrive at ?(?).
Figure 8.4 shows the difference in predicted phase from the TEOB model and
the TaylorF2 model (8.56) for two nearly equal mass binary NS (BNS) systems.
378
2
1
0
-1
-2
100 1000
Figure 8.4: Dephasing between the TEOB and tidal TaylorF2 models for
non-spinning BNS systems including adiabatic quadrupolar tidal effects.
The curves end at the prediction for the merger from NR simulations
described in Ref. [458]. The labels denote the masses (in units of M)
and EoS of the NSs.
For our analysis, we consider two representative equations of state (EoS) for NSs:
the relatively soft SLy model [456] and the stiff MS1b EoS [457]. Figure 8.4 il-
lustrates that the dephasing between the TaylorF2 and TEOB waveforms remains
small compared to the size of tidal effects, which is on the order of & 20 rad for
MS1b (1.4 + 1.4)M. Thus, we conclude that the TaylorF2 approximant is suffi-
ciently accurate for our purposes and leave an investigation of the measurability of
tidal parameters with more sophisticated waveform models for future work.
Besides the waveform model, the computation of the Fisher matrix also re-
quires a model of the detector noise. We consider here the Advanced LIGO Zero-
Detuned High Power configuration [389]. To assess the prospects for measurements
with third-generation detectors we also use the ET-D [388] and Cosmic Explorer
379
[367] noise curves.
To compute the measurement errors we specialize to the restricted set of sig-
nal parameters ? = {?c, tc,M, ?, ??}. The extrinsic parameters of the signal such as
orientation on the sky enter only into the waveform?s amplitude and can be treated
separately; they are irrelevant for our purposes. Spin parameters are omitted be-
cause the TaylorF2 approximant inadequately captures these effects and one would
instead need to use a more sophisticated model such as SEOBNR. We restrict our
analysis to systems with low masses M . 12M [259] for which the merger occurs
at frequencies fmerger > 900Hz so that the information is dominated by the inspiral
signal. The termination conditions for the inspiral signal employed in our analysis
are the predicted merger frequencies from NR simulations: for BNSs the formula
from Ref. [458], and for BBH that from Ref. [459].
Ultimately, we need to convert our measurements of {M, ?, ??} into estimates
of the individual masses and tidal deformabilities {m1,m2,?1,?2}. Comparing the
dimensions of these two parameter spaces, one can immediately see that this trans-
formation is underdetermined; any given measurement of {M, ?, ??} corresponds to a
one-dimensional subspace of compatible choices for ?1 and ?2. However, this infinite
range of ?1 and ?2 can be constrained through physically motivated assumptions
on the relative size of ?1 and ?2. While we remain unable to estimate each body?s
tidal deformability precisely, we can at least place bounds on these quantities. The
details of this analysis are presented below.
We adopt the convention thatm1 ? m2. For any realistic, stable self-gravitating
body, we expect an increase in mass to also increase the body?s compactness. Be-
380
cause the tidal deformability scales as ? ? 1/C5, we assume that ?1 ? ?2 provided
that both bodies are the same type of compact object (e.g. NS, massive BS, solitonic
BS, etc.). Furthermore, we assume that both tidal deformabilities are non-negative,
as is the case for all compact objects we consider here.
Next, we consider the combinations of tidal deformabilities ?1,?2 that are
consistent with a particular set of measurements {M?, ??, ???}, or equivalently, a
particular set of measurements {m?,m?, ???1 2 }. Employing the assumption that ?1 ?
?2, one finds that the deformability of the more massive object ?1 takes its maximal
value when it is exactly equal to ?2, i.e. when ??
? = ??(m?1,m
?
2,?1,?1). Conversely,
?2 takes its maximal value when ?1 vanishes exactly so that ??
? = ??(m?,m?1 2, 0,?2).
Substituting the expression for ?? from Eq. (8.58) and using that m1,2 = M(1 ?
?
1? 4?)/2 leads to the following bounds on the individual deformabilities
?1 ? g1(?)??, ?2 ? g2(?)??, (8.59)
where the functions gi are given by
g1(?) ?
13
? , (8.60a)16(1 + 7? 31?2)
13
g2(?) ? [ ? ]
8 1 + 7? ? 31?2 ? 1? ? , (8.60b)4? (1 + 9? 11?2)
and where we have dropped the asterisks for simplicity. Thus, the expected mea-
surement precision of ? and ?? provide an estimate of the precision with which ?1
and ?2 can be measured thr[ough( )2 ( ) ]2 1/2
??1 ? [(g1(?)???) + (g
?
1(?)????) ] , (8.61a)2 2 1/2
??2 ? g2(?)??? + g?2(?)???? , (8.61b)
381
For simplicity, we have assumed in Eq. (8.61) that the statistical uncertainty in ?
and ?? is uncorrelated. Note that for BBH signals, this assumption is unnecessary
because ?? = 0, and thus the second terms in Eqs. (8.61a) and (8.61b) vanish.
In the following subsections, we outline two tests to distinguish conventional
GW sources from BSs and discuss the prospects of successfully differentiating the
two with current- and third-generation detectors. First, we investigate whether
one could accurately identify each body in a binary as a BH/NS rather than a
BS. This test is only applicable to objects whose tidal deformability is significantly
smaller than that of a BS, e.g., BHs and very massive NSs. For bodies whose
tidal deformabilities are comparable to that of BSs, we introduce a novel analysis
designed to test the slightly weaker hypothesis: can the binary system of BHs or NSs
be distinguished from a binary BS (BBS) system? For both tests, we will assume
that the true waveforms we observe are produced by BBH or BNS systems and then
assess whether the resulting measurements are also consistent with the objects being
BSs. In our analyses we consider only a single detector and assume that the sources
are optimally oriented; to translate our results to a sky- and inclination-averaged
?
ensemble of signals, one should divide the expected SNR by a factor of 2 and thus
multiply the errors on ??? by the same factor.
We consider two fiducial sets of binary systems in our analysis. First, we
consider BBHs at a distance of 400 Mpc (similar to the distances at which GW150914
and GW151226 were observed [73, 184]) with total masses in the range 8M ?M ?
12M. This range is determined by the assumption that the lowest BH mass is 4M
and the requirement that the merger occurs at frequencies above ? 900 Hz so the
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information in the signal is dominated by the inspiral. The SNRs for these systems
range from approximately 20 to 49 given the sensitivity of Advanced LIGO. The
second set of systems that we consider are BNSs at a distance of 200 Mpc and with
total masses 2M ? M ? Mmax, where Mmax is twice the maximum NS mass for
each equation of state. The lower limit on this mass range comes from astrophysical
considerations on NS formation [460]. The BNS distance was chosen to describe
approximately one out of every ten events within the expected BNS range of ? 300
Mpc for Advanced LIGO and translates to SNR ? 12? 22 for the SLy equation of
state.
8.6.2 Distinguishability with a single deformability measurement
A key finding from Sec. 8.5 is that the tidal deformability is bounded below
by ? & 280 for massive BSs and ? & 1.3 for solitonic BSs. By comparison, the
deformability of BHs vanishes exactly, i.e. ? = 0, whereas for nearly-maximal mass
NSs, the deformability can be of order ? ? O(10). Thus, a BH or high-mass NS
could be distinguished from a massive BS provided that a measurement error of
?? ? 200 can be reached with GW detectors. Similarly, to distinguish a BH from
a solitonic BS requires a measurement precision of ?? ? 1.
The results for the measurement errors with Advanced LIGO for BBH systems
at 400 Mpc are shown in Fig. 8.5, for a starting frequency of 10 Hz. The left panel
shows the error in the combination ?? that is directly computed from the Fisher
matrix as a function of total mass M and mass ratio q = m1/m2. As discussed above,
383
the ranges of M and q we consider stem from our assumptions on the minimum BH
mass and a high merger frequency. The right panel of Fig. 8.5 shows the inferred
bound on the less well-measured individual deformability in the regime of unequal
masses. We omit the region where the objects have nearly equal masses q ? 1
because in this regime, the 68% confidence interval ? + 2?? exceeds the physical
bound ? ? 1/4. Inferring the errors on the parameters of the individual objects
requires a more sophisticated analysis [394] than that considered here. The coloring
ranges from small errors in the blue shaded regions to large errors in the orange
shaded regions; the labeled black lines are representative contours of constant ??.
Note that the errors on the individual deformability ?2 are always larger than those
on the combination ??.
We find that the tidal deformability of our fiducial BBH systems can be mea-
sured to within ?? . 100 by Advanced LIGO, which indicates that BHs can be
readily distinguished from massive BSs. However, even for ideal BBHs?high mass,
low mass-ratio binaries?the tidal deformability of each BH can only be measured
within ?? & 15 by Advanced LIGO. Therefore one cannot distinguish BHs from
solitonic BSs using estimates of each bodies? deformability alone. Given these find-
ings, we also estimate the precision with which the tidal deformability could be
measured with third-generation instruments. Compared to Advanced LIGO, the
measurement errors in the tidal deformability decrease by factors of ? 13.5 and
? 23.5 with Einstein telescope and Cosmic Explorer, respectively. Thus, the more
massive BH in the binary would be marginally distinguishable from a solitonic BS
with future GW detectors, as ??1 ? ??? . 1. These findings are consistent with
384
20 40 60 80 100
2.0 2.0
1.8 1.8
1.6 1.6
1.4 1.4
1.2 1.2
1.0 1.0
8 9 10 11 12 8 9 10 11 12
Figure 8.5: Estimated measurement error with Advanced LIGO of
(left) the weighted average tidal parameter ?? and (right) the less well-
constrained individual tidal parameter ?2 for BBH systems at 400 Mpc.
The black lines are contours of constant ??? and ??2 in the left and
right plots, respectively.
the conclusions of Cardoso et al [416], although these authors considered only equal-
mass binaries at distances D = 100Mpc with total masses up to 50M. However,
we find that in an unequal-mass BBH case, the less massive body could not be
differentiated from a solitonic BS even with third-generation detectors.
Next, we consider the measurements of a BNS system, shown in Fig. 8.6 as-
suming the SLy EoS. We restrict our analysis to systems with individual masses
1M ? mNS ? mmax, where mmax ? 2.05M is the maximum mass for this EoS.
Similar to Fig. 8.5, the left panel in Fig. 8.6 shows the results for the measurement
error in the combination ?? directly computed from the Fisher matrix, and the right
panel shows the error for the larger of the individual deformabilities. The slight
385
warpage of the contours of constant ??? compared to those in Fig. 8.5, best visible
for the ??? = 50 contour, is due to an additional dependence of the merger fre-
quency on ?? for BNSs that is absent for BBHs, and a small difference in the Fisher
matrix elements when evaluated for ?? =6 0. We see that the deformability of NSs
of nearly maximal mass in BNS systems can be measured to within ?? . 200, and
thus can be distinguished from massive BSs. However, the measurement precision
worsens as one decreases the NS mass, rendering lighter NSs indistinguishable from
massive BSs using only each bodies deformability alone. In the next subsection, we
discuss how combining the measurements of ? for each object in a binary system
can improve distinguishability from BSs even when the criteria discussed above are
not met.
For completeness, we also computed how well third-generation detectors could
measure the tidal deformabilities in BNS systems. As in the BBH case, we find that
measurement errors in ? decrease by factors of ? 13.5 and ? 23.5 with the Einstein
Telescope and Cosmic Explorer, respectively. However, the conclusions reached
above concerning the distinguishability of BHs or NSs and BSs remain unchanged.
8.6.3 Distinguishability with a pair of deformability measurements
In the previous subsection we determined that compact objects whose tidal
deformability is much smaller than that of BSs could be distinguished as such with
Advanced LIGO, e.g., BHs versus massive BSs. In this subsection, we present a more
refined analysis to distinguish compact objects from BSs when the deformabilities
386
200 400 600 800 1000
2.0 2.0
50
1.8 1.8
400
1.6 100 1.6
1.4 150 1.4 600
200
1.2 1.2 800
200 100
1.0 1.0
2.0 2.5 3.0 3.5 4.0 2.0 2.5 3.0 3.5 4.0
Figure 8.6: Estimated measurement error with Advanced LIGO of
(left) the weighted average tidal parameter ?? and (right) the less well-
constrained individual tidal parameter ?2 for BNS systems at 200 Mpc
with the SLy equation of state. The black lines are contours of constant
??? and ??2 in the left and right plots, respectively.
387
of each are of approximately the same size. In particular, we focus on the prospects
of distinguishing NSs between one and two solar masses from massive BSs and
distinguishing BHs from solitonic BSs. Throughout this section, we only consider
the possibility that a single species of BS exists in nature; differentiation between
multiple, distinct complex scalar fields goes beyond the scope of this work. We
show that combining the tidal deformability measurements of each body in a binary
system can break the degeneracy in the BS model associated with choosing the
boson mass ?. Utilizing the mass and deformability measurements of both bodies
allows one to distinguish the binary system from a BBS system.
In Figs. 8.2 and 8.3, the tidal deformability of BSs was given as a function of
mass rescaled by the boson mass and self-interaction strength. By simultaneously
adjusting these two parameters of the BS model, one can produce stars with the same
(unrescaled) mass and deformability. This degeneracy presents a significant obstacle
in distinguishing BSs from other compact objects with comparable deformabilities.
For example, the boson mass can be tuned for any value of the coupling ? (?0) so
that the massive (solitonic) BS model admits stars with the exact same mass and
tidal deformability as a solar mass NS. However, combining two tidal deformability
measurements can break this degeneracy and improve the distinguishability between
BSs and BHs or NSs. As an initial investigation into this type of analysis, we pose
the following question: given a measurement (m1,?1) of a compact object in a
binary, can the observation (m2,?2) of the companion exclude the possibility that
both are BSs? We stress that our analysis is preliminary and that only qualitative
conclusions should be drawn from it; a more thorough study goes beyond the scope
388
of this work.
From the Fisher matrix estimates for the errors in (M, ?, ??) we obtain bounds
on the uncertainty in the measurement (mi,?i) for each body in a binary, which we
approximate as being characterized by a bivariate normal distribution with covari-
ance matrix ? = diag(?mi,??i). Figure 8.7 depicts such potential measurements
by Advanced LIGO of (m1,?1) and (m2,?2), shown in black, for a (1.55 + 1.35)M
BNS system at a distance of 200 Mpc with two representative equations of state
for the NSs: the SLy and MS1b models discussed above. The dashed black curves
in Figure 8.7 show the ?(m) relation for these fiducial NSs. Figure 8.8 shows the
corresponding measurements in a 6.5?4.5M BBH measured at 400 Mpc made by
Advanced LIGO, Einstein Telescope, and Cosmic Explorer in blue, red, and black,
respectively.
The strategy to determine if the objects could be BSs is the following. Consider
first the measurement (m2,?2) of the less massive body. For each point x = (m,?)
within the 1? ellipse, we determine the combinations of theory parameters (?, ?)[x]
or (?, ?0)[x] that could give rise to such a BS, assuming the massive or solitonic BS
model, respectively. As discussed above, in general, ? or ?0 can take any value by
appropriately rescaling ?. Finally, we combine all mass-deformability curves from
Figs. 8.2 or 8.3 that?pass through the 1? e?llipise, that is we consider the model
parameters (?, ?) ? x(?, ?)[x] or (?, ?0) ? x(?, ?0)[x] for massive and solitonic
BS, respectively. These portions of BS parameter space are shown as the shaded
regions in Figs. 8.7 and 8.8. If the tidal deformability measurements (m1,?1) of
the more massive body?indicated by the other set of crosses?lie outside of these
389
104
102
100
1 2 1 2
Figure 8.7: Dimensionless tidal deformability as a function of mass.
Black points indicate hypothetical measurements of a (1.55 + 1.35)M
binary NS system with the (left) MS1b and (right) SLy EoS; the er-
ror bars are estimated for a system observed at 200 Mpc. The shaded
regions depict all possible massive BSs (i.e., all possible values of the
boson mass ? and coupling ?) consistent with the measurement of the
smaller compact object. For the MS1b EoS, the tidal deformabilities of
the binary are ?1.55 = 714 and ?1.35 = 1516. For the SLy EoS, the tidal
deformabilites are ?1.55 = 150 and ?1.35 = 390.
shaded regions, one can conclude that the measurements are inconsistent with both
objects being BSs.
Figure 8.7 demonstrates that an asymmetric BNS with masses 1.55?1.35M
can be distinguished from a BBS with Advanced LIGO by using this type of analysis.
When considered individually, either NS measurement shown here would be consis-
tent with a possible massive BS; by combining these measurements we improve our
ability to differentiate the binary systems. This type of test can better distinguish
BBSs from conventional GW sources than the analysis performed in the previous
section because it utilizes measurements of both the mass and tidal deformability
390
rather than just using the deformability alone. However the power of this type of
test hinges on the asymmetric mass ratio in the system; with an equal-mass system,
this procedure provides no more information than that described in Section 8.6.2.
A similar comparison between a BBH with masses 6.5?4.5M and a binary
solitonic BS system is illustrated in Fig. 8.8. For simplicity, the yellow shaded region
depicts all possible solitonic BSs for a particular choice of coupling ?0 = 0.05mPlanck
that are consistent with the measurement of the smaller mass by Advanced LIGO
(rather than all possible values of the coupling ?0). We see that in contrast to the
massive BS case, after fixing the boson mass ? with the measurement of one body,
the measurement of the companion remains within that shaded region. As with the
more simplistic analysis performed in Section 8.6.2, we again find that Advanced
LIGO will be unable to distinguish solitonic BSs from BHs.
In the previous section, we showed that third-generation GW detectors will be
able to distinguish marginally at least one object in a BBH system from a solitonic
BS and thus determine whether a GW signal was generated by a BBS system. Us-
ing the analysis introduced in this section, we can now strengthen this conclusion.
We repeat the procedure described above for a 6.5?4.5M BBH at 400 Mpc but in-
stead use the 3? error estimates in the measurements of the bodies? mass and tidal
deformability. In Fig. 8.8, all possible solitonic BSs consistent with the measure-
ment of the smaller mass are shown in green and pink for Einstein Telescope and
Cosmic Explorer, respectively. We see that while the deformability measurements
of each BH considered individually are consistent with either being solitonic BSs,
they cannot both be BSs. Thus, we can conclude with much greater confidence
391
that third-generation detectors will be able to distinguish BBH systems from binary
systems of solitonic BSs.
To summarize, the precision expected from Advanced LIGO is potentially
sufficient to differentiate between massive BSs and NSs or BHs, particularly in
systems with larger mass asymmetry. Advanced LIGO is not sensitive enough to
discriminate between solitonic BSs and BHs, but next-generation detectors like the
Einstein Telescope or Cosmic Explorer should be able to distinguish between BBS
and BBH systems. However, we emphasize again that our conclusions are based on
several approximations and further studies are needed to make these precise. We
also note that we have deliberately restricted our analysis to the parameter space
where waveforms are inspiral-dominated in Advanced LIGO. Tighter constraints on
BS parameters are expected for binaries where information can also be extracted
from the merger and ringdown portion, provided that waveform models that include
this regime are available.
8.7 Conclusions
Gravitational waves can be used to test whether the nature of BHs and NSs is
consistent with GR and to search for exotic compact objects outside of the standard
astrophysical catalog. A compact object?s structure is imprinted in the GW signal
produced by its coalescence with a companion in a binary system. A key target for
such tests is the characteristic ringdown signal of the final remnant. However, the
small SNR of that part of the GW signal complicates such efforts. Complementary
392
102
101
100
0 2 4 6 8 10 12
Figure 8.8: Dimensionless tidal deformability as a function of mass. Hy-
pothetical measurements of a (6.5+4.5)M binary BH system with error
bars estimated for a system observed at 400 Mpc by Advanced LIGO,
the Einstein Telescope, and Cosmic Explorer are given in blue, red, and
black, respectively. The shaded regions depict all possible solitonic BSs
with coupling ?0 = 0.05mPlanck that are consistent with the measure-
ments of the smaller compact object by each detector.
information can be obtained by measuring a small but cumulative signature due to
tidal effects in the inspiral that depend on the compact object?s structure through
its tidal deformability. This quantity may be measurable from the late inspiral and
could be used to distinguish BHs or NSs from exotic compact objects.
In this work, we computed the tidal deformability ? for two models of BSs:
massive BSs, characterized by a quartic self-interaction, and solitonic BSs, whose
scalar self-interaction is designed to produce very compact objects. For the quartic
interaction, our results span the entire two-dimensional parameter space of such a
model in terms of the mass of the boson and the coupling constant in the potential.
For the solitonic case, our results span the portion of interest for BH mimickers.
We presented fits to our results for both cases that can be used in future data
393
analysis studies. We find that the deformability of massive BSs is markedly larger
than that of BHs and very massive NSs; in particular, we showed that the tidal
deformability ? & 280 irrespective of the boson mass and the strength of the quartic
self-interaction. The tidal deformability of solitonic BSs is bounded below by ? &
1.3.
To determine whether ground-based GW detectors can distinguish NSs and
BHs from BSs, we first computed a lower bound on the expected measurement errors
in ? using the Fisher matrix formalism. We considered BBH systems located at 400
Mpc and BNS systems at 200 Mpc with generic mass ratios that merge above 900
Hz. We found that, with Advanced LIGO, BBHs could be distinguished from binary
systems composed of massive BSs and that BNSs could be distinguished provided
that the NSs were of nearly-maximal mass or of sufficiently different masses (i.e. a
high mass ratio binary). We also demonstrated that the prospects for distinguishing
solitonic BSs from BHs based only on tidal effects are bleak using current-generation
detectors; however, third-generation detectors will be able to discriminate between
BBH and BBS systems. We presented two different analyses to determine whether
an observed GW was produced by BSs: the first relied on the minimum tidal de-
formability being larger than that of a NS or BH, while the second combined mass
and deformability measurements of each body in a binary system to break degen-
eracies arising from the (unknown) mass of the fundamental boson field.
Work by Cardoso et al. [416] also investigated the tidal deformabilities of BSs
and the prospects of distinguishing them from BHs and NSs. Despite the topic being
similar, the work in this chapter is complementary: Cardoso et al. [416] performed
394
a broad survey of tidal effects for different classes of exotic objects and BHs in
modified theories of gravity, while our work focuses on an in-depth analysis of BSs.
Additionally, these authors computed the deformability of BSs to both axial and
polar tidal perturbations with l = 2, 3, whereas our results are restricted to the l = 2
polar case. The l = 2 effects are expected to leave the dominant tidal imprint in
the GW signal, with the l = 3 corrections being suppressed by a relative factor of
125?3/(351?2)(M?)
4/3 ? 4(M?)4/3 [461] using the values from Table I of Ref. [416],
where ? is the orbital frequency of the binary. For reference, M? ? 5? 10?3 for a
binary with M = 12M at 900Hz.
We also cover several aspects that were not considered in Ref. [416], where
the study of BSs was limited to a single example for a particular choice of theory
parameters for each potential (quartic and solitonic). Here, we analyzed the entire
parameter space of self-interaction strengths for the quartic potential and the regime
of interest for BH mimickers in the solitonic case. Furthermore, we developed fitting
formulae for immediate use in future data analysis studies aimed at constraining
the BS parameters with GW measurements. Cardoso et al. [416] also discussed
prospective constraints obtained from the Fisher matrix formalism for a range of
future detectors, including the space-based detector LISA that we did not consider
here. However, their analysis was limited to equal-mass systems, to bounds on ??,
and to the specific examples within each BS models. We went beyond this study
by delineating a strategy for obtaining constraints on the BS parameter space from
a pair of measurements and considering binaries with generic mass ratio. We also
restricted our results to the regime where the signals are dominated by the inspiral.
395
Although this choice significantly reduces the parameter space of masses surveyed
compared to Cardoso et al., we imposed this restriction because full waveforms that
include the late inspiral, merger and ringdown are not currently available. Another
difference is that we took BBH or BNS signals to be the ?true? signals around
which the errors were computed and used results from NR for the merger frequency
to terminate the inspiral signals, whereas the authors of Ref. [416] chose BS signals
for this purpose and terminated them at the Schwarzschild ISCO.
The purpose of this work was to compute the tidal properties of BSs that could
mimic BHs and NSs for GW detectors and to estimate the prospects of discriminat-
ing between such objects with these properties. Our analysis hinged on a number
of simplifying assumptions. For example, the Fisher matrix approximation that we
employed only yields lower bounds on estimates of statistical uncertainty. Addition-
ally, we considered only a restricted set of waveform parameters, whereas including
spins could also worsen the expected measurement accuracy. On the other hand, im-
proved measurement precision is expected if one uses full inspiral-merger-ringdown
waveforms or if one combines results from multiple GW events. Our conclusions
should be revisited using Bayesian data analysis tools and more sophisticated wave-
form models, such as the EOB model. Tidal effects are a robust feature for any
object, meaning that the only change needed in existing tidal waveform models is
to insert the appropriate value of the tidal deformability parameter for the object
under consideration. However, the merger and ringdown signals are more difficult
to predict, and further developments and NR simulations are needed to model them
for BSs or other exotic objects.
396
8.8 Acknowledgments
N.S. acknowledges support from NSF Grant No. PHY-1208881. We thank
Ben Lackey for useful discussions. We are grateful to Vitor Cardoso and Paolo Pani
for helpful comments on this manuscript.
397
Chapter 9: Parameterized tests of general relativity with generic fre-
quency-domain waveform models
Authors: Noah Sennett1
Abstract: Theory-agnostic and theory-specific gravitational-wave tests of
gravity rely on ?generalized? waveform models that introduce parameterized de-
viations to pre-established waveform models in general relativity (GR). We develop
a new flexible theory-agnostic (FTA) framework that allows such deviations to be
added to the inspiral of any frequency-domain, non-precessing baseline waveform.
This infrastructure is well-adapted to testing GR with LIGO and Virgo observa-
tions and to investigating the systematic biases that may arise in such tests through
the particular construction of generalized waveforms. As part of the LIGO Scientific
Collaboration, we use this infrastructure to bound phenomenological deviations from
GR in the phase evolution of BBH and BNS events observed during the first two
observing runs of the Advanced LIGO and Virgo detectors; by comparing the results
of these tests with those conducted with other generalized waveforms, we verify that
1Contains results prepared separately in: Phys. Rev. D102, 044056 (2020) [8]; Phys. Rev.
Lett. 123, 011102 (2019) [9]; Phys. Rev. D100, 104036 (2019) [10]; Sennett, Buonanno, Gergely,
Isi, and Sathyaprakash, ?Constraining Jordan-Fierz-Brans-Dicke gravity with GW170817,? (in
prep) [11].
398
waveform systematics have not significantly affect these tests. We also employ the
FTA framework to design and carry out tests of specific alternative theories of gravity.
Using GW170817, the first observed BNS, we place the first GW bounds on Jordan-
Fierz-Brans-Dicke gravity, finding that the scalar-tensor coupling ?0 . 4 ? 10?1.
Additionally, using two low mass BBH events?GW151226 and GW170608?we
constrain a higher-order curvature corrections that arise naturally from a class of
effective-field-theory-inspired extensions of GR, ruling out new physics that enters
on distance scales between 70 and 200 km.
9.1 Introduction
Theory-agnostic tests of general relativity (GR) with gravitational-waves (GWs)
often utilize waveform models that have been ?generalized? from an existing GR
template. The basic strategy for such tests is as follows. One first constructs a
generalized waveform model by introducing deviations to a GR waveform model
controlled by some set of parameters {???n}. These deviations are defined such that
(GR)
there exists some choice of parameters {???n } for which the generalized waveform
model exactly reproduces the underlying GR waveform, i.e. the generalized model
(GR)
restricted to the hypersurface of parameter space ???n = ???n can be identified
with the GR model. Without loss of generality, one can reparameterize the devia-
(GR)
tions such that that the GR prediction corresponds with zero, i.e. ???n = 0; we
adopt this convention throughout the remainder of this chapter.
Then, given a detected GW, one performs standard parameter estimation using
399
this generalized waveform model to assess the consistency of the observed signal with
the predictions of GR. This consistency can be quantified in a number of ways. One
oft-used technique is to examine the posterior distribution of binary parameters and
measure the proximity of this distribution?s mode to the GR hypersurface of the
generalized model. As a specific example, the LIGO and Virgo collaborations adopt
this method for certain tests of GR performed on observed binary coalescences,
testing whether the GR predictions ???n = 0 fall within the 90% credible regions
of the marginalized posterior distributions for ???n [9, 10, 73, 122]. Another way
to quantify the consistency between observation and GR prediction is through the
relative odds between two competing hypotheses: the detected signal does (does
not) agree with the predictions of GR HGR (Hnon-GR) [120, 176, 462]. This approach
offers a more easily quantified p-value, but such estimates require proper context
for correct interpretation. Due to the different dimensionality of the hypotheses
(???n = 0 vs. ???n 6= 0) and biases inherited from experimental design optimized for
GW detection in GR, one can only assess the significance of the aforementioned
odds ratio for a particular detection by comparing against a theoretical background
distribution estimated independently.
The overarching framework outlined above subsumes a large swath of tests
of GR that have been performed with GWs [9, 10, 73, 122, 187]. And yet, these
tests are not all exactly the same; they differ in either (i) the choice of generalized
waveform model and/or (ii) how this model is applied to data. In this work, we
develop a framework to examine how the details of the construction of generalized
waveform models systematically affect the tests of GR in which they are employed.
400
This infrastructure allows one to add generic corrections to the inspiral portion of
any non-precessing, frequency-domain waveform, thereby allowing theory-agnostic
tests of GR to be performed with a broader range of generalized waveform models
than previously possible. Accordingly, this framework is denoted as the flexible
theory-agnostic (FTA) approach. In addition to this original use, the easily adaptable
design of the FTA framework also allows for easy construction of waveform models
for theory-specific tests.
This work is organized as follows. Section 9.2 details the construction of
the FTA infrastructure. Section 9.3 presents the results of theory-agnostic tests
conducted as part of the LIGO Scientific Collaboration employing the FTA framework
on binary black hole (BBH) and binary neutron star (BNS) events detected during
their first and second observing runs. In Sec. 9.4, we use the FTA construction
to test Jordan-Fierz-Brans-Dicke gravity (JFBD) with GW170817?the first GW
observation of a BNS. These results represent the first (albeit weak) GW constraints
on this very well-known alternative theory of gravity. Similarly, in Sec. 9.5, we test
the higher-order curvature extension of GR recently proposed in Ref. [351] with the
lowest-mass BBH observations during the first and second observing runs of LIGO
and Virgo. Using the FTA infrastructure, we rule out the appearance of new physics
in this theory entering at the scale of 70-200 km.
401
9.2 The FTA construction of generalized waveform models
The construction of any generalized waveform model begins with a baseline
model in GR, which we express in the Fourier domain as hGR(f ;?) for the binary
parameters ? (e.g. masses, spins, etc.). Restricting our attention to the dominant
` = m = 2 mode of signals from non-precessing systems, the complex phase ?GR of
the waveform can be well appro[ximated during the early inspiral by [3]0, 259]?7 ?6
?(GR) ? 3 (GR)(f ;?) ?(GR)n (?)vn + ?n(l) (?)vn log v , (9.1)128?v5
n=0 n=5
1/3
where v ? (GM?f/c3) is the standard post-Newtonian (PN) parameter, M is
(GR) (GR)
the total mass of the binary, ? is its symmetric mass ratio, and ?n and ?n(l) are
the (n/2)-PN coefficients, which depend on the binary parameters. Note that the
logarithmic terms in Eq. (9.1) arise from so-called ?hereditary contribution? to the
inspiral, i.e. terms that depend on the full past history of the binary; see Chapter 3.4
for an detailed discussion of such terms.
To generalize this waveform model, one must add parameterized deviations to
the baseline waveform hGR(f). Though not the only possible option, a compelling
choice to test the inspiral behavior of the signal is to consider corrections to the
phase that take a similar PN form
( ) [? ]7 ?63
?? f ;?, {???n, ???n(l)} ? ??n(?, ??? nn)v + ??n(l)(?, ??? nn(l))v log v ,
128?v5
n=?2 n=5
(9.2)
where ??n and ??n(l) are deviations to the (n/2)-PN phase coefficients that, in
addition to ?, each depend on the corresponding deviation parameter ???n or ???n(l),
402
respectively. We include possible deviations at ?pre-Newtonian? orders (n < 0) as
these are predicted in many alternative theories of gravity, e.g. through the emission
of dipole radiation. Parameterized deviations of this form can be mapped onto
the predictions of any hypothetical alternative theory provided that (i) the theory
admits a weak-field, slow-velocity PN expansion as in GR and (ii) the deviations
from GR are parametrically smaller than the PN expansion parameter v2/c2; note
that this excludes theories that admit non-perturbative phenomena like dynamical
scalarization, for which the PN expansion breaks down. 2
In the FTA framework, we adopt the following definitions for ??n and ??n(l)
(GR)
??n(?, ?? ) ? ?? ?(GR)n n n (?), ??n(l)(?, ??n(l)) ? ??n(l)?n(l) (?), (9.3)
that is, each deviation parameter represents a fractional deviation to the correspond-
ing PN coefficient in GR. We handle PN orders for which the GR coefficient vanishes
slightly differently (e.g. at 0.5PN order); for those cases, we let ??n instead represent
2Despite the generality of Eq. (9.2), in most practical settings, the majority of the deviation
parameters are explicitly set to zero when performing tests of GR. Our null hypothesis for these
tests is that GR is correct, and accordingly that the deviation parameters recovered using a gen-
eralized waveform are consistent with zero. Under this assumption, each independent deviation
parameter allowed to vary freely would tend to worsen the statistical significance of the conclusions
that can be made. Fortunately, if this assumption is incorrect, many deviations from GR can still
be identified with a reduced number of deviation parameters; for example, Ref. [121] showed that
a signal containing deviations at several PN orders will lead to measurement of non-zero deviation
using a model with only single deviation parameter. Because of the scientific cost of introducing
new free parameters, deviations to the GW amplitude are neglected, as GW detectors are more
sensitive to the evolution of a signal?s phase than its amplitude.
403
an absolute deviation at that order.
While Eq. (9.2) unambiguously details how to generalize GR waveforms con-
taining only the inspiral, additional care must be taken for waveforms that contain
later portions of the GW signal. For the FTA approach, we require that the param-
eterized deviation observe the following properties:
1. The early inspiral (low frequency) waveform has a phase ?(f ;?) = ?(GR)(f ;?)+
??(f ;?), where ?? takes the form of Eq. (9.2).
2. The post-inspiral (high frequency) waveform has a phase ?(f ;?) = ?GR(f ;?)+
??(?) that exactly reproduces the underlying GR model up to some constant
shift (which represents the total dephasing from the GR model accumulated
over the complete inspiral).
3. The waveform is C2 smooth over all frequencies.
We construct ??(f ;?) starting from the total ?PN-like? phase correction given
by {???n, ???n(l)} [ ]
7 6
3 ? ? (GR)
??(PN)(f ;?) ? ?(GR)(?)??? nnv + ? (?)??? nn(l)v log v . (9.4)
128?v5 n n(l)
n=?2 n=5
To smoothly apply this correction over only the inspiral, we use a windowing function
W (f ; fwin,?fwin) given by [ ( )]?1
f ? fwin
W (f ; fwin,?fwin) ? 1 + exp , (9.5)
?fwin
which smoothly transitions between one and zero at fwin over a frequency range
of ? ?fwin. We construct the total phase correction by combining this windowing
404
function with the second derivative of ??(PN) and re-integrating with appropriate
integration constants to ensue C2 smoothness
? f ? f ?
??(f ; f ,?f (PN) ?? ??win win) = ?? (f )W (f ; fwin,?fwin), (9.6)
fref fref
where fref is an arbitrarily chosen reference frequency at which the phase of the
original waveform model vanishes, i.e. ?GR(fref) = 0.
9.3 Theory-agnostic inspiral tests with LIGO/Virgo events
One of the primary scientific objectives of the LIGO Scientific and Virgo Col-
laborations is to test the accuracy of the predictions of GR in the highly dynam-
ical, strong-field regime of gravity reached during the coalescence of BHs and/or
NSs. Several types of theory-agnostic tests are performed on observed GW events;
amongst those are tests of the inspiral portion of those signals, for which the FTA
infrastructure has been employed. These tests provide a consistency check on the
inspiral dynamics of the sources that generated the observed GW signals. To date,
no statistically significant deviations from GR have been uncovered with these tests.
The overarching design of these tests closely follows the description laid out in
Sec. 9.1: using a generalized waveform model, one performs parameter estimation
to measure (or constrain) the parameterized deviations {??n, ??n(l)} most consistent
with an observed GW event. Consistency with GR is indicated by measurements
of the parameterized deviations consistent with zero. Due to limitations in the
sensitivity of current detectors, we do not allow the full set of parameterized de-
viations {??n, ??n(l)} to vary freely in our model; each additional free parameter in
405
one?s model worsens the measurement errors on the deviation parameters (see, e.g.
Refs. [120, 122]). Instead, we allow only one deviation parameter to vary and set
the remainder to zero, repeating this test for each such deviation parameter.
