ABSTRACT
Title of dissertation: LOW-LATENCY SEARCHES FOR
GRAVITATIONAL WAVES AND THEIR
ELECTROMAGNETIC COUNTERPARTS
WITH ADVANCED LIGO AND VIRGO
Min-A Cho, Doctor of Philosophy, 2019
Dissertation directed by: Professor Peter Shawhan
Department of Physics
For the first time in history, advanced detectors are available to observe
the stretching and squeezing of space—gravitational waves—from violent as-
trophysical events. This opens up the prospect of joint detections with in-
struments of traditional astronomy, creating the new field of multi-messenger
astrophysics. Joint detections allow us to form a coherent picture of the un-
folding event as told by the various channels of information: mass and energy
dynamics from gravitational waves, charged particle environments (along with
magnetic field and specific element environments) from electromagnetic radia-
tion, and thermonuclear reactions/relativistic particle outflows from neutrinos.
In this work, I motivate low-latency electromagnetic and neutrino follow-
up of sources known to emit gravitational radiation in the sensitivity band of
ground-based interferometric detectors, Advanced LIGO and Advanced Virgo.
To this end, I describe the low-latency infrastructure I developed with col-
leagues to select and enable successful follow-up of the first few gravitational-
wave candidate events in history, including the first binary black hole merger,
named GW150914, and binary neutron star coalescence, named GW170817,
from the first and second observing runs.
As a review, I outline the theory behind gravitational waves and explain
how the advanced detectors, low-latency searches, and data quality vetting
procedures work. To highlight the newness of the field, I also share results
from an offline search for a more speculative source of gravitational waves,
intersecting cosmic strings, from the second observing run.
Finally, I address how LIGO/Virgo is prepared to adapt to challenges
that will arise during the upcoming third observing run, an era that will be
marked by near-weekly binary black hole candidate events and near-monthly
binary neutron star candidate events. To handle this load, we made several
improvements to our low-latency infrastructure, including a new, streamlined
candidate event selection process, expansions I helped develop for temporal
coincidence searches with electromagnetic/neutrino triggers, and data quality
products on source classification and probability of astrophysical origin to
provide to our observing partners for potential compact binary coalescences.
These measures will further our prospects for multi-messenger astrophysics
and increase our science returns.
LOW-LATENCY SEARCHES FOR
GRAVITATIONAL WAVES AND THEIR
ELECTROMAGNETIC COUNTERPARTS WITH
ADVANCED LIGO AND VIRGO
by
Min-A Cho
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
2019
Advisory Committee:
Professor Peter Shawhan, Chair/Advisor
Professor Paulo Bedaque
Professor Sarah Eno
Professor Jordan Goodman
Professor M. Coleman Miller, Dean’s Representative
©c 2019
Min-A Cho
All Rights Reserved
ii
For my umma and appa, Inseon Cho-Kim and Kijo Cho,
and for my unni, Min Jeong Cho.
“My heart is a traitor,” the boy said to the alchemist, when they had paused
to rest the horses. “It doesn’t want me to go on.”
“That makes sense. Naturally it’s afraid that, in pursuing your dream, you
might lose everything you’ve won.”
“Well, then, why should I listen to my heart?”
“Because you will never again be able to keep it quiet. Even if you pretend
not to have heard what it tells you, it will always be there inside you,
repeating to you what you’re thinking about life and about the world.”
“You mean I should listen, even if it’s treasonous?”
“Treason is a blow that comes unexpectedly. If you know your heart well, it
will never be able to do that to you. Because you’ll know its dreams and
wishes, and will know how to deal with them.”
Paulo Coelho, the Alchemist
iii
Acknowledgments
Professor Peter Shawhan, thank you for mentoring me and advising me
through the past 6 years of my life. When I applied to graduate school at the
University of Maryland, I remember thinking, “Wow! It would be so cool if I
could work for this guy and work on detecting gravitational waves...” Thus, I
have been truly fortunate to be one of your graduate students. Aside from the
physics, seeing you interact with your colleagues, students, and family, I’ve
also learned how to become a better and kinder person under your advisory.
Thank you.