During the first and second observing runs of the Advanced LIGO and Virgo
detectors, ten GW events from BBHs [463] and one GW event from a BNS [186]
were observed. However, not all of these events are amenable to tests of the in-
spiral dynamics of the corresponding sources. The inspiral ends at comparatively
lower frequencies for high-mass sources; because the detectors were only sensitive
above 20-30 Hz during the first two observing runs, the inspiral portion of these
signals was severely truncated. Following the criteria laid out in Ref. [10], we con-
sider only events for which the inspiral was recovered with SNR > 6, which are
the BBHs GW150914 [184], GW151226 [182], GW170104 [123], GW170608 [183],
GW170814 [185] and the BNS GW170817 [186].
The particular choice of generalized waveform used in each test is based off
of the type source observed. For the BBH events listed above, we performed the
aforementioned parameterized test of GR using the aligned-spin effective-one-body
(EOB) model SEOBNRv4 [45]. A fast, frequency-domain surrogate model is con-
structed from SEOBNRv4 using reduced-order modeling techniques [464], and then
finally, we apply the FTA procedure to construct a generalized waveform. For the
BNS, we instead use a baseline GR waveform model that includes additional phys-
ical effects absent in pure vacuum binaries, such as the tidal interactions between
the stars during the late inspiral. We employ the NRTidal model introduced in
Refs. [465?467], which allows one to convert a BBH waveform model to a BNS
406
0.01 0.4 0.8 12
SEOBNRv4
IMRPhenomPv2
0.005 0.2 0.4 6
0 0 0 0
-0.005 -0.2 -0.4 -6
-0.01 -0.4 -0.8 -12
?1PN 0PN 0.5PN 1PN 1.5PN 2PN 2.5PN(l) 3PN 3PN(l) 3.5PN
4e-05 0.4 0.8 12
SEOBNRT
PhenomPNRT
2e-05 0.2 0.4 6
0 0 0 0
-2e-05 -0.2 -0.4 -6
-4e-05 -0.4 -0.8 -12
?1PN 0PN 0.5PN 1PN 1.5PN 2PN 2.5PN(l) 3PN 3PN(l) 3.5PN
Figure 9.1: Marginalized posterior distributions of fractional deviations
to PN coefficients. (Top): Combined posterior distributions from all
BBHs observed during first and second observing run of LIGO and Virgo
with sufficiently measurable inspiral; adapted from Ref. [10]. (Bottom):
Posterior distribution from GW170817; adapted from Ref. [9].
waveform by adding an analytic tidal term to the frequency-domain phase evolution
fit from PN and NR results. This tidal correction is added prior to generalizing
the model using the FTA framework; this tidal model is denoted as SEOBNRT. Note
that SEOBNRT is parameterized by two additional quantities relative to the BBH
waveform SEOBNRv4: the tidal deformabilities of each NS. Because of this difference
in dimensionality of their respective parameter spaces, one must take care when
directly comparing constraints measured with BBH and BNS waveform models; for
example, degeneracies between the tidal deformabilities and deviation parameters
???n produce features in the posterior distributions recovered for BNS signals that
are absent for BBH signals.
As part of the LIGO Scientific Collaboration, we performed theory-agnostic
407
???n ???n
101
100
10?1
10?2
GW150914
10?3 GW151226
GW170104
? GW17060810 4 GW170814
Combined BBH
10?5
GW170817
(l) (l)
1 P
N PN N N N0 5 P 1 P 5 P 2 P
N N PN N PN
? 0. 1. 5 P 3 3 P 3.52.
Figure 9.2: 90% upper bounds on |???n| from individual BBH obser-
vations, combining these BBH observations, and the BNS observation
GW170817. Plot adapted from Refs. [9] and [10].
408
|???n|
tests of the inspiral dynamics for the restricted set of BBH and BNS observations
listed above with generalized SEOBNRv4 and SEOBNRT waveform models, respectively.
These tests were carried out using the LALInference software package [78], a com-
ponent of the LIGO Algorithm Library Suite (LALSuite) [468]. Figures 9.1 and 9.2
summarize the results of the tests; these figures have been adapted from analogous
plots in Refs. [9] and [10]. The top panel of Fig. 9.1 shows the marginalized posterior
distributions on ???n combined across all BBH events. Assuming all of the observed
signals are statistically independent, the combined posterior distribution is simply
given by the product of the posterior distributions across all events (suitably normal-
ized). The results using SEOBNRv4 in conjunction with the FTA framework are shown
in solid black lines; the 90% credible regions are demarcated with horizontal lines.
To compare, we also show the posteriors recovered using a different baseline GR
waveform?IMRPhenomPv2 [412, 469, 470]?generalized using a procedure other than
FTA [120, 176] in gray. Whereas the aforementioned EOB models are constructed by
resumming PN calculations and calibrating against NR, the IMRPhenomPv2 model
is a phenomenological fit to a combination of PN, EOB, and NR results; addition-
ally, this latter model includes precession, whereas the EOB models we consider
do not. Despite the different baseline waveform and generalization procedure, we
recover consistent measurements with both the EOB and phenomenological wave-
form models, indicating that waveform systematics do not significantly affect these
tests. Additionally, because the measured ???n are consistent with zero at a 90%
confidence level, our tests indicate that the BBH inspirals observed are consistent
with GR. We can convert this consistency check into a more quantitative measure
409
in Fig. 9.2; here we plot the 90% upper bounds on |???n| measured using SEOBNRv4
for each event
Similarly, the bottom panel of Fig. 9.1 shows the marginalized posterior dis-
tributions on the deviation parameters for GW170817. For comparison, we plot
the measurements recovered with SEOBNRT in blue and with an alternative model
PhenomPNRT?-constructed by applying the NRTidal model to IMRPhenomPv2 and
then generalizing using the procedure of Ref. [120, 176]?in orange. Again, we find
good agreement between the two waveform models, indicating that waveform sys-
tematics do not significantly bias the measurement of deviation parameters. Con-
sistency with GR at a 90% confidence level is found for all deviation parameters
except for at 3PN and 3.5PN, where the GR value falls at the 95th percentile of the
marginalized posterior. At present, we have no reason to believe that these offsets
have anything other than a statistical origin. The upper bounds on each deviation
parameter are also shown in Fig. 9.2. Note in particular the significant improve-
ment in the constraint on -1PN deviations made possible with GW10817. We use
this particularly strong constraint in the following section to place constraints on a
JFBD?a well-known alternative to GR.
9.4 Constraints on Jordan-Fierz-Brans-Dicke gravity from GW170817
Jordan-Fierz-Brans-Dicke gravity is one of the most well-known alternatives to
Einstein?s theory of general relativity [177?179]. Initially formulated in the mid-20th
century, JFBD was the very first ST theory?a theory in which gravity is mediated
410
by both a tensor (the metric) and a scalar. Since then, significant work has been
done to extend this notion beyond JFBD to broader, more generic classes of ST
theories (e.g. Horndeski theories [90], Beyond Horndeski theories [93], Degenerate
Higher-Order Scalar-Tensor theories [471, 472], etc.). Yet, despite its simplicity,
JFBD remains relevant today, though more as a pedagogical archetype of modified
gravity than as a truly viable alternative to GR. In this vein, constraining JFBD
with a particular experiment offers an easily understood benchmark of its sensitivity
to deviations from GR. In this work, we present the first bounds on JFBD from the
measurement of gravitational waves by the Advanced LIGO [63] and Virgo [64]
detectors.
The action for JFBD written in the Jordan frame is given by
? ?? ( )4 g? ? ?BDS = d x ?R? g????????? + Sm[g??? , ?], (9.7)
16? ?
where ? is a massless scalar field, ?BD is a dimensionless coupling constant
3, and
Sm represents the action for matter fields ? minimally coupled to the metric g??? .
Alternatively, the action can be rewritten in the Einstein frame by performing the
conformal transformation g?? ? ?g?? ???
4 ?gS = d x (R? 2g????????) + Sm[e?2?0?g?? , ?], (9.8)
16?
where we have defined the dimensionless parameter ? ? (3 + 2? )?1/20 BD and intro-
duced the scalar field ? ? log(?)/(2?0); note that ?0 is non-negative and that we
have implicitly assumed that ?BD > ?3/2. In the limit that ?0 ? 0 (?BD ? ?),
3JFBD is also commonly known as simply Brans-Dicke gravity (BD); following the standard
convention in the literature, we adopt this abbreviation when denoting the coupling constant ?BD.
411
the scalar field decouples from the metric and matter, and JFBD reduces to GR
with an additional free, massless scalar. Doppler tracking of the Cassini space-
craft through the Solar System [180] provides the best current constraints on this
parameter: ? < 4? 10?30 (?BD > 4? 104).
The recent advent of GW astronomy offers a new avenue to test gravity in
the relativistic regime. The majority of GWs observed by LIGO and Virgo thus far
were generated by the coalescence of BBHs; several tests of GR have already been
conducted using these observations [10, 73, 122, 187]. However, Hawking famously
showed that stationary BHs in JFBD must have a trivial scalar profile, and thus are
indistinguishable from the analogous solutions in GR [23]. Although there are some
possible scenarios that evade this no-hair theorem (see Ref. [98] for details), binary
systems composed of BHs are generally expected to behave identically in JFBD and
GR, and thus GWs from such systems are unable to constrain this ST theory.
Unlike BHs, NSs source a non-trivial scalar field in JFBD, and thus BNS sys-
tems can be used to constrain ?0. Using this fact, precise timing of binary pulsars
constrains ?0 . 10?2 [329]. In this section, we use the first GW observation of a
coalescing BNS?GW170817 [186]?to constrain ? ?10 . 4? 10 at a 68% confidence
level. Though the constraint from GW170817 is not as strong as those previously
quoted from other experiments, this result represents the first bound directly from
the highly dynamical (orbital velocities v ? 10?1) and strong-field (Newtonian po-
tential ? ?1Newt = M/R ? 10 ) regime of gravity.
This section is organized as follows. In Section 9.4.1, we detail the GW sig-
nature of JFBD in BNSs. Then, in Section 9.4.2, we present two Bayesian analyses
412
to constrain ?0 with GW170817: the first directly uses the theory-agnostic analyses
presented in Section 9.3, while the second is tailored specifically to test JFBD.
9.4.1 Gravitational-wave signature of Jordan-Fierz-Brans-Dicke grav-
ity
The predominant differences in GWs produced in JFBD as compared to GR
stem from the fact that only the latter respects the strong equivalence principle. This
principle extends the universality of free fall by test particles implied by the Einstein
equivalence principle introduced in Chapter 1 to also include self-gravitating bod-
ies; unlike in GR, the motion of a body through spacetime depends on its internal
gravitational interactions (i.e. its composition) in ST theories like JFBD. This sec-
tion details how this violation of the strong equivalence principle impacts the GWs
produced by binary systems in JFBD. This alternative theory of gravity falls within
the class of ST theories examined in Chapter 3, and thus we can employ PN predic-
tions computed there to the task at hand. Though those results were computed at
next-to-next-to-leading PN order (and even higher order PN calculations have been
completed recently [210, 473]), we will assume that ?0 is sufficiently small that we
can neglect all but the leading-order PN effects when describing the signature of
JFBD in a gravitational waveform.
The dominant effect on the inspiral from the new scalar introduced in JFBD
is the emission of dipole radiation, which enters into the phase evolution at -1PN
413
order. In the notation of the FTA framework, this contribution is given by
5(? ? ? )21 2 ( )
?? 4?2 = ? +O ?0 , (9.9)168
where ?A is the scalar charge of body A, defined as
? ?d logmA(?)?A , (9.10)
d?
where mA(?) is the gravitational mass of body A measured in the Einstein frame.
That a body?s mass depends on the local value of the scalar field is unsurprising
given the form of Eq. (9.8); a shift in ? modulates the physical metric e?2?0?g??
that effects gravity upon the matter fields ?, and thus also modulates any body?s
gravitational mass. This dependence is an explicit manifestation of violation of the
strong equivalence principle. Note that in the limit that a body has no self-gravity
(i.e. the test-body limit), the functional form of mA(?) simplifies significantly to
(test body) (test body)
mA (?) = e
??0?mA (? = 0), (9.11)
and thus its scalar charge reduces to
(test body)
?A = ?0. (9.12)
9.4.1.1 Isolated neutron star solutions in Jordan-Fierz-Brans-Dicke
gravity
As strongly self-gravitating bodies, violations of the strong equivalence princi-
ple are particularly pronounced in NSs. This violation manifests as a scalar charge
that differs significantly from the test-body charge ?0. As the scalar charges of a
414
binary?s constituents?or rather their difference, the scalar dipole?control the dom-
inant effect on the GW signal in JFBD, we devote the remainder of this subsection
to computing these quantities for various NSs.
We consider spherically symmetric, static solutions sourced by a perfect fluid
as a model for an isolated, non-rotating NS. Under these assumptions, the field
equations for Eq. (9.8) reduce to the Tolman-Oppenheimer-Volkoff (TOV) equations,
given in the Einstein frame in Ref. [153]. These solutions are parameterized by three
degrees of freedom; for our purposes, these are most clearly manifested as the (i)
background scalar field ?0, i.e. the scalar field far from the NS, (ii) the NS equation
of state (EOS), and (iii) the NS mass4. In fact, though, the asymptotic scalar field
?0 can be set to zero without loss of generality by rescaling the Jordan-frame bare
gravitational constant G? accordingly, i.e. ? ? 0 ? G? ? G?e2?0?00 . The remaining
degrees of freedom can be mapped to boundary conditions for the matter and scalar
field at the origin with a numerical shooting method [430]. These conditions are
parameterized by the central pressure Pc and scalar field ?c, which serve as the
inputs for numerically integrating the TOV equations. The details for extracting
the mass and scalar charge from the numerical solutions of the NS interior are given
in Ref. [153].
4We define the NS mass as the tensor mass mT introduced in Ref. [474] because?as shown in
that reference?it obeys the same conservation laws as the ADM mass in GR
415
Table 9.1: One-dimensional polynomial fits of the normalized NS coupling ?A/?0
as a function of NS mass mA (in units of M) for various equations of state.
EOS [?A/?0](mA)
sly[456] ?0.726798? 0.749029mA + 1.270944m2A ? 0.72871m3A + 0.161002m4A
eng[475] ?0.817884? 0.393375mA + 0.772615m2A ? 0.435306m3A + 0.095059m4A
H4[476] ?0.613880? 1.210074mA + 1.836631m2A ? 1.056595m3A + 0.228102m4A
9.4.1.2 Polynomial fits of the neutron star scalar charge
Ultimately, we would like to combine the numerical calculations of NS scalar
charge outlined above with their anticipated effect on the GW signal (9.9) to con-
strain JFBD. However, evaluating the scalar charge directly for every point visited
by the stochastic sampling algorithms used for parameter estimation would require
an unreasonable amount of computational resources. Instead, in this subsection, we
compute polynomial fits for the scalar charge, which, after having been derived, can
be evaluated quickly and with little computational overhead.
We first construct solutions for various choices of EOS, NS mass, and ST
coupling ?0. We interpolate tabulated EOS data for the sly [456], eng [475], and
H4 [476] EOSs; sly is a soft EOS (compact stars) whereas H4 is relatively stiff
(diffuse stars). Then, we numerically construct NSs with masses ranging between
mA ? [0.5M, 2.0M] and scalar coupling ?0 ? [0.001, 1.0] and compute their scalar
charge.
We calculate two types of polynomial fits of the scalar charge for each EOS.
For the first, we factor out the dominant linear dependence of ?A on ?0, fitting
their quotient as a fourth-order polynomial in mA. We compute the polynomial fits
416
Table 9.2: Two-dimensional polynomial fits of NS coupling ?A as functions of of
Brans-Dicke parameter ?0 and NS mass mA (in units of M) for various equations
of state.
EOS ?A(?0,mA)
sly[456] ?0(?0.92569 + 0.22258?0mA + 0.13329m2A ? 0.15151? m20 A)
eng[475] ?0(?0.97423 + 0.15584mA + 0.18527?0mA? 0.11739?0m2A + 0.024333m3A)
H4[476] ? (?0.93341 + 0.19073? m + 0.10270m20 0 A A ? 0.11284?0m2A)
with least-squares regression; the fits are given in Table 9.1 for each EOS that we
consider. These fits match all of our data sets within 5% relative error, with the
greatest discrepancy arising for masses close to 2M.
Although these (effectively) one-dimensional polynomial fits are crucial for
some of the analysis contained in this chapter, it is possible to construct two-
dimensional fits that are simpler (fewer terms) and more accurate using more so-
phisticated methods. We compute these fits using the greedy-multivariate-rational
regression method developed in Ref. [477]. This method relies on a greedy algorithm
to construct a multivariate fit: during each iteration, it adds a polynomial term to
the current fit (up to a pre-specified maximum degree) so as to best improve the
agreement with the inputted data. This process is repeated until sufficient accuracy
is achieved, and then terms are systematically removed from the polynomial until
the accuracy goal is saturated. Using this method, we construct fits that agree to
within 1% relative error for each EOS?these are listed in Table 9.2.
417
9.4.2 Constraining ?0 with GW170817
Next, we use the tools introduced in the previous subsections to place con-
straints on the ST coupling ?0 in JFBD with GW170817?the first GW event from a
coalescing BNS. We present two complementary analyses based off of the FTA infras-
tructure to achieve this result. These two methods follow the same overall approach,
but adopt different statistical assumptions, utilize different waveform models, and
use different numerical fits for the NS scalar charge ?A. In both approaches, we
employ a generalized waveform model that allows for additional contribution to the
phase evolution at -1PN order, so as to reproduce the behavior seen in Eq. (9.9);
however, the parameterization of this -1PN deviation from GR differs in each ap-
proach. Ultimately, both analyses provide a bound on ?0 of the same order of
magnitude.
The first approach we adopt directly uses the theory-agnostic constraints on a
-1PN deviation discussed in Section 9.3 and originally published in Ref. [9]. Recall
that this analysis used a generalization of the SEOBNRT baseline waveform model
in which -1PN deviations were parameterized by the deviation parameter ????2.
By assuming a particular NS EOS, we can use the polynomial fit in Table 9.1 in
conjunction with Eq. (9.9) to map a measured value of ????2 to an inferred value
on ?0; schematically, this mapping takes the form ?0(????2,m1,m2; EOS). Note
that this mapping is infeasible using the multivariate fit in Table 9.2 because of the
nonlinear dependence of ?A on ?0. Though the exact NS EOS remains unknown,
we can repeat this analysis for the three candidate EOSs detailed earlier, and then
418
use the variance in bounds on ?0 recovered each time as an estimate of systematic
error arising from our ignorance of the true NS EOS.
In practice, one does not measure the masses and deviation parameter ????2
with perfect accuracy, but instead uses Bayesian inference to reconstruct the poste-
rior distribution P (?|d) on these parameters given some assumed prior distribution
P (?). So, rather than map a single point from one parameterization to another,
one instead maps the appropriate distributions to their counterparts in the new
parameterization. These prior and p?osterior? distributions transform respectively as??? ????1??0P (?0,m1,m2) = P (????2,m1,m2), (9.13)?????2
and ?? ??1??0 ?
P (?0,m1,m2|d) = ?? ?? P (????2,m1,m2|d), (9.14)?????2
where the first term on the righthand side is the inverse of the Jacobian of the
aforementioned transformation.
In the analysis of Section 9.3 (and Ref. [9]), a flat prior (i.e. a uniform distri-
bution over a bounded region) was assumed on the component masses and deviation
parameter ????2. These choices reflect the theory-agnostic nature of that test; with-
out a preferred alternative, this choice represents the simplest prior in terms of these
binary parameters. Figure 9.3 depicts with dashed lines how this choice of prior dis-
tribution maps to an assumed prior on ?0 through Eq. (9.13); here the different
colors correspond to different assumed EOSs. Similarly, Fig. 9.4 shows the corre-
sponding marginalized posteriors on ?0, transformed from the posterior on ????2
and component masses (which, when marginalized, is shown in the bottom panel of
419
1.0
SLy
0.8 ENG
0.6 H4
0.4
0.2
0.0
0 1 2 3 4 5
Figure 9.3: Marginalized prior distributions on the JFBD parameter ?0
used in the two analyses discussed in this chapter. The dashed colored
curves depict the prior distribution equivalent to the flat prior distri-
bution on component masses and deviation parameter ????2 assumed in
the theory-agnostic analysis of Section 9.3. The solid black dashed curve
depicts the flat prior on ?0 assumed in the second analysis presented in
this section.
Fig. 9.1) via Eq. (9.14). This analysis provides a bound of ? ?10 . 2? 10 at a 68%
confidence level; note that the systematic error arising from our ignorance of the NS
EOS does not impact our estimate at this level of precision.
The second approach we employ to constrain ?0 relies instead on a waveform
model design specifically to test JFBD. Using the FTA infrastructure, we construct
a generalized waveform model from SEOBNRT in which the deviation parameter is
precisely ?0. The appropriate form of the -1PN correction to the phase evolution
is obtained by inserting the polynomial fit for ?A(?0,mA) found in Table 9.2 for a
particular choice of EOS into Eq. (9.9). Additionally, unlike the previous theory-
agnostic analysis in which the tidal parameters were allowed to vary freely, for this
analysis, we express these parameters as functions of the respective NS masses and
420
8
SLy
6 ENG
4 H4
2
0
0.0 0.2 0.4 0.6 0.8 1.0
Figure 9.4: Marginalized posterior distributions on the JFBD parameter
?0 recovered from GW170817 from both analyses discussed in the text.
The dashed colored curves show the posterior recovered directly from
the theory-agnostic analysis of Section 9.3. The solid colored curves
depict the posterior using the theory-specific test of JFBD assuming
a flat prior for ?0. For both analyses, different colors correspond to
different assumed EOSs. Colored ticks on the horizontal axes represent
the 68% upper bounds on ?0 for each analysis.
421
assumed EOS.5 This step reduces the dimensionality of the waveform model by
two parameters while ensuring that all matter effects are handled self consistently.
We assume a flat prior on ?0 ? [0, 1] for this analysis; beyond this upper bound,
our assumption that JFBD effects, which scale as ?20, are subdominant to the PN
effects in GR is no longer valid. This prior distribution is depicted in Fig. 9.3 with
a solid black curve. Using the generalized waveform described above, we perform
parameter estimation to construct the marginalized posterior distribution on ?0,
shown in Fig. 9.4 with solid colored curves corresponding to the assumed EOS. We
obtain the upper bound of ?0 . 4? 10?1 where, as before, the systematic error due
to ignorance of the true NS EOS does not contribute at this level of precision.
Comparing the bounds set by the two analyses, we see that the theory-agnostic
test provides a stronger bound on ?0. At first glance, this result may appear coun-
terintuitive, as Fig. 9.3 shows that this test assumed a marginalized prior on ?0
with greater support away from zero. The predominant cause for this discrepancy
stems from how the tidal parameters are handled by each waveform model. For the
theory-agnostic test, these parameters are allowed to vary freely, independent of the
masses of the NSs. However, in the theory-specific test, the tidal parameters are
linked directly to the component masses. This latter restriction significantly affects
the recovered posterior distribution on the component masses, placing much greater
5We use polynomial fits to the tidal parameters as a function of NS mass that are constructed
in GR, not in JFBD. However, the differences between these two relations scale as ?20, and thus can
be neglected in favor of the simpler GR relation by the same reasoning that the other sub-dominant
PN effects (e.g. at 0PN order, 0.5PN order, etc.) can be ignored.
422
weight near equal-mass configurations than in the previous case. As can be seen in
Eq. (9.9), in very symmetric configurations, the total deviation from the baseline
GR waveform remains small even when ?0 is relatively large; as a result, the JFBD
parameter is more poorly measured when the tidal parameters cannot vary freely,
and thus we recover a weaker bound with this theory-specific test.
9.5 Constraints on higher-order curvature corrections from GW151226
and GW170608
The previous two sections demonstrate two very different ways in which the
FTA framework can be used to test GR with GW observations. These investigations
represent two opposite ends of the spectrum of theory-agnostic (Sec. 9.3) vs. theory-
specific (Sec. 9.4) tests. In this final section, we again use the FTA framework,
but this time to perform tests that fall between these two extremes. Using the
powerful tools of effective field theory (EFT), we place constraints on higher-order
curvature corrections expected to arise in many possible UV-completions of GR.
The assumptions made in constructing the extensions of GR are minimal; the most
conspicuous of them is that we restrict our attention to theories for which deviations
from GR are conceivably observable with ground-based GW detectors.
In the interest of brevity, we limit our discussion concerning the construction
of the EFT in this thesis and instead direct interested readers to Ref. [181]. We
consider the most general extension to GR under the following assumptions: (i)
locality, causality, Lorentz invariance, unitarity and diffeomorphism invariance are
423
preserved by new physics; (ii) no new particle lighter than the cutoff scale of the
theory is introduced; (iii) the extension to GR impacts phenomena observable with
the LIGO and Virgo detectors. The r(esulting EFT takes the? )
form
? 2 2
S = 2M2
C C? C C?
eff pl d
4x ?g ?R + + + + . . . , (9.15)
?6 6 ?6
? ?? ?
where Mpl = ~c/G is the Planck mass,
C ? R ???????? R , C? ? R ?????? ?? R???? , (9.16)
?, ??, ?? ?1? ? O(km ) are various cutoff scales of the EFT, the dots in Eq. (9.15)
denote terms with powers in the Riemann tensor beyond four, and the Levi-Civita
?
tensor ???? is defined such that 0123 = 1/ ?g. The cutoff scales are taken to be no
greater than km?1, as this is approximately the maximum mass scale (shortest dis-
tance scale) to which ground-based GW detectors would be sensitive; for reference,
the Schwarzschild radius of a one solar-mass BH is ? 3km.
Naively, one might expect corrections to gravity occurring on distance scales
of kilometers to be already strongly constrained by existing laboratory and Solar
System tests. However, the curvature scales involved in those experiments are much
smaller than those reached near solar-mass BHs, and thus the possibility remains
that new physics could remain at the km?1 scale that only emerges in the high-
curvature regime. This possibility can be formalized into an assumption of a ?soft
UV completion? of the theory, which states that all EFT effects saturate at the
cutoff scale, i.e. they do not continue to grow as some power of E/?c at energies E
above the cutoff ?c?see Ref. [181] for more detail.
424
9.5.1 Gravitational-wave signature of higher-order curvature correc-
tions
Having established the assumptions and action (9.15) describing our EFT, we
next turn to the signature of this extension to GR in the inspiral signal of a BBH.
For simplicity, we restrict our attention to only the C2 term in (9.15), as this provides
the dominant (lowest PN order) effect in the waveform. Furthermore, we assume
that ? . 1/M , where M is the total mass of the binary; other effects are known to
play an important role in the regime wherein ? & 1/M (such as tidal effects) [478],
but we have checked that these are not observable for the BBH systems that we
consider.
During the early inspiral, the dynamics of a binary system of two objects with
mass m1 and m2 can be characterized by an effective action that in center-of-mass
frame take?s the(form [181] )
1 1
S = dt m1 +m2 + ?(t)|v|2 ? V (r(t)) + Qij(t)Ri0j0 + . . . , (9.17)
2 2
where ? is the reduced mass of the system, |v| is the relative velocity between the
inspiraling objects, V (r(t)) is the potential energy, Qij(t) is the mass quadrupole
moment of the system and the dots represent higher-order multipole moments.
As shown in Ref. [181], the leading-order PN corrections to the gravitational
potential of a non-spinning or slowly-sp(innin)g binary system take the form6
? 2 m1m
2 2
2 2? 4(m1 +mV = 2
)
. (9.18)
?6 r ?r r2
The non-GR terms in the action (9.15) also modify the emission of GW radiation.
425
The leading-order effect manifests as a correction to the Newtonian quadrupole
moment QNewtij of the bi[nary system( ) ]6( )2
? 21 2? 2(m1 +m2)Q = 1 + QNewtij ij . (9.19)2?6 ?r r
In the following subsections, we compute the effect of these corrections on the con-
servative and dissipative sectors independently. We then use the balance equation
to relate the two and to compute the leading-order correction to the GW phase.
9.5.1.1 Conservative dynamics
Working on the orbital timescale, the various time-dependent quantities in the
action (9.17) can be treated as constant and the radiative terms can be neglected.
Restricting to quasi-circular orbits and varying the action (9.17) with respect to
r(t), the equations of motion of the binary system can be written as:
2 dV dVNewt dV??? r = = + + . . . , (9.20)
dr dr dr
where . . . denote higher PN corrections. This relation can be inverted to get a
relation between the orbital radius r and the orbital frequency ?. To leading order
in 1/? one gets
( )
M ? 1536 (m
2 2
r = 1 1
+m2) (M?)16/3 . (9.21)
(M?)2/3 M8?6
Using the equations above one finds that the binding energy (per unit total
mass) is given by
( )6
1
E = ?|v|2 + V (r)/M = ?1?v2 ? ? d?2560?(1 2?) v18, (9.22)
2 2 M
426
where M = m1 + m2 is the total mass of the system, ? = m1m
2
2/M is the sym-
metric mass ratio and we have defined v ? (M?)1/3. For convenience we have also
introduced the parameter d? ? 1/?. Restoring the higher PN corrections in GR we
have ( )6
? d?E (v) = EGR ? 2560? (1? 2?) v18 . (9.23)
M
where EGR denotes the PN expression for the binding energy in GR.
9.5.1.2 Dissipative dynamics
The renormalized quadrupole moment (9.19) leads to corrections to the GW
flux. As in GR, Eq. (9.17) predicts a leading-order GW flux given by the quadrupole
formula
F 1
... ...ij
= < QijQ > , (9.24)5
where < ? ? ? > indicates the average over an orbit. The resulting GW flux can be
written as
F(v) = FGR(v) + F?(v) , (9.25)
where FGR(v) is the PN expression for the flux in GR and the leading-order correc-
tion to the flux F?(v) reads
( )( )6
F? 393216 24576 d?(v) = ?3 ? ?2 v26 . (9.26)
5 5 M
427
9.5.1.3 Gravitational waveform in the stationary phase approxima-
tion
We are now in the position to compute the leading-order correction to the
gravitational-wave phase due to the non-GR corrections computed above. In the
PN regime one can compute the Fourier representation of the GW waveform using
the stationary phase approximation (see e.g. [259, 479]). In this approximation the
waveform in the frequency domain can be written as [259]
?A(tf )h?(f) = ei[?f (tf )??/4] ,
F? (tf )
?f (t) ? 2?ft? ?(t) , (9.27)
where A(t) is the amplitude of the time-domain waveform, ?(t) is the orbital phase
of the binary and ?F (t) = d?(t)/dt defines the instantaneous GW frequency F (t).
The quantity tf is the saddle point where d?f (t)/dt = 0, i.e. the time when F (t) is
equal to the Fourier variable f . In the adiabatic approximation ?f and tf are given
by ? vref E ?(v)
tf = tref +M F ? dv , (9.28)v (v)f vref E ?(v)
?f (t
3 3
f ) = 2?ftref ? ?ref + 2 (vf ? v )
v F
dv , (9.29)
(v)
f
where we defined v 1/3f ? (?Mf) , tref and ?ref are integration constants and vref
is and arbitrary reference velocity, commonly taken to be the velocity at the last
stable orbit.
Using the PN expansions of the energy and flux and expanding the ratio
E ?(v)/F(v) at consistent PN order, the integral in Eq. (9.29) can be solved explicitly.
428
We find that the leading-order to the GW phase is given by
( )( )6
3 234240 522240 d?
?(f) = ?GR(f) + ? ? v16 , (9.30)
128?v5f 11 11 M
f
where ?GR(f) represents the GW phase in GR. An important conclusion of this cal-
culation is that although the non-GR corrections enter formally at 8PN order, when
d v2/M ? (?r)?1? ? 1 the corrections are numerically of 2PN order in magnitude.
9.5.1.4 On the validity of the post-Newtonian waveform model
By construction, the EFT action (9.15) is valid only for orbital separations
r & d?; below this separation, terms containing higher powers of ??1 can impact
the binary dynamics. Therefore, one must be careful to only employ this model in
this limited regime of validity. For the purposes of parameter estimation, this lower
limit on orbital separations r is more conveniently expressed as an upper limit on
the frequency content of the signal. However, the conversion between these limits in
not unique because r is not a gauge-invariant quantity. A natural choice is to first
relate r to the orbital frequency ? through the relation r = (M/?2)1/3 +O(v2/c2).
Then, because the dominant contribution to the GW signal comes at twice the
orbital fre?quency during the adiabatic inspiral, we can define a cutoff frequency
f? ? 1/? M/d3?, such that the waveform model given by Eqs. (9.27) and (9.30)
is expected to be valid for GW frequencies f  f?.
Additionally, because we employ the PN approximation to compute our wave-
form, we must also impose that the typical velocity of the system v  1. Again,
there is no single well-defined point at which the PN approximation is no longer
429
valid, but a reasonable benchmark is the innermost stable circular orbit (ISCO) of an
equivalent BH with mass M . We use this point as a canonical cutoff point for the PN
?
approximation, restricting our waveforms to frequencies below fISCO = 1/(6 6?M).
Combining these two conditions, our model is only valid for frequencies f <
fcutoff ' min[fISCO, f?].
9.5.2 Constraining d? with GW151226 and GW170608
Having calculated the impact of higher-order curvature corrections on the
phase evolution of the inspiral, we now establish a statistical framework to con-
strain such deviations from GR with GW observations. As before, the first step
is to construct a generalized waveform that can represent these deviations. We
use the FTA framework to add deviations of the form given in Eq. (9.30) to a PN
waveform model in GR. As a baseline waveform model, we use the TaylorF2 wave-
form approximant, an inspiral-only PN model given in the frequency domain that
is valid for spinning, non-precessing systems up to 3.5PN order.6 However, unlike
the approach in Secs. 9.3 and 9.4, we use Bayesian model selection to constrain the
effective distance-scale of new physics d?.
As discussed in Chapter 1, the goal of Bayesian model selection is to calculate
the odds ratio between two competing (1.26) hypotheses by computing the Bayes
factor?the ratio of evidences?for either proposal. Here, the two hypotheses we
6As a check of consistency, we also repeat this analysis using the inspiral-merger-ringdown model
IMRPhenomPv2 (restricted to non-precessing systems) and find similar results. For this separate
analysis, we do not impose the frequency cutoff of fISCO required by the PN approximation.
430
consider are: (i) Hnon-GR: higher-order curvature corrections enter at some fixed
scale d? = d
?
? ? km ; and (ii) HGR: higher-order curvature corrections are not
present or (equivalently) appear at infinitesimal distance scales d? = 0. The odds
ratio between these hypotheses is given by
Onon-GR p(Hnon-GR)= BFnon-GRGR p(H ) GR , (9.31)GR
where the Bayes factor BF is determined by
non-GR ? p(d|Hnon-GR)BFGR (9.32)p(d|HGR)
given a particular GW observation d. Normally, one must be careful when using
Bayesian model selection to perform tests of GR because the null and alternative
hypothesis typically have parameter spaces of different dimensions. We avoid this
particular obstacle by fixing d? in Hnon-GR to some particular value; we perform this
test repeatedly for different values of d? to scan the full parameter space. To remain
agnostic, we always assume prior odds p(Hnon-GR)/p(HGR) of unity.
The evidence for each hypothesis is computed as the marginalization of the
likelihood over the space of binary parameters
?
P (d|H) = d?P (d|?,H). (9.33)
The total likelihood is given by the (normalized) product of likelihoods for the signal
di observed at each detector i
?
P (d|?,H) ? Pi(di|?,H), (9.34)
i
431
where Pi(di|?,H) takes the same form as introduced in Chapter 1:
P (d |?,H) ? e(di?h(?)|di?h(?))i i , (9.35)
with the inner product weighted by the power spectral density Sin(f) of each detector? fhigh
| ? a(f)b
?(f) + a?(f)b(f)
(a b) 2 . (9.36)
f S
i (f)
low n
The evidence for each hypothesis is evaluated using the nested sampling algorithm
implemented in the LALInference code package [78].
We consider the bounds that can be set by GW151226 [182] and GW170608 [183],
the two lowest mass (i.e. longest) BBH events detected by LIGO and Virgo during
the first two observing runs. Because of their low mass, the vast majority of the
SNR of these signals is concentrated in their inspiral phase, which minimizes the
measurement biases inherited from using a inspiral-only waveform model. For each
event, we use flow = 20 Hz, as this is the lowest frequency for which Sn(f) has been
released for these events.7 We choose fhigh so as to respect the range of validity of
the waveform, i.e. fhigh < fISCO and fhigh < f?. The first of these conditions re-
stricts the waveform to the PN regime; it is automatically imposed in the LALSuite
implementation of the TaylorF2 waveform mode, as the integrand of Eq. (9.36)
always evaluates to zero for frequencies above fISCO. The second condition restricts
the waveform to the regime in which the EFT is valid. However, because the fre-
quency at which this approximation breaks down is unknown a priori, there remains
7This research has made use of data, software and/or web tools obtained from the Gravitational
Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the
LIGO Scientific Collaboration and the Virgo Collaboration.
432
f?[Hz] f?[Hz]
950 450 300 200 150 100 80 950 450 300 200 150 100 80 60
0 0
?20 ?20
GW151226,TaylorF2 GW170608,TaylorF2
?40 fhigh = 0.5f? ?40 fhigh = 0.5f?
fhigh = 0.35f? fhigh = 0.35f?
fhigh = 0.25f? fhigh = 0.25f?
?60 ?60
100 200 300 400 100 200 300 400
d?[km] d?[km]
Figure 9.5: The log Bayes factors of non-GR versus GR hypotheses with
(left) GW151226 and (right) GW1?70608 for different choices of d? (in
km). The corresponding f? ? 1/? M 3tot/d? shown on the top x-axis.
Different choices of cutoff criteria fhigh are shown in different colors.
some ambiguity in how to choose fhigh; this ambiguity comprises a source of system-
atic error in our analysis. To gauge the impact of this ambiguity, we perform our
analysis with the different choices of cutoff frequency fhigh ? [0.25f?, 0.35f?, 0.5f?].
Smaller choices for fhigh represent more conservative usage of our EFT, whereas
larger choices assume that it?remains valid over a larger range of scales. For sim-
plicity, we compute f? ? 1/? M 3tot/d? by fixing the total BH mass to the median
value for the total BH mass obtained from parameter estimation with a full IMR
waveform model in GR as given in Refs. [182, 183].
Figure 9.5 depicts the logarithm of the Bayes factors in opposition of versus
in support of GR from GW151226 (left) and GW170608 (right) as a function of
d?. Results computed using different cutoff criteria are depicted with different
colors?red, green, and blue corresponding to the least to most conservative cutoff,
respectively. For each event, the value of f? corresponding to d? is shown on the top
433
log BFnon?GRGR
log BFnon?GRGR
0
?20
?40
GW151226 + GW170608
?60 fhigh = 0.5f?
fhigh = 0.35f?
? f = 0.25f80 high ?
100 200 300 400
d?[km]
Figure 9.6: Same as Fig. 9.5 but computed by combining information
from both GW151226 and GW170608.
axis. We observe that log BFnon-GRGR < 0 across a range of values of d?, indicating that
the non-GR hypothesis is disfavored in those regions. These regions largely overlap
across the two events due to their similar total mass; however, one obtains slightly
stronger evidence in support of GR from GW170608 due to the event?s larger SNR
relative to GW151226.
The qualitative features of our results can be understood in terms of two ef-
fects. First, the Bayes factor asymptotically approaches Bnon?GRGR ? 1 as d? tends
to zero because the non-GR contribution to the waveform scales with d6?. Second,
the Bayes factor also asymptotically approaches Bnon?GRGR ? 1 for large d? because
this limit corresponds to a very small cutoff frequency fhigh. By restricting our-
selves to only the very low-frequency content of the observed GW, we no longer
have sufficient data to distinguish the signal from noise regardless of which model
(non-GR vs. GR) we assume, and thus neither hypothesis is preferred. Assuming
434
log BFnon?GRGR
statistical independence between the two observed events, we can compute Bayes
factor using both observations by simply multiplying the Bayes factors from each
event individually; this combined result is shown in Fig. 9.6.
The Bayes factors discussed above indicate the odds for/against higher-order
curvature corrections at a given scale d?. These can be translated into a constraint
on d? by setting some significance threshold beyond which one unequivocally accepts
either hypothesis. We adopt as our threshold | log BF| & 5, shown as a horizontal
gray line in Figs. 9.5 and 9.6. We note that this threshold is substantially larger
than standard choices suggested in the literature [480]. We adopt this conservative
threshold because we find that systematic effects?such as changing the choice of
prior distributions?can shift log BF by as much as ?3; we suspect that these sys-
tematic errors also cause the small fluctuations seen in Figs. 9.5 and 9.6. Imposing
this significance threshold, we rule out the existence of higher-order curvature con-
tributions on the distance scales of d? ? [70, 200] km using the most conservative
cutoff criteria we considered; relaxing this cutoff criteria increases the width of the
excluded region of parameter space.
435
Chapter 10: Conclusions and future work
In this thesis, I have presented work directed towards extending and improving
tests of GR with GWs. This is an incredibly multi-faceted endeavor; even when
focusing on just one portion of only a single type of GW observation?the inspiral
of a coalescing compact binary system?I addressed this central objective from a
variety of directions, employing an assortment of different techniques. In this final
chapter, I wish to reiterate the common threads that connect the work described in
the previous chapters of this thesis. Then, inspired by this summary, I conclude by
addressing directions for future research.
One key theme of this thesis was modeling the behavior of binary inspirals
and the GW signals they produce in alternative theories of gravity. Not only are
these modeling efforts necessary for any theory-specific test of GR, they can also
provide a conceptual foundation for more theory-agnostic approaches as well. I
carried out work of this nature in great detail in Chapters 2 and 3 for two particular
scalar extensions of GR: Einstein-Maxwell-dilaton gravity (EMd) [1] and massless
scalar-tensor (ST) theories [2], respectively. These theories represent perturbative
deviations from GR, in the sense that there exists a limit in which the scalar degrees
of freedom smoothly decouple from gravity and matter, and binaries behave as
436
in GR. By contrast, in Chapters 4 and 5, I examine a ST theory in which non-
perturbative deviations from GR manifest in the form of scalarization of NSs [3, 4].
In the strong-coupling regime, NSs (either in isolation or in binary systems) can
undergo a phase transition during which the scalar field is spontaneously excited;
I study in particular detail the behavior of binary systems during the onset of this
transition.
Another focus of this thesis was modeling specific phenomenology known to
occur across a range of alternatives to GR and then understanding how this can be
connected to specific examples of such theories. In contrast to the previous chap-
ters, this type of approach embodies a more bottom-up philosophy. Taking lessons
learned in these earlier chapters, Chapter 6 presented a theory-agnostic description
of scalarization and offered a simple yet robust parameterization suitable for GW
searches for such non-perturbative effects [5]. Chapter 7 described how the generic
signatures of spontaneous and dynamical scalarization could be translated to bounds
on a specific ST theory that manifests such phenomena [6]. Chapter 8 examined
a different type of signature for new strong-field physics: the tidal deformations of
compact objects close to the merger of a binary system [7]. This phenomenon is
known to occur for mundane sources (e.g. NSs) as well as exotic alternatives (e.g.
?BH mimickers?); here I estimated the extent to which such observations can be
used to distinguish boson stars from NSs and BHs.
Finally, in Chapter 9, I developed tools to test non-GR phenomenology in
observed GWs not linked to any particular alternative theory. The primary result
of this work was the introduction of the flexible theory-agnostic (FTA) framework
437
that allowed one to adapt any frequency-domain waveform model for use in such
tests. As a member of the LIGO Scientific Collaboration, I used this framework to
perform theory-agnostic tests on BBH and BNS inspirals observed with Advanced
LIGO and Virgo [9, 10]. However, I then circled back to the efforts presented at the
start of the thesis by using the FTA infrastructure to perform theory-specific tests
for two specific alternatives to GR. [11, 138]
A crucial takeaway from the previous discussion is that many of the exact
same tools can be used for theory-specific and theory-agnostic (or top-down and
bottom-up) tests of GR. This fact has been used often to convert the bounds from
theory-agnostic tests (e.g. those produced by the LIGO and Virgo collaborations)
to constraints on particular modified theories of gravity; see, for example Refs. [187,
481]. However, theory-specific and theory-agnostic analyses test slightly different
statistical hypotheses, and within a fully Bayesian framework, converting results
from one to the other requires care. To my knowledge, this issue has not been studied
in detail in the context of GW tests of GR, and thus offers an interesting new avenue
for future work. The ultimate goal of such a study would be to understand how
to systematically translate results between theory-specific and theory-agnostic (as
well as top-down and bottom-up) tests. I provide some speculative starting points
for such an investigation below, using the theory-agnostic and theory-agnostic tests
presented in Chapter 9 as prototypical examples.
Recall that in Chapter 9 a theory-agnostic test of dipole radiation (or equiva-
lently, an anomalous -1PN contribution to the GW phase evolution) employed the
same type of generalized waveform model as a theory-specific test of Jordan-Fierz-
438
Brans-Dicke gravity (JFBD). In fact, there exists an invertible map between the
-1PN deviation parameter ????2 used in the former and the JFBD parameter ?0
used in the latter that allows the two waveform models to be identified. Thus,
within a Bayesian context, the likelihood function P (d|?) in either test would be
identical [see Eq. (1.23)]; the only difference between these two tests are the choice
of prior functions P (?). Note that any choice of prior distribution can be identified
with a family of parameterizations in which that prior is uniform over its support,
i.e. one can always find a reparameterization ? ? ??? such that the Jacobian??????a ???? ? 1 (10.1)??b P (?)
over the support of P (?), and thu?s??????a ??P (??) ? ??P (?) = constant. (10.2)??b
In the theory-agnostic test of dipole radiation presented in Chapter 9, the param-
eterization corresponding to a flat prior used ????2, whereas in the (second) test of
JFBD, this parameterization used ?0. So, an open question for future study is how
should one choose the prior?or equivalently, preferred parameterization?for tests
of GR?
One approach, in line with a top-down philosophy, would be to choose a pre-
ferred parameterization that leaves ?physical? parameters unbiased by the prior.
For example, in a theory-specific test, these parameters could be the fundamen-
tal masses and couplings that enter at the level of the action. In contrast, a
more theory-agnostic approach could follow the principles of effective field theory
and include all relevant operators in an action up to some dimension. Then, the
439
masses/couplings of each operator would serve as the natural parameters to test
GR. Similar approaches have been adopted for cosmological tests of modified grav-
ity for theories in which the current accelerated expansion of the Universe is caused
by a single scalar field [482?486]. However, these models have been used primarily
to study Friedmann-Lema??tre-Robertson-Walker solutions and their linear pertur-
bations, whereas modeling the inspiral of compact objects could require greater
understanding of nonlinear interactions. In the most ambitious scenario, one could
hope to formulate GW tests of GR with an approach like the Standard Model Ex-
tension [487?489], which provides a complete effective-field-theoretical framework
for studying CPT and Lorentz violations on curved backgrounds.
Another approach, more in line with a bottom-up philosophy, would be to
choose a parameterization based off of our understanding of waveform models in GR
and the characteristics of our detectors. For example, adopting a Jeffreys prior [490]
would eliminate the parameterization-dependence of one?s statistical analysis; with
this prior, the likelihood function P (d|?) takes the same form for any parameteri-
zation ?. This prior assigns greater weight to regions of parameter space in which
detectors are most sensitive, as determined by the determinant of the Fisher infor-
mation matrix ?ij(?). For the generalized waveforms conventionally used for GW
tests of GR?such as all of those used in Chapter 9?one can always reparameter-
ize the deviation parameters of the waveform such that Jeffreys prior appears flat
(i.e. is uniform over some finite bounded region of parameter space). Though not
tied directly to any underlying physical parameter, this new parameterization of-
fers more appealing statistical properties (e.g. parameterization-independence) than
440
that currently used in standard theory-agnostic tests.
441
Appendix A: The 1PN two-body Lagrangian in Einstein-Maxwell-
dilaton theory
In this appendix, we derive the 1PN two-body Lagrangian in EMd theory using
the Fokker action method [207] (see also Refs. [208?210]). To derive the Lagrangian,
we expand the EMd action in Eq. (2.2), together with the matter action for point
particles in Eq. (2.3) and the mass expansion from Eq. (2.7). After that, we obtain
the field equations for the potentials, solve them, and plug the solutions back into
the action to get the Lagrangian. Throughout, we work in the harmonic gauge
g?????? = 0 and the Lorenz gauge ? A
?
? = 0. Also, in this appendix and the next,
we explicitly write c and G for bookkeeping.
A.1 Expanding the metric and connection coefficients
Before expanding the EMd action, we start by expanding the metric in powers
of v/c [491],
g00 = ?1 + 2V ? 2V 2 + . . . ,
g0i = ?4Vi + . . . ,
gij = ?ij + 2V ?ij + . . . , (A.1)
442
where the potentials V ? O(1/c2), and V 3i ? O(1/c ). The inverse metric satisfies
g??g ??? = ? ? .
The connection coefficients in terms of the metric are given by
??
1
= g???? (??g?? + ??g?? ? ??g??) . (A.2)
2
Plugging the metric expansion in terms of the potentials yields the connection co-
efficients to O(1/c4)
?000 = ??0V ,
?00i = ??iV ,
?i00 = ??iV + 2?iV 2 ? 4?0Vi ,
?0ij = 2 (?jVi ? ?iVj) + ?ij?0V ,
?i0j = 2 (?iVj ? ?jVi) + ?ij?0V ,
?ijk = ?(1 + 2V ) (?ij?kV + ?ik?jV ? ?jk?iV ) . (A.3)
A.2 Expanding the action
The action of EMd theory is given by Eq. (2.2). We can divide that action
into four pieces
S = Sg + S? + Sem + Sm , (A.4)
where Sg is the gravitational action, S? is the dilaton action, Sem is the electromag-
netic action with the dilaton coupling, and Sm is the matter action.
In the Einstein frame, the gravitational action is the same as in GR. The
443
Einstein-Hilbert gravitational?action can be written in the Landau-Lifshitz form
c4 ? ( )
S = dtd3 ??g x ?gg ?? ? ? ?????? ? ?????? . (A.5)16?G
Substituting the connection coefficients from Eq. (A.3) in terms of the potentials
leads to
4 ?c [ ]
S 3g = dtd x ? 2?iV ?iV ? 16?iV ?0Vi ? 6?0V ?0V + 8?iVj?iVj ? 8?iVj?jVi .
16?G
(A.6)
Imposing the harmonic gauge condition g?????? = 0 gives ?0V + ?iVi = 0. Applying
that condition in the act?ion and integrating by parts yields
c4 [ ]
Sg = dtd
3x ? 2?iV ?iV + 2?0V ?0V + 8?iVj?iVj . (A.7)
16?G
The dilaton action is given by
c4
?
? ?S = dtd3x ?gg??? ?????? . (A.8)
8?G
Since ? is of order 1/c2, then to O? (1/c
2)
? c
4
S? = dtd
3x (??0??0?+ ?i??i?) . (A.9)
8?G
The electromagnetic action i?ncluding the dilaton coupling is given by?1 ?
S = dtd3x ?ge?2a?F F ??em ?? , (A.10)
16?
with the electromagnetic field F?? = ??A? ???A? = ??A? ? ??A? and the vector
potential A? = (A
2
0, Ai). The component A0 = O(1) + O(1/c ) + . . . , while the
components Ai = O(1/c) + . . . . Therefore, expanding F F ???? to O(1/c2) leads to
F ????F =? 2?iA0?iA0 + 2?jAi?jAi + 4?0Ai?iA0 ? 2?iAj?jAi . (A.11)
444
Because the last two terms in Eq. (A.11) are of order 1/c2, we can use integration
by parts and the Lorentz gauge condition (? A?? = 0) to replace these last two terms
?
by 2?0A0?0A0. Since ?g = 1+2V , and e?2a? ' 1?2a?+ . . . , the action becomes?
1 [ ]
Sem = dtd
3x (1 + 2V ? 2a?)?iA0?iA0 ? ?jAi?jAi ? ?0A0?0A0 . (A.12)
8?
The matter action Sm for point particles at monopolar order (dipole/spin and
higher multipoles neglected[) is given by?? ? ]?
Sm = ? dt mA(?)c2 ?g ? ?
1 dx
?? vAvA/c
2 ? qAA? , (A.13)
c dt
A
where the field-dependent m[ass of each body has the expansion]given by Eq. (2.7)
1 ( )
m(?) = m 1 + ??+ (?2 + ?)?2 +O 1/c6 . (A.14)
2
Defining the mass density ?g in terms of the constant masses
?
? ? m ?3g A (x? xA), (A.15)
A
and defining the electric charge density by
?
?e ? q 3A? (x? xA), (A.16)
A
then the matter a[ction to O(1/c
2
( ) can be written as? )
3 ? 1 1 v
4 3 1
Sm = dtd x (?g c
2 + v2 + V c2 + + V v2) ? V
2c2 ? 4V vic
2 8 c2
i
2 2 ( )]
? 2 1+ ? ?? c + v2 1 1g + V c2 ? c2?g(?2 + ?)?2 + ?e A0 + A iiv .
2 2 c
(A.17)
The parameters ? and ? will be assigned a subscript when multiplied by the delta
functions in ?g.
445
A.3 The field equations
Combining the expansion of the action from the previous subsection, the total
action?at 1PN {order is given by
c4
S = dtd3x [[?2?iV ?iV + 2?0V ?0V + 8?iVj?iVj]16?G ]
4
+ ? ? 2 1c + v2 2 1 v 3 1g + V c + + V v2 ? V 2c2( ? 4Viv
ic
2 8 c2 2 2 )
? c
4
(? 1?0??0?+ ?i??(i?) + ?g?? )?c
2 + v2 + V c2
8?G 2
? 1c2? (?2 + ?)?2 1 ig + ?e A0 + A2 iv2 c }
1
+ [(1 + 2V ? 2a?)?iA0?iA0 ? ?jAi?jAi ? ?0A0?0A0] . (A.18)
8?
Varying the action with respect to the potentials Vi, Ai, V , ?, and A0 respec-
tively yields the field equations
?2 ?4?GV = ? vii g , (A.19)
c3
?2 ?4?Ai = ? iev , ( ) (A.20)c
?4?G ? 4?G 3 2 ? 2 ? 4?G GV = ?g [ ?g v V c ?]g??? ?2 4 2 iA0?iA0 , (A.21)c c 2 c c4
?4?G ? 1 Ga? = ?g ? + ?v2 + ?V ? (?2 + ?)? + ?iA ? A4 4 0 i 0 , (A.22)c 2 c
A0 = 4??e ? 2V?2A0 ? 2?iV ?iA0 + 2a??2A0 + 2a?i??iA0 , (A.23)
where  = ??2 +?20 is the flat d?Alembertian.
The first two equations can(be solved directly for)Vi, and Ai
G m i i1v m2v
Vi =
1 2
( | ? | + | ? |) , (A.24)c3 x x1 x x2
1 q1v
i i
A = 1
q2v
i | ? +
2 . (A.25)
c x x1| |x? x2|
446
To solve the other three equations, we first rewrite the terms ?iA0?iA0, ?i??iA0,
and ?iV ?iA0 using the identity
?2(??) = ??2? + ??2?+ 2?i??i? , (A.26)
where ? and ? are any scalar functions. Using that identity, Eqs. (A.21), (A.22),
and (A.23) can be written as
( )
?4?G ? 4?G 3 2 ? 4?G G GV = ?g ?g v V c2 ? ? ??? ?2(A )2 + A ?2A ,
c2 c4
g
2 c2 2c4
0
c4
0 0
[ ] (A.27)
?4?G 1? = ? ?c2? + ?v2g + c2?V ? c2
Ga Ga
(?2 + ?)? ? A0?2A0 + ?2(A0)2 ,
c4 2 c4 2c4
(A.28)
A0 = 4??e ??2(V A0)? V?2A0 + A ?20 V + a?2(?A0) + a??2A0 ? aA ?20 ? .
(A.29)
At this point, one could split the fields into separate PN orders, followed by
further simplifications of the action through partial integrations and use of the field
equations. Eventually one would only need an explicit expression for the leading
order solution to the field equations here in order to obtain the 1PN Fokker action.
This is essentially the ?n+2? method from Ref. [209]. However, at this order this
does overall not provide a big simplification, and we need a solution for the 1PN
scalar field for Figs. 2.4 and 2.5. We therefore proceed by solving the 1PN field
equations and straightforwardly insert the solution into the complete action.
To solve those three equations, we first solve for the leading order terms of
V , ?, and A0, and then insert that solution back into the right hand side of the
447
equations.(Equation (A.27) yield)s ( )
G m1 m2 G ?
2 ?2
V = | (? | + | ? | +) m1 |(x? x1|+m2 |x? x2|c2 x x x x 4 2 21 2 2c ?t ?t )
3G m v21 m v
2 2
2
+ 1 + 2| ? |( | ? | ?
G 1 1
m1m2
2c4 x x 41 x x2 c ) r|x?( +x1| r|x? x2| )
2 2
? G 1 1 G q1 q2?1?2m1m2 + ? +
c4 ( r|x? x1| r|x)? x2| 2c4 |x? x1| |x? x2|
G 1 1
+ q1q2 + , (A.30)
c4 r|x? x1| r|x? x2|
where r ? |x1 ? x2| and
?2 2 2|x? v1 (n1 ? v1)x
2 1
| =
?t | ? | ? n1 ? a1 ? | ? | , (A.31)x x1 x x1
with n1 ? (x? x1)/|x? x1|, and a1 = dv1/dt is the acceleration.
Solving Eq.((A.28) and using Eq.)(A.31), we ge(t )
? G ?1m ? m G (n
2
1 2 2 ? 1 ? v1)? = + + ? m n a +
c2 | ?( 4 1 1 1 1x x1| |x? x2| 2)c ( |x? x1|)2
G ? (n
2
2 ? v2) a q1 q2
+ ?2m2 (n2 a2 + | ? | + | ? | +2c4 x x2 2 x x1 )|x? x2|
G2 ? + ? (?2 + ? ) ? + ? 21 2 1 1 2 1(?2 + ?2)+ m1m2 +
c4 ( r|x? x1| ) r|x? x2|
? Ga 1 1q1q2 | ? | +c4 r x x1 r|x? x2|
. (A.32)
The s(olution of Eq. (A.29) )for A0 is(given by )
? q1 q2 q
2
1 v1 (n1 ? v 21)A0 = | (? | + | ? ? n1 ? a1 ?x x1 x? x 22| 2c |x?)x1| |x? x1|
? q2 v
2 2
2
( | ? | ? ? ?
(n2 ? v2)
n2 a2
2c2 x x2 |x? x2| )( )
G m1 m2 q1 q2
+ (1 + a?1) + (1 + a?2) +
c2 ( |x? x1| |x? x2| )|x? x1| |x? x2|
G q1m2 q2m1
+ ((1 + a?2) | ? | + (1 + a?1)c2 r x x1 r|x? x2|)
? G m1q2 m2q1(1 + a?1) | + (1 + a?2) . (A.33)c2 r x? x1| r|x? x2|
448
A.4 The 1PN Lagrangian
The total action, after using the field equations and integrating by parts, can
be w?ritten as[ ( ) ( )
1 v43 ? 2 2 1 2 3 1 1S = dtd x ?g c +( v + )+ V c + V v
2 ? 2V vii c + ? A i2 e 0 + Aiv2 8c 2 4 2 2c]
1 ? 2 1+ ? ?? c + v2 1g + ?eA0V ?
1 G
a?eA0?+ ? (1 + a?)A
2 .
2 2 2 2 4c2
g 0
(A.34)
Substituting the potentials gives acceleration terms that can be eliminated using
integ?ration by parts?in th(e action
v2 (n ? v )2? ? 1 1 ? (n ? v1)(n ?
)
v2) v1 ? v2
dt (n a1) = dt + + , (A.35)
r r r r
where n ? (x1 ? x2)/|x1 ? x2|, and a1 = v?1. Finally, integrating over space term
by term and simplifying leads to the 1PN Lagrangian
L = ?m c2 ?m c2 11 2 + L0 + L1 , (A.36)
c2
with
1 1 m1m2 q1q2
L0 = m v
2 2
1 1 + m2v2 +G(1 + ?1?2) ? ,2 2 r r
1 1 q1q2
L1 = m
4
1v1 + [m2v
4
2 + [v1 ? v2 + (n ? v1)(n ? v2)]8 8 2r
Gm1m2 ]
+ [(3? ?1?2)(v
2
1 + v
2
2)? (7? ?1?2)(v1 ? v2)? (1 + ?1?2)(n ? v1)(n ? v2)2r
? G
2m1m2 ]
(1 + 2?1?2)(m1 +m2) +m
2
1?1(?
2 2 2
2 [2 + ?2) +m2?2(?1 + ?1)2rGq1q2 G ]
+ [m1(1 + a?1) +m
2 2
2 2
(1 + a?2)]? m1q2(1 + a?1) +m2q1(1 + a?2) .r 2r2
(A.37)
449
Appendix B: Energy flux to next-to-leading PN order in Einstein-
Maxwell-dilaton theory
In this appendix, we derive the next-to-leading order scalar, vector, and tensor
energy fluxes for general orbits. The derivation follows the one used in Ref. [147] in
the context of ST theory.
B.1 Scalar energy flux
The scalar field in a radiative coordinate system(can)be written as
? 1 O 1?(X ) = ?0 + ?(U,N ) + , (B.1)
R R2
where R ? |X|, U ? T ? R/c, N ? X/R, and the Einstein-frame radiative scalar
multipole moments are defined by
? 1 (`)
?(U,N ) = G NL?L (U). (B.2)`!c`+2
`?0
In this notation, an uppercase index denotes a multi-index, such asNL = N i1N i2 . . . N i` .
A superscript in parentheses denotes derivative, such as ?(`)(U) = d`?/dU `.
Next, to relate the radiative moments to the source moments, one defines
?algorithmic? moments that serve as functional parameters for a general external
metric. Based on the arguments in Refs. [147, 492], the radiative moments coincide
450
with the algorithmic ones to O(1/c3), and the algorithmic moments agree with the
source moments KL to order O(1/c4)
(alg)
?L = ?L +O(1/c3), (B.3)
(alg)
? 4L = KL +O(1/c ). (B.4)
The source moments are defined by
? [ ]2
K = d3
1 ? S
L x x?LS + x
2x?L , (B.5)
2(2`+ 3)c2 ?t2
where the hat on xL denotes a symmetric trace-free projection on the ` indices. The
source function S is defined by the field equation for ? as
4?G
? = ? S. (B.6)
c2
The scalar energy flux
?
F = ?cR2 T S iS 0iN d?, (B.7)
where the scalar part of the stress-energy tensor is given by
[ ]
S c
4 1
T?? = ??????? g??(??)2 . (B.8)4?G 2
In the far zone,
c4 c4
T S0i ' ? ?? ? ' ? N (? ?)20 i i 0 , (B.9)4?G 4?G
where, in the last step, we used the relation
?i? = ?Ni?0?+O(r/R2). (B.10)
451
The scalar flux becomes
? ( )3 2
F c ??S = d?
4??G ?U?1 d? L P (`+1) (`+1)= G N N ? (U)? (U). (B.11)
c2`+1(`!)2 4? L P
`?0
To integrate over the solid angle, we use the integration formula given by Eq. (A
29a) in Ref. [493], which yields
?
F 1 (`+1) (`+1)S = G[ ?L (U)? (U)c2`+1`!(2`+ 1)!! L`?0 ]
?(1) (1)
(2) (2) (3) (3)
? ? ? ?i ?i ij ij= G + + + . . . , (B.12)
c c3 c5
where the first term is the monopole flux, the second is the dipole flux, and the third
is the quadrupole flux. In terms of the source function S, those multipole moments
needed for the calculation of the next-to-leading order flux are given by
? [ ]
1 d
? = d3? x [S + (x
2S) , ] (B.13)6c2 dt ( )
? = d3x xi
1 d
S + x2xii ? ( S , (B.14)10c2 dt)
1
?ij = d
3x xixj ? x2?ij S. (B.15)
3
The 1PN field equation for ? is given by Eq. (A.28)
[ ]
? 4?G ? 1 2 Ga Ga? = ?g ? + ?v + ?V ? (?2 + ?)? ? A0?2A0 + ?2(A0)2 .
c2 2c2 c4 2c4
(B.16)
The last term in that equation can be moved to the left hand side by a redefinition
of the field, and since A20 ? 1/R2, we can neglect that term to O(1/R). The other
terms are expressed in terms of delta functions. Hence, we can write the source
452
function S as ?
S(x, t) = ? 3A? (x? xA), (B.17)
A
with ( )
v2 m1m2 ( ) aq1q2
?1 =?m1?1 1? 1 + ? 21 + ?1?2 + ?1?2 ? , (B.18)2c2 c2r c2r
and similarly for ?2, where r ? x1?x2. In the center-of-mass coordinates, we define
v ? dr dv, a ? ,
dt ( d)t
m2 1
x1 = r +O ,
M (c2 )
?m1 O 1x2 = r + . (B.19)
M c2
Thus, ?1 can be written as [ ]
? ? M
2? aq1q2
?1 = m1?1 + m1?
2 2
2 1
v + ?1 + ?1? + ?2 2 1?2 ? . (B.20)2c c r M?
The multipole moments can now be written in terms of ? after integrating the delta
functions
(1) d?1 ? m1?1 d
2
? = x21 + 1? 2, (B.21)dt 6c2 dt2
(2) d
2 m ? d41 1
?i = (x
i
1?1)?( x2xi + 1? 2, (B.22)dt2 10c2 dt4 1 1)
3
(3) d 1
?ij = ?m1?1 xi j1x1 ? x21?ij + 1? 2, (B.23)dt3 3
where, in the higher order terms, we used ?1 = ?m1?1.
For the monopole and quadrupole fluxes, the multipole moments in the center-
of-mass coordinates can be written as
3 2
?(1)
d ? ? d r= (?1 + ?2) (m2 2(?1 +m1?2)dt 6c dt) , (B.24)3
3
(3) d i j 1? 2ij = ??(m2?1 +m1?2) r r ? r ?ij . (B.25)dt3 3
453
Differentiating, and using the relations
( )
d2r ?G12M 1= n+O ,
dt2 r2 c2
dn v ? r?n
= , (B.26)
dt r
where ( )
G12 ? G 1 + ?1?2 ?
q1q2
, (B.27)
M?
we get
?(1) =? 2G12M?r? (X ? +X
2 [2 2 1 1?2)3 c r ]
? M? 2aq1q2r? ?1 + ?2 + ?21?2 +[?
2
2?1 + ? ? + ?2 2 1 2 2?1 ? ,c r M? ]
(3) ? G12M?? = (X ? +X ? ) 6r?ninj ? 4(nivj + njvi 2ij 1 2 2 1 ) + r??ij . (B.28)r2 3
Squaring leads to the monopole and quadrupole scalar fluxes
(1) (1)
FMon ? ?S =G ( c )2 [
G G12M? 2
= r?2 ((X2?1 +X5 2 1?2)c r 3 )]2
+ 1 ? + ? + ?2 2
2aq1q2
q q ?
1+? ? ? 1 2 1 2 1 2 + ?2?1 + ?1?2 + ?2?1 ? .1 2 M? M?
(B.29)
(3) (3)
FQuad
?ij ?
S =G ( ijc5 )2 ( )
G G12M? 88
= (X1?2 +X2?1)
2 32v2 ? r?2 . (B.30)
30c5 r2 3
For the dipole flux, we need to write x1 and x2 in the center-of-mass coor-
dinates to 1PN order. From the boost invariance of the Lagrangian, we obtain
454
[147] ( )
?2 1
x1 = r +O ,
?1 + ? 42 (c )
?1 1
x2 = ? r +O , (B.31)
? 41 + ?2 c
where ( ) ( )
? v
2
? m 1 + 1 ? G12m2 11 1( +O2c2 2c2r ) c4 ( )
?v2 G12M? 1
= M X1 +X2 ? +O , (B.32)
2c2 2c2r c4
and similarly for ?2. This leads to the dipole moment ?i in the center-of-mass
coordinates( ) ( )
2
(2) d ?2 i ? d
2 ? 41 ? ( ) d
?i = r ?1 r
i?2 + X
2?2 ?X2?1 (r2ri).
dt2 ?1 + ?2 dt2 ?1 + ? 10c2
1 2 dt42
(B.33)
To calculate the dipole flux, we also need the 1PN acceleration, which can be derived
from the 1PN Lagra{ngian, an[d we obtain ]
d2r 2? G12M v 1? ?1?2 + q1q2/2M? 3 G12M= n 1 + 3? + ? ?r?2 ? 2?
dt2 r2 c2 1[+ ?1(?2 ? q1q2/M? ) 2c2 c2r
? G12M
2 ( 1 ) 2q1q2 q22 2? 1 + ?1?2 ? +X 12 (1 + a?2)c r M? M?
1 + ? q1q21?2 ? M?
q22 ? q1q2 q1q2+X1 (1 + a?1) 2a (X1?1]+}X2?2)? (5? ?1?2)M? M? M?
+ 4(1 + ? ? ) +X ? ?2[ 1 2 2 1 2 +X1?
2
2?1
? G12M ? 4?
] ( )
q1q2/M? 1
r?v 2? +O . (B.34)
c2r2 1 + ?1?2 ? q1q2/M? c4
Finally, w(e obtain t)he{dipole scalar flux }2 2 2
Fdip G G12M?S = (?1 ? ? )2
r? v G12M
2 + f
S S
r?2 + fv2 + f
S , (B.35)
3c3 r2 c2 c2 1/r c2r
455
with the coefficients
S ?
{
1
f = 8(? ? ? )2r?2 ? q q 1 2 ? 2?(?1 ? ? )
2
2 (1 + ?1 2 1?2)1 + ?1?2 M?
+ (?1 ?[?2)(1 + ?1?2) [X1(?1 + 3?2)?X2(?2 + 3?1)] ]
? q1q2 2(1? ?)(? 21 ? ?2) + (X1}?X2) (?1 ? ?2)(2a+ ?1 + ?2)M?
+ 2(?1 ? ?2) (X1?1?{2 ?X2?2?1) , (B.36a)
S ( 2fv2 = ) 5(?1 ? ?2)2(1? ?1?2) + 5(?1 ? ?2) (X1?1?2 ?X2?2?1)
5 1 + ? ? ? q1q21 2 M? [
25 11 ( )
+ (?1 ? ?2)(1 + ?1?2) ?] (1? ?)(? ? ? ) + X
2
1 2 2?1 ?X21?22 2
35
+ (X1?1 ?X2?2)
2
q1q2 [ ( )]
+ (?1 ? ?2) 5(1? 5?)(?1 ? ?2)? 10a (X1 ?X2)? 11 X2 2
2M? } 2?1 ?X1?2
? 5q1q2{ (?1 ? ?2) [2 (X1?1 ?X2?2) + 3 (X1?2 ?X2?1)] , (B.36b)2M?
2 ( )
fS1/r = ? 5?(? ? ? )21 2 + 5[(?
2 ? ?21 2) (X1 ?X2) + 6(?1 ? ?2) X22?1 ?X25 1?2
+ ( 5(?1 ? ? 22) ) ?q1q22 (5? ?1?2 + 2aX1?1 + 2aX2?2)
1 + ? ? ? q1q2 M?1 2 M? ]
q2 2
+X 1
q
2 (1 + a?
2
2) +X1 (1 + a?1)
M? M?
( }5(?1 ? ?2)2 ) [ ]+ 2 22 4(1 + ?1?2) +X2?2?1 +X1?1?2 . (B.36c)
1 + ? ? ? q1q21 2 M?
This flux reduces to the ST dipole flux derived in Ref. [147] in the limit where the
electric charges are zero.
456
B.2 Vector energy flux
The calculation of the vector flux is similar to that of the scalar flux. The
vector potential can be written in terms of radiative multipole moments as [494]
1 ? 1 L (`)A0(X, T ) = N Q
R `!c`[ L
(U),
?`?0 ]1 1 (`) ` (`)
Ai(X, T ) = N` L?1QR `!c iL?1
(U)? ?iabNaL?1M
(`+ 1)c bL?1
. (B.37)
`?1
As was done in the previous subsection, the radiative moments can be related to
the source moments using algorithmic moments. At leading order, the three agree,
and we can express the electric and magnetic multipole moments directly in terms
of the source moments
? [ ]
1 d2? 2`+ 1 dJa
QL(U) = d
3x x?L?+ x
2x?L ? x? , ` ? 0 ,
2(2`+ 3)c2 dt2 (`+ 1)(2`+ 1)c2
aL
dt
? [ ] (B.38)
1 d
ML(U) = d
3x x? 2?L?1mi ? + x x?` 2 ?L?1 mi ? , ` ? 1 , (B.39)2(2`+ 3)c dt2 `
where the magnetization density m = x ? J . The source functions ? and Ji are
defined by
4?
A0 = 4?? , Ai = ? Ji. (B.40)
c
The vector flux ?
FV = ?cR2 N iTEM0i d?, (B.41)
where the electromagnetic part of the stress-energy tensor is given by
( )
TEM
1 ?2a? 1
?? = e 2F F
? ? g F 2?? ? ?? . (B.42)
8? 2
457
In the far zone,
TEM
1
0i = F
j
0jFi
4?
1
= (?0Aj ? ?jA0) (?iAj ? ?jAi) . (B.43)
4?
The vector flux becomes
? [ ]
F R
2 ?Ai ?Ai ? i ?Ai ?AjV = d? N N j . (B.44)
4?c ?U ?U ?U ?U
The vector potential Ai, to the required order, has the multipole expansion[ ]
1 1 (1) ? 1 (1) 1 (2)Ai = Qi ?ijkN jMk + N jQR c 2c2 2c2 ij , (B.45)
which leads to
? ( )2 2 (2) (2) (2) (2) (3) (3) ( )R ?Ai Q Q Q Q
d? = i i
M
+ i
Mi ij ij O 1+ +
4?c ?U ?c3 6c5 [ 12c5 c71 2`+ 1 (`+1) (`+1)
= Q
c2`+1`!(2`+ 1)!! ` L
Q
]L`?1
` (`+1) (`+1)
+ M
c2(`+ 1) L
ML , (B.46)
and
2 ? (2) (2) (3) (3) ( )R i j ?Ai ?Aj Qi Q Qi ij Qij 1d?N N = + +O
4?c ?U ?U ?3c3 30c5 c71 (`+1) (`+1)
= Q Q . (B.47)
c2`+1`!(2`+ 1)!! L L
`?1
Hence, the vector flux
? [ ]
F 1 `+ 1 (`+1) (`+1) ` (`+1) (`+1)V = QL QL + M Mc2`+1`!(2`+ 1)!! ` c2(`+ 1) L L
`?1
(2) (2) (2) (2) (3) (3)
2Q Q M M Qij Qi i ij= + i i + + . . . . (B.48)
3c3 6c5 20c5
458
The first two terms give the dipole flux, and the third term is the quadrupole flux.