Professors Paulo Bedaque, Sarah Eno, Jordan Goodman, and M. Coleman
Miller, thank you for encouraging *cough terrorizing* me while being a part of
my doctoral committee. I appreciate your incisive questions and detailed feed-
back during my examination. In particular, thank you Professor Sarah Eno,
because you told me to seek support early on in my graduate studies when
my mom’s health was deteriorating. I appreciate that to this day because I
needed it!
Dr. Timothy Edberg, Pauline Rirksopa, and Cregg Yancey, you made my
day-to-day life at work more fulfilling and enjoyable, so thank you. Tim, it
is always a pleasure to say hello to you. It also makes me happy to look at
your nature photography. Pauline, I adore you! I cherish our 5 or 10 minute
conversations and your laughter. I hope you are having fun traveling with
your husband in your retirement. Cregg, my one and only office mate. Thank
you for your whiteboard explanations, your stories, and your friendship. I will
miss all of these things.
Paulina Alejandro, Lorraine DeSalvo, Donna Hammer, and Sally Megoni-
gal, thank you for making life manageable—not just for me, but also for many
iv
other graduate students, professors, and researchers. I would not have had
access to my office after hours or health insurance or travel reimbursement or
opportunities to do public outreach if it weren’t for you. Thank you.
Professors Tom Antonsen, Alessandra Buonanno, Adil Hassam, Ted Ja-
cobson, Ed Ott, and Min Ouyang, thank you because I either enjoyed taking
your graduate level course or being your teaching assistant. To Professor Ed
Ott, I would like to point out that I found your doppelgänger. He is also a
professor albeit in the animated series Futurama.
Professors Christoph Boehme, John DeFord, Henryk Hecht, Stefan LeBo-
hec, and Nicholas Korevaar, thank you because you shaped my undergraduate
studies. Professor Boehme, you once told me to study something I am natu-
rally curious about, because it will keep me going—especially when the times
get hard... It proved to be profound advice; I have passed it along to other
students as well.
Sarah Antier, Deep Chatterjee, Shaon Ghosh, Giuseppe Greco, Barbara
Patricelli, Karelle Siellez, and Koh Ueno (a.k.a. the O2 EM Follow-up Paper
Writing Team)—Yay, we did it! Thank you because it’s been an honor working
with you guys. I cherish our friendship and am inspired by your scholarship.
Don’t ever hesitate to reach out!
Imène Belahcene, Kipp Cannon, Florent Robinet, and Daichi Tsuna (the
O2 Cosmic Strings Search Analysis Team), thank you for accepting me into
the team and for your patience while teaching me. Imène and Daichi, I am
also thankful for our friendship; I hope we keep in touch.
Patrick Brady, Marica Branchesi, Gaby González, Jonah Kanner, Erik
Katsavounidis, Laura Nuttall, and Leo Singer, thank you for entrusting me
with important tasks and for believing in me. With great trust comes great
responsibility... I also want to thank you Leo, for elevating my standards,
v
whether it came to coding practices and style or dissertation and presentation
aesthetics. I was definitely inspired!
Imani Herring, Nishat Mhamud, and Arul Teimouri, thank you for filling
our house with laughter. Arul, my fellow INFP and book enthusiast! Thank
you for being my friend through all the ups and downs this past year. I look
forward to more swimming adventures with you and Melanie, perhaps in Cuba
next time. *Wink*
Amanda Alexander, Adella Croft, Connie Eng Fulkerson, Maria Garcia,
Julia Schneider, Olha Krechkivska, and Izzy Velez, thank you for your love
and support throughout the years. I miss and love you guys!
Chaka Freeman, Tingting Liu, Joe Murray, Dalia Ornelas, Holly Tinkey,
Ana Valdés, Chiao-Hsuan Wang, Hongcheng Xu, and Vitaley Zaretsky. Thank
you for your friendship, I am glad we got to meet and get to know each other.
I adore you all. Chaka, one word: all-sky! Holly, you have a great sense of
humor. (Are skin pockets still a thing?) Tingting, my fellow ballet enthusiast!
Hongcheng, my National Geographics worthy birder.
Renee Ren-Patterson, I look up to you for so many different reasons—each
time I talk with you I learn something new, and I love it. Thank you. You are
my beauty and brains inspo.