There is no monopole flux because of the conservation of the total electric charge.
The 1PN field equations are given by Eqs. (A.20) and (A.29), which are
4?
Ai = ? ?evi , (B.49)
c
A0 = 4?? ? V?2A + A ?2e 0 0 V + a??2A0
? aA0?2???2(V A ) + a?20 (?A0). (B.50)
The last two terms in the above equation are of order 1/R2, and hence do not
contribute to the next-to-leading order flux. The source functions ? and J i are then
given by
? = ?e = q1?
3(x? x1) + q 32? (x? x2), (B.51)
J i = ? vie = q
i
1v1?
3(x? x i 31) + q2v2? (x? x2). (B.52)
The function ? is simply the electric charge density because the higher order terms
from the field equation cancel when summed over the two bodies.
For the dipole flux, we need Qi and M to O(1/c2i )
? [
1 d2 ( ) ]3 d
Qi =( d
3x x ? + ? x2xii e e
10c2 dt2) ? (x?ijJj)10(c2 dt )
? 3 3 22 ( )
= q ? ?1q ri 1 m2 ? m1 d1 2 + q1 q2 rir2
? 2 3 3 21 +(?2 ?1 + ?2) ( 10c M ) M dt
3 3
? 3 mq 2 ? m dq 1 rirj1 2 ?
1
r2?ij vj , (B.53)
10c2 M3 M3 dt 3
M = q(? xjvki 1 ijk 1 1 + q2? )xj kijk 2v2
m2 2
= q 2
m1 j k
1 + q ? r v . (B.54)
M2
2 2 ijkM
459
Differentiating and using the 1PN acceleration from Eq. (B.34), we obtain the next-
to-leading order vector dipo[le flux( )2 ( ) ]2 2 2
FDip 2 G12M? q1 ? q2 V v V r? V G12MV = + fv2 + fr?2 + f1/r , (B.55)3c3 r2 m m c2 21 2 c c2r
with the coefficients
( )( ) ( )
fV
2 q1
v2 = X
2
2 ?X2
q2 q1 q2 2
1 ( ?) ( + (X1 ?
q1 q2
X2) (q1 + q2) ?
5 m1 m2 m1 m2 M m1 m2
2 )
1 q1 ? q2 q1q2+ ? q q 2 + 6? + 2?1?2(3? ? 1) + (1? 6?) ,1 + ? ? 1 21 2 mM? 1 m2 M?
( ) ? (B.56a)2
V ? q1 ? q2 ? (X1 ?(X2)(q1 + q2)fr?2 = 3? + )m1 m2 M q1 ?
m1 ] q2m2
8? 4?(1 + ? ? )? 2 q1q21 2 (1? 2?)M?
+ , (B.56b)
( 1 + ?)1?[2 ? q1q2M?2
V ? q1 ? q2 2 X
2q1/m ?X21 q2/m2
f1/r = 2 2? +
2 1
m1 m2 5 q1/m1 ? q2/m2
? q1q2 5? ?(1?2 + 2a (X1?1 +)X2?2) (X1 ?(X2)(q1 ++ )q2)
M? 2 q q
1 + ? ? ? q1q2 M 1 ? 21 2 M? m1 m2
q2 q2 ]
4(1 + ?1?2) +X 12 (1 +(a?2) +X 2 2 2M? 1 (1 +) a?1) +X2?2?1 +XM? 1?1?2+ 2 .
1 + ?1?2 ? q1q2M?
(B.56c)
For the quadrupole flux,
? ( ) ( ) ( )(3) 3 1 3Q = d x x x ? x2 d 1ij i j ?ij ?e = X2q +X2q rirj ? r2?ij2 1 1 2 , (B.57)3 dt3 3
which leads to
(3) (3) ( )2( )2( )
FQuad
Qij Qij 1 G12M? q1 q2 88
V = = X2 +X
2 2
1 32v ? r? . (B.58)
30c5 30c5 r2 M m2 3
460
B.3 Tensor energy flux
The metric in radiative coordinates ( )
1 1
G ???(X ) = ??? + H??(U,N ) +O , (B.59)
R R2
where the radiative multipole m[ oments ML and SL are defined by? ]TT
TT 1 (`) 2` (`)Hij (U,N ) = 4G NL?2MijL?2(U)? N`+2 hL?2?hk(iS`!c (`+ 1)c j)kL?2 .
`?2
(B.60)
The radiative multipoles agree with the source multipoles IL and JL up to order
M 3L = IL +O(1/c ), SL = JL +O(1/c2), (B.61)
where [147,?494] [ ]2
3 1 2 ? ? ? 4(2`+ 1) ??
s
IL(t) = ? d x x?L? + x x?L x?Ls , (B.62)2(2`+ 3)c2 ?t2 (`+ 1)(2`+ 3)c2 ?t
J 3L(t) = d x ?
k
hk?i x?L?1?h? . (B.63)`
In terms of the mu?ltipole(momen)ts, the tensor flux is given by2
c3 ?HTTij
Fg = d?
32??G ?U [1 (`+ 1)(`+ 2) (`+1) (`+1)
= G ? ML (U)ML (U)c2`+1`!(2`+ 1)!! `(` 1)
`?2 ]
4`(`+ 2) `+1 (`+1)+ ? S (U)Sc2(` 1)(`+ 1) L L (U)
G (3) (3) G (4) (4) 16G (3) (3) ( )
= Mij M
9
5c5 ij
+ M M + S S
189c7 ijk ijk 45c7 ij ij
+O 1/c , (B.64)
where the first term is the mass quadrupole flux, the second is the mass octopole,
and the third is the current quadrupole.
461
The source functions ? and ?i are given by
T 00 + T ss T 0i
? ? , ?i ? , (B.65)
c2 c
and from the 1PN field equations (A.27) and (A.19)
4?G
V = ? ?, V i 4?G= ? ?i, (B.66)
c2 c3
with
?i = [m vi?3(x? x ) +m vi?31 1 1 2 2 (x ]? x2), (B.67)
3 2 ? G12m1m2? = m1 + m1v1 ?(x? x1) + 1? 2. (B.68)2c2 c2r
The multipole moments needed for the next-to-leading order flux are Mij,
Mijk, and Sij, which are given by( )
3 2 ? G
2
12m1m2 m1 d
M = m + m v x?ij + x2x?ijij 1 1 1 1 1 1 ?
20m1 d
vkx?ijk + 1? 2,
2c2 c2r 14c2 dt2 21c2 dt 1 1
(B.69)
M ijk ijkijk = m1x?1 +m2x?2 , (B.70)
i?
S = m hk?j h kij 1? x1 x1v1 + 1? 2. (B.71)
In the center-of-mass coordinates, this becomes
[ ]
3 ? G12MMij = ? 1 + (1 3?)v2 ? (1? 2?)
2c2 c2r
? d2
+[ (1? 3?) r2 ij ?
20? d
r? (1? 3?) vkr?ijk, (B.72)
14c2 ]dt2 21c2 dt
m2 m2
M = ? 2 ? 1 r?ijkijk [ , (B.73)M2 M2]
m2 2
S = ? 2 ? m1 ?hk?j i? hij r r vk, (B.74)
M2 M2
462
where
r?ij = rirj ? 1r2?ij(, ) (B.75)3
ijk i j k ? r
2
r? = r r r [ ri?jk + rj?ik + rk?ij , ] (B.76)5
?hk?jri?rh k
1 hkj i 1v = ? r + ?hkirj ? 1?hkmrm rhvk. (B.77)
2 2 3
Taking the time derivatives of the multipole moments and squaring, we obtain the
tensor flux ( )2 ( )2 [
F 8G G12M?
[ ]
2 ? 2 8G G12M?T = 12v 11r? + fTv]4v
4 + fT 2v2r?2v r?
2
15c5 r2 420c7 r2
T 4 T G12Mv
2
T G
2 2 2
12Mr?
+ f r? + f T
G12M
r?4 v2/r + fr?2/r + f1/r2 , (B.78)r r r2
with the c[oefficients ]
785 + 113? q1q2
T 1
?2 ? 281M?
fv4 = [ ? 852? , (B.79a)1 + ?1?2 ? q1q2M? ]
1487 + 255? q1q2
T 1
?2 ? 563M?
fv2r?2 = ?[2 ? q q ? 13]92? , (B.79b)1 + ? ? 1 21 2 M?
687 + 127?1?2 ? 267 q1q2T M?fr?4 = 3 1 + ?1?2 ? q q
? 620? , (B.79c)
1 2
M?
fT1/r2 = 16(1? 4?), [ (B.79d)
T 8fv2/r = ?( )2 20(1 + ?1?2)(17? ?) + 4?1?2(1 + ?1?2)(22? 5?)
1 + ?1?2 ? q1q2M?
q2 q2 q2q2
+ 84 1 X2(1 + a?2) + 84
2 X1(1 + a?1) +
1 2 (67? 20?)
M? M? M2?2
? aq1q2 q1q2168 (X1?1 +X2?2)? (491? 40?)
M?
q1q2 ( M? ) ]? ?1?2 (71? 40[?) + 84 X ?
2
1 1?2 +X2?
2
2?1 , (B.79e)M?
fT
8
r?2/r = ( )2 (1 + ?1?2)(367? 15?) + 3?1?2(1 + ?1?2)(29? 5?)
1 + ?1? ? q1q22 M?
463
q2 q2 q2q2
+ 84 1 X2(1 + a?
2 1 2
2) + 84 X1(1 + a?1) + (73? 15?)
M? M? M2?2
? aq1q2 ? 2q1q2168 (X1?1 +X2?2) (262? 15?)
M? M? ]
? q1q2
( )
2 ?1?2(38? 15?) + 84 X1?2 21?2 +X2?2?1 . (B.79f)M?
This flux reduces to the one derived in Ref. [151], in the context of ST theory, when
the electric charges are zero and after converting the notation to the Jordan-Fierz
frame.
B.4 Energy flux for circular orbits
In this section, we express the energy flux for circular orbits in terms of the
gauge-independent parameter x, whi(ch is defin)ed by2/3
? G12M?x , (B.80)
c3
where ? is the orbital frequency. To do that, we need to find the relation between r
and ? to 1PN order (Kepler?s third law). We start by writing the Lagrangian (2.31)
in the center-of-mass coordinates
L = ?Mc{2
1 2 G12M?+ ?v +
2 r [( ) ]
1 1 ? 4 G12M? 3? ?1?2+ [(1 3?)?v +
2 2
c2 8 2r 1 + ? ? ? q q + ? v + ?r?1 21 2 M?
? M
2? q1q2
(1 + ? ? )2 +X ?2? +X ?21 2 2 1 1 ?2 ? 2 (1 + a?1X1 + a?2X2)
2r2 2 1]} M?
q22 q
2
+ X (1 + a? ) + 11 1 X2(1 + a?2) . (B.81)
M? M?
Applying the Euler-Lagrange e[quation and us(ing )r? ]= 0 and v = r? leads to
2 G12M? = 1? 3f?? +O
1
, (B.82)
r3 c4
464
where the parameter ? is defined by
? G12M? , (B.83)
c2r
and the the coefficient f? is defined by[
f? ?
1
G212(1? 2?) +G12(3? ?1?2) + 2(1 + ?1?2)2 + 2X ?22 2? 22 1 + 2X1?1?26G12 ]
q2 22 q q1q2+ 2 X1(1 + a?1) + 2
1 X2(1 + a?2)? 4 (1 + aX1?1 + aX2?2) .
M? M? M?
(B.84)
Substituting x instead of ? and inverting Eq. (B.82), we obtain
[ ( )]
? = x 1 + f?x+O 1/c4 . (B.85)
To express the flux for circular orbits in terms of ?, we set r? = 0 and v = r?
and then use Eqs. (B.82) to obtain
[ ]
5
F Gc 2 4 ? 2 Gc
5 16
= ? ? (? ? ) + ?2?5 fS S 2S 2 1 2 2 + f3G 3G2 v 1/r
+ (X1?2 +X2?1) ,
12 12 5
( ) [ ( ) (B.8]6a)
5 2 5 2
F 2Gc 2 4 q1 ? q2 2Gc 2 5 8 q1 q2= ? ? + ? ? X +X + fV + fVV 2 1 v2 1/r ,3G2 212 m1 m2 3G12 5 m1 m2
( ) (B.86b)
F 32Gc
5
2 5 2Gc
5
T = ? ? + ?
2?6 fT T T
5G2 105G2 v
4 + fv2/r + f1/r2 + 1008f? . (B.86c)
12 12
Using Eq. (B.85) to express the energy flux in terms of x instead of ? leads to
Eq. (2.43a).
465
Appendix C: Translating between scalar-tensor theory notations
This appendix contains the conversion between notation introduced in Table
3.1 and that employed by Damour and Esposito-Fare?se [495, 496]. Note that these
authors defined ?A with a relative minus sign compared to Eq. (3.11) and defined
G? as the bare gravitational constant in the Einstein frame.
Weak-field parameters :
( )
G ? G? = G 2 2?A0 1 + ?0 , (C.1)
?2
? ? 0 , (C.2)
1 + ?20
?1 ? ?
?0?0
, (C.3)
(1 + ?2)A20 0
?2 (? ??? ? 0 0 0 ? 3?
2
0)?2 2 . (C.4)
(1 + ?2 40) A0
Strong-field parameters :
? 1 ? ?AsA , (C.5)
2 2?0
s?A ?
?0?A ?A
+ , (C.6)
2?2 2 20A0 4?0
? ? 2
s?? ? ?A?0 ?A ? ?+ 0?A ?0?A ? 3?0?AA + . (C.7)2?2A4 8?3 ?3A4 2?2 2 30 0 0 0 0 0A0 4?0A20
466
Binary parameters :
? 1 + ?1?2? , (C.8)
1 + ?20
? ? 2?1?2?12 = ? , (C.9)
1 + ?1?2
?1?
2
? 1 21 ? ?22 = , (C.10)2(1 + ?1?2)2
? 2 ?2?
2
? ? = 12 11 , (C.11)2(1 + ?1? 22)
?2
? ? 11 , (C.12)
(1 + ?1? )22
2
?2 ?
?2 , (C.13)
(1 + ?1?2)2
1 ???3
?1 ? ? 1 = ? 1 2222 , (C.14)4 4(1 + ? ? )31 2
? ?1
? 3
? 2 = ? ?2?12 111 . (C.15)4 4(1 + ? 31?2)
Our conversions agree with those presented in Table II of Ref. [497] with one
exception: we find G? = G?12 rather than G? = G12.
467
Appendix D: Explicit formulas for the Fourier-domain phasing of
waveforms in scalar-tensor theories
This appendix gathers explicit formulae for the coefficients entering our results
that were too voluminous to be kept in the main text.
In the dipole-driven case of Section 3.5.1, we obtained for the flux
FDD 4S
2 ??2x4 [ ]
(x) = ? 1 + fDDx + fDDx3/22 3 + f
DD 2
4 x +O(5) (D.1)3G?
with the coefficients
DD ? 14 4S
2
+ ? 4 4S+?? 4?+f2 = + S ?+ + S + ?
2 24
? +
5 5 2? 3 ?? ? 3 5S2??
12? ? 4 4 ? 4???+ S ? + ??? ?
4S+?+?
5 2?? 3 3 ? S
, (D.2)
??
fDD3 =2? + ??, (D.3)
DD ? 29 ? 97S
2 2 2 2 2
+ ? 4S+?? ? 4 2 2 ? 32S ?+f4 = S S ?? + ?
+
+ S ?
4
?2
4?? 4S+??
28 28 2 3 3 15 15 2 3 +
+ +
? ? ? ?
2 S2 2??
16S 2 2 2 2 2+???+ 4?+ 4S ? 4S+?? 8? 36?+ 16S ?+ 16S+???++ S + +
+ + ? +
??2 ?2 S
? S ? ? +2 ?2 ? 3? 5? 5S ?2? ? ?? 3S??
8?2 2 16S2? + ? 1 2 1247 64?+ 2143? 32?+?+ ? ? +S + ?
2 + ?+ ? ? ? ?
3? 5 15 2 2 2 2 2? 2 3 70S?? 5S?? 140S?? 5S??
16?2 55 7S2? ? +? 16 40 48?
2 ? 32S+???+? 32?2 ?
S + ? S + ?
2 ? + ? ? ? ? ? + +
5 2 ? 6 3 2 ?
+
3 3 ?2 S??2 ?2? ?
16S2?2? + +? ? 56S+??? 80?
2
?? ? 56?+? 32S+???+? 16?
2 ? 4
S S + + ?
+ ? ??
2 ?2? 3 ?? 3? 3? 3S?? ? 3
468
? 1 2 ? 4 ? 14? ? 7?? 2 2 4S+?? 8S+??? 4 4?+ 8? ? ?+? S S + ? + S ? S ? ?+ + + ??+3 3 2 2?? ?? 3 ?? ?? 3 ? 3
? 8??+ 11S
2 2
+? ? 2 32S ??? 4S+?+? 8 8S+? ?+ S ??? +
+
S + S + ?
?
2 ??+? ?? 2 ? 15 15 ? 3 ? 3 S??2
8? ? ? 8S2? ? ? 8S ?2? 36? ? 16S2? ? 8S ?2? ? + ? + ? + ? + + ?+ ? + ? ++ ??
?2 S2 ?2? S??2 5? 5S2?? 3S??
4S 2+?+? 16???+? 8S+? ? 3S+?? 2 64???
+ +S + + + + ??? + 2?? 3? 3S?? S? 3 5S??
32???? ? 16 20???? 20S+?+?? 4 4??? 4S+?+?+ S ???? + + S + ??? ? ? .5 2?? 3 3? 3 ?? 3 ? S??
(D.4)
Note that fd3 is determined entirely by tail terms of the form of Eq. (3.36);
this is the only hereditary contribution to the flux.
For the ratio ?(x), we obtained
[ ]
?DD
3
(x) = DD DDS 1 + ?2 x + ?3 x
3/2 + ?DDx24 +O(5) (D.5)8 2 4???x
with the coefficients
DD 13 ? 4S
2
+ 8 ? 4S+?? 4?+?2 = S + ?+ S ? ?
2
?
10 5 2? 3 ?? ? 3
? 24 ? 12? 7 ? 8 4??? 4S+?+?S + ? ??? + +5 2?? 5S2?? 6 3 ? S
, (D.6)
??
?DD3 =? 2? ? ??, (D.7)
DD 7629 129S2 16S4 4S+?? 80 181 16S2?+ 80?4 =? + + + + + + ?2 + ?+ ? + + ?21400 700S2 4 ? 2 +? 25S? 3S? 9 15 15S? 9
12?2 12S2?2? + ? 48S+? 2 2 2 3??+ 12?+ 12S+?+ ? 62S+?? 32S ??+ + S + S + + S S +
+
?2 2 2 2 2 2 2 3?? ?? ? ?? 5 ?? 5S??
2 2 2 2
? 40?? ? 46?+ 16S+?+ ? 80S+???+ ? 40?+ ? 77 16S ? 44+ S S ? +
+ + ? ?
3? 5? 5 2 ? 3 ? 3? 5 15S2 +? ? ? 9
? 205 2 1 576 576? 144?
2 2
? + ?+ + + + ?
653 192S
+ +
96S+??
+
36 3 25S4 ?2 25S4? ??2 25S4??2 350S2?? 25S4?? 5S3??
64?+ 192S+?? 192?+ 3827? 96S2 ? 16? 2+? 16? 71
+ S + S + S + S +
+ ?
5 2 ? 5 3 ?? 5 2 ?? 700 2 ? 25S4? ? ? ? ?? 5S
+ + ?
2 2
?? 5S?? 24
469
S2+? ? 320 2 ? 245
2
? 16??? ? 96S+???+? ? 32?
2
+? ? 48S
2?2+ +?+
3S ?2 9 ?? ?+?9 ?2 S??2 ?2 S2 2? ??
26S+??? ? 32?
2
?? 26?+? 160S+???+? 64?2+? 110 4+ S + + S + + ?? + ?
2?
3 ?? 3? 3? 3 ?? ? 9 3
16 2? ?? 55 2 ? 4S+?? 8S+??? 16 ? 4?+ 32+ ?+? + S + S + ? S + S + ?+ ? ??3 2 2 +?? ?? 72 ?? ?? 3 ? 3
8??+ ? 11S+? ? 181 16S
2
+??? ? 4S+?+? ? 160 ? 24S
2
+???+ S ??? + S S ??? ?? 2 2 +? 15 15 ? 3 ? 9 S??2
? 24???+? ? 24S
2
+??? ? 24S 2 2 2+ ? +?+? 46??? 16S+??? 40S+? ?S S + ? S +
?
?2 2 2 2 2?? ?? 5? 5 ?? 3S??
62S+?+? ? 32S
3
+?+? 80???+? 40S ?2++ + + +? ? 3S+?? ? 44 1S S ???? + ???5 3?? 5 ?? 3? 3S?? S? 9 3
? 64??? ? 96S+?+? ? 192??? ? 192S+?+? 16???? 11 10????S + + ? ?? +5 2 ? 5S3 ? 5S2 ?? 5S3 ?? ? ? ??? 5S2?? 9 3?
10S+?+?? ? 16 4??? 4S+?+?+ S ??? + + S . (D.8)3 ?? 3 ? ??
In the quadrupole-driven case of Section 3.5.2, we obtained for the dipolar
part of the flux
F 4S
2
???
2x4 [ ]
(x) = 1 + fdx + fdx3/2 + fddip 2 3 4 x
2 +O(5) (D.9)
3G?
with coefficients given by
fd ? 14 ? 4 4S+?? 4?+ ? 22 = ?+ + S + ? ?
4 4 ? 4??? ? 4S+?+?? + ??? ,
5 3 ?? ? 3 3 3 ? S??
(D.10)
fd3 =2? + ??, (D.11)
d ? 29 ? 4S+?? ? 4 2 2 ? 4 2 4?
2
? 16S+???+ 4?2 4S+??f4 = ? + ?+ ? + + + + ?28 3S ?? 3 15 3 + ?2 S??2 ?2 S??
8?2? ? ? 36?+ ? 16S+???+ ? 8?
2
+ 2 1 2 2 55 16
S + ? + ? + ?+ + ? + ?
2 ?
3? 5? 3 ?? 3? 5 2 3 6 3
?
40 48?2 ? 32S ? 2 2+ ??+? 32? ? 56S+??? 80? ? 56?+?
+ ?+? ? ? ? +S + ? +
? ?
3 ?2 ??2 ?2 3S?? 3? 3?
32S+???+? ? 16?
2
+ +
? ? 4 ? 1?? ?2S ? ?
4 2 4S+?? 8S+???
?+? + ?
2 + ?
3 ?? ? 3 3 3 3 S?? S??
470
? 4 4?+ 8 ? 8??+ 11S+? 2 4S+?+? 8?+ + + ??+ +
3 ? 3 ? 2S ? ??? + S + ???+?? 15 3 ? 3
8S ?2? + ?? ? 8???
2
+? ? 8S+?+? 36??? 8S+?
2
?? 4S+?+? 16???+?
S S + + + +??2 ?2 ??2 5? 3S?? S?? 3?
8S 2+?+? 3S+?? 2 ? 16 20???? 20S+?+?? 4+ S + S + ??? ???? + + S + ???3 ?? ? 3 3 3? 3 ?? 3
? 4??? ? 4S+?+?S . (D.12)? ??
For the non-dipolar part, we obtained
F 32?
2?x5 [ ]nd
non-dip(x) = 1 + f2 x +O(3) (D.13)5G?
with the coefficient
nd ? 1247 ? 8?+ ? 2143? ? 4?+? ? 2?
2
? 485S
2
+? ? 4S
2
+?+? 5S2+?2f ??2 = +336? 3? 672? 3? 3? 672? 9? 6?2?
5S2+?2+? 2S2+?+? 2S2? +?? ? 35? ? 35?? ? 35S
2 2 2
+?? ? 10S+?+??+ +
6?2? 3?? 9? 12? 24? 72? 3?2?
8? ? 4? ?? 4S2? ?? 5S2 2? ? + ? ? +???+?? ? 2S+????+ + + . (D.14)
3? 3? 9? 3?2? 3??
For the non-dipolar ratio ?non-dip(x), we obtain
5 [ ]
? nd nd 3/2non-dip(x) = 1 + ?2 x + ?3 x + ?
nd
4 x
2 +O(5) (D.15)
64??x5
with the coefficients
nd 743 4?+ 743? 2?+? 317S2 ? 2S2? 2 2 2 2+ + +? 5S ? ? 5S ? ??2 = + + + + + ? + ? ? + +336? ? 672? ? 672? 3? 6?2? 6?2?
? 2S
2? ? 11? 11?? 11S2+ + +?? 10S2 2+ + + + +?+?? ? 4??? ? 2????
3?? 4? 8? 24? 3?2? ? ?
? 2S
2
+???? 5S2 2+???+?? 2S+????+ + , (D.16)
3? 3?2? 3??
?nd
4?
3 =? fST3 ? , (D.17)?
nd ? 81 ? ST 19 57?4 = f4 + 4?2? + 3?+ + 4?2+ ? 14? + 4? 2+? ? ? + ?+ + ? ? 16?2 ?8 4 8 ?
471
? ? 48?
2 2
?? 48?+? 2 ? 1 2 1555009 64?
2
19?+? + + 11?? + ? ? + 4?+? ? + +
?
? ? 8 112896?2 9?2
1247? 2+ 64?+ 2672321? 64?
2
?? 565?+? 64?
2 ? 2275691?2
+ + + + + + + +
63?2 9?2 112896?2 9?2 21?2 9?2 150528?2
16?2??
2 1013? ?2+ 16?
2 2
+? 2143?
3 16? ?3+ 4?
4 604795S2 ?
+ + + + + + + +
9?2 84?2 9?2 504?2 9?2 9?2 112896?2
14S2?2 ? 4379S2 2 2 2 2 2 2? + ? +?+? ? 14S+?+ +? ? 6235S+??? ? 40S+???+?
27?2 1512?2 27?2 1008?2?2 3?2?2
6235S2?2 ? 40S2?3 ? 2 2 2 2 2? + + ? + + ? 1987S+??? ? 1247S+?+? ? 20S+???+?
1008?2?2 9?2?2 224??2 252??2 3??2
? 1987S
2
+?
2 ? 20S2 3 2 2 2 2+ ? +?+? 4235377S+?? 32S+???? 91S+?+??+ + +
224??2 9??2 677376?2 27?2 18?2
32S2?2 ?? 257S2 ?2? 32S2? ?2? 8S2 ?3? 235225S4 ?2 16S4?2 2
+ + +
+ ?
+ + + + + + + + + + ?
27?2 108?2 27?2 27?2 451584?2 81?2
29S4+?+?2 16S4 2+?+?2 25S4?4 ?2 25S4?2 ?2 ?2 25S4?4 ?2+ + + + ? + + ? + + + +
84?2 81?2 36?4?2 6?4?2 36?4?2
10S4?2 ? ?2 10S4?3 ?2 1529S4?2 ?2 20S4?2 ? ?2 1529S4?2 ?2
+ + ?
+
+ + + ? + ? ? + ? + ? + +
3?3?2 9?3?2 2016?2?2 9?2?2 2016?2?2
20S4?3 ?2 26S4?2 2? + + ? + ?? ? 485S
4 2 4 2 2 4 2
+?+? ? 26S+?+? 485S+ +??
27?2?2 27??2 504??2 27??2 1512?2
16S4+?+??2 4S4+?2?2 6235? 256?2? ?? 140?+? 2825?? 256?
2 ??
+ + + + + ? ?
81?2 81?2 288?2 9?2 9?2 96?2 9?2
140? 2+?? 5065? ? ? 64?
2 2 2 3
?? ? 35?+? ? 35? ? 6755S2+ + + + + +??
9?2 384?2 9?2 9?2 18?2 864?2
? 64S
2
+?
2
??? 175S2+?+?? 40S2?2 ?? 175S2+ + ? +?
2
??? 320S2?2+ ??+??+ + +
27?2 54?2 9?2 36?2?2 9?2?2
835S2 2 2+?+?? 160S+?3+?? 283S2?2 2+ ??? ? 35S+?+?? 160S
2 2
+ + + + +
???+??
42?2?2 9?2?2 24??2 9??2 9??2
4745S2 2+?+?? 80S2?3+ +?? 3745S2+??? ? 128S
2 2 2
+ + + +
????? 70S+?+???+
252??2 9??2 576?2 27?2 27?2
35S2 ?2?? 2425S4 ?2? 64S4?2 ?2? 35S4? ?2? 50S4?2 ?2 2
+ + + + ? + ? + + ? + ? +? ?+
27?2 3456?2 81?2 81?2 3?4?2
50S4?4 ?2? 80S4?2 ? ?2? 40S4?3 ?2? 559S4 2 2? + + ? + ? + ? + + ? +??? ?
9?4?2 9?3?2 9?3?2 216?2?2
160S4+?2 2??+? ? 3025S4 2 2 4 3 2 4 2 2 4 2+ + +?+? ? 80S ?+ + +? ? 64S+??? ? 35S ?+? ?+ ? +
27?2?2 756?2?2 27?2?2 27??2 54??2
472
40S4?2 ?2? 35S4 2 2 2 2 2+ + +?? ? 1225? 1225?? 1225? ? 1225S2 ??2+ + + + + + +
27??2 162?2 144?2 144?2 576?2 432?2
175S2?2 ??2+ + 175S2 2+?+??2 1225S2+???2 1225S4 2+? ?2 100S4+?4+?2?2+ + + + +
9?2?2 18??2 864?2 5184?2 9?4?2
175S4?2 ?2+ + ?2 2 2? 1655 32?? 239?+ 32?+ ? 39239? 16?
2 ?
+ + + + + ?
54?2?2 2592? 9? 252? 9? 4032? 9?
? 73?
2
+? 16?+? ? 2647?
2
? 8?+?
2
? 8?
3 2 2 2 2
+ ? 485S+? 16S+??? 199S ?+?+ + +
56? 9? 504? 9? 9? 448? 27? 168?
16S2?2 ? 5S2?2 2 2 2 2 2 3 2 2
+ + + + + ?
? ? 10S+???+? 5S+ +?+? ? 10S+?+? 2S ? ?+ + ?
27? 4?2? 3?2? 4?2? 9?2? 9??
S2 2 2 2 2 2 2 2
+ +
?+? 2S+?+? ? 653S+?? ? 8S+?+?? ? 8S+? ? ? 5239?+ ? 128???
?? 9?? 504? 27? 27? 224? 9?
31?+? ? 8881?? ? 64?
2
??? 31?+?? ? 37?
2? ? 3425S
2 ?? 64S2?2
+ + + ? + ???
9? 1344? 9? 18? 18? 4032? 27?
31S2? ?? 5S2 2 2 2 2 2 2 3
+ +
+
+ +
???? 80S+???+?? ? 175S+?+?? 40S ? ??+ + + +
54? 36?2? 9?2? 36?2? 9?2?
32S2?2 ?? S2? ?? 2 2 2 2 2 2 2
+ + ?
+
+ + ? 40S+?+?? ? 37S+??? ? 295? ? 35?? ? 35S+??
9?? 9?? 9?? 54? 72? 144? 432?
5S2?2 ??2? + + + 4?+ ? 8??+ ? 3?2 ?? ? 8???+? ? 4???? + ??? + ????9? ?
? 1247??? ? 128???+? ? 565???? ? 128?
2
??+?? ? 1013??? ?
63?2 9?2 21?2 9?2 84?2
32? ? ?2? 16? ?3? ? + ? ? ? ? 4379S
2? ?? 28S2? ? ?? 40S2+ ? + ? + +?3???+ +
9?2 9?2 1512?2 27?2 9?2?2
6235S2 2 2 2 2 3+???+?? 40S+???+?? 1247S+???? 20S ? ??+ + + + + ?
504?2?2 3?2?2 252??2 9??2
1987S2? ? ?? 20S2? ?2 2+ ? + ? ?? 91S ????? 64S2???+???+ + + + ? + ? +
112??2 3??2 18?2 27?2
? 32S
2
+???
2?? ? 29S
4
+? ?
2
? ? ? 32S
4
+??? ?
2? 25S4?3 ? ?2+ ? + ? + ?
27?2 84?2 81?2 9?4?2
25S4? ?3 ?2? 10S4?3 ?2? 10S4? ?2 ?2? 20S4?3 ?2? + ? + ? + ? ? + ? + + + ? ?
9?4?2 9?3?2 3?3?2 27?2?2
1529S4 2 4 2 2 4 2 4 2+???+? ? 20S+???+? ? 485S ??? ? 52S ???+? ?+ + + + + +
1008?2?2 9?2?2 504??2 27??2
4
? 16S+????
2? ? 140???? ? 140????? ? 35? ?
2
? ?? ? 175S
2
+?????
81?2 9?2 9?2 9?2 54?2
473
175S2? ? ??? 160S2? ?2+ ? + ? + ? +??? 35S
2
+????? 175S2+???+???+ + +
18?2?2 9?2?2 9??2 36??2
80S2? +?
2
??+???
2
? 70S+?????? 35S
4? ?2?? 100S4? ?3 ?2? + ? + + ? + ??
9??2 27?2 81?2 9?4?2
40S4 2 2+???+? ?? 175S4? ? ?2?? 80S4? ?2 ?2+ ? + ? + ? + ?? 35S
4
+???
2??
+ + +
9?3?2 108?2?2 27?2?2 54??2
? 239??? ? 64???+? 73???? ? 32?
2 2
??+?? 8??? ? 199S ????
+ + ? +
252? 9? 56? 9? 9? 168?
32S2? ? ?? 10S2?3 ?? 5S2? ? ?? 10S2? ?2 2? + ? + + ? ? + ? + + ?+ + +?? ? S+????
27? 9?2? 2?2? 3?2? ??
2 2 2
? 4S+???+?? 8S+????? ? 31???? ? 31????? ? 31S+?????+
9?? 27? 9? 18? 54?
? 5S
2
+???+??? 40S2? ?2? + ? +??? S
2
? +????? ? 4???. (D.18)
18?2? 9?2? 9??
The result for the dipolar ratio ?dip(x) is
? 25S
2
?? [ ]? (x) = 1 + ?dx + ?dx3/2 + ?dx2dip 2 3 4 +O(5) (D.19)1536??2x6
with the coefficients
2 2
?d
2623 2S+?? 22?+ 4S+?? 4?+ 3743? 8?+? ? 407S ?
2 = + S + + + + + + +
+
840? ?? 3? S??? ?? 1680? 3? 3? 560?
8S2? ? 5S2?2 2 2 3 2 2
+ +
+ ? + ?? ? 5S+?+? 2S+??? ? 2S+?+? S+?? 13?+ + +
9? 3?2? 3?2? 3S??? 3?? 9? 3?
13?? 13S2+?? 20S2 2+?+?? ? 22??? ? 2S+?+? 4??? 4S+?+?+ + +
6? 18? 3?2? 3? S ? ??? ?? S???
8? ?? 8S2? ?? 10S2? ? ? + ? +???
2
+?? 2S+???? ? 2S
3
+?+??+ + , (D.20)
3? 9? 3?2? 3?? 3S???
?d =? 2fST 8?3 3 + 2? + ?? ? , (D.21)?
d ? 1949 ? ST ? 20S+?? 8 2 ? 59 8 2 4?
2
? 16S+???+ 4?2?4 = 2f4 S + ?? ?+ + ?+ + + S +
+
280 3 ? 9 15 9 ?2 ??2 ?2
? 10S+? 8?
2
? ? ? 66?+ 16S+???+ 8?
2 133 44 121
+ + + + ? ? + ? ? ? ?2S +?? 3? 5? 3S?? 3? 15 9 36
5 2251 ? 32 2 ? 65 ? 48?
2 ? 2
+ ? ?+ + ? ??? ?+? ?
32S+???+? 32?+? ? 58S+???+
3 120 9 9 ?2 S??2 ?2 3S??
474
? 128?
2
?? ? 58?+? ? 32S+???+? 32?
2
+ +
? 104 2 2 8 55
S + ?? + ? ? + ?+? + ?
2
3? 3? 3 ?? ? 9 3 3 72
1555009 64?2 1247?+ 64?
2 2672321? 64?2 ? 565?+?
+ + ? + + + + + ? +
37632?2 3?2 21?2 3?2 37632?2 3?2 7?2
64?2+? 2275691?
2 16?2 ?2? 1013?+?
2 16?2 ?2 2143?3 16? 3+?
+ + + + + + + +
3?2 50176?2 3?2 28?2 3?2 168?2 3?2
4?4 604795S2+? ? 14S
2
+?
2 2 2 2
?? 4379S+?+? ? 14S+?+? ? 6235S
2?2 ?
+ + + + ?
3?2 37632?2 9?2 504?2 9?2 336?2?2
? 40S
2
+?
2
??+? ? 6235S
2?2 ? 40S2?3+ + ? + +? ? 5961S
2?2 ? 1247S2+ ? ? +?+?
?2?2 336?2?2 3?2?2 224??2 84??2
20S2?2? + ??+? ? 5961S
2 2
+?+? ? 20S
2 3
+?+? 4235377S2 2 2+ +?? 32S+????+
??2 224??2 3??2 225792?2 9?2
91S2? ?? 32S2?2 2 2+ + + +?? 257S+? ? 32S2+? ?2+ ? 8S2 ?3? 235225S4 ?2+ + + + + + + +
6?2 9?2 36?2 9?2 9?2 150528?2
16S4?2 ?2 29S4? ?2 16S4 2 2 4 4 2 4 2 2 2
+ + ? + +
+
+ +
?+? 25S+??? 25S+???+?+ +
27?2 28?2 27?2 12?4?2 2?4?2
25S4?4 ?2 10S4?2 ? ?2 10S4?3 ?2+ + + ? + + + ? 1529S
4 2 2 4 2 2
+ + + +
??? ? 20S+???+?
12?4?2 ?3?2 3?3?2 672?2?2 3?2?2
? 1529S
4?2 ?2 20S4?3 ?2 4 2 2 4 2 4 2 2+ + ? + + ? 26S+??? ? 485S+?+? ? 26S+?+?
672?2?2 9?2?2 9??2 168??2 9??2
485S4 2 4 2 4 2 2 2
+ +
?? 16S+?+?? 4S+? ? 6235?+ + + ? 256??? 140?+? 2825??+ +
504?2 27?2 27?2 96?2 3?2 3?2 32?2
? 256?
2
??? 140?
2
+?? 5065? ? ? 64?
2 ?2? ? 35?
2 3
+? ? 35? ?
+ + + +
3?2 3?2 128?2 3?2 3?2 6?2
6755S2+?? 64S2?2? + ??? 175S
2? ?? 40S2?2 ?? 175S2?2+ ++ + + + + ? + ???
288?2 9?2 18?2 3?2 12?2?2
320S2?2+ ??+?? 835S2 2 2 3+?+?? 160S+?+?? 283S2 2 2+ + + + +???? ? 35S+?+??
3?2?2 14?2?2 3?2?2 8??2 3??2
160S2?2 ? ?? 4745S2?2 ?? 80S2?3 2 2 2+ ? ++ + + + + + +?? 3745S ??? 128S ? ???+ + ? + ?
3??2 84??2 3??2 192?2 9?2
70S2? ??? 35S2 ?2?? 2425S4 ?2? 64S4?2 2 4 2+ ? ? 35S
+ + + + + + ? + ? + +?+? ?
9?2 9?2 1152?2 27?2 27?2
50S4?2 ?2 ?2? 50S4?4 ?2? 80S4?2 ? ?2? 40S4?3 ?2? 559S4?2 2? + ? + ? + + ? + ? + ? + + ? + ?? ?
?4?2 3?4?2 3?3?2 3?3?2 72?2?2
160S4 2 2+???+? ? 3025S4?2 ?2? 80S4?3 2 4 2 2 4 2+ + + + + +? ? 64S+? ? ?+ + ? ? 35S+?+? ?
9?2?2 252?2?2 9?2?2 9??2 18??2
475
40S4?2 ?2? 35S4 ??2+ + + ? 1225?2 1225??2 1225?2?2 1225S2 ??2+ + + + + + +
9??2 54?2 48?2 48?2 192?2 144?2
175S2?2 ??2 175S2?2 ??2+ + + + 1225S2 2 4 2 2 4 4+??? 1225S+? ? 100S+?+?2?2+ + + + +
3?2?2 6??2 288?2 1728?2 3?4?2
175S4?2 ?2?2+ + ? 142951 2143S+?? 32?
2 2
+ + + ?
361?+ 64S+???+ 32?
+ + + +
18?2?2 6480? 84S?? 3? 140? 3S?? 3?
1247S+?? 64?2? 1247?+ 128S+???+ 64?2+ ? 47343? 16S+???+ S + + + + +42 ??? 3?? 42?? 3S??? 3?? 1120? 3S??
? 84? ?
2 2 3 2 3 2
+ ? 5177? ? 16?+? ? 8? ? 4171S+? 16S ??? 28S ?+?+ + + +
5? 280? 3? 3? 672? 9S?? 45?
? 20S
3
+?
3
?? ? 20S
2 2 3
+???+? ? 20S+???
2
+? ? 20S
2
+?
3 2 2 3
+? 11S+??? ? 32S+???+?+
3S??3? ?3? S??3? 3?3? 6?2? 3S??2?
11S2?2 ? 485S3 2 2 2 3 2 2
+ + + + +
??? 62S+??? 1611S+?+? 64S+ + + +???+? 62S+?+?+
6?2? 84S??? 9?? 140?? 9S??? 9??
? 12091S
2
+?? ? 16S
2
+?+??
2 2
? 8S+? ? ? 10513? 35S+??? 128?
2
? ?? 11?+?+ +
2520? 9? 9? 144? 3S?? 3? 3?
? 128S+???+? 70S
2
+??? ? 256??? 70?+? ? 256S+???+? ? 22697??S + +3 ?? 3S??? 3?? 3?? 3S??? 672?
? 4?
2 2 2
+?? ? 47? ? ? 12793S+?? ? 4S+?+?? 160S
2?2+ ??+?? 80S3+???2+ + +??
? 6? 2016? 3? 3?3? S??3?
80S2?3 ?? 143S2+ + +?2??? 64S3? 2 2??+?? 157S ? ?? 35S3????+ + + + ? + + + +
3?3? 6?2? 3S??2? 6?2? 9S???
? 128S
2 2
+???? 53S2+?+?? ? 128S
3
+???+?? ? 40S
2
+?
2
+??
2 2
+ S ?
47S+??? ? 575?
9?? 9?? 9 ??? 3?? 18? 36?
? 35??
2
? 35S
2 ??2+ ? 10S
2 2 2
+?+?? 4S+?? ? 8S+??? 8 4?++ S S + ?+ + ?
16
??+
8? 24? ?2? ?? ?? 3 ? 3
2
? 8??+ 11S+? 59 20S+?+? 16+ S + ??? + S ? ???+? ?
8S+??? ? 8???+?
? 2 ? 15 3 ? 9 S??2 ?2
? 8S
2 2 2
+?+? 66??? ? 8S+??? 10S+?+? ? 16???+? ? 8S+?+? 3S+??S + S + S S +??2 5? 3 ?? ?? 3? 3 ?? S?
? 44 5 ? 25 22???? 22S+?+?? 1247??? 128???+????? + ??? ???? + + ? ?
9 3 9 3? 3S?? 21?2 3?2
? 565???? ? 128???+?? ? 1013?
2 2 3
?? ? ? 32???+? ? ? 16??? ?
7?2 3?2 28?2 3?2 3?2
4379S2? ?? 28S2? ? ?? 40S2?3? + ? + ? + + ??? 6235S
2
+???+?? 40S2? ?2? ??+ + + + + +
504?2 9?2 3?2?2 168?2?2 ?2?2
476
1247S2? ?? 20S2?3 ?? 5961S2 2 2+ ? + ? +???+?? 20S+???+?? ? 91S
2?????
+ + + + +
84??2 3??2 112??2 ??2 6?2
? 64S
2 2 2 4 2 4 2
+???+??? ? 32S+??? ?? ? 29S+??? ? ? 32S+???+? ? ? 25S
4?3 2+ ??+? ?
9?2 9?2 28?2 27?2 3?4?2
25S4? ?3 ?2? 10S4?3 2? + ? + ? + ?? ? ? 10S
4
+?
2 2 4 3 2
??+? ? 20S+??? ? 1529S4+??? 2+? ?+ +
3?4?2 3?3?2 ?3?2 9?2?2 336?2?2
20S4? ?2 2+ ? +? ? 485S4+???2? 52S4? ? ?2? 16S4? ??2+ ? + ? + ? ? ? 140????+ + +
3?2?2 168??2 9??2 27?2 3?2
140? ??? 35? ?2? ? ? ? ?? ? 175S
2
+????? 175S2+???+??? ? 160S
2
+?
2
??+???+
3?2 3?2 18?2 6?2?2 3?2?2
35S2+????? 175S2+???+??? 80S2? ?2 ??? 70S2? + ? + ? +??????
4
? 35S+???
2??
+ +
3??2 12??2 3??2 9?2 27?2
100S4+? 3 2 4 2 2 4 2 4 2 2??+ +? ?? 40S+???+? ?? 175S ???+? ?? 80S ??? ? ??+ + + ? + +
3?4?2 3?3?2 36?2?2 9?2?2
35S4+? 2 2 2?? ?? ? 361??? ? 32S+??? ? 2143S+?+? ? 64???+? ? 32S+?+ +?
18??2 140? 3S?? 84S?? 3? 3S??
? 1247??? ? 64S+?
2 2
?? ? 1247S+?+? ? 128???+? ? 64S+?+? 84????+
42?? 3S??? 42S??? 3?? 3S??? 5?
? 16S ?
2 2 3 2 3 3 2
+ +?? 16??? ? ? 28S+???? ? 16S+?+?? 20S+ + +???? 20S+???+??+
3S?? 3? 45? 9S?? 3?3? S??3?
20S2+???2+?? 20S3+?3+?? 16S3+?2??? ? 11S
2
+?
3 2
??+?? 16S ? ??
+ + S + +
+ +
?3? 3 ??3? 3S??2? 3?2? 3S??2?
? 1611S
2
+????
3 2 3 2
? 32S+???? ? 485S+?+?? ? 124S+???+?? ? 32S
3?2+ +??
140?? 9S??? 84S??? 9?? 9S???
16S2+????? ? 11???? ? 35S+?+?? ? 70???? ? 70S+?+?? 4?????+
9? 3? 3S?? 3?? 3S
+
??? ?
4S2 2 2 3 3+????? ? 80S+???+??? ? 80S+?+??? ? 5S
2
+???+???
2
+ S ?
53S+?????
3? 3?3? 3 ??3? ?2? 9??
35S3? +?+??? ? 8 ? 4??? ? 4S+?+?S ??? . (D.22)9 ??? 3 ? S??
477
Appendix E: Post-Dickean expansion of mass and scalar charge
The exact form of the mass m and the scalar charge ? defined in Eq. (4.79)
depend on how the mass is resummed, that is, on our choice of m(?, ?) and F (?).
For example, using the expressions in m(RJ) and F (?) given in Table 4.1, the mass
and scalar charge are given at 1PD order by
(RJ,?)
mA =mA((?), ) ( )( )(E.1)
(RJ,?) ? d logmA B?0 ? d logmA d logmB GmB?A =?0 1 (2?0 )( + 1 2?)0 (1?d? 2 d? )2?0 d? ? 20rc
? 3 d logmA ? d logmB GmB 14?0 1 2?0 +O ,
d? d? rc2 c4
(E.2)
where A 6= B, while the choice of m(RE)( and F
(??) gives
? )(E) ( )
(RE,?) (E) Gm ?0?B 1
mA = ?
B
0mA (?) 1 + ? +O , (E.3)? 2 40rc c
(E)
(RE,?) ? d logmA (?)?A = . (E.4)d?
Note that the expressions in Eqs. (E.2) and (E.3) receive higher-order corrections,
while Eqs. (E.1) and (E.4) are exact. In general, whenever the function m(?, ?) can
be factored into
m(?, ?) = m?(?)m?(?), (E.5)
478
the quantities
(m?(?) ? m?(?), )/ (E.6)
? ?d logm? d logm? D logm(?, ?) dFq = ? , (E.7)
d? d? D? d?
are exact at all orders in the PD expansion. These quantities represent the resummed
piece of the mass and scalar charge; for the resummation schemes defined in Table
4.1, these quantities are listed in Table E.1. When using a particular resummation
scheme, it is most convenient to work with these variables instead of m and ? so
as to avoid the additional bookkeeping required to track the PD corrections to the
mass and scalar charge.
Table E.1: Resummed piece of the mass m and scalar charge ? for the resumma-
tion schemes given in Table 4.1. We denote the differential operator D with the
D?
abbreviation D.
Resummation Scheme m?(?) q(?)
m( ) F ( )
RJ ? m ( ?D l)ogm
RJ ?? m ?2? B log ? 1/2D logm
2
RE ? m(E) ( )1 ?D logm2?
RE ?? m(E) B log ?
1/2
(1? 2?D logm)
2
479
Appendix F: Two-body potentials at post-Dickean order
The sources defined in Eqs. (4.53)?(4.56) computed at 1PD order are
( ) ( )
3v21 ? Gm2(1? 5?0?2) 1? =m 1 + ?(3)1 ( (x? x1) +O + (1
 2) , (F.1)2c2 ?0rc2 ) c4 ( )
m1?
2 2
1 v1 Gm2(6?0 +B?2 ? 6?0?2)? (3) 1s = 1? ? ? (x? x1) +O + (1
 2) ,
?0 2c2 2?0?0rc2 c4
( ) (F.2)
i
?i
m1v
= 1 ?(3)(x? x1) +O
1
( )+ (1
 2) , (F.3)c c3
2
ii m1v? = 1 ?(3)(x? 1x1) +O + (1
 2) , (F.4)
c2 c4
where r = |x1 ? x2| and we have suppressed the expansions in mi and ?i using
the notation of Eq. (4.79). Hence, the two body potentials needed to compute the
equations of motion and scalar mass in Secs. 4.5 and 4.7 are given by
? (
? ?(t,x
?) 3 ? m1 3v
2
1 ? Gm2(1?
) ( )
5?0?2) 1
U | ? ?|d x = 1 + +O + (1
 2) ,x x r 2 2 41 2c ?0rc c
? ( ) ((F.)5)
? ?s(t,x
?) 3 ? m
2 2
1?1 ? v1 ? Gm2(6?0 +B?2 ? 6?U d x = 1 0?2) 1s |x? x?| +O? r 20 1 2c 2? 2 40?0rc c
+ (1
 2) , (F.6)
480
? ( )
M ? ? 3 ? ?1 1s (?s(t),x )d x = ?0 m1?1 +O + (1
 2) ,c2 (F.7)
1
M?s = O? ,c3 ? ( )i
i ? ? (t,x ) 3 ? m
i
1 v 1
V ? | ? ?|d x =
1 +O (+ ()1
 2) , (F.8)x x r1 c c3? ?i i
i ? ?s(t,x )v m1?1 v 1V 3 ? 1s ? | ? d x = +O + (1
 2) , (F.9)x x?| ? 30r1 c ( )c
?ii? (t,x
?) m v23 ? 1? 1
1
1 ? | ? ?| d x = +Ox x r c2 c(31? j )
+ (1
 2) , (F.10)
ij
ij ? ? (t,x ) m v
i
1 v 1
?1 ? | ? ?| d
3x? = 1 1 +O +((1
) 2) , (F.11)x x r1 c2 c3?
s ? ?s(t,x )U(t,x
?) m1m2?1 1
?2 ? | ? ?| d
3x? = +O + (1
 2) , (F.12)
x x ? r 20 1r ( c )
? ?(t,x
?)Us(t,x
?) m1m2?2 1
? 3 ?2s ? | ? ?| d x = +O ( +)(1
 2) , (F.13)x x ?0r 21r c?
s ? ?s(t,x )Us(t,x
?)
? d3 ?
m1m2?1?2 1
2s ? | ? ?| x = ( +)O + (1
 2) , (F.14)x x ? 2r r c20 1
X ? 1?(t,x?)|x? x?|d3x? = m1r1 +O ( )+ (1
 2) ,c2
d2m dm dr d21 1 1 r1 O 1X? = ( r1 + 2 +m1 )+ ( + (1
 2) , (F.15)dt2 dt dt dt2 c2 )
? v
2 (v1 ? n 21) 1
= m? 1 a1 n1 +
1 ? +O + (1
 2) ,
r 21 r1 c ( )
Xs ? ? ?
1
s[(t,x )|x? x
?|d3x? = ? ?10 m1?1r1 +O ]+ (1
 2) .c2
d2?1 (m1?1) d(m1?1) dr
2
1 d r1
X?s = ?0 ( r1 + 2dt2 dt d? )
+m(1?1 ) + (1
 2) , (F.16)t dt2
m1?1 ? v
2 2
1 ? (v1 n1) O 1= a1 n1 + + + (1
 2) ,
? 20 r1 r1 c
where the time derivatives of the masses and scalar charges are pushed to higher
PD order because ( )
dmA DmA 1
= v????(xA) ? O , (F.17)
dt D? A c3
where v? ? 0A ? uA/uA.
481
Appendix G: A closer look at the dynamical scalarization model of
Palenzuela et. al.
Another analytic model of dynamical scalarization was proposed in Ref. [173].
This model augments the PN equations of motion with a feedback mechanism that
simulates the non-perturbative growth of the scalar field around each body (see Sec.
4.8.2 for more detail). This prescription is uniquely defined when working at leading
order but becomes ambiguous when extended to higher PN orders. The construction
given in Ref. [173] uses the 2.5PN equations of motion (given in Ref. [497]) and
a Newtonian order feedback mechanism [Eqs. (4.142) and (4.143)]. The authors
also set to zero all derivatives of the scalar charge [the first of which is given in Eq.
(4.93)].
While this particular set of choices leads to predictions consistent with numerical-
relativity, we would like to explore other realizations of this model for two reasons.
First, we want to understand the impact of these algorithmic decisions; if a par-
ticular choice greatly impacts the model?s performance, understanding its physical
significance is important. Second, we would like to track the changes to the model
at each order so as to check the best way to improve the results of Ref. [173] with
future PN calculations. We address these two concerns by investigating the effects
482
fGW [Hz] fGW [Hz]
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200
10-5 ? ? ? ? ? ? ? ? ? ? ? ? ? 10-5 ? ? ? ? ? ??
- ?
??? ?? ?
10 6 ? ??((n))?=0 10-6n 0
10-7 ???
? QE
10-7
? ??
10-
?
8 Newtonian 1PN 10-8 ? ?
?
1PN MS+2PN EOM ???? ???
102 102 
101 101 
100 0 
10-
10
1
- 10
-1
10 2 10-2
0.00 0.01 0.02 ?/ 0.03 0.04 0.05 0.00 0.01 0.02 ?/ 0.03 0.04 0.05GM c3 GM c3
Figure G.1: Scalar mass predicted by the model of Ref. [173] for a
(1.35 + 1.35)M neutron-star binary system on a circular orbit as a
function of the orbital frequency and gravitational wave frequency. The
scalar mass at Newtonian and 1PN order computed without (with) the
derivatives of scalar charge is plotted in red (blue) using the Newtonian,
1PN, and 2PN equations of motion (dashed, dot-dashed, and solid lines,
respectively) and the Newtonian order feedback mechanism given in Eq.
(4.142). We also plot the quasi-equilibrium configurations (QE) reported
in Ref. [174] (dotted). The bottom panels depict the magnitude of the
fractional error between the PD and quasi-equilibrium results. We use
the APR4 equation of state with (left) B = 9, ?? = 3.33 ? 10?110 and
(right) B = 8.4, ??0 = 3.45? 10?11.
of including derivatives of the scalar charge and using a higher-order feedback mech-
anism. The authors of Ref. [173] briefly mention these two modifications and argue
that they do not significantly impact the model; we expand on this discussion here,
offering a precise, quantitative description of their effects.
Including derivatives of the scalar charge: The derivatives of the scalar charge
enter this model through the equations of motion [see Eqs. (4.89) and (4.90)] and
the feedback mechanism [see Eqs. (G.2) and (G.3)] beginning at 1PN order. The
483
Fractional Error MS [M?]
Fractional Error MS [M?]
decision to set these derivatives to zero was made in Ref. [173] to ensure that m?(?)
and ??(?) were evaluated at each star rather than expanded about the background
value ?0. However, this procedure is problematic, as simply setting the derivatives
of ?? to zero does not properly resum these expansions. For example, ?? itself appears
in every term of the Einstein-frame mass [analogous to Eq. (4.15)]
[ ]
m?(E)(??) =m(E)
1
(??0) 1 + ???? (??2 ? ???)?2 + ? ? ? , (G.1)
2
where ? ? (?? ? ??0). Removing the derivatives of ?? eliminates most of the terms
in the expansions of m?(?) and ??(?) but leaves certain higher-order terms propor-
tional to powers of the scalar charge. A fully consistent treatment should absorb
these extraneous terms into the definitions of m?(?) and ??(?); instead, this model?s
treatment of m? and ?? as unexpanded quantities ensures that the surviving terms in
the expansion are effectively double counted.
Even without a mathematically rigorous motivation, the choice to drop the
derivatives of the charge still yields predictions in qualitative agreement with nu-
merical relativity. However, there are many other equally valid ways to alter the
coefficients in expansions like that of Eq. (G.1) ? for example, the coefficients con-
taining ??? could be halved rather than set to zero. To provide some bound on the
effect of these choices, in Fig. G.1 we compare the total scalar mass predicted by the
model of Ref. [173] when all derivatives of the scalar charge are dropped (red) and
when all are kept (blue). We restrict to circular orbits using the Newtonian, 1PN,
and 2PN equations of motion (dashed, dot-dashed, and solid lines, respectively) ?
the exact prescription used in Ref. [173] is the solid, red curve.
484
fGW [Hz] fGW [Hz]
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200
-5 ? ? ? ? ? ? ?10
????
? ? ? ? ? ? 10-5 ? ? ? ? ? ?
? ??
?
10-6 ? Newtonian ?B 10-6? 1PN ?BQE
10-7 ?? 10-7??? ???
10-8 Newtonian+ 1PN ?10-8 ?1PN MS 2PN EOM ???? ???
102 102 
101 101 
10-0 10010-1 10--
 
1
10 2 10 2
10-3 10-3
0.00 0.01 0.02 ?/ 0.03 0.04 0.05 0.00 0.01 0.02 ?/ 0.03 0.04 0.05GM c3 GM c3
Figure G.2: Same as Fig. G.1 but also including the predictions of the
model of Ref. [173] computed using the 1PN extension of the feedback
model (green) given in Eq. (G.2). The derivatives of the scalar charge
were dropped in computing all of the plotted curves.
The inclusion of these terms increases the scalar mass by approximately 20?
50% both before and after scalarization. This result is consistent with our previ-
ous observation that higher-order terms left in the PN expansion should produce
extraneous contributions when the mass and charge are resummed. In addition, we
note that the model of Ref. [173] overestimates the scalar mass compared to the PD
approximation at the same order. Combining these two observations, we argue that
double counting can become a significant issue when using a simple feedback mech-
anism like that of Ref. [173], and that while simply dropping particular terms from
the PN expansion can help remedy these issues, it is not the ideal solution. Instead,
the corresponding resummation should be accounted for in a more systematic way,
as is done with the PD approach.
485
Fractional Error MS [M?]
Fractional Error MS [M?]
Extending the feedback mechanism to 1PN order: The feedback mechanism
used in Ref. [173] contains only the leading order contributions to the scalar field
despite being paired with the 2.5PN equations of motion computed.1 The impact
of using approximants of such different order is unclear, but the mismatch may
lead to certain unintuitive predictions. We consider instead using the natural 1PN
extension of the feedback mech[anism ( ) ]
(1) Gm?2??2 Gm?2
?B =??0 + + ?
1 ? 3 3 Gm?1??2(v2 n)2 ? ?? + ?? ??2 ?2 1 2 ? ??1??2 ,? rc2 ? rc40 0 2 2 2 ?0r
(G.2)
(2)
?B = (1
 2) , (G.3)
(i)
where m?i, ??i, and ??
?
i are evaluated at ?B .
We compare the total scalar mass computed using the Newtonian (red) and
1PN feedback models (green) with equations of motion at Newtonian, 1PN, and
2PN order in Fig. G.2, making the additional choice to set all derivatives of the
scalar charge to zero, as was done in Ref. [173]. The inclusion of higher-order effects
in these two aspects of the model produces competing shifts in the predicted onset
of DS: the choice of 1PN feedback system over the Newtonian system pushes this
transition point to higher frequency, while the 1PN terms in the equations of motion
push the transition to lower frequency. These two effects nearly cancel each other in
1The authors of Ref. [173] considered the effect of adding to this feedback mechanism the order
O(1/r2) terms from the field felt by a static test mass far from an isolated body. These terms
were shown to have negligible impact on their model. Here, we consider the 1PN corrections to
the scalar field felt by each body in a comparable-mass binary system, which comprise a more
comprehensive set of O(1/r2) corrections.
486
such a way that the predictions when working consistently at Newtonian order (i.e.
Newtonian order feedback and equations of motion) are very close to those when
working consistently at 1PN order. We observe that working consistently at one
order generally improves the agreement with the quasi-equilibrium configuration
calculations of Ref. [174]. The most accurate model depicted in Fig. G.2 uses
the 1PN feedback model in conjunction with the 2PN equations of motion, but in
line with the previous observation, we suspect that adding the 2PN corrections to
Eqs. (G.2) and (G.3) will improve these predictions; we leave the calculation and
implementation of these higher-order terms for future work.
To recap, some of the technical aspects in the construction of the model pro-
posed in Ref. [173] are ambiguous; the prescription for these options is only precisely
specified when working at Newtonian order. These choices arise because the model
splices a non-linear feedback mechanism on to independently computed PN equa-
tions of motion. We find that the model is most accurate when one uses a feedback
mechanism and equations of motion of the same order and when one drops some of
the higher-order terms in the PN expansions (e.g. the derivatives of the charge) to
minimize double counting.
The PD formalism avoids these issues by performing a resummation of the
post-Newtonian expansion at the level of the action. By carrying through this
resummation consistently, a non-linear feedback mechanism analogous to Eqs. (G.2)
and (G.3) organically arises alongside the equations of motion. Thus, the PD model
gives results at a consistent order while also avoiding double counting.
487
Appendix H: Effective action for compact objects away from critical
point
A crucial assumption made in the construction of the effective action (6.7) is
the near-criticality of the scalar mode q. The power counting used in the main text to
formulate this relatively simple effective theory is not valid without this assumption.
For example, if c(2) > 0 (i.e., the object does not spontaneously scalarize), then away
from the critical point one finds q ? ?IR(y) ? O(R/r) and must include terms like
[?IR(y)]2 to the action to work consistently at the given order in R/r. For scalarized
compact objects c(2) < 0 far from the critical point, the scalar field ?
IR(y) and
mode q reach values too large for our polynomial expansion around zero to be valid.
If one expands the fields around their true (nonzero) equilibrium values instead,
the effective action no longer respects the spontaneously broken scalar-inversion
symmetry of the underlying theory.
In this appendix, we relax this assumption that the scalar mode q is nearly
critical. We construct an effective action valid in this broader context, and then
show that the model (6.7) is recovered as one approaches the critical point. We
still ignore derivative couplings involving ???
IR(y) since they belong to multipoles
above the monopole (` > 0) and we are only interested in a monopolar mode q. The
488
most generic effective action in the scalar-inversion-symmetric (unbroken) phase
then reads
? [ ]
Sunbroken
cq?2
= d? q?2CO ? c(0,0) ? V ? V ? ? V q? + c AIRA ? (y)y?? , (H.1)2
c(0,2) 2 c(0,4)V = q + q4 + . . . , (H.2)
2 4!
? c(2,0) c(4,0)V = [?IR(y)]2 + [?IR(y)]4 + time derivatives + . . . , (H.3)
2 4!
V q? = ??IR c(1,3) c(2,2)(y)q + ?IR(y)q3 + [?IR(y)]2q2
3! 4 (H.4)
c(3,1)
+ [?IR(y)]3q + time derivatives + . . . ,
3!
where the subscripts in c(i,j) indicate the powers of ?
IR and q, respectively; the
coefficients c(n) in the main text correspond to c(0,n) in this notation. All terms
must be even polynomials in ?IR, q due to scalar-inversion symmetry and must
contain an even number of time derivatives due to time-reversal symmetry. Higher
time derivatives that would appear in V can always be removed by appropriate
redefinition of q [498]; we assume that such field redefinition has been done. This
allows for an interpretation of V as an ordinary potential for the mode q in the
absence of an external driving field ?IR.
To fix all coefficients in the action, one needs to match against the exact
solution for an isolated body in a generic time-dependent external scalar field. In
general, this is a complicated endeavor, and we do not attempt it here.1 Instead, we
explore what information can be gleaned from the sequences of equilibrium solutions
1Note that our assumption of time-reversal symmetry needs to be imposed on the exact solution
as well, so, in the case of a BH, one must impose somewhat unphysical (reflecting) boundary
conditions at the horizon.
489
considered in the main text. Using only this restricted class of solutions, we do not
expect to find a unique match for all coefficients in the effective action above, but
rather a series of relations relating them to the exact solutions.
Solutions for compact objects in equilibrium are manifestly time independent,
and thus cannot inform the terms containing time derivatives in the effective action;
we omit these terms from the action below for brevity. We perform the same pro-
cedure outlined in the main text to match the IR fields of the effective theory (H.1)
to the IR projection of the UV solutions (6.11) and find cA = E , ?0 = ?IR(y), and
q? IR ? IR
Q(?0) = ?
?V (q, ? ) ? dV (? ) , (H.5)
??IR d?IR
M(?0) = c(0,0) + V + V ? + V q?. (H.6)
Together with the equation of motion for q,
dV (q) ?V q?(q, ?IR)
0 = + , (H.7)
dq ?q
we see that
[ ] [ ]
M dV (q) ?V
q?(q, ?IR) ?V q?(q, ?IR) dV ?(?IR)
d = + dq + + d?IR (H.8)
dq ?q ??IR d?IR
= ?Qd?0, (H.9)
in agreement with the first law of BH thermodynamics [333, 334]. We see that ?0
and Q are conjugate variables, and therefore we can construct the ?gravitational
free energy? M(Q) via a Legendre transformation of M(?0),
M(Q) ?M(?0) + ?0Q, (H.10)
490
such that ?0 = dM/dQ. As in the main text, from a sequence of exact compact-
object solutions in the full theory, one obtains M(?0) and Q(?0) and then can
compute M(Q) numerically from Eq. (H.10).
To aid comparison, we also define
V(Q) ?M(Q)? c(0,0), (H.11)
which represents the component of the ?free energy? due to the scalar charge Q. It
admits an expansion around Q = 0
C
V (2)
C(4)
(Q) = Q2 + Q4 + . . . , (H.12)
2 4!
whose coefficients can be extracted numerically. Unlike V , this quantity does not
correspond to the potential of any dynamical variable, but instead simply represents
the energetics of a sequence of equilibrium solutions. While these two quantities are
not directly related in general, in the vicinity of the critical point it is possible to
reconstruct the potential V from the energetics V (as we found in the main text).
In the remainder of this appendix, we demonstrate this connection explicitly by
expressing the C(n) in terms of the coefficients in the effective action c(i,j), and then
take the limit that q becomes unstable c(0,2) ? 0, i.e., approaches the critical point.
Working perturbatively in ? = ?IR0 , we solve the equation of motion (H.7) for
q, relate this solution to Q and M via Eqs. (H.5) and (H.6), and then insert these
solutions into Eq. (H.11) and read off the coefficients C(n) from the expansion in
491
Eq. (H.12). We find that
c(0,2)
C(2) = , (H.13)
1? c(0,2)c(2,0)
1 [ ]
C 2 3 4(4) = ? c4 (0,4) + 4c(0,2)c(1,3) + 6c c(2,2) + 4c(1 c c ) (0,2) (0,2)c(3,1) + c(0,2)c(4,0) .(0,2) (2,0)
We see that, in general, an instability in the mode q cannot be inferred directly from
V(Q), i.e., C(2) < 0 ; c(0,2) < 0 and c(0,2) < 0 ; C(2) < 0. However, close to the
critical point c(0,2) ? 0, we find
C(2) = c(0,2) +O(c2(0,2)), (H.14)
C(4) = c(0,4) +O(c(0,2)), (H.15)
which confirms the link between the dynamical mode potential V (q) and the ener-
getics of equilibrium solutions V(Q), close to the critical point.
The relation between the mode q and the scalar charge Q along a sequence of
equilibrium solutions reads
Q Q3 [
q = ? + c(1,3) + c4 (2,0)c(0,4) + 3c(0,2)(c(2,2) + c(2,0)c(1,3))1 c(0,2)c(2,0) 3!(1? c(0,2)c(2,0)) ]
+ 3c2 3(0,2)(c(3,1) + c(2,0)c(2,2)) + c(0,2)(c(4,0) + c(2,0)c(3,1)) + . . . (H.16)
Note that Eq. (H.16) does not lead to Q = q as in the main text when c(0,2) = 0.
But full agreement with the main text (ignoring higher orders in c(0,2) throughout)
is achieved if we redefine
3 [ ]
q ? qq + c 5(1,3) + c(2,0)c(0,4) +O(c(0,2), Q ). (H.17)
3!
492
Appendix I: Numerical calculation of the effective point-particle ac-
tion V (q)
In this appendix, we detail the numerical calculation of equilibrium BH so-
lutions in EMS theory used to construct V (q) through the matching procedure
discussed in the main text. We consider the class of theories in Eq. (6.1) and re-
strict our attention to static, spherically symmetric configurations. Starting with
the ansatz for vector potential and metric
A = ?(r) dt, (I.1)
2
ds2 = ?N(r)e?2?(r) 2 drdt + + r2(d?2 + sin2 ?d?2), (I.2)
N(r)
where N(r) ? 1? 2m(r)/r and m(r) is the Misner-Sharp mass (not to be confused
with m(q) introduced in the main text), the field equations reduce to
e?? E
?? = ? , (I.3a)
f(?) r2
? 1
2
m = r2N??2
E
+ , (I.3b)
2 2f(?)r2
?? + r??2 = 0, (I.3c)
? 2
(e??
f (?)E
r2N??)? = ?e?? , (I.3d)
2(f(?))2r2
where ? = d/dr and E is the electric charge of the BH. Here, the field equation for
the electric potential ? was already integrated once, introducing the electric charge
493
E as an integration constant. We impose the following boundary conditions at the
horizon
rH
m(rH) = , ?(rH) =
2
? (?H , ?(rH) = )?H ,
? ? f (?H) E
2
? (rH) = . (I.4)
2f(?H)rH f(?H)r2H ? E2
Note that ?H represents a simple rescaling of the time coordinate, and thus can be
chosen arbitrarily; we ultimately rescale t such that ?(r = ?) = 0. For computa-
tional simplicity, we rescale all dimensional quantities by the horizon radius rH , i.e.,
r? ? r/rH , m? ? m/rH , and E? = E/rH , and then compactify the domain over which
they are solved using the variable
? r ? rH r? ? 1x = , (I.5)
r + brH r? + b
where the constant b is chosen to adequately resolve the solution.
As discussed in the main text, we consider sequences of BH solutions with
fixed electric charge E and entropy S (or horizon area), which is equivalent to fixed
E? and rH . Fixing these two parameters, we generate a sequence of solutions by
solving Eqs. (I.3) and (I.4) for several values of ?H . We then extract the mass
M, asymptotic field ?0, and scalar charge Q from the asymptotic behavior of the
solution
( ) ( )
?M O 1 ? Q 1m(r) + , ?(r) ?0 + +O , (I.6)
r r r2
allowing us to implicitly construct the functionsM(?0) and Q(?0) used in the main
text to compute the effective potential V (q).
494
Appendix J: Ingredients for the Fisher matrix analysis
For a GW detector with the one-side PSD, Sn(f), the SNR of a Fourier-domain
waveform, h?(f), is ( ?? )1/2
? = h?(f)? h?(f) , (J.1)
where the inner product is defined to be [393, 394],
( ??? ) ? fmax h??? 1(f)h?2(f) + h?1(f)h??2(f)h?1(f) h?2(f) 2 df . (J.2)
f S (f)min n
For all calculations in Sec. 7.4.1, we use the design zero-detuned high-power noise
PSD, starting from 10 Hz for aLIGO [389], the target noise PSD, starting from 5 Hz
for CE [367], and the ET-D noise PSD, starting from 1 Hz for ET [388]. Thus,
in Eq. (J.2) we choose fmin = 10 Hz, 5 Hz, and 1 Hz for aLIGO, CE, and ET,
respectively. Somewhat arbitrarily, we choose for fmax twice the innermost stable
circle orbit (ISCO) frequency computed from the binary?s binding energy at 2 PN
order. It reads [499], [ ( ? )]3/2
c3 3 14
fmax = 1? 1? ? . (J.3)
?GM 14? 9
Because the three detectors do not have good sensitivity at high frequency, say
& kHz, the choice of fmax influences the result very marginally.
For the nonspinning BNS inspiraling waveform in the Fourier domain, we use a
495
restricted waveform with the leading-order term in amplitude, A, and up to 3.5 PN
terms in the phase, ?(f) [240, 259, 266],
h?(f) =Af?7/6ei?(f) , { ( ) (J.4)
? ? 3 ?5/3 5 2 ?2/3 3715 55?(f) =2?ftc ? 2/3c(? + u 1? (??) u + + ? u4 128? 168 ) 756 9
? 15293365 27145 308516(?u + ) + ? + ?