Omar Ortiz, I’m so glad we became friends! Thank you for all of the
laughter and tears; I’m looking forward to more memories and can’t wait to
see what the future holds!
Sharon Wall and Meghan Hughes. Four words: annual Christmas cookie-
baking parties! Sharon, because of you I had family away from family my first
few years of graduate school. Thank you for letting me get to know them.
Mr. and Mrs. Wall, Leeroy, Jess, Aunt Cheryl, Audrey and Mike, Ashlee and
Susie, thank you for accepting me with open arms. Leeroy, thank you for being
vi
my brother (I’ve always wanted one!). Also Sharon, because of you I started
eating much healthier and am aware of nutrition—thank you babe!
Yao Odamtten, Vincent W. Williams, and Julieane ‘Julie’ Hill. Yao, thank
you for your LARGE LOVE AND SPIRIT. There is no other way to put it.
You are a shining star in my heart. You are a shining star, PERIOD! Vin-
cent, the other shining star in my heart. How did I get so lucky? Thank you
for being my friend, boo! Julie, my sweet, smart, kind, and beautiful friend!
Sometimes I smile and think to myself, “Wow, there are people like Julie in
the world!”, and I know I’m not alone in thinking that *cough Vincent and
Yao*; I love you guys! Let’s keep on dancing.
Naasir Ali, Alex ‘Axiz’ Benitez, Geoffrey ‘Toyz’ Chang, Carrington Lewis-
Sweeney, Jonathon ‘Frenzy’ Marie, Kerrie ‘Kerrie-Sauce’ Marie, Nelson Men-
cho, Fasil Sheta, Julie ‘Ju-Ju’ Stoessel, Grisha Tikhonov, and James Ulmer. I
am so lucky to have met and danced with you guys! You are all so mad tal-
ented. Fasil, please keep drawing! Carrington, I consider you one of my closest
friends although I hardly ever see you. Alex, eres mi tesoro, the Hobbes to
my Calvin. Thank you for sharing life’s little moments with me, and making
everything a little bit sweeter. *Mwah*
John (cat), Piyali ‘Pipi’ Das, Davy Foote, John Giannini, Prasoon Gupta,
and Luke ‘Lukey’ Robertson (a.k.a. The Perch Family), I love you guys! I
petition that we meet up at least once a year. Pipi and Lukey... I could not
have made it through this past year without your support. *Squeeze*
Benjamin Chung (정복헌) and Christine Chung (위ᅥᄉᆼᅮᄉᆨ), thank you for
reaching out when my mom passed. I know that your family was important
to her and that she cherished your friendship.
Suok Pak (ᄇᆨᅡ수옥) and Kyongsook Kim (ᄀᆷᅵᆼᅧᄀᄉᆨᅮ), I consider you both
like my godmothers, and I’m so lucky! I love you both, and thank you for
vii
helping me become who I am today. I also want to say thank you for being
close friends with my mom, and for being there for her. That means the world
to me. On a similar note, I would like to thank my mom’s dear friends, Samuel
Roh and Jungsooki-eemo... Thank you from the bottom of my heart.
Weh-halmunee (선귀ᅪᄒ) and weh-halabujee (김ᆼᅧ액ᄐ), thank you for loving
me and my sister unconditionally throughout the years. We felt it even from
overseas, especially with your homemade kimchi and side dishes, halmunee!
I would love to meet you again—soon—to cook with you and listen to your
stories. Halabujee, I wish I could buy you pizza one more time and go hiking!
Rest In Peace.
Umma (ᄀᆷᅵ인선) and appa (ᅩᄌ기조). Minnie (조민ᆼᅥᄌ) and Darami.
Darami, sweet baby sister. When you were five (six?), you asked me, “Min-A,
what is treasure?” and I said, “Treasure is something you love and hold dear
in your heart.” Wherever you are, you are so loved—and treasured! Umma,
I miss you and think about you all the time... Appa! I will always remember
your words, “힘들겠지ᅡᄆᆫ ᅳᄌᆯ겁게 ᅢᄉᆼᅡᄀᆨᅡᄒ고 하ᅡᄌ.” Words to live by.
Umma and appa, the most important thing you’ve taught me is to be a
promise keeper to myself. I evaluate daily whether my action or inaction for
the day has measured up to what I say I want for my life. Thank you for
guiding me to become a more accountable and self-actualized human being.