2 u4/3
508032 504 72[
38645 ? 65 11583231236531 640 6848+? ?( (1 + ln u) u
5/3 + ? ?2 ? ?E
756 9 4694)215680 3 21 ]
?6848 15737765635 2255 76055 127825( ln (64u) + ? + ?
2 ? + ?2 ? ?3 u2
63 3048192 )12 } 1728 1296
77096675 378515 74045
+? + ? ? ?2 u7/3 , (J.5)
254016 1512 756
where u ? ?GMf/c3, A ? M5/6/DL with the chirp mass M ? ?3/5M and the
luminosity distance DL, tc and ?c are reference time and phase respectively, and
?E = 0.577216 . . . is the Euler constant. Note that in Eq. (J.5) the gothic u is equal
to ??3/5u where u ? ?GMf/c3, as defined in Ref. [240]. We include in Eq. (J.5) only
the leading dipole term for the scalar contribution. Furthermore, since the spins of
BNS systems are supposed to be small, we do not include them in the analysis (see
Table 7.3 where we give a rough estimation of the spin terms in the GW phasing).
To calculate the Fisher matrix (7.17), we need to compute partial derivatives
of the frequency-domain waveform (J.4). They read (notice that, when calculating
496
derivatives, u depends on both ? and M),
?h?(f)
=h?(f) , (J.6)
? lnA { ( )
?h?(f) i ?5/3 ? 1 2 ?2/3 ? 743 11 9= u ?? u + + ? u2/3( ) + ?u? ln ? ? 3584 16128 128 40
?3058673 5429 617+ + ? + ?2 u4/3( 5419008 21504 512 ) [
?7729 ? 38645 13+? ln u + ? u5/3 ?11328104339891 321
4032 32256 ( + ) + ?E128 166905446400 35 ]
2 107 3147553127 ? 451 2 15211 25565+6? + ln (64u) + ? ? + ?2 ? ?3 u2( 35 130056192 )512 } 18432 6144
?15419335 ? 75703 ? 14809+? 2 7/3{ ? ? u h?(f) , (J.7)1548288 32256 1075(2 )
?h?(f) i
= u?5/3M ( ?
5 5 2 ?2/3 ? 3715 ? 55+ ?? u + ? u2/3 ?+ u
? ln ? 128 3072 ) 32256 (384 4)
?15293365 ? 27145 ? 3085 2 4/3 38645 65+ ? ? u + ? ? ? u5/3[ 65028096 64512 9216 ( 32256 384 )
10052469856691 ? 5+ ?2 ? 107 ?15737765635 2255? + + ?2E ?
600859607040 3 42 ] 390168576 1536
?107 76055 2 ? 127825( ln (64u) + ? ?
3 u2
126 221184 165888 ) }
77096675 378515 ? 74045+? + ? ?2 u7/3 h?(f) , (J.8)
16257024 96768 48384
?h?(f)
=i2?fh?(f) , (J.9)
?tc
?h?(f)
=? ih?(f) , (J.10)
??c
?h?(f)
=? 5i u?7/3h?(f) . (J.11)
?(??)2 7168?
497
Appendix K: Dynamical scalarization in ultra-relativistic binary neu-
tron stars
In this appendix, we discuss the sharp feature observed in the averaged effective
scalar coupling of a very massive BNS that undergoes dynamical scalarization (see
the 1.9M?1.9M case in Fig. 7.10). We find that generically NSs of very high
mass can scalarize more abruptly than their less massive counterparts.
From Fig. 7.5, we observe that very massive NSs exhibit very small effective
scalar couplings ?A. In these stars, the effective scalar coupling is nonperturbatively
suppressed below the non-relativistic (low-mass) limit ?A ? ?0. The cores of these
stars are ultra-relativistic, with a negative trace of the stress-energy tensor T? = ??
3p? < 0. The mass at which NSs become ultra-relativistic in this sense depends on
the EOS and can be read off from Fig. 7.7 as the mass at which the best constraint on
|?| drops to zero. Recall that spontaneous scalarization stems from a large, positive
source on the righthand side of Eq. (7.3) that grows with ?. In ultra-relativistic
stars, this source term becomes negative, causing the star to spontaneously ?de-
scalarize?.
When placed in a binary system, ultra-relativistic NSs can dynamically scalar-
ize, but the transition occurs very abruptly (e.g., the 1.9M?1.9M system with
498
50 f = 180 Hz
f = 130 Hz f = 230 Hz
20
f = 560 Hz f = 608 Hz f = 608.2 Hz
10
mA = 1.8M5 
mA = 1.9M
10?5 10?4 10?3 10?2 10?1
?
9 9
6 6
3 3
0 0
?3 mA = 1.8M ? m = 1.9M3 A 
0 2 4 6 8 0 2 4 6 8
rc2/G [M] rc2/G [M]
Figure K.1: (Top) Evolution of ??A for NSs with (dashed) and without
(solid) ultra-relativistic cores as a function of the background scalar field.
The blue, red, and green annotations indicate the scalar field at each star
at frequencies before, during, and after dynamical scalarization in an
equal-mass binary system respectively. (Bottom) Profile of the trace of
the stress-energy tensor within each NS at each of the annotated points
in the top panel.
the EOS WFF1 shown in Fig. 7.10). As the system scalarizes, the massive NSs
transition to a state in which T? is everywhere positive. Figure K.1 depicts this
transition in comparison to dynamical scalarization in less massive systems. The
top panel shows ??A?the PD equivalent of the quantity defined in Eq. (7.22)?for
1.8M (solid) and 1.9M (dashed) stars with the EOS WFF1 plotted as a function
of scalar field. The highlighted points indicate the field at each NS in an equal-mass
binary before (blue), during (red), and after (green) dynamical scalarization. The
bottom panels depict the profile of T? within each star at each of these points.
499
G3T?/c8 [10?4M
?2 ??
 ] A
The top panel of Fig. K.1 demonstrates why dynamical scalarization occurs
abruptly for ultra-relativistic NSs. Recall that ?A (and consequently ??A) quantifies
how easily a NS can scalarize, with larger values indicating that the star is more
susceptible to dynamical scalarization. As expected, when immersed in a weak scalar
field (i.e., during the early inspiral), ??A is significantly smaller for ultra-relativistic
stars than less massive stars, indicating that the latter will dynamically scalarize
at a lower frequency. However, unlike for less massive stars, ??A increases slightly
as the scalar field reaches larger values (? ? 0.002 in Fig. K.1) for ultra-relativistic
NSs. This triggers a run-away process in a binary system as a small increase in
the scalar field produced by one star causes the other star to scalarize more easily,
which in turn allows the second star to produce a larger scalar field for the first.
For the ultra-relativistic BNS depicted in the top panel of Fig. K.1, this transition
is completed after an evolution of only 0.2 Hz.
500
Bibliography
[1] M. Khalil, N. Sennett, J. Steinhoff, J. Vines, and A. Buonanno, ?Hairy binary
black holes in Einstein-Maxwell-dilaton theory and their effective-one-body
description,? Phys. Rev. D98, 104010 (2018), arXiv:1809.03109 .
[2] N. Sennett, S. Marsat, and A. Buonanno, ?Gravitational waveforms in
scalar-tensor gravity at 2PN relative order,? Phys. Rev. D94, 084003 (2016),
arXiv:1607.01420 .
[3] N. Sennett and A. Buonanno, ?Modeling dynamical scalarization with a
resummed post-Newtonian expansion,? Phys. Rev. D93, 124004 (2016),
arXiv:1603.03300 .
[4] N. Sennett, L. Shao, and J. Steinhoff, ?Effective action model of dynam-
ically scalarizing binary neutron stars,? Phys. Rev. D96, 084019 (2017),
arXiv:1708.08285 .
[5] M. Khalil, N. Sennett, J. Steinhoff, and A. Buonanno, ?Theory-agnostic
framework for dynamical scalarization of compact binaries,? Phys. Rev. D
100, 124013 (2019), arXiv:1906.08161 [gr-qc] .
[6] L. Shao, N. Sennett, A. Buonanno, M. Kramer, and N. Wex, ?Constraining
nonperturbative strong-field effects in scalar-tensor gravity by combining pul-
sar timing and laser-interferometer gravitational-wave detectors,? Phys. Rev.
X7, 041025 (2017), arXiv:1704.07561 .
[7] N. Sennett, T. Hinderer, J. Steinhoff, A. Buonanno, and S. Ossokine, ?Dis-
tinguishing Boson Stars from Black Holes and Neutron Stars from Tidal In-
teractions in Inspiraling Binary Systems,? Phys. Rev. D96, 024002 (2017),
arXiv:1704.08651 .
[8] N. Sennett, R. Brito, A. Buonanno, V. Gorbenko, and L. Senatore,
?Gravitational-Wave Constraints on an Effective Field-Theory Extension of
General Relativity,? Phys. Rev. D 102, 044056 (2020), arXiv:1912.09917 [gr-
qc] .
501
[9] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Tests of General Relativity with
GW170817,? Phys. Rev. Lett. 123, 011102 (2019), arXiv:1811.00364 .
[10] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Tests of General Relativity with
the Binary Black Hole Signals from the LIGO-Virgo Catalog GWTC-1,? Phys.
Rev. D100, 104036 (2019), arXiv:1903.04467 .
[11] N. Sennett, A. Buonanno, L. Gergely, M. Isi, and B. Sathyaprakash,
?Constraining Jordan-Fierz-Brans-Dicke gravity with GW170817,? (2019, in
prep.).
[12] A. Einstein, ?On the General Theory of Relativity,? Sitzungsber. Preuss.
Akad. Wiss. Berlin (Math. Phys.) 1915, 778?786 (1915), [Addendum:
Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.)1915,799(1915)].
[13] A. Einstein, ?The Field Equations of Gravitation,? Sitzungsber. Preuss. Akad.
Wiss. Berlin (Math. Phys.) 1915, 844?847 (1915).
[14] H. Minkowski, ?Raum un Zeit,? Phys. Z. 10, 104?111 (1908).
[15] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman,
San Francisco, 1973).
[16] K. Schwarzschild, ?U?ber das Gravitationsfeld eines Massenpunktes nach der
Einsteinschen Theorie,? Sitzungsberichte der Ko?niglich Preu?ischen Akademie
der Wissenschaften (Berlin), 1916, Seite 189-196 (1916).
[17] K. S. Thorne, ?Nonspherical Gravitational Collapse: A Short Review,? in
Magic Without Magic: John Archibald Wheeler: A Collection of Essays in
Honor of His Sixtieth Birthday, edited by J. R. Klauder (W. H. Freeman, San
Francisco, 1972).
[18] Y. B. Zel?dovich and I. D. Novikov, ?The Hypothesis of Cores Retarded during
Expansion and the Hot Cosmological Model,? Sov. Astr. 10, 602 (1967).
[19] S. Hawking, ?Gravitationally collapsed objects of very low mass,? Mon. Not.
Roy. Astron. Soc. 152, 75 (1971).
[20] M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, ?Primordial black
holes?perspectives in gravitational wave astronomy,? Class. Quant. Grav.
35, 063001 (2018), arXiv:1801.05235 .
[21] J. E. Chase, ?Event horizons in static scalar-vacuum space-times,? Commun.
Math. Phys. 19, 276?288 (1970).
[22] J. D. Bekenstein, ?Transcendence of the law of baryon-number conservation
in black hole physics,? Phys. Rev. Lett. 28, 452?455 (1972).
[23] S. W. Hawking, ?Black holes in the Brans-Dicke theory of gravitation,? Com-
mun. Math. Phys. 25, 167?171 (1972).
502
[24] M. S. Volkov and D. V. Galtsov, ?NonAbelian Einstein Yang-Mills
black holes,? JETP Lett. 50, 346?350 (1989), [Pisma Zh. Eksp. Teor.
Fiz.50,312(1989)].
[25] R. Penrose, ?Gravitational collapse and space-time singularities,? Phys. Rev.
Lett. 14, 57?59 (1965).
[26] A. Einstein, ?Approximative Integration of the Field Equations of Gravita-
tion,? Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916, 688?696
(1916).
[27] A. Einstein, ?U?ber Gravitationswellen,? Sitzungsber. Preuss. Akad. Wiss.
Berlin (Math. Phys.) 1918, 154?167 (1918).
[28] M. Maggiore, Gravitational Waves. Vol. I: Theory and Experiments , Oxford
Master Series in Physics (Oxford University Press, 2007).
[29] E. Poisson, A. Pound, and I. Vega, ?The Motion of point particles in curved
spacetime,? Living Rev. Rel. 14, 7 (2011), arXiv:1102.0529 .
[30] L. Blanchet, ?Gravitational Radiation from Post-Newtonian Sources and In-
spiralling Compact Binaries,? Living Rev. Rel. 17, 2 (2014), arXiv:1310.1528
.
[31] T. Damour, ?The problem of motion in Newtonian and Einsteinian gravity.?
in Three hundred years of gravitation (1987) pp. 128?198.
[32] P. C. Peters and J. Mathews, ?Gravitational radiation from point masses in a
Keplerian orbit,? Phys. Rev. 131, 435?439 (1963).
[33] S. F. Portegies Zwart and S. McMillan, ?Black hole mergers in the universe,?
Astrophys. J. 528, L17 (2000), arXiv:astro-ph/9910061 .
[34] M. L. Lidov, ?The evolution of orbits of artificial satellites of planets under
the action of gravitational perturbations of external bodies,? Planet. Space
Sci. 9, 719?759 (1962).
[35] Y. Kozai, ?Secular perturbations of asteroids with high inclination and eccen-
tricity,? AJ 67, 591?598 (1962).
[36] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Search for Eccentric Bi-
nary Black Hole Mergers with Advanced LIGO and Advanced Virgo dur-
ing their First and Second Observing Runs,? Astrophys. J. 883, 149 (2019),
arXiv:1907.09384 [astro-ph.HE] .
[37] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields , 4th ed.
(Butterworth-Heinemann, 1980).
503
[38] A. Buonanno and T. Damour, ?Effective one-body approach to general rel-
ativistic two-body dynamics,? Phys. Rev. D59, 084006 (1999), arXiv:gr-
qc/9811091 .
[39] A. Buonanno and T. Damour, ?Transition from inspiral to plunge in binary
black hole coalescences,? Phys. Rev. D62, 064015 (2000), arXiv:gr-qc/0001013
.
[40] L. Barack, T. Damour, and N. Sago, ?Precession effect of the gravitational
self-force in a Schwarzschild spacetime and the effective one-body formalism,?
Phys. Rev. D82, 084036 (2010), arXiv:1008.0935 .
[41] D. Bini and T. Damour, ?Gravitational self-force corrections to two-body tidal
interactions and the effective one-body formalism,? Phys. Rev. D90, 124037
(2014), arXiv:1409.6933 .
[42] S. Akcay and M. van de Meent, ?Numerical computation of the effective-one-
body potential q using self-force results,? Phys. Rev. D93, 064063 (2016),
arXiv:1512.03392 .
[43] D. Bini, T. Damour, and A. Geralico, ?Spin-orbit precession along eccentric
orbits: improving the knowledge of self-force corrections and of their effective-
one-body counterparts,? Phys. Rev. D97, 104046 (2018), arXiv:1801.03704
.
[44] A. Taracchini et al., ?Effective-one-body model for black-hole binaries
with generic mass ratios and spins,? Phys. Rev. D89, 061502 (2014),
arXiv:1311.2544 .
[45] A. Bohe? et al., ?Improved effective-one-body model of spinning, nonprecessing
binary black holes for the era of gravitational-wave astrophysics with advanced
detectors,? Phys. Rev. D95, 044028 (2017), arXiv:1611.03703 .
[46] R. L. Arnowitt, S. Deser, and C. W. Misner, ?Dynamical Structure and
Definition of Energy in General Relativity,? Phys. Rev. 116, 1322?1330 (1959).
[47] R. Arnowitt, S. Deser, and C. Misner, ?The dynamics of general relativity,? in
Gravitation: An Introduction to Current Research, edited by L. Witten (John
Wiley & Sons, Inc., New York, 1962).
[48] F. Pretorius, ?Evolution of binary black hole spacetimes,? Phys. Rev. Lett.
95, 121101 (2005), arXiv:gr-qc/0507014 .
[49] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, ?Accurate
evolutions of orbiting black-hole binaries without excision,? Phys. Rev. Lett.
96, 111101 (2006), arXiv:gr-qc/0511048 .
504
[50] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, ?Grav-
itational wave extraction from an inspiraling configuration of merging black
holes,? Phys. Rev. Lett. 96, 111102 (2006), arXiv:gr-qc/0511103 .
[51] J. A. Font, ?Numerical Hydrodynamics and Magnetohydrodynamics in Gen-
eral Relativity,? Living Rev. Rel. 11, 7 (2008).
[52] L. Lehner and F. Pretorius, ?Numerical Relativity and Astrophysics,? Ann.
Rev. Astron. Astrophys. 52, 661?694 (2014), arXiv:1405.4840 .
[53] M. Shibata, ?Constraining nuclear equations of state using gravitational
waves from hypermassive neutron stars,? Phys. Rev. Lett. 94, 201101 (2005),
arXiv:gr-qc/0504082 .
[54] M. Shibata, K. Taniguchi, and K. Uryu, ?Merger of binary neutron stars with
realistic equations of state in full general relativity,? Phys. Rev. D71, 084021
(2005), arXiv:gr-qc/0503119 .
[55] C. V. Vishveshwara, ?Scattering of Gravitational Radiation by a Schwarzschild
Black-hole,? Nature 227, 936?938 (1970).
[56] S. Chandrasekhar and S. L. Detweiler, ?The quasi-normal modes of the
Schwarzschild black hole,? Proc. Roy. Soc. Lond. A344, 441?452 (1975).
[57] J. Weber, ?Detection and Generation of Gravitational Waves,? Phys. Rev.
117, 306?313 (1960).
[58] J. Weber, General Relativity and Gravitational Waves (Interscience Publish-
ers, New York, 1961).
[59] P. Astone et al. (International Gravitational Event), ?Methods and results of
the IGEC search for burst gravitational waves in the years 1997 - 2000,? Phys.
Rev. D68, 022001 (2003), arXiv:astro-ph/0302482 .
[60] L. Stodolsky, ?Matter and light wave interferometry in gravitational fields,?
Gen. Relativ. Gravit. 11, 391?405 (1979).
[61] S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajen-
dran, ?An Atomic Gravitational Wave Interferometric Sensor (AGIS),? Phys.
Rev. D78, 122002 (2008), arXiv:0806.2125 .
[62] P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, ?A New
Method for Gravitational Wave Detection with Atomic Sensors,? Phys. Rev.
Lett. 110, 171102 (2013), arXiv:1206.0818 .
[63] J. Aasi et al. (LIGO Scientific), ?Advanced LIGO,? Class. Quant. Grav. 32,
074001 (2015), arXiv:1411.4547 .
505
[64] F. Acernese et al. (Virgo), ?Advanced Virgo: a second-generation interfero-
metric gravitational wave detector,? Class. Quant. Grav. 32, 024001 (2015),
arXiv:1408.3978 .
[65] B. J. Meers, ?Recycling in Laser Interferometric Gravitational Wave Detec-
tors,? Phys. Rev. D38, 2317?2326 (1988).
[66] J. Mizuno, K. Strain, P. Nelson, J. Chen, R. Schilling, A. Ru?diger, W. Winkler,
and K. Danzmann, ?Resonant sideband extraction: a new configuration for
interferometric gravitational wave detectors,? Phys. Lett. A 175, 273 ? 276
(1993).
[67] S. A. Hughes and K. S. Thorne, ?Seismic gravity gradient noise in interferomet-
ric gravitational wave detectors,? Phys. Rev. D58, 122002 (1998), arXiv:gr-
qc/9806018 .
[68] S. Klimenko, I. Yakushin, A. Mercer, and G. Mitselmakher, ?Coherent
method for detection of gravitational wave bursts,? Proceedings, 18th Inter-
national Conference on General Relativity and Gravitation (GRG18) and 7th
Edoardo Amaldi Conference on Gravitational Waves (Amaldi7), Sydney, Aus-
tralia, July 2007, Class. Quant. Grav. 25, 114029 (2008), arXiv:0802.3232 .
[69] S. Klimenko et al., ?Method for detection and reconstruction of gravitational
wave transients with networks of advanced detectors,? Phys. Rev. D93, 042004
(2016), arXiv:1511.05999 .
[70] R. Lynch, S. Vitale, R. Essick, E. Katsavounidis, and F. Robinet,
?Information-theoretic approach to the gravitational-wave burst detection
problem,? Phys. Rev. D95, 104046 (2017), arXiv:1511.05955 .
[71] N. J. Cornish and T. B. Littenberg, ?BayesWave: Bayesian Inference for
Gravitational Wave Bursts and Instrument Glitches,? Class. Quant. Grav.
32, 135012 (2015), arXiv:1410.3835 .
[72] B. J. Owen, ?Search templates for gravitational waves from inspiraling bi-
naries: Choice of template spacing,? Phys. Rev. D53, 6749?6761 (1996),
arXiv:gr-qc/9511032 .
[73] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Binary Black Hole Mergers in
the first Advanced LIGO Observing Run,? Phys. Rev. X6, 041015 (2016),
[erratum: Phys. Rev.X8,no.3,039903(2018)], arXiv:1606.04856 .
[74] T. Dal Canton and I. W. Harry, ?Designing a template bank to observe com-
pact binary coalescences in Advanced LIGO?s second observing run,? (2017),
arXiv:1705.01845 .
[75] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and
E. Teller, ?Equation of state calculations by fast computing machines,? J.
Chem. Phys. 21, 1087?1092 (1953).
506
[76] W. K. Hastings, ?Monte carlo sampling methods using markov chains and
their applications,? Biometrika 57, 97?109 (1970).
[77] J. Skilling, ?Nested sampling for general bayesian computation,? Bayesian
Anal. 1, 833?859 (2006).
[78] J. Veitch et al., ?Parameter estimation for compact binaries with ground-
based gravitational-wave observations using the LALInference software li-
brary,? Phys. Rev. D91, 042003 (2015), arXiv:1409.7215 .
[79] C. M. Will, ?The Confrontation between General Relativity and Experiment,?
Living Rev. Rel. 17, 4 (2014), arXiv:1403.7377 .
[80] M. N. R. Wohlfarth, ?Gravity a la Born-Infeld,? Class. Quant. Grav. 21, 1927
(2004), [Erratum: Class. Quant. Grav.21,5297(2004)], arXiv:hep-th/0310067 .
[81] T. Biswas, A. Mazumdar, and W. Siegel, ?Bouncing universes in string-
inspired gravity,? JCAP 0603, 009 (2006), arXiv:hep-th/0508194 .
[82] M. Banados and P. G. Ferreira, ?Eddington?s theory of gravity and its
progeny,? Phys. Rev. Lett. 105, 011101 (2010), [Erratum: Phys. Rev.
Lett.113,no.11,119901(2014)], arXiv:1006.1769 .
[83] K. S. Stelle, ?Renormalization of Higher Derivative Quantum Gravity,? Phys.
Rev. D16, 953?969 (1977).
[84] D. Lovelock, ?The Einstein tensor and its generalizations,? J. Math. Phys. 12,
498?501 (1971).
[85] D. Lovelock, ?The four-dimensionality of space and the einstein tensor,? J.
Math. Phys. 13, 874?876 (1972).
[86] M. Ostrogradsky, ?Sur les inte?grales des e?quations ge?ne?rales de la dynamique,?
Mem. Ac. St. Petersbourg VI 4, 385 (1850).
[87] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, ?Modified Gravity and
Cosmology,? Phys. Rept. 513, 1?189 (2012), arXiv:1106.2476 .
[88] N. Yunes and X. Siemens, ?Gravitational-Wave Tests of General Relativity
with Ground-Based Detectors and Pulsar Timing-Arrays,? Living Rev. Rel.
16, 9 (2013), arXiv:1304.3473 .
[89] E. Berti et al., ?Testing General Relativity with Present and Future Astrophys-
ical Observations,? Class. Quant. Grav. 32, 243001 (2015), arXiv:1501.07274
.
[90] G. W. Horndeski, ?Second-order scalar-tensor field equations in a four-
dimensional space,? Int. J. Theor. Phys. 10, 363?384 (1974).
507
[91] M. Zumalaca?rregui and J. Garc??a-Bellido, ?Transforming gravity: from deriva-
tive couplings to matter to second-order scalar-tensor theories beyond the
Horndeski Lagrangian,? Phys. Rev. D89, 064046 (2014), arXiv:1308.4685 .
[92] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, ?Exploring gravitational
theories beyond Horndeski,? JCAP 1502, 018 (2015), arXiv:1408.1952 .
[93] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, ?Healthy theories beyond
Horndeski,? Phys. Rev. Lett. 114, 211101 (2015), arXiv:1404.6495 .
[94] J. Alsing, E. Berti, C. M. Will, and H. Zaglauer, ?Gravitational radiation
from compact binary systems in the massive Brans-Dicke theory of gravity,?
Phys. Rev. D85, 064041 (2012), arXiv:1112.4903 .
[95] S. S. Yazadjiev, D. D. Doneva, and D. Popchev, ?Slowly rotating neutron
stars in scalar-tensor theories with a massive scalar field,? Phys. Rev. D93,
084038 (2016), arXiv:1602.04766 .
[96] S. S. Yazadjiev, D. D. Doneva, and K. D. Kokkotas, ?Tidal Love num-
bers of neutron stars in f(R) gravity,? Eur. Phys. J. C78, 818 (2018),
arXiv:1803.09534 .
[97] C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge Uni-
versity Press, Cambridge, 1993).
[98] T. P. Sotiriou, ?Black Holes and Scalar Fields,? Class. Quant. Grav. 32, 214002
(2015), arXiv:1505.00248 .
[99] O. Sarbach, E. Barausse, and J. A. Preciado-Lo?pez, ?Well-posed Cauchy for-
mulation for Einstein-?ther theory,? Class. Quant. Grav. 36, 165007 (2019),
arXiv:1902.05130 .
[100] D. Blas, O. Pujolas, and S. Sibiryakov, ?Consistent Extension of Horava
Gravity,? Phys. Rev. Lett. 104, 181302 (2010), arXiv:0909.3525 .
[101] P. Horava, ?Quantum Gravity at a Lifshitz Point,? Phys. Rev. D79, 084008
(2009), arXiv:0901.3775 .
[102] T. Jacobson and D. Mattingly, ?Einstein-Aether waves,? Phys. Rev. D70,
024003 (2004), arXiv:gr-qc/0402005 .
[103] K. Yagi, D. Blas, E. Barausse, and N. Yunes, ?Constraints on Einstein-?ther
theory and Horava gravity from binary pulsar observations,? Phys. Rev. D89,
084067 (2014), [Erratum: Phys. Rev.D90,no.6,069901(2014)], arXiv:1311.7144
.
[104] O. Ramos and E. Barausse, ?Constraints on Horava gravity from binary black
hole observations,? Phys. Rev. D99, 024034 (2019), arXiv:1811.07786 .
508
[105] C. Eling and T. Jacobson, ?Black Holes in Einstein-Aether Theory,?
Class. Quant. Grav. 23, 5643?5660 (2006), [Erratum: Class. Quant.
Grav.27,049802(2010)], arXiv:gr-qc/0604088 .
[106] L. Barack et al., ?Black holes, gravitational waves and fundamental physics:
a roadmap,? Class. Quant. Grav. 36, 143001 (2019), arXiv:1806.05195 .
[107] C. de Rham and G. Gabadadze, ?Generalization of the Fierz-Pauli Action,?
Phys. Rev. D82, 044020 (2010), arXiv:1007.0443 .
[108] C. de Rham, G. Gabadadze, and A. J. Tolley, ?Resummation of Massive
Gravity,? Phys. Rev. Lett. 106, 231101 (2011), arXiv:1011.1232 .
[109] C. de Rham, ?Massive Gravity,? Living Rev. Rel. 17, 7 (2014),
arXiv:1401.4173 .
[110] A. I. Vainshtein, ?To the problem of nonvanishing gravitation mass,? Phys.
Lett. 39B, 393?394 (1972).
[111] C. M. Will, ?Bounding the mass of the graviton using gravitational wave
observations of inspiralling compact binaries,? Phys. Rev. D57, 2061?2068
(1998), arXiv:gr-qc/9709011 .
[112] M. Maggiore and M. Mancarella, ?Nonlocal gravity and dark energy,? Phys.
Rev. D90, 023005 (2014), arXiv:1402.0448 .
[113] Y. Dirian, S. Foffa, M. Kunz, M. Maggiore, and V. Pettorino, ?Non-
local gravity and comparison with observational datasets. II. Updated re-
sults and Bayesian model comparison with ?CDM,? JCAP 1605, 068 (2016),
arXiv:1602.03558 .
[114] A. Kehagias and M. Maggiore, ?Spherically symmetric static solutions in a
nonlocal infrared modification of General Relativity,? JHEP 08, 029 (2014),
arXiv:1401.8289 .
[115] E. Belgacem, Y. Dirian, S. Foffa, and M. Maggiore, ?Gravitational-wave
luminosity distance in modified gravity theories,? Phys. Rev. D97, 104066
(2018), arXiv:1712.08108 .
[116] E. Belgacem, Y. Dirian, S. Foffa, and M. Maggiore, ?Modified gravitational-
wave propagation and standard sirens,? Phys. Rev. D98, 023510 (2018),
arXiv:1805.08731 .
[117] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, ?Probing
the non-linear structure of general relativity with black hole binaries,? Phys.
Rev. D74, 024006 (2006), arXiv:gr-qc/0604067 .
[118] N. Yunes and F. Pretorius, ?Fundamental Theoretical Bias in Gravitational
Wave Astrophysics and the Parameterized Post-Einsteinian Framework,?
Phys. Rev. D80, 122003 (2009), arXiv:0909.3328 .
509
[119] C. K. Mishra, K. G. Arun, B. R. Iyer, and B. S. Sathyaprakash, ?Parametrized
tests of post-Newtonian theory using Advanced LIGO and Einstein Tele-
scope,? Phys. Rev. D82, 064010 (2010), arXiv:1005.0304 .
[120] T. G. F. Li, W. Del Pozzo, S. Vitale, C. Van Den Broeck, M. Agathos,
J. Veitch, K. Grover, T. Sidery, R. Sturani, and A. Vecchio, ?Towards a
generic test of the strong field dynamics of general relativity using compact
binary coalescence,? Phys. Rev. D85, 082003 (2012), arXiv:1110.0530 .
[121] J. Meidam et al., ?Parametrized tests of the strong-field dynamics of general
relativity using gravitational wave signals from coalescing binary black holes:
Fast likelihood calculations and sensitivity of the method,? Phys. Rev. D97,
044033 (2018), arXiv:1712.08772 .
[122] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Tests of general relativity with
GW150914,? Phys. Rev. Lett. 116, 221101 (2016), [Erratum: Phys. Rev.
Lett.121,no.12,129902(2018)], arXiv:1602.03841 .
[123] B. P. Abbott et al. (LIGO Scientific, Virgo), ?GW170104: Observation of a
50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2,? Phys. Rev.
Lett. 118, 221101 (2017), arXiv:1706.01812 .
[124] K. G. Arun and C. M. Will, ?Bounding the mass of the graviton with gravita-
tional waves: Effect of higher harmonics in gravitational waveform templates,?
Class. Quant. Grav. 26, 155002 (2009), arXiv:0904.1190 .
[125] A. Stavridis and C. M. Will, ?Bounding the mass of the graviton with gravita-
tional waves: Effect of spin precessions in massive black hole binaries,? Phys.
Rev. D80, 044002 (2009), arXiv:0906.3602 .
[126] S. Mirshekari, N. Yunes, and C. M. Will, ?Constraining Generic Lorentz
Violation and the Speed of the Graviton with Gravitational Waves,? Phys.
Rev. D85, 024041 (2012), arXiv:1110.2720 .
[127] T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller, and I. Saw-
icki, ?Strong constraints on cosmological gravity from GW170817 and GRB
170817A,? Phys. Rev. Lett. 119, 251301 (2017), arXiv:1710.06394 .
[128] P. Creminelli and F. Vernizzi, ?Dark Energy after GW170817 and
GRB170817A,? Phys. Rev. Lett. 119, 251302 (2017), arXiv:1710.05877 .
[129] J. Sakstein and B. Jain, ?Implications of the Neutron Star Merger GW170817
for Cosmological Scalar-Tensor Theories,? Phys. Rev. Lett. 119, 251303
(2017), arXiv:1710.05893 .
[130] J. M. Ezquiaga and M. Zumalaca?rregui, ?Dark Energy After GW170817:
Dead Ends and the Road Ahead,? Phys. Rev. Lett. 119, 251304 (2017),
arXiv:1710.05901 .
510
[131] D. Langlois, R. Saito, D. Yamauchi, and K. Noui, ?Scalar-tensor theories and
modified gravity in the wake of GW170817,? Phys. Rev. D97, 061501 (2018),
arXiv:1711.07403 .
[132] A. Dima and F. Vernizzi, ?Vainshtein Screening in Scalar-Tensor Theories
before and after GW170817: Constraints on Theories beyond Horndeski,?
Phys. Rev. D97, 101302 (2018), arXiv:1712.04731 .
[133] S. Boran, S. Desai, E. O. Kahya, and R. P. Woodard, ?GW170817 Falsifies
Dark Matter Emulators,? Phys. Rev. D97, 041501 (2018), arXiv:1710.06168 .
[134] C. de Rham and S. Melville, ?Gravitational Rainbows: LIGO and Dark Energy
at its Cutoff,? Phys. Rev. Lett. 121, 221101 (2018), arXiv:1806.09417 .
[135] K. Chatziioannou, N. Yunes, and N. Cornish, ?Model-Independent Test of
General Relativity: An Extended post-Einsteinian Framework with Com-
plete Polarization Content,? Phys. Rev. D86, 022004 (2012), [Erratum: Phys.
Rev.D95,no.12,129901(2017)], arXiv:1204.2585 .
[136] M. Isi and A. J. Weinstein, ?Probing gravitational wave polarizations with
signals from compact binary coalescences,? (2017), arXiv:1710.03794 .
[137] T. B. Littenberg, J. B. Kanner, N. J. Cornish, and M. Millhouse, ?Enabling
high confidence detections of gravitational-wave bursts,? Phys. Rev. D94,
044050 (2016), arXiv:1511.08752 .
[138] N. Sennett, R. Brito, A. Buonanno, V. Gorbenko, and L. Senatore, ?Con-
straints on an effective field theory extension to General Relativity using
gravitational-wave observations,? (2019, in prep.).
[139] A. De Rujula, S. L. Glashow, and U. Sarid, ?Charged dark matter,? Nucl.
Phys. B333, 173?194 (1990).
[140] M. L. Perl and E. R. Lee, ?The search for elementary particles with fractional
electric charge and the philosophy of speculative experiments,? Am. J. Phys.
65, 698?706 (1997).
[141] S. D. McDermott, H.-B. Yu, and K. M. Zurek, ?Turning off the Lights: How
Dark is Dark Matter?? Phys. Rev. D83, 063509 (2011), arXiv:1011.2907 .
[142] L. Ackerman, M. R. Buckley, S. M. Carroll, and M. Kamionkowski, ?Dark
Matter and Dark Radiation,? Phys. Rev. D79, 023519 (2009), arXiv:0810.5126
.
[143] J. L. Feng, H. Tu, and H.-B. Yu, ?Thermal Relics in Hidden Sectors,? JCAP
0810, 043 (2008), arXiv:0808.2318 .
[144] J. L. Feng, M. Kaplinghat, H. Tu, and H.-B. Yu, ?Hidden Charged Dark
Matter,? JCAP 0907, 004 (2009), arXiv:0905.3039 .
511
[145] E. Barausse, N. Yunes, and K. Chamberlain, ?Theory-Agnostic Constraints
on Black-Hole Dipole Radiation with Multiband Gravitational-Wave Astro-
physics,? Phys. Rev. Lett. 116, 241104 (2016), arXiv:1603.04075 .
[146] H. W. Zaglauer, ?Neutron stars and gravitational scalars,? Astrophys. J. 393,
685?696 (1992).
[147] T. Damour and G. Esposito-Farese, ?Tensor multiscalar theories of gravita-
tion,? Class. Quant. Grav. 9, 2093?2176 (1992).
[148] T. P. Sotiriou and V. Faraoni, ?Black holes in scalar-tensor gravity,? Phys.
Rev. Lett. 108, 081103 (2012), arXiv:1109.6324 .
[149] S. Mirshekari and C. M. Will, ?Compact binary systems in scalar-tensor grav-
ity: Equations of motion to 2.5 post-Newtonian order,? Phys. Rev. D87,
084070 (2013), arXiv:1301.4680 .
[150] R. N. Lang, ?Compact binary systems in scalar-tensor gravity. II. Tensor grav-
itational waves to second post-Newtonian order,? Phys. Rev. D89, 084014
(2014), arXiv:1310.3320 .
[151] R. N. Lang, ?Compact binary systems in scalar-tensor gravity. III. Scalar
waves and energy flux,? Phys. Rev. D91, 084027 (2015), arXiv:1411.3073 .
[152] T. Damour and G. Esposito-Farese, ?Tensor - scalar gravity and binary pulsar
experiments,? Phys. Rev. D54, 1474?1491 (1996), arXiv:gr-qc/9602056 .
[153] T. Damour and G. Esposito-Farese, ?Nonperturbative strong field effects in
tensor - scalar theories of gravitation,? Phys. Rev. Lett. 70, 2220?2223 (1993).
[154] I. Z. Stefanov, S. S. Yazadjiev, and M. D. Todorov, ?Phases of 4D scalar-
tensor black holes coupled to Born-Infeld nonlinear electrodynamics,? Mod.
Phys. Lett. A23, 2915?2931 (2008), arXiv:0708.4141 .
[155] D. D. Doneva, S. S. Yazadjiev, K. D. Kokkotas, and I. Z. Stefanov, ?Quasi-
normal modes, bifurcations and non-uniqueness of charged scalar-tensor black
holes,? Phys. Rev. D82, 064030 (2010), arXiv:1007.1767 .
[156] C. Palenzuela and S. L. Liebling, ?Constraining scalar-tensor theories of grav-
ity from the most massive neutron stars,? Phys. Rev. D93, 044009 (2016),
arXiv:1510.03471 .
[157] R. F. P. Mendes and N. Ortiz, ?Highly compact neutron stars in scalar-tensor
theories of gravity: Spontaneous scalarization versus gravitational collapse,?
Phys. Rev. D93, 124035 (2016), arXiv:1604.04175 .
[158] C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Font, ?Spontaneous
Scalarization of Charged Black Holes,? Phys. Rev. Lett. 121, 101102 (2018),
arXiv:1806.05190 .
512
[159] P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu, and N. Sanchis-
Gual, ?Spontaneous Scalarisation of Charged Black Holes: Coupling De-
pendence and Dynamical Features,? Class. Quant. Grav. 36, 134002 (2019),
arXiv:1902.05079 .
[160] P. Chen, T. Suyama, and J. Yokoyama, ?Spontaneous scalarization: asym-
metron as dark matter,? Phys. Rev. D92, 124016 (2015), arXiv:1508.01384
.
[161] F. M. Ramazanoglu and F. Pretorius, ?Spontaneous Scalarization with Mas-
sive Fields,? Phys. Rev. D93, 064005 (2016), arXiv:1601.07475 .
[162] L. Sagunski, J. Zhang, M. C. Johnson, L. Lehner, M. Sakellariadou, S. L.
Liebling, C. Palenzuela, and D. Neilsen, ?Neutron star mergers as a probe of
modifications of general relativity with finite-range scalar forces,? Phys. Rev.
D97, 064016 (2018), arXiv:1709.06634 .
[163] D. D. Doneva and S. S. Yazadjiev, ?New Gauss-Bonnet Black Holes with
Curvature-Induced Scalarization in Extended Scalar-Tensor Theories,? Phys.
Rev. Lett. 120, 131103 (2018), arXiv:1711.01187 .
[164] H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, ?Spon-
taneous scalarization of black holes and compact stars from a Gauss-Bonnet
coupling,? Phys. Rev. Lett. 120, 131104 (2018), arXiv:1711.02080 .
[165] G. Antoniou, A. Bakopoulos, and P. Kanti, ?Evasion of No-Hair Theorems
and Novel Black-Hole Solutions in Gauss-Bonnet Theories,? Phys. Rev. Lett.
120, 131102 (2018), arXiv:1711.03390 .
[166] G. Antoniou, A. Bakopoulos, and P. Kanti, ?Black-Hole Solutions with Scalar
Hair in Einstein-Scalar-Gauss-Bonnet Theories,? Phys. Rev. D97, 084037
(2018), arXiv:1711.07431 .
[167] D. D. Doneva and S. S. Yazadjiev, ?Neutron star solutions with curvature
induced scalarization in the extended Gauss-Bonnet scalar-tensor theories,?
JCAP 1804, 011 (2018), arXiv:1712.03715 .
[168] Y. Brihaye and B. Hartmann, ?Spontaneous scalarization of charged black
holes at the approach to extremality,? Phys. Lett. B792, 244?250 (2019),
arXiv:1902.05760 .
[169] E. Barausse, C. Palenzuela, M. Ponce, and L. Lehner, ?Neutron-star merg-
ers in scalar-tensor theories of gravity,? Phys. Rev. D87, 081506 (2013),
arXiv:1212.5053 .
[170] C. Palenzuela, E. Barausse, M. Ponce, and L. Lehner, ?Dynamical scalar-
ization of neutron stars in scalar-tensor gravity theories,? Phys. Rev. D89,
044024 (2014), arXiv:1310.4481 .
513
[171] M. Shibata, K. Taniguchi, H. Okawa, and A. Buonanno, ?Coalescence of
binary neutron stars in a scalar-tensor theory of gravity,? Phys. Rev. D89,
084005 (2014), arXiv:1310.0627 .
[172] K. Taniguchi, M. Shibata, and A. Buonanno, ?Quasiequilibrium sequences of
binary neutron stars undergoing dynamical scalarization,? Phys. Rev. D91,
024033 (2015), arXiv:1410.0738 .
[173] C. Palenzuela, E. Barausse, M. Ponce, and L. Lehner, ?Dynamical scalar-
ization of neutron stars in scalar-tensor gravity theories,? Phys. Rev. D 89,
044024 (2014).
[174] K. Taniguchi, M. Shibata, and A. Buonanno, ?Quasiequilibrium sequences of
binary neutron stars undergoing dynamical scalarization,? Phys. Rev. D 91,
024033 (2015).
[175] L. Sampson, N. Yunes, N. Cornish, M. Ponce, E. Barausse, A. Klein, C. Palen-
zuela, and L. Lehner, ?Projected Constraints on Scalarization with Gravita-
tional Waves from Neutron Star Binaries,? Phys. Rev. D90, 124091 (2014),
arXiv:1407.7038 .
[176] M. Agathos, W. Del Pozzo, T. G. F. Li, C. Van Den Broeck, J. Veitch, and
S. Vitale, ?TIGER: A data analysis pipeline for testing the strong-field dy-
namics of general relativity with gravitational wave signals from coalescing
compact binaries,? Phys. Rev. D89, 082001 (2014), arXiv:1311.0420 .
[177] P. Jordan, Schwerkraft und Weltall (F. Vieweg, Braunschweig, 1955).
[178] M. Fierz, ?On the physical interpretation of P.Jordan?s extended theory of
gravitation,? Helv. Phys. Acta 29, 128?134 (1956).
[179] C. Brans and R. H. Dicke, ?Mach?s principle and a relativistic theory of grav-
itation,? Phys. Rev. 124, 925?935 (1961).
[180] B. Bertotti, L. Iess, and P. Tortora, ?A test of general relativity using radio
links with the Cassini spacecraft,? Nature 425, 374?376 (2003).
[181] S. Endlich, V. Gorbenko, J. Huang, and L. Senatore, ?An effective formalism
for testing extensions to General Relativity with gravitational waves,? JHEP
09, 122 (2017), arXiv:1704.01590 .
[182] B. P. Abbott et al. (LIGO Scientific, Virgo), ?GW151226: Observation of
Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence,?
Phys. Rev. Lett. 116, 241103 (2016), arXiv:1606.04855 .
[183] B. P. Abbott et al. (LIGO Scientific, Virgo), ?GW170608: Observation of a
19-solar-mass Binary Black Hole Coalescence,? Astrophys. J. 851, L35 (2017),
arXiv:1711.05578 .
514
[184] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Observation of Gravitational
Waves from a Binary Black Hole Merger,? Phys. Rev. Lett. 116, 061102
(2016), arXiv:1602.03837 .
[185] B. P. Abbott et al. (LIGO Scientific, Virgo), ?GW170814: A Three-Detector
Observation of Gravitational Waves from a Binary Black Hole Coalescence,?
Phys. Rev. Lett. 119, 141101 (2017), arXiv:1709.09660 .
[186] B. Abbott et al. (LIGO Scientific, Virgo), ?GW170817: Observation of Grav-
itational Waves from a Binary Neutron Star Inspiral,? Phys. Rev. Lett. 119,
161101 (2017), arXiv:1710.05832 .
[187] N. Yunes, K. Yagi, and F. Pretorius, ?Theoretical Physics Implications of the
Binary Black-Hole Mergers GW150914 and GW151226,? Phys. Rev. D94,
084002 (2016), arXiv:1603.08955 .
[188] G. W. Gibbons and K.-i. Maeda, ?Black Holes and Membranes in Higher Di-
mensional Theories with Dilaton Fields,? Nucl. Phys. B298, 741?775 (1988).
[189] D. Garfinkle, G. T. Horowitz, and A. Strominger, ?Charged black
holes in string theory,? Phys. Rev. D43, 3140 (1991), [Erratum: Phys.
Rev.D45,3888(1992)].
[190] J. H. Horne and G. T. Horowitz, ?Rotating dilaton black holes,? Phys. Rev.
D46, 1340?1346 (1992), arXiv:hep-th/9203083 .
[191] V. P. Frolov, A. I. Zelnikov, and U. Bleyer, ?Charged Rotating Black Hole
From Five-dimensional Point of View,? Annalen Phys. 44, 371?377 (1987).
[192] G. W. Gibbons, ?Vacuum Polarization and the Spontaneous Loss of Charge
by Black Holes,? Commun. Math. Phys. 44, 245?264 (1975).
[193] D. M. Eardley and W. H. Press, ?Astrophysical processes near black holes,?
Ann. Rev. Astron. Astrophys. 13, 381?422 (1975).
[194] V. Cardoso, C. F. B. Macedo, P. Pani, and V. Ferrari, ?Black holes and
gravitational waves in models of minicharged dark matter,? JCAP 1605, 054
(2016), arXiv:1604.07845 .
[195] S. Davidson, S. Hannestad, and G. Raffelt, ?Updated bounds on millicharged
particles,? JHEP 05, 003 (2000), arXiv:hep-ph/0001179 .
[196] C. Burrage, J. Jaeckel, J. Redondo, and A. Ringwald, ?Late time CMB
anisotropies constrain mini-charged particles,? JCAP 0911, 002 (2009),
arXiv:0909.0649 .
[197] M. Ahlers, ?The Hubble diagram as a probe of mini-charged particles,? Phys.
Rev. D80, 023513 (2009), arXiv:0904.0998 .
515
[198] K. Kadota, T. Sekiguchi, and H. Tashiro, ?A new constraint on millicharged
dark matter from galaxy clusters,? (2016), arXiv:1602.04009 .
[199] E. W. Hirschmann, L. Lehner, S. L. Liebling, and C. Palenzuela, ?Black
Hole Dynamics in Einstein-Maxwell-Dilaton Theory,? Phys. Rev. D97, 064032
(2018), arXiv:1706.09875 .
[200] M. Zilha?o, V. Cardoso, C. Herdeiro, L. Lehner, and U. Sperhake, ?Collisions
of charged black holes,? Phys. Rev. D85, 124062 (2012), arXiv:1205.1063 .
[201] M. Zilha?o, V. Cardoso, C. Herdeiro, L. Lehner, and U. Sperhake, ?Colli-
sions of oppositely charged black holes,? Phys. Rev. D89, 044008 (2014),
arXiv:1311.6483 .
[202] F.-L. Julie?, ?On the motion of hairy black holes in Einstein-Maxwell-dilaton
theories,? JCAP 1801, 026 (2018), arXiv:1711.10769 .
[203] E. E. Flanagan, ?The Conformal frame freedom in theories of gravitation,?
Class. Quant. Grav. 21, 3817 (2004), arXiv:gr-qc/0403063 .
[204] D. M. Eardley, ?Observable effects of a scalar gravitational field in a binary
pulsar,? Astrophys. J. 196, L59?L62 (1975).
[205] A. Taracchini, A. Buonanno, G. Khanna, and S. A. Hughes, ?Small mass
plunging into a Kerr black hole: Anatomy of the inspiral-merger-ringdown
waveforms,? Phys. Rev. D90, 084025 (2014), arXiv:1404.1819 .
[206] J. S. Read, B. D. Lackey, B. J. Owen, and J. L. Friedman, ?Constraints on a
phenomenologically parameterized neutron-star equation of state,? Phys. Rev.
D79, 124032 (2009), arXiv:0812.2163 .
[207] A. D. Fokker, ?Ein invarianter Variationssatz fu?r die Bewegung mehrerer elek-
trischer Massenteilchen,? Z. Phys. 58, 386?393 (1929).
[208] T. Damour and G. Esposito-Farese, ?Testing gravity to second postNewtonian
order: A Field theory approach,? Phys. Rev. D53, 5541?5578 (1996), arXiv:gr-
qc/9506063 .
[209] L. Bernard, L. Blanchet, A. Bohe?, G. Faye, and S. Marsat, ?Fokker action of
nonspinning compact binaries at the fourth post-Newtonian approximation,?
Phys. Rev. D93, 084037 (2016), arXiv:1512.02876 .
[210] L. Bernard, ?Dynamics of compact binary systems in scalar-tensor theo-
ries: I. Equations of motion to the third post-Newtonian order,? (2018),
arXiv:1802.10201 .
[211] L. Blanchet and B. R. Iyer, ?Third postNewtonian dynamics of compact bina-
ries: Equations of motion in the center-of-mass frame,? Class. Quant. Grav.
20, 755 (2003), arXiv:gr-qc/0209089 .
516
[212] N. Yunes and S. A. Hughes, ?Binary Pulsar Constraints on the Param-
eterized post-Einsteinian Framework,? Phys. Rev. D82, 082002 (2010),
arXiv:1007.1995 .
[213] W. Del Pozzo, J. Veitch, and A. Vecchio, ?Testing General Relativity
using Bayesian model selection: Applications to observations of gravita-
tional waves from compact binary systems,? Phys. Rev. D83, 082002 (2011),
arXiv:1101.1391 .
[214] N. Cornish, L. Sampson, N. Yunes, and F. Pretorius, ?Gravitational Wave
Tests of General Relativity with the Parameterized Post-Einsteinian Frame-
work,? Phys. Rev. D84, 062003 (2011), arXiv:1105.2088 .
[215] S. Gossan, J. Veitch, and B. S. Sathyaprakash, ?Bayesian model selection
for testing the no-hair theorem with black hole ringdowns,? Phys. Rev. D85,
124056 (2012), arXiv:1111.5819 .
[216] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, ?Frequency domain P
approximant filters for time truncated inspiral gravitational wave signals from
compact binaries,? Phys. Rev. D62, 084036 (2000), arXiv:gr-qc/0001023 .
[217] ?LIGO-T1800042-v5: Updated Advanced LIGO sensitivity design curve,?
https://dcc.ligo.org/LIGO-T1800044/public.
[218] R. Coquereaux and G. Esposito-Farese, ?The Theory of Kaluza-Klein-Jordan-
Thiry revisited,? Ann. Inst. H. Poincare Phys. Theor. 52, 113?150 (1990).
[219] F.-L. Julie?, ?Reducing the two-body problem in scalar-tensor theories to the
motion of a test particle : a scalar-tensor effective-one-body approach,? Phys.
Rev. D97, 024047 (2018), arXiv:1709.09742 .
[220] T. Damour and A. Nagar, ?Effective One Body description of tidal effects in in-
spiralling compact binaries,? Phys. Rev. D81, 084016 (2010), arXiv:0911.5041
.
[221] F.-L. Julie? and N. Deruelle, ?Two-body problem in Scalar-Tensor theories as a
deformation of General Relativity : an Effective-One-Body approach,? Phys.
Rev. D95, 124054 (2017), arXiv:1703.05360 .
[222] T. Damour, P. Jaranowski, and G. Scha?fer, ?Fourth post-Newtonian effective
one-body dynamics,? Phys. Rev. D91, 084024 (2015), arXiv:1502.07245 .
[223] A. Buonanno, ?Reduction of the two-body dynamics to a one-body descrip-
tion in classical electrodynamics,? Phys. Rev. D62, 104022 (2000), arXiv:hep-
th/0004042 .
[224] T. Damour, ?Gravitational scattering, post-Minkowskian approximation and
Effective One-Body theory,? Phys. Rev. D94, 104015 (2016), arXiv:1609.00354
.
517
[225] J. Vines, ?Scattering of two spinning black holes in post-Minkowskian gravity,
to all orders in spin, and effective-one-body mappings,? Class. Quant. Grav.
35, 084002 (2018), arXiv:1709.06016 .
[226] Y. Pan, A. Buonanno, L. T. Buchman, T. Chu, L. E. Kidder, H. P. Pfeiffer,
and M. A. Scheel, ?Effective-one-body waveforms calibrated to numerical rel-
ativity simulations: coalescence of non-precessing, spinning, equal-mass black
holes,? Phys. Rev. D81, 084041 (2010), arXiv:0912.3466 .
[227] P. Jai-akson, A. Chatrabhuti, O. Evnin, and L. Lehner, ?Black hole merger
estimates in Einstein-Maxwell and Einstein-Maxwell-dilaton gravity,? Phys.
Rev. D96, 044031 (2017), arXiv:1706.06519 .
[228] V. Ferrari, M. Pauri, and F. Piazza, ?Quasinormal modes of charged, dilaton
black holes,? Phys. Rev. D63, 064009 (2001), arXiv:gr-qc/0005125 .
[229] B. P. Abbott et al. (KAGRA, LIGO Scientific, Virgo), ?Prospects for Ob-
serving and Localizing Gravitational-Wave Transients with Advanced LIGO,
Advanced Virgo and KAGRA,? Living Rev. Rel. 21, 3 (2018), arXiv:1304.0670
.
[230] B. P. Abbott et al. (LIGO Scientific, Virgo), ?GW150914: The Advanced
LIGO Detectors in the Era of First Discoveries,? Phys. Rev. Lett. 116, 131103
(2016), arXiv:1602.03838 .
[231] B. P. Abbott et al. (LIGO Scientific, Virgo), ?The Rate of Binary Black
Hole Mergers Inferred from Advanced LIGO Observations Surrounding
GW150914,? Astrophys. J. 833, L1 (2016), arXiv:1602.03842 .
[232] T. Accadia et al. (Virgo), ?Virgo: a laser interferometer to detect gravitational
waves,? JINST 7, P03012 (2012).
[233] Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi,
D. Tatsumi, and H. Yamamoto (KAGRA), ?Interferometer design of the
KAGRA gravitational wave detector,? Phys. Rev. D88, 043007 (2013),
arXiv:1306.6747 .
[234] B. Iyer, T. Souradeep, C. S. Unnikrishnan, S. Dhurandhar, S. Raja, and
A. Sengupta, ?LIGO-India, Proposal of the Consortium for Indian Initiative
in Gravitational-wave Observations (IndIGO),? LIGO-India Tech. rep. (2011),
(unpublished).
[235] P. Amaro-Seoane et al., ?eLISA/NGO: Astrophysics and cosmology in
the gravitational-wave millihertz regime,? GW Notes 6, 4?110 (2013),
arXiv:1201.3621 .
[236] Y. Xie, W.-T. Ni, P. Dong, and T.-Y. Huang, ?Second post-Newtonian ap-
proximation of scalar-tensor theory of gravity,? Adv. Space Res. 43, 171?180
(2009), arXiv:0704.2991 .
518
[237] L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha, ?The Third post-Newtonian
gravitational wave polarisations and associated spherical harmonic modes
for inspiralling compact binaries in quasi-circular orbits,? Class. Quant.
Grav. 25, 165003 (2008), [Erratum: Class. Quant. Grav.29,239501(2012)],
arXiv:0802.1249 .
[238] G. Faye, S. Marsat, L. Blanchet, and B. R. Iyer, ?The third and a half post-
Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling
compact binaries,? Class. Quant. Grav. 29, 175004 (2012), arXiv:1204.1043 .
[239] G. Faye, L. Blanchet, and B. R. Iyer, ?Non-linear multipole interactions and
gravitational-wave octupole modes for inspiralling compact binaries to third-
and-a-half post-Newtonian order,? Class. Quant. Grav. 32, 045016 (2015),
arXiv:1409.3546 .
[240] C. M. Will, ?Testing scalar - tensor gravity with gravitational wave obser-
vations of inspiraling compact binaries,? Phys. Rev. D50, 6058?6067 (1994),
arXiv:gr-qc/9406022 .
[241] P. Jordan, Schwerkraft und Weltall (F. Vieweg, Braunschweig, 1955).
[242] D. M. Eardley, ?Observable effects of a scalar gravitational field in a binary
pulsar,? 196, L59?L62 (1975).
[243] C. M. Will and H. W. Zaglauer, ?Gravitational Radiation, Close Binary Sys-
tems, and the Brans-dicke Theory of Gravity,? Astrophys. J. 346, 366 (1989).
[244] H. M. Antia, S. M. Chitre, and D. O. Gough, ?Temporal Variations in the
Sun?s Rotational Kinetic Energy,? Astron. Astrophys. (2007), 10.1051/0004-
6361:20078209, [Astron. Astrophys.477,657(2008)], arXiv:0711.0799 .
[245] N. D. R. Bhat, M. Bailes, and J. P. W. Verbiest, ?Gravitational-radiation
losses from the pulsar-white-dwarf binary PSR J1141-6545,? Phys. Rev. D77,
124017 (2008), arXiv:0804.0956 .
[246] J. Antoniadis, C. G. Bassa, N. Wex, M. Kramer, and R. Napiwotzki, ?A white
dwarf companion to the relativistic pulsar PSR J1141-6545,? Mon. Not. Roy.
Astron. Soc. 412, 580 (2011), arXiv:1011.0926 .
[247] P. C. C. Freire, N. Wex, G. Esposito-Farese, J. P. W. Verbiest, M. Bailes,
B. A. Jacoby, M. Kramer, I. H. Stairs, J. Antoniadis, and G. H. Janssen, ?The
relativistic pulsar-white dwarf binary PSR J1738+0333 II. The most stringent
test of scalar-tensor gravity,? Mon. Not. Roy. Astron. Soc. 423, 3328 (2012),
arXiv:1205.1450 .
[248] J. Antoniadis et al., ?A Massive Pulsar in a Compact Relativistic Binary,?
Science 340, 6131 (2013), arXiv:1304.6875 .
519
[249] N. Wex, ?Testing Relativistic Gravity with Radio Pulsars,? (2014),
arXiv:1402.5594 .
[250] L. Shao et al., ?Testing Gravity with Pulsars in the SKA Era,? Advancing
Astrophysics with the Square Kilometre Array (AASKA14): Giardini Naxos,
Italy, 2014, PoS AASKA14, 042 (2015), arXiv:1501.00058 .
[251] L. Blanchet, ?Radiative gravitational fields in general relativity. 2. Asymp-
totic behaviour at future null infinity,? Proc. Roy. Soc. Lond. A409, 383?399
(1987).
[252] L. Blanchet and T. Damour, ?Hereditary effects in gravitational radiation,?
Phys. Rev. D46, 4304?4319 (1992).
[253] L. Blanchet and T. Damour, ?Radiative gravitational fields in general relativ-
ity I. general structure of the field outside the source,? Phil. Trans. Roy. Soc.
Lond. A320, 379?430 (1986).
[254] C. M. Will and A. G. Wiseman, ?Gravitational radiation from compact binary
systems: Gravitational wave forms and energy loss to second postNewtonian
order,? Phys. Rev. D 54, 4813?4848 (1996).
[255] M. E. Pati and C. M. Will, ?PostNewtonian gravitational radiation and equa-
tions of motion via direct integration of the relaxed Einstein equations. 1.
Foundations,? Phys. Rev. D 62, 124015 (2000).
[256] L. Blanchet and G. Schaefer, ?Gravitational wave tails and binary star sys-
tems,? Class. Quant. Grav. 10, 2699?2721 (1993).
[257] D. Christodoulou, ?Nonlinear nature of gravitation and gravitational wave
experiments,? Phys. Rev. Lett. 67, 1486?1489 (1991).
[258] K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, ?The 2.5PN
gravitational wave polarisations from inspiralling compact binaries in circular
orbits,? Class. Quant. Grav. 21, 3771?3802 (2004), [Erratum: Class. Quant.
Grav.22,3115(2005)], arXiv:gr-qc/0404085 .
[259] A. Buonanno, B. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash,
?Comparison of post-Newtonian templates for compact binary inspiral sig-
nals in gravitational-wave detectors,? Phys. Rev. D80, 084043 (2009),
arXiv:0907.0700 .
[260] K. S. Thorne, ?Multipole Expansions of Gravitational Radiation,? Rev. Mod.
Phys. 52, 299?339 (1980).
[261] L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. Wiseman, ?Gravita-
tional radiation damping of compact binary systems to second postNewtonian
order,? Phys. Rev. Lett. 74, 3515?3518 (1995), arXiv:gr-qc/9501027 .
520
[262] K. Thorne, ?Gravitational radiation,? in Three hundred years of gravitation,
edited by S. Hawking and W. Israel (Cambridge University Press, Cambridge,
1987) pp. 330?458.
[263] N. Yunes, P. Pani, and V. Cardoso, ?Gravitational Waves from Quasicircular
Extreme Mass-Ratio Inspirals as Probes of Scalar-Tensor Theories,? Phys.
Rev. D85, 102003 (2012), arXiv:1112.3351 .
[264] P. D. Scharre and C. M. Will, ?Testing scalar tensor gravity using space grav-
itational wave interferometers,? Phys. Rev. D65, 042002 (2002), arXiv:gr-
qc/0109044 .
[265] C. M. Will and N. Yunes, ?Testing alternative theories of gravity using LISA,?
Class. Quant. Grav. 21, 4367 (2004), arXiv:gr-qc/0403100 .
[266] E. Berti, A. Buonanno, and C. M. Will, ?Estimating spinning binary param-
eters and testing alternative theories of gravity with LISA,? Phys. Rev. D71,
084025 (2005), arXiv:gr-qc/0411129 .
[267] K. Yagi and T. Tanaka, ?Constraining alternative theories of gravity by
gravitational waves from precessing eccentric compact binaries with LISA,?
Phys. Rev. D81, 064008 (2010), [Erratum: Phys. Rev.D81,109902(2010)],
arXiv:0906.4269 .
[268] J. R. Gair, M. Vallisneri, S. L. Larson, and J. G. Baker, ?Testing General
Relativity with Low-Frequency, Space-Based Gravitational-Wave Detectors,?
Living Rev. Rel. 16, 7 (2013), arXiv:1212.5575 .
[269] T. G. F. Li, W. Del Pozzo, S. Vitale, C. Van Den Broeck, M. Agathos,
J. Veitch, K. Grover, T. Sidery, R. Sturani, and A. Vecchio, ?Towards a
generic test of the strong field dynamics of general relativity using compact
binary coalescence: Further investigations,? Gravitational waves. Numerical
relativity - data analysis. Proceedings, 9th Edoardo Amaldi Conference, Amaldi
9, and meeting, NRDA 2011, Cardiff, UK, July 10-15, 2011, J. Phys. Conf.
Ser. 363, 012028 (2012), arXiv:1111.5274 .
[270] L. Sampson, N. Cornish, and N. Yunes, ?Gravitational Wave Tests of Strong
Field General Relativity with Binary Inspirals: Realistic Injections and Opti-
mal Model Selection,? Phys. Rev. D87, 102001 (2013), arXiv:1303.1185 .
[271] L. Sampson, N. Cornish, and N. Yunes, ?Mismodeling in gravitational-wave
astronomy: The trouble with templates,? Phys. Rev. D89, 064037 (2014),
arXiv:1311.4898 .
[272] T. Damour and K. Nordtvedt, ?General relativity as a cosmological attractor
of tensor scalar theories,? Phys. Rev. Lett. 70, 2217?2219 (1993).
[273] T. Damour and K. Nordtvedt, ?Tensor - scalar cosmological models and their
relaxation toward general relativity,? Phys. Rev. D48, 3436?3450 (1993).
521
[274] S. DeDeo and D. Psaltis, ?Towards New Tests of Strong-field Gravity with
Measurements of Surface Atomic Line Redshifts from Neutron Stars,? Phys.
Rev. Lett. 90, 141101 (2003), arXiv:astro-ph/0302095 .
[275] H. Sotani and K. D. Kokkotas, ?Probing strong-field scalar-tensor gravity
with gravitational wave asteroseismology,? Phys. Rev. D70, 084026 (2004),
arXiv:gr-qc/0409066 .
[276] T. Damour and G. Esposito-Farese, ?Gravitational wave versus binary - pul-
sar tests of strong field gravity,? Phys. Rev. D58, 042001 (1998), arXiv:gr-
qc/9803031 .
[277] S. DeDeo and D. Psaltis, ?Testing strong-field gravity with quasiperiodic oscil-
lations,? (2004), submitted to: Phys. Rev. D, arXiv:astro-ph/0405067 [astro-
ph] .
[278] D. D. Doneva, S. S. Yazadjiev, N. Stergioulas, K. D. Kokkotas, and T. M.
Athanasiadis, ?Orbital and epicyclic frequencies around rapidly rotating com-
pact stars in scalar-tensor theories of gravity,? Phys. Rev. D90, 044004 (2014),
arXiv:1405.6976 .
[279] K. Just, Z. Phys. 157 (1959).
[280] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, ?The Equation of state
of nucleon matter and neutron star structure,? Phys. Rev. C58, 1804?1828
(1998), arXiv:nucl-th/9804027 .
[281] T. Damour and J. H. Taylor, ?Strong field tests of relativistic gravity and
binary pulsars,? Phys. Rev. D45, 1840?1868 (1992).
[282] G. Esposito-Fare?se, ?Motion in alternative theories of gravity,? Mass and Mo-
tion in General Relativity, Fundam. Theor. Phys. 162, 461?489 (2011).
[283] W. Scherrer, ?Zur theorie der elementarteilchen,? Verh. Schweiz. Naturf. Ges.
121, 86?87 (1941).
[284] Y. Thiry, J. Math. Pures et Appl. 30, 275?396 (1951).
[285] R. Epstein and R. V. Wagoner, ?Post-Newtonian generation of gravitational
waves,? Astrophys. J. 197, 717 (1975).
[286] R. V. Wagoner and C. M. Will, ?Post-Newtonian gravitational radiation from
orbiting point masses,? Astrophys. J. 210, 764 (1976).
[287] C. M. Will and A. G. Wiseman, ?Gravitational radiation from compact binary
systems: Gravitational wave forms and energy loss to second postNewtonian
order,? Phys. Rev. D54, 4813?4848 (1996), arXiv:gr-qc/9608012 .
522
[288] M. E. Pati and C. M. Will, ?PostNewtonian gravitational radiation and equa-
tions of motion via direct integration of the relaxed Einstein equations. 1.
Foundations,? Phys. Rev. D62, 124015 (2000), arXiv:gr-qc/0007087 .
[289] M. E. Pati and C. M. Will, ?PostNewtonian gravitational radiation and equa-
tions of motion via direct integration of the relaxed Einstein equations. 2.
Two-body equations of motion to second postNewtonian order, and radia-
tion reaction to 3.5 postNewtonia order,? Phys. Rev. D65, 104008 (2002),
arXiv:gr-qc/0201001 .
[290] A. Buonanno, Y.-b. Chen, and M. Vallisneri, ?Detection template fami-
lies for gravitational waves from the final stages of binary?black-hole inspi-
rals: Nonspinning case,? Phys. Rev. D67, 024016 (2003), [Erratum: Phys.
Rev.D74,029903(2006)], arXiv:gr-qc/0205122 .
[291] G. ?Scha?fer, ?The gravitational quadrupole radiation reaction force and the
canonical formalism of ADM,? Annals Phys. 161, 81?100 (1985).
[292] T. Hinderer et al., ?Effects of neutron-star dynamic tides on gravitational
waveforms within the effective-one-body approach,? Phys. Rev. Lett. 116,
181101 (2016), arXiv:1602.00599 .
[293] J. Steinhoff, private communication.
[294] M. Kramer, ?Pulsars as probes of gravity and fundamental physics,? Int. J.
Mod. Phys. D25, 1630029 (2016), arXiv:1606.03843 .
[295] D. Psaltis, ?Probes and Tests of Strong-Field Gravity with Observations in the
Electromagnetic Spectrum,? Living Rev. Rel. 11, 9 (2008), arXiv:0806.1531 .
[296] T. Baker, D. Psaltis, and C. Skordis, ?Linking Tests of Gravity On All Scales:
from the Strong-Field Regime to Cosmology,? Astrophys. J. 802, 63 (2015),
arXiv:1412.3455 .
[297] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Upper Limits on the Rates of
Binary Neutron Star and Neutron Star?Black Hole Mergers From Advanced
Ligo?s First Observing run,? Astrophys. J. 832, L21 (2016), arXiv:1607.07456
.
[298] J. E. Vines and E?. E?. Flanagan, ?Post-1-Newtonian quadrupole tidal inter-
actions in binary systems,? Phys. Rev. D88, 024046 (2013), arXiv:1009.4919
.
[299] J. Steinhoff, T. Hinderer, A. Buonanno, and A. Taracchini, ?Dynamical Tides
in General Relativity: Effective Action and Effective-One-Body Hamiltonian,?
Phys. Rev. D94, 104028 (2016), arXiv:1608.01907 .
[300] P. Jordan, ?Formation of the Stars and Development of the Universe,? Nature
164, 637?640 (1949).
523
[301] M. Abernathy et al. (ET Science Team), ?Einstein gravitational wave tele-
scope: Conceptual design study,? (2011).
[302] D. Anderson, N. Yunes, and E. Barausse, ?The Effect of Cosmological Evo-
lution on Solar System Constraints and on the Scalarization of Neutron
Stars in Massless Scalar-Tensor Theories,? Phys. Rev. D94, 104064 (2016),
arXiv:1607.08888 .
[303] T. A. de Pirey Saint Alby and N. Yunes, ?Cosmological Evolution and So-
lar System Consistency of Massive Scalar-Tensor Gravity,? Phys. Rev. D96,
064040 (2017), arXiv:1703.06341 .
[304] R. F. P. Mendes, ?Possibility of setting a new constraint to scalar-tensor the-
ories,? Phys. Rev. D91, 064024 (2015), arXiv:1412.6789 .
[305] L. Shao and N. Wex, ?Tests of gravitational symmetries with radio pulsars,?
Sci. China Phys. Mech. Astron. 59, 699501 (2016), arXiv:1604.03662 .
[306] P. Ajith et al., ?Phenomenological template family for black-hole coalescence
waveforms,? Class. Quant. Grav. 24, S689?S700 (2007), arXiv:0704.3764 .
[307] T. Damour and A. Nagar, ?Relativistic tidal properties of neutron stars,?
Phys. Rev. D80, 084035 (2009), arXiv:0906.0096 .
[308] E?. E?. Flanagan and T. Hinderer, ?Constraining neutron star tidal Love num-
bers with gravitational wave detectors,? Phys. Rev. D77, 021502 (2008),
arXiv:0709.1915 .
[309] S. Chakrabarti, T. Delsate, and J. Steinhoff, ?New perspectives on neutron
star and black hole spectroscopy and dynamic tides,? (2013), arXiv:1304.2228
.
[310] D. M. Eardley, ?Observable effects of a scalar gravitational field in a binary
pulsar,? Astrophys. J. 196, L59?L62 (1975).
[311] M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory
(Addison-Wesley, Boston, 1995).
[312] S. Weinberg, ?Effective field theory, past and future,? Int. J. Mod. Phys. A31,
1630007 (2016).
[313] T. Jacobson, ?Primordial black hole evolution in tensor scalar cosmology,?
Phys. Rev. Lett. 83, 2699?2702 (1999), arXiv:astro-ph/9905303 .
[314] M. W. Horbatsch and C. P. Burgess, ?Cosmic Black-Hole Hair Growth and
Quasar OJ287,? JCAP 1205, 010 (2012), arXiv:1111.4009 .
[315] W. D. Goldberger and I. Z. Rothstein, ?An Effective field theory of gravity
for extended objects,? Phys. Rev. D73, 104029 (2006), arXiv:hep-th/0409156
.
524
[316] T. Damour, P. Jaranowski, and G. Schafer, ?Nonlocal-in-time action for the
fourth post-Newtonian conservative dynamics of two-body systems,? Phys.
Rev. D89, 064058 (2014), arXiv:1401.4548 .
[317] K. Taniguchi and M. Shibata, ?Binary Neutron Stars in Quasi-equilibrium,?
Astrophys. J. Suppl. 188, 187 (2010), arXiv:1005.0958 .
[318] L. D. Landau, ?Theory of phase transformations. I,? Zh. Eksp. Teor. Fiz. 7,
19?32 (1937).
[319] S. W. Hawking and R. Penrose, ?The Singularities of gravitational collapse
and cosmology,? Proc. Roy. Soc. Lond. A314, 529?548 (1970).
[320] F. Englert and R. Brout, ?Broken Symmetry and the Mass of Gauge Vector
Mesons,? Phys. Rev. Lett. 13, 321?323 (1964).
[321] P. W. Higgs, ?Broken Symmetries and the Masses of Gauge Bosons,? Phys.
Rev. Lett. 13, 508?509 (1964).
[322] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, ?Global Conservation
Laws and Massless Particles,? Phys. Rev. Lett. 13, 585?587 (1964).