And more than anything, thank you for your love and support. It gave me the
courage to dream and to go after my dreams.
Lastly, to my older sister Minnie. You are the person I look up to the
most! There is no one more grounding than you. Thank you for being able
to tell me the hard things I need to hear in life. You are my rock and keeper,
and I am your rock and keeper too. Love you.
viii
Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
I. Multi-Messenger Sources and Motivations . . . . . . . . . . . . . . . 1
1.1 Core-Collapse Supernovae . . . . . . . . . . . . . . . . . . . . . 2
Progenitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
The Supernova Problem . . . . . . . . . . . . . . . . . . . . . 7
Detection Prospects . . . . . . . . . . . . . . . . . . . . . . . . 10
Gravitational Wave Signatures . . . . . . . . . . . . . . . . . . 11
Electromagnetic/Neutrino Signatures . . . . . . . . . . . . . . 14
1.2 Magnetars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Electromagnetic/Neutrino Signatures . . . . . . . . . . . . . . 19
Gravitational Wave Signatures . . . . . . . . . . . . . . . . . . 23
1.3 Compact Binary Coalescences . . . . . . . . . . . . . . . . . . 24
Progenitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Constraining the Nuclear Equation of State . . . . . . . . . . 30
II. Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1 General Theory of Relativity in a Nutshell . . . . . . . . . . . 32
2.2 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Sources of Gravitational Waves . . . . . . . . . . . . . . . . . . 41
ix
2.4 The Hulse-Taylor Binary . . . . . . . . . . . . . . . . . . . . . 44
Pulsars Are Neutron Stars . . . . . . . . . . . . . . . . . . . . 45
Timing with Pulsar Profiles . . . . . . . . . . . . . . . . . . . 50
Pulsar Timing Formula for Isolated Pulsars . . . . . . . . . . . 52
Full Timing Formula and Results . . . . . . . . . . . . . . . . 66
III. Advanced Ground-Based Laser Interferometric Gravitational-Wave
Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.1 A Simple Michelson interferometer . . . . . . . . . . . . . . . . 70
3.2 Interaction with Gravitational Waves in the Transverse Trace-
Free Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Advanced Detectors . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 The Noise Power Spectral Density . . . . . . . . . . . . . . . . 78
Optical Read-Out Noise . . . . . . . . . . . . . . . . . . . . . 80
Seismic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Newtonian Noise . . . . . . . . . . . . . . . . . . . . . . . . . 85
Suspension Thermal Noise . . . . . . . . . . . . . . . . . . . . 86
Coating Thermo-Optic and Brownian/Substrate Brownian Noise 86
Excess Gas Noise . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Interferometer Antenna Response . . . . . . . . . . . . . . . . 87
IV. Low-Latency Searches for Gravitational-Wave Candidate Events . . 95
4.1 Compact Binary Coalescence Searches . . . . . . . . . . . . . . 95
GstLAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
MBTAOnline . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
PyCBC Live . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
SPIIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Burst Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
cWB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
x
oLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 GraceDb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4 Supervised Electromagnetic/Neutrino Follow-Up Process . . . 107
4.5 Online Automated Data Vetting . . . . . . . . . . . . . . . . . 108
4.6 Human Vetting . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
V. O1: The First Observing Run . . . . . . . . . . . . . . . . . . . . . 114
5.1 approval_processor: The First of the Advanced Detector Era
Gravitational-Wave Candidate Event Annotators . . . . . . . . 115
5.2 Information Sent to MOU Partners . . . . . . . . . . . . . . . 116
False Alarm Rate . . . . . . . . . . . . . . . . . . . . . . . . . 116
Two-Dimensional Sky Localization Probability Maps . . . . . 117
VI. O2: The Second Observing Run . . . . . . . . . . . . . . . . . . . . 121
6.1 Information Sent to MOU Partners . . . . . . . . . . . . . . . 123
EM-Bright Source Classification . . . . . . . . . . . . . . . . . 123
Three-Dimensional Sky Localization Probability Maps . . . . 127
6.2 The Advanced Virgo Detector . . . . . . . . . . . . . . . . . . 130
VII. Cosmic Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Strings vs. Superstrings . . . . . . . . . . . . . . . . . . . . . 134
Loops, Cusps, and Kinks . . . . . . . . . . . . . . . . . . . . . 135
7.2 Cosmic String Cusps Search Algorithm . . . . . . . . . . . . . 136
7.3 Search Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
VIII. En Route to O3: The Third Observing Run . . . . . . . . . . . . 144
8.1 Low-Latency Gravitational Wave-Electromagnetic and Neutrino
Counterpart Coincidence Searches . . . . . . . . . . . . . . . . 144
Temporal Coincidence Searches . . . . . . . . . . . . . . . . . 144
xi
Gravitational Wave-Gamma-Ray Burst Coincidences . . . . . 145
8.2 P_astro: The Probability of Astrophysical Origin . . . . . . . 157
8.3 Public Alerts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
IX. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . 163
A. Appendix A: LIGO/Virgo Notices for GW150914 . . . . . . . . . . 167
B. Appendix B: LIGO/Virgo Notices for GW170817 . . . . . . . . . . 173
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
xii
List of Figures
Number Page
1.1 Net heating and cooling regions between the shock and proto-
neutron star ‘surface’ which is defined by the neutrinospheres.
(Figure from Mezzacappa et al., 2006) . . . . . . . . . . . . . . 9
1.2 Sample gravitational-wave strain (h+) times the distance D vs.
time after bounce. This signal was extracted from a 2-dimensional
15 M simulation. (Figure from Murphy et al., 2009) . . . . . 12
1.3 Comparisons of plume frequencies, fp, with the gravitational-
wave spectrogram. This signal was extracted from a 2-dimensional
15 M simulation. (Figure from Murphy et al., 2009) . . . . . 13
1.4 Schematic diagram of magnetar magnetic field lines. Uniform
poloidal field lines thread the liquid core and the solid, outer
crust of the star while toroidal field lines are created by twisted
field lines inside the core. (Figure from Thompson & Duncan,
2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Possible electromagnetic counterparts to neutron star binaries.
In blue is the cross section of the centrifugally supported disk
around the final black hole, in red is the surrounding circumburst
medium, and in yellow is the more isotropic kilonova. (Figure
from Metzger & Berger, 2012) . . . . . . . . . . . . . . . . . . 28
xiii
1.6 A possible blue bump in the observed kilonova emission could be
indicative of a long-lived hypermassive neutron star phase where
the electron fraction is raised to a high enough value (Ye ∼ 0.4)
that no Lanthanides are produced. On the other hand, if the
remnant black hole is formed promptly after the merger, both
the dynamical ejecta before the merger and disk outflows after
the merger will be highly neutron rich (Ye < 0.1) generating
a kilonova emission that peaks later and in the near-infrared.
(Figure from Metzger & Fernández, 2014) . . . . . . . . . . . . 29
2.1 A circular ring of freely falling masses distorted by + (blue, top)
and × (red, bottom) polarizations of a gravitational wave prop-
agating out of the page. A rotation of 45◦ takes a + mode into
a × mode and vice versa. For the figure, a complete wave cycle
is shown from left to right. . . . . . . . . . . . . . . . . . . . . 40
2.2 Schematic model of a pulsar. (Figure from Lorimer & Kramer,
2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Pulse profile at 430 MHz for the pulsar in Hulse-Taylor binary.
Observed during July 1977 (dotted line), June 1978 (dashed
line), and October 1978 (solid line). The central component has
been gradually moving to the left and becoming broader, while
the third component has shifted to the right. All profiles have
been smoothed to the resolution indicated by the horizontal bar,
400 µs. (Figure from Taylor et al., 1979) . . . . . . . . . . . . . 51
2.4 A possible geometry to account for pulse shape changes in the
Hulse-Taylor binary. (Figure from Taylor et al., 1979) . . . . . 51
xiv
2.5 Geometry of the orbit with orbital parameters. The periastron
of a binary system has been labeled as pericenter here. (Figure
from Weisstein, 2018) . . . . . . . . . . . . . . . . . . . . . . . 64
2.6 Constraints on the pulsar mass, mp, and companion mass, mc,
from extracted values of 〈ω̇〉 and γ