[323] A. Coates, M. W. Horbartsch, and T. P. Sotiriou, ?Gravitational Higgs
Mechanism in Neutron Star Interiors,? Phys. Rev. D95, 084003 (2017),
arXiv:1606.03981 .
[324] F. M. Ramazanog?lu, ?Spontaneous growth of gauge fields in gravity through
the Higgs mechanism,? Phys. Rev. D98, 044013 (2018), arXiv:1804.03158 .
[325] F. M. Ramazanog?lu, ?Spontaneous growth of vector fields in gravity,? Phys.
Rev. D96, 064009 (2017), arXiv:1706.01056 .
[326] F. M. Ramazanog?lu, ?Spontaneous tensorization from curvature coupling and
beyond,? Phys. Rev. D99, 084015 (2019), arXiv:1901.10009 .
[327] F. M. Ramazanog?lu, ?Spontaneous growth of spinor fields in gravity,? Phys.
Rev. D98, 044011 (2018), arXiv:1804.00594 .
[328] P. Zimmerman, ?Gravitational self-force in scalar-tensor gravity,? Phys. Rev.
D92, 064051 (2015), arXiv:1507.04076 .
[329] D. Anderson, P. Freire, and N. Yunes, ?Binary Pulsar constraints on massless
scalar-tensor theories using Bayesian statistics,? (2019), arXiv:1901.00938 .
[330] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Properties of the binary neutron
star merger GW170817,? Phys. Rev. X9, 011001 (2019), arXiv:1805.11579 .
[331] Y. S. Myung and D.-C. Zou, ?Instability of Reissner-Nordstro?m black hole in
Einstein-Maxwell-scalar theory,? (2018), arXiv:1808.02609 .
525
[332] F. M. Ramazanog?lu, ?Regularization of instabilities in gravity theories,? Phys.
Rev. D97, 024008 (2018), arXiv:1710.00863 .
[333] G. W. Gibbons, R. Kallosh, and B. Kol, ?Moduli, scalar charges, and the first
law of black hole thermodynamics,? Phys. Rev. Lett. 77, 4992?4995 (1996),
arXiv:hep-th/9607108 .
[334] M. Ca?rdenas, F.-L. Julie?, and N. Deruelle, ?Thermodynamics sheds light on
black hole dynamics,? (2017), arXiv:1712.02672 .
[335] J. L. Bla?zquez-Salcedo, D. D. Doneva, J. Kunz, and S. S. Yazadjiev, ?Ra-
dial perturbations of the scalarized Einstein-Gauss-Bonnet black holes,? Phys.
Rev. D98, 084011 (2018), arXiv:1805.05755 .
[336] M. Minamitsuji and T. Ikeda, ?Scalarized black holes in the presence of
the coupling to Gauss-Bonnet gravity,? Phys. Rev. D99, 044017 (2019),
arXiv:1812.03551 .
[337] H. O. Silva, C. F. B. Macedo, T. P. Sotiriou, L. Gualtieri, J. Sakstein, and
E. Berti, ?Stability of scalarized black hole solutions in scalar-Gauss-Bonnet
gravity,? Phys. Rev. D99, 064011 (2019), arXiv:1812.05590 .
[338] Y. S. Myung and D.-C. Zou, ?Stability of scalarized charged black holes
in the Einstein?Maxwell?Scalar theory,? Eur. Phys. J. C79, 641 (2019),
arXiv:1904.09864 .
[339] Y. S. Myung and D.-C. Zou, ?Quasinormal modes of scalarized black holes
in the Einstein?Maxwell?Scalar theory,? Phys. Lett. B790, 400?407 (2019),
arXiv:1812.03604 .
[340] L. D. Landau, Ukr. J. Phys 53, 25?35 (2008), (English translation).
[341] M. W. Choptuik, ?Universality and scaling in gravitational collapse of a mass-
less scalar field,? Phys. Rev. Lett. 70, 9?12 (1993).
[342] A. M. Abrahams and C. R. Evans, ?Critical behavior and scaling in vacuum
axisymmetric gravitational collapse,? Phys. Rev. Lett. 70, 2980?2983 (1993).
[343] C. R. Evans and J. S. Coleman, ?Observation of critical phenomena and self-
similarity in the gravitational collapse of radiation fluid,? Phys. Rev. Lett. 72,
1782?1785 (1994), arXiv:gr-qc/9402041 .
[344] M. W. Choptuik, T. Chmaj, and P. Bizon, ?Critical behavior in gravitational
collapse of a Yang-Mills field,? Phys. Rev. Lett. 77, 424?427 (1996), arXiv:gr-
qc/9603051 .
[345] S. E. Gralla and P. Zimmerman, ?Critical Exponents of Extremal Kerr Per-
turbations,? Class. Quant. Grav. 35, 095002 (2018), arXiv:1711.00855 .
526
[346] S. E. Gralla and P. Zimmerman, ?Scaling and Universality in Extremal Black
Hole Perturbations,? JHEP 06, 061 (2018), arXiv:1804.04753 .
[347] W. D. Goldberger and A. Ross, ?Gravitational radiative corrections from ef-
fective field theory,? Phys. Rev. D81, 124015 (2010), arXiv:0912.4254 .
[348] A. Kuntz, F. Piazza, and F. Vernizzi, ?Effective field theory for gravitational
radiation in scalar-tensor gravity,? JCAP 1905, 052 (2019), arXiv:1902.04941
.
[349] H. Witek, L. Gualtieri, P. Pani, and T. P. Sotiriou, ?Black holes and binary
mergers in scalar Gauss-Bonnet gravity: scalar field dynamics,? Phys. Rev.
D99, 064035 (2019), arXiv:1810.05177 .
[350] V. Cardoso, S. Chakrabarti, P. Pani, E. Berti, and L. Gualtieri, ?Floating and
sinking: The Imprint of massive scalars around rotating black holes,? Phys.
Rev. Lett. 107, 241101 (2011), arXiv:1109.6021 .
[351] S. Endlich and R. Penco, ?A Modern Approach to Superradiance,? JHEP 05,
052 (2017), arXiv:1609.06723 .
[352] L. K. Wong, A.-C. Davis, and R. Gregory, ?Effective field theory for black
holes with induced scalar charges,? (2019), arXiv:1903.07080 .
[353] S. Reyes and D. A. Brown, ?Constraints on non-linear tides due to p-g mode
coupling from the neutron-star merger GW170817,? (2018), arXiv:1808.07013
.
[354] B. P. Abbott et al. (LIGO Scientific, Virgo), ?Constraining the p-Mode?g-
Mode Tidal Instability with GW170817,? Phys. Rev. Lett. 122, 061104 (2019),
arXiv:1808.08676 .
[355] E. Gaertig, K. Glampedakis, K. D. Kokkotas, and B. Zink, ?The f-mode
instability in relativistic neutron stars,? Phys. Rev. Lett. 107, 101102 (2011),
arXiv:1106.5512 .
[356] N. Andersson and K. D. Kokkotas, ?The R mode instability in rotating neu-
tron stars,? Int. J. Mod. Phys. D10, 381?442 (2001), arXiv:gr-qc/0010102
.
[357] K. Yagi and N. Yunes, ?Approximate Universal Relations for Neutron Stars
and Quark Stars,? Phys. Rept. 681, 1?72 (2017), arXiv:1608.02582 .
[358] K. Yagi and N. Yunes, ?I-Love-Q,? Science 341, 365?368 (2013),
arXiv:1302.4499 .
[359] T. Binnington and E. Poisson, ?Relativistic theory of tidal Love numbers,?
Phys. Rev. D80, 084018 (2009), arXiv:0906.1366 .
527
[360] B. Kol and M. Smolkin, ?Black hole stereotyping: Induced gravito-static po-
larization,? JHEP 02, 010 (2012), arXiv:1110.3764 .
[361] T. Kaluza, ?On the Problem of Unity in Physics,? Sitzungsber. Preuss. Akad.
Wiss. Berlin (Math. Phys.) 1921, 966?972 (1921).
[362] P. Jordan, Schwerkraft und Weltall:, Die Wissenschaft, Bd. 107 (Braunschweig,
1952).
[363] L. Shao, ?Testing the strong equivalence principle with the triple pulsar PSR
J0337+1715,? Phys. Rev. D93, 084023 (2016), arXiv:1602.05725 .
[364] E. Berti, A. Buonanno, and C. M. Will, ?Testing general relativity and prob-
ing the merger history of massive black holes with LISA,? Class. Quant. Grav.
22, S943?S954 (2005), arXiv:gr-qc/0504017 .
[365] K. Yagi and T. Tanaka, ?DECIGO/BBO as a probe to constrain alter-
native theories of gravity,? Prog. Theor. Phys. 123, 1069?1078 (2010),
arXiv:0908.3283 .
[366] L. Shao et al., ?Testing Gravity with Pulsars in the SKA Era,? Proceed-
ings, Advancing Astrophysics with the Square Kilometre Array (AASKA14):
Giardini Naxos, Italy, June 9-13, 2014, PoS AASKA14, 042 (2015),
arXiv:1501.00058 .
[367] B. P. Abbott et al. (LIGO Scientific), ?Exploring the Sensitivity of Next Gener-
ation Gravitational Wave Detectors,? Class. Quant. Grav. 34, 044001 (2017),
arXiv:1607.08697 .
[368] T. Damour, ?Binary Systems as Test-beds of Gravity Theories,? (2007),
arXiv:0704.0749 .
[369] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, ?Shapiro
Delay Measurement of A Two Solar Mass Neutron Star,? Nature 467, 1081?
1083 (2010), arXiv:1010.5788 .
[370] G. Esposito-Fare?se, ?Binary pulsar tests of strong field gravity and gravita-
tional radiation damping,? in Proceedings of the Tenth Marcel Grossmann
Meeting. On recent developments in theoretical and experimental general rela-
tivity, gravitation and relativistic field theories (2004) pp. 647?666, arXiv:gr-
qc/0402007 .
[371] G. Esposito-Fare?se, ?Tests of scalar-tensor gravity,? Phi in the Sky: The Quest
for Cosmological Scalar Fields, AIP Conf. Proc. 736, 35?52 (2004), arXiv:gr-
qc/0409081 .
[372] D. Anderson and N. Yunes, ?Solar System constraints on massless scalar-
tensor gravity with positive coupling constant upon cosmological evolution of
the scalar field,? Phys. Rev. D96, 064037 (2017), arXiv:1705.06351 .
528
[373] K. Lazaridis et al., ?Generic tests of the existence of the gravitational dipole
radiation and the variation of the gravitational constant,? Mon. Not. R. As-
tron. Soc. 400, 805?814 (2009), arXiv:0908.0285 .
[374] D. J. Reardon et al., ?Timing analysis for 20 millisecond pulsars in the Parkes
Pulsar Timing Array,? Mon. Not. Roy. Astron. Soc. 455, 1751?1769 (2016),
arXiv:1510.04434 .
[375] I. Cognard et al., ?A Massive-born Neutron Star with a Massive White Dwarf
Companion,? Astrophys. J. 844, 128 (2017), arXiv:1706.08060 .
[376] P. J. McMillan, ?The mass distribution and gravitational potential of the
Milky Way,? Mon. Not. Roy. Astron. Soc. 465, 76?94 (2017), arXiv:1608.00971
.
[377] P. J. Callanan, P. M. Garnavich, and D. Koester, ?The mass of the neutron
star in the binary millisecond pulsar PSR J1012 + 5307,? Mon. Not. Roy.
Astron. Soc. 298, 207?211 (1998).
[378] J. Antoniadis, T. M. Tauris, F. Ozel, E. Barr, D. J. Champion, and P. C. C.
Freire, ?The millisecond pulsar mass distribution: Evidence for bimodality and
constraints on the maximum neutron star mass,? (2016), arXiv:1605.01665 .
[379] J. Antoniadis, M. H. van Kerkwijk, D. Koester, P. C. C. Freire, N. Wex, T. M.
Tauris, M. Kramer, and C. G. Bassa, ?The relativistic pulsar-white dwarf
binary PSR J1738+0333 I. Mass determination and evolutionary history,?
Mon. Not. Roy. Astron. Soc. 423, 3316 (2012), arXiv:1204.3948 .
[380] W. Del Pozzo and A. Vecchio, ?On tests of general relativity with binary radio
pulsars,? Mon. Not. Roy. Astron. Soc. 462, L21?L25 (2016), arXiv:1606.02852
.
[381] J. M. Lattimer, ?The nuclear equation of state and neutron star masses,? Ann.
Rev. Nucl. Part. Sci. 62, 485?515 (2012), arXiv:1305.3510 .
[382] J. M. Lattimer and M. Prakash, ?Neutron star structure and the equation of
state,? Astrophys. J. 550, 426 (2001), arXiv:astro-ph/0002232 .
[383] J. Goodman and J. Weare, ?Ensemble samplers with affine invariance,?
Comm. App. Math. and Comp. Sci. 5, 65?80 (2010).
[384] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, ?em-
cee: The MCMC Hammer,? Publ. Astron. Soc. Pac. 125, 306?312 (2013),
arXiv:1202.3665 .
[385] P. C. Peters, ?Gravitational Radiation and the Motion of Two Point Masses,?
Phys. Rev. 136, B1224?B1232 (1964).
529
[386] A. Gelman and D. B. Rubin, ?Inference from Iterative Simulation Using Mul-
tiple Sequences,? Statist. Sci. 7, 457?472 (1992).
[387] S. Brooks, A. Gelman, G. Jones, and X. Meng, Handbook of Markov Chain
Monte Carlo (CRC Press, 2011).
[388] S. Hild et al., ?Sensitivity Studies for Third-Generation Gravitational Wave
Observatories,? Class. Quant. Grav. 28, 094013 (2011), arXiv:1012.0908 .
[389] D. Shoemaker (LIGO Collaboration), ?Advanced LIGO anticipated sensitivity
curves,? (2010), LIGO Document T0900288-v3.
[390] M. Kramer et al., ?Tests of general relativity from timing the double pulsar,?
Science 314, 97?102 (2006), arXiv:astro-ph/0609417 .
[391] J. W. T. Hessels, S. M. Ransom, I. H. Stairs, P. C. C. Freire, V. M. Kaspi,
and F. Camilo, ?A radio pulsar spinning at 716-hz,? Science 311, 1901?1904
(2006), arXiv:astro-ph/0601337 .
[392] W. Del Pozzo, K. Grover, I. Mandel, and A. Vecchio, ?Testing general relativ-
ity with compact coalescing binaries: comparing exact and predictive meth-
ods to compute the Bayes factor,? Class. Quant. Grav. 31, 205006 (2014),
arXiv:1408.2356 .
[393] L. S. Finn, ?Detection, measurement and gravitational radiation,? Phys. Rev.
D46, 5236?5249 (1992), arXiv:gr-qc/9209010 .
[394] C. Cutler and E. E. Flanagan, ?Gravitational waves from merging compact
binaries: How accurately can one extract the binary?s parameters from the
inspiral wave form?? Phys. Rev. D49, 2658?2697 (1994), arXiv:gr-qc/9402014
.
[395] M. Vallisneri, ?Use and abuse of the Fisher information matrix in the assess-
ment of gravitational-wave parameter-estimation prospects,? Phys. Rev. D77,
042001 (2008), arXiv:gr-qc/0703086 .
[396] P. A. R. Ade et al. (Planck), ?Planck 2015 results. XIII. Cosmological param-
eters,? Astron. Astrophys. 594, A13 (2016), arXiv:1502.01589 .
[397] P. Lazarus et al., ?Einstein@Home discovery of a Double-Neutron Star Binary
in the PALFA Survey,? Astrophys. J. 831, 150 (2016), arXiv:1608.08211 .
[398] K. G. Arun and A. Pai, ?Tests of General Relativity and Alternative theories
of gravity using Gravitational Wave observations,? Int. J. Mod. Phys. D22,
1341012 (2013), arXiv:1302.2198 .
[399] X. Zhang, J. Yu, T. Liu, W. Zhao, and A. Wang, ?Testing Brans-
Dicke gravity using the Einstein telescope,? Phys. Rev. D95, 124008 (2017),
arXiv:1703.09853 .
530
[400] F. Mignard and S. A. Klioner, ?Gaia: Relativistic modelling and testing,?
in Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and
Data Analysis , IAU Symposium, Vol. 261, edited by S. A. Klioner, P. K.
Seidelmann, and M. H. Soffel (2010) pp. 306?314.
[401] R. Nan, D. Li, C. Jin, Q. Wang, L. Zhu, W. Zhu, H. Zhang, Y. Yue,
and L. Qian, ?The Five-Hundred-Meter Aperture Spherical Radio Telescope
(FAST) Project,? Int. J. Mod. Phys. D20, 989?1024 (2011), arXiv:1105.3794
.
[402] B. Iyer et al. (LIGO Collaboration), ?LIGO-India, Proposal of the Consor-
tium for Indian Initiative in Gravitational-wave Observations,? (2011), LIGO
Document M1100296-v2.
[403] N. Itoh, ?Hydrostatic Equilibrium of Hypothetical Quark Stars,? Prog. Theor.
Phys. 44, 291 (1970).
[404] F. E. Schunck and E. W. Mielke, ?General relativistic boson stars,? Class.
Quant. Grav. 20, R301?R356 (2003), arXiv:0801.0307 .
[405] S. L. Liebling and C. Palenzuela, ?Dynamical Boson Stars,? Living Rev. Rel.
15, 6 (2012), arXiv:1202.5809 .
[406] P. O. Mazur and E. Mottola, ?Gravitational condensate stars: An alternative
to black holes,? (2001), arXiv:gr-qc/0109035 .
[407] P. O. Mazur and E. Mottola, ?Gravitational vacuum condensate stars,? Proc.
Nat. Acad. Sci. 101, 9545?9550 (2004), arXiv:gr-qc/0407075 .
[408] J. Eby, M. Leembruggen, J. Leeney, P. Suranyi, and L. C. R. Wijewardhana,
?Collisions of Dark Matter Axion Stars with Astrophysical Sources,? JHEP
04, 099 (2017), arXiv:1701.01476 .
[409] A. Iwazaki, ?A Possible origin of gamma-ray bursts and axionic boson stars,?
Phys. Lett. B455, 192?196 (1999), arXiv:astro-ph/9903251 .
[410] K. D. Kokkotas and B. G. Schmidt, ?Quasinormal modes of stars and black
holes,? Living Rev. Rel. 2, 2 (1999), arXiv:gr-qc/9909058 .
[411] P. Ajith et al., ?A Template bank for gravitational waveforms from coalescing
binary black holes. I. Non-spinning binaries,? Phys. Rev. D77, 104017 (2008),
[Erratum: Phys. Rev. D79, 129901 (2009)], arXiv:0710.2335 .
[412] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pu?rrer, X. Jime?nez Forteza, and
A. Bohe?, ?Frequency-domain gravitational waves from nonprecessing black-
hole binaries. II. A phenomenological model for the advanced detector era,?
Phys. Rev. D93, 044007 (2016), arXiv:1508.07253 .
531
[413] P. Pani, ?I-Love-Q relations for gravastars and the approach to the
black-hole limit,? Phys. Rev. D92, 124030 (2015), [Erratum: Phys.
Rev.D95,no.4,049902(2017)], arXiv:1506.06050 .
[414] N. Uchikata, S. Yoshida, and P. Pani, ?Tidal deformability and I-Love-Q
relations for gravastars with polytropic thin shells,? Phys. Rev. D94, 064015
(2016), arXiv:1607.03593 .
[415] R. F. P. Mendes and H. Yang, ?Tidal deformability of dark matter clumps,?
(2016), arXiv:1606.03035 .
[416] V. Cardoso, E. Franzin, A. Maselli, P. Pani, and G. Raposo, ?Testing strong-
field gravity with tidal Love numbers,? Phys. Rev. D95, 084014 (2017), [Ad-
dendum: Phys. Rev.D95,no.8,089901(2017)], arXiv:1701.01116 .
[417] N. V. Krishnendu, K. G. Arun, and C. K. Mishra, ?Testing the binary black
hole nature of a compact binary coalescence,? Phys. Rev. Lett. 119, 091101
(2017), arXiv:1701.06318 .
[418] A. Maselli, P. Pani, V. Cardoso, T. Abdelsalhin, L. Gualtieri, and V. Fer-
rari, ?Probing Planckian corrections at the horizon scale with LISA binaries,?
(2017), arXiv:1703.10612 .
[419] C. Palenzuela, I. Olabarrieta, L. Lehner, and S. L. Liebling, ?Head-on col-
lisions of boson stars,? Phys. Rev. D75, 064005 (2007), arXiv:gr-qc/0612067
.
[420] V. Cardoso, S. Hopper, C. F. B. Macedo, C. Palenzuela, and P. Pani,
?Gravitational-wave signatures of exotic compact objects and of quan-
tum corrections at the horizon scale,? Phys. Rev. D94, 084031 (2016),
arXiv:1608.08637 .
[421] C. Palenzuela, L. Lehner, and S. L. Liebling, ?Orbital Dynamics of Binary
Boson Star Systems,? Phys. Rev. D77, 044036 (2008), arXiv:0706.2435 .
[422] S. Yoshida, Y. Eriguchi, and T. Futamase, ?Quasinormal modes of boson
stars,? Phys. Rev. D50, 6235?6246 (1994).
[423] S. H. Hawley and M. W. Choptuik, ?Boson stars driven to the brink of black
hole formation,? Phys. Rev. D62, 104024 (2000), arXiv:gr-qc/0007039 .
[424] C. F. B. Macedo, P. Pani, V. Cardoso, and L. C. B. Crispino, ?Astrophysical
signatures of boson stars: quasinormal modes and inspiral resonances,? Phys.
Rev. D88, 064046 (2013), arXiv:1307.4812 .
[425] C. B. M. H. Chirenti and L. Rezzolla, ?How to tell a gravastar from a black
hole,? Class. Quant. Grav. 24, 4191?4206 (2007), arXiv:0706.1513 .
532
[426] P. Pani, E. Berti, V. Cardoso, Y. Chen, and R. Norte, ?Gravitational wave
signatures of the absence of an event horizon. I. Nonradial oscillations of a
thin-shell gravastar,? Phys. Rev. D80, 124047 (2009), arXiv:0909.0287 .
[427] C. Chirenti and L. Rezzolla, ?Did GW150914 produce a rotating gravastar??
Phys. Rev. D94, 084016 (2016), arXiv:1602.08759 .
[428] M. Colpi, S. L. Shapiro, and I. Wasserman, ?Boson Stars: Gravitational
Equilibria of Selfinteracting Scalar Fields,? Phys. Rev. Lett. 57, 2485?2488
(1986).
[429] R. Friedberg, T. D. Lee, and Y. Pang, ?Scalar Soliton Stars and Black Holes,?
Phys. Rev. D35, 3658 (1987).
[430] W. Press, Numerical Recipes in C: The Art of Scientific Computing, Numerical
Recipes in C book set No. v. 1 (Cambridge University Press, 1992).
[431] S. Chatrchyan et al. (CMS), ?Observation of a new boson at a mass of 125
GeV with the CMS experiment at the LHC,? Phys. Lett. B716, 30?61 (2012),
arXiv:1207.7235 .
[432] G. Aad et al. (ATLAS), ?Observation of a new particle in the search for the
Standard Model Higgs boson with the ATLAS detector at the LHC,? Phys.
Lett. B716, 1?29 (2012), arXiv:1207.7214 .
[433] L. Roszkowski, ?Supersymmetric dark matter: A Review,? in 1993 Aspen
Winter Physics Conference Elementary Particle Physics: Particle Physics
from Supercolliders to the Planck Scale Aspen, Colorado, January 10-16, 1993
(1993) arXiv:hep-ph/9302259 .
[434] D. J. Kaup, ?Klein-Gordon Geon,? Phys. Rev. 172, 1331?1342 (1968).
[435] D. A. Feinblum and W. A. McKinley, ?Stable States of a Scalar Particle in Its
Own Gravational Field,? Phys. Rev. 168, 1445 (1968).
[436] R. Ruffini and S. Bonazzola, ?Systems of selfgravitating particles in general
relativity and the concept of an equation of state,? Phys. Rev. 187, 1767?1783
(1969).
[437] F. S. Guzman, ?Accretion disc onto boson stars: A Way to supplant black
holes candidates,? Phys. Rev. D73, 021501 (2006), arXiv:gr-qc/0512081 .
[438] P. Amaro-Seoane, J. Barranco, A. Bernal, and L. Rezzolla, ?Constraining
scalar fields with stellar kinematics and collisional dark matter,? JCAP 1011,
002 (2010), arXiv:1009.0019 .
[439] H. A. Buchdahl, ?General Relativistic Fluid Spheres,? Phys. Rev. 116, 1027
(1959).
533
[440] S. Yoshida and Y. Eriguchi, ?Rotating boson stars in general relativity,? Phys.
Rev. D56, 762?771 (1997).
[441] F. D. Ryan, ?Spinning boson stars with large selfinteraction,? Phys. Rev. D55,
6081?6091 (1997).
[442] B. Kleihaus, J. Kunz, and M. List, ?Rotating boson stars and Q-balls,? Phys.
Rev. D72, 064002 (2005), arXiv:gr-qc/0505143 .
[443] P. Grandclement, C. Some?, and E. Gourgoulhon, ?Models of rotating boson
stars and geodesics around them: new type of orbits,? Phys. Rev. D90, 024068
(2014), arXiv:1405.4837 .
[444] Y. Kobayashi, M. Kasai, and T. Futamase, ?Does a boson star rotate?? Phys.
Rev. D50, 7721?7724 (1994).
[445] K. D. Kokkotas and G. Scha?fer, ?Tidal and tidal resonant effects in coalescing
binaries,? Mon. Not. Roy. Astron. Soc. 275, 301 (1995), arXiv:gr-qc/9502034
.
[446] D. Lai, ?Resonant oscillations and tidal heating in coalescing binary neutron
stars,? Mon. Not. Roy. Astron. Soc. 270, 611 (1994), arXiv:astro-ph/9404062
.
[447] T. Hinderer, ?Tidal Love numbers of neutron stars,? Astrophys. J. 677, 1216?
1220 (2008), arXiv:0711.2420 .
[448] T. Regge and J. A. Wheeler, ?Stability of a Schwarzschild singularity,? Phys.
Rev. 108, 1063?1069 (1957).
[449] K. S. Thorne, ?Tidal stabilization of rigidly rotating, fully relativistic neutron
stars,? Phys. Rev. D58, 124031 (1998), arXiv:gr-qc/9706057 .
[450] C.-W. Lai, A Numerical study of boson stars, Ph.D. thesis, British Columbia
U. (2004), arXiv:gr-qc/0410040 .
[451] B. Kleihaus, J. Kunz, and S. Schneider, ?Stable Phases of Boson Stars,? Phys.
Rev. D85, 024045 (2012), arXiv:1109.5858 .
[452] E. W. Mielke and F. E. Schunck, ?Boson stars: Alternatives to primordial
black holes?? Nucl. Phys. B564, 185?203 (2000), arXiv:gr-qc/0001061 .
[453] J. Vines, E?. E?. Flanagan, and T. Hinderer, ?Post-1-Newtonian tidal effects
in the gravitational waveform from binary inspirals,? Phys. Rev. D83, 084051
(2011), arXiv:1101.1673 .
[454] M. Favata, ?Systematic parameter errors in inspiraling neutron star binaries,?
Phys. Rev. Lett. 112, 101101 (2014), arXiv:1310.8288 .
534
[455] L. Wade, J. D. E. Creighton, E. Ochsner, B. D. Lackey, B. F. Farr,
T. B. Littenberg, and V. Raymond, ?Systematic and statistical errors in
a bayesian approach to the estimation of the neutron-star equation of state
using advanced gravitational wave detectors,? Phys. Rev. D89, 103012 (2014),
arXiv:1402.5156 .
[456] F. Douchin and P. Haensel, ?A unified equation of state of dense matter
and neutron star structure,? Astron. Astrophys. 380, 151 (2001), arXiv:astro-
ph/0111092 .
[457] H. Mu?ller and B. D. Serot, ?Relativistic mean field theory and the high density
nuclear equation of state,? Nucl. Phys. A606, 508?537 (1996), arXiv:nucl-
th/9603037 .
[458] S. Bernuzzi, T. Dietrich, and A. Nagar, ?Modeling the complete gravitational
wave spectrum of neutron star mergers,? Phys. Rev. Lett. 115, 091101 (2015),
arXiv:1504.01764 .
[459] A. Taracchini, Y. Pan, A. Buonanno, E. Barausse, M. Boyle, T. Chu,
G. Lovelace, H. P. Pfeiffer, and M. A. Scheel, ?Prototype effective-one-body
model for nonprecessing spinning inspiral-merger-ringdown waveforms,? Phys.
Rev. D86, 024011 (2012), arXiv:1202.0790 .
[460] K. Belczynski, G. Wiktorowicz, C. Fryer, D. Holz, and V. Kalogera, ?Missing
Black Holes Unveil The Supernova Explosion Mechanism,? Astrophys. J. 757,
91 (2012), arXiv:1110.1635 .
[461] T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read, ?Tidal deformability of
neutron stars with realistic equations of state and their gravitational wave sig-
natures in binary inspiral,? Phys. Rev. D81, 123016 (2010), arXiv:0911.3535
.
[462] A. Zimmerman, C.-J. Haster, and K. Chatziioannou, ?On combining infor-
mation from multiple gravitational wave sources,? Phys. Rev. D99, 124044
(2019), arXiv:1903.11008 .
[463] B. P. Abbott et al. (LIGO Scientific, Virgo), ?GWTC-1: A Gravitational-
Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and
Virgo during the First and Second Observing Runs,? Phys. Rev. X9, 031040
(2019), arXiv:1811.12907 .
[464] M. Pu?rrer, ?Frequency domain reduced order models for gravitational waves
from aligned-spin compact binaries,? Class. Quant. Grav. 31, 195010 (2014),
arXiv:1402.4146 .
[465] T. Dietrich, S. Bernuzzi, and W. Tichy, ?Closed-form tidal approximants
for binary neutron star gravitational waveforms constructed from high-
resolution numerical relativity simulations,? Phys. Rev. D96, 121501 (2017),
arXiv:1706.02969 .
535
[466] T. Dietrich et al., ?Matter imprints in waveform models for neutron star
binaries: Tidal and self-spin effects,? Phys. Rev. D99, 024029 (2019),
arXiv:1804.02235 .
[467] T. Dietrich, A. Samajdar, S. Khan, N. K. Johnson-McDaniel, R. Dudi, and
W. Tichy, ?Improving the NRTidal model for binary neutron star systems,?
Phys. Rev. D100, 044003 (2019), arXiv:1905.06011 .
[468] LIGO Scientific Collaboration and Virgo Collaboration, ?LALSuite software,?
(2018).
[469] S. Husa, S. Khan, M. Hannam, M. Pu?rrer, F. Ohme, X. Jime?nez Forteza, and
A. Bohe?, ?Frequency-domain gravitational waves from nonprecessing black-
hole binaries. I. New numerical waveforms and anatomy of the signal,? Phys.
Rev. D93, 044006 (2016), arXiv:1508.07250 .
[470] M. Hannam, P. Schmidt, A. Bohe?, L. Haegel, S. Husa, F. Ohme, G. Prat-
ten, and M. Pu?rrer, ?Simple Model of Complete Precessing Black-Hole-
Binary Gravitational Waveforms,? Phys. Rev. Lett. 113, 151101 (2014),
arXiv:1308.3271 .
[471] D. Langlois and K. Noui, ?Degenerate higher derivative theories beyond
Horndeski: evading the Ostrogradski instability,? JCAP 1602, 034 (2016),
arXiv:1510.06930 .
[472] D. Langlois, ?Degenerate Higher-Order Scalar-Tensor (DHOST) theories,? in
Proceedings, 52nd Rencontres de Moriond on Gravitation (Moriond Gravita-
tion 2017): La Thuile, Italy, March 25-April 1, 2017 (2017) pp. 221?228,
arXiv:1707.03625 .
[473] L. Bernard, ?Dynamics of compact binary systems in scalar-tensor theories:
II. Center-of-mass and conserved quantities to 3PN order,? Phys. Rev. D99,
044047 (2019), arXiv:1812.04169 .
[474] D. L. Lee, ?Conservation laws, gravitational waves, and mass losses in the
Dicke-Brans-Jordan theory of gravity,? Phys. Rev. D10, 2374?2383 (1974).
[475] L. Engvik, G. Bao, M. Hjorth-Jensen, E. Osnes, and E. Ostgaard, ?Asym-
metric nuclear matter and neutron star properties,? Astrophys. J. 469, 794
(1996), arXiv:nucl-th/9509016 .
[476] B. D. Lackey, M. Nayyar, and B. J. Owen, ?Observational constraints on
hyperons in neutron stars,? Phys. Rev. D73, 024021 (2006), arXiv:astro-
ph/0507312 .
[477] L. London and E. Fauchon-Jones, ?On modeling for Kerr black holes: Ba-
sis learning, QNM frequencies, and spherical-spheroidal mixing coefficients,?
(2018), arXiv:1810.03550 .
536
[478] V. Cardoso, M. Kimura, A. Maselli, and L. Senatore, ?Black holes in an
Effective Field Theory extension of GR,? (2018), arXiv:1808.08962 .
[479] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, ?Improved filters for gravi-
tational waves from inspiralling compact binaries,? Phys. Rev. D57, 885?907
(1998), arXiv:gr-qc/9708034 .
[480] H. Jeffreys, The Theory of Probability (Clarendon Press, Oxford, England,
1961).
[481] R. Nair, S. Perkins, H. O. Silva, and N. Yunes, ?Fundamental Physics Im-
plications for Higher-Curvature Theories from Binary Black Hole Signals in
the LIGO-Virgo Catalog GWTC-1,? Phys. Rev. Lett. 123, 191101 (2019),
arXiv:1905.00870 .
[482] P. Creminelli, G. D?Amico, J. Noren?a, and F. Vernizzi, ?The effective theory of
quintessence: the w ? -1 side unveiled,? Journal of Cosmology and Astroparticle
Physics 2009, 018?018 (2009).
[483] G. Gubitosi, F. Piazza, and F. Vernizzi, ?The effective field theory of dark en-
ergy,? Journal of Cosmology and Astroparticle Physics 2013, 032?032 (2013).
[484] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, ?Essential building blocks
of dark energy,? Journal of Cosmology and Astroparticle Physics 2013, 025?
025 (2013).
[485] J. Bloomfield, E?. E?. Flanagan, M. Park, and S. Watson, ?Dark energy or
modified gravity? an effective field theory approach,? Journal of Cosmology
and Astroparticle Physics 2013, 010?010 (2013).
[486] E. Bellini and I. Sawicki, ?Maximal freedom at minimum cost: linear large-
scale structure in general modifications of gravity,? JCAP 1407, 050 (2014),
arXiv:1404.3713 .
[487] D. Colladay and V. A. Kostelecky, ?CPT violation and the standard model,?
Phys. Rev. D55, 6760?6774 (1997), arXiv:hep-ph/9703464 .
[488] D. Colladay and V. A. Kostelecky, ?Lorentz violating extension of the standard
model,? Phys. Rev. D58, 116002 (1998), arXiv:hep-ph/9809521 .
[489] V. A. Kostelecky, ?Gravity, Lorentz violation, and the standard model,? Phys.
Rev. D69, 105009 (2004), arXiv:hep-th/0312310 .
[490] H. Jeffreys, ?An invariant form for the prior probability in estimation prob-
lems,? Proc. R. Soc. Lond. A 186 (1946), 10.1098/rspa.1946.0056.
[491] T. Damour, M. Soffel, and C. Xu, ?General relativistic celestial mechanics.
1. Method and definition of reference systems,? Phys. Rev. D43, 3273?3307
(1991).
537
[492] L. Blanchet and T. Damour, ?Postnewtonian Generation of Gravitational
Waves,? Ann. Inst. H. Poincare Phys. Theor. 50, 377?408 (1989).
[493] L. Blanchet and T. Damour, ?Radiative gravitational fields in general relativ-
ity i. general structure of the field outside the source,? Phil. Trans. R. Soc.
Lond. A 320, 379?430 (1986).
[494] T. Damour and B. R. Iyer, ?Multipole analysis for electromagnetism and lin-
earized gravity with irreducible cartesian tensors,? Phys. Rev. D43, 3259?3272
(1991).
[495] T. Damour and G. Esposito-Fare?se, ?Tensor multiscalar theories of gravita-
tion,? Class. Quant. Grav. 9, 2093?2176 (1992).
[496] T. Damour and G. Esposito-Fare?se, ?Tensor-scalar gravity and binary-pulsar
experiments,? Phys. Rev. D 54, 1474?1491 (1996).
[497] S. Mirshekari and C. M. Will, ?Compact binary systems in scalar-tensor grav-
ity: Equations of motion to 2.5 post-Newtonian order,? Phys. Rev. D 87,
084070 (2013).
[498] T. Damour and G. Scha?fer, ?Redefinition of position variables and the reduc-
tion of higher order Lagrangians,? J. Math. Phys. 32, 127?134 (1991).
[499] M. Favata, ?Conservative self-force correction to the innermost stable circular
orbit: comparison with multiple post-Newtonian-based methods,? Phys. Rev.
D83, 024027 (2011), arXiv:1008.4622 .
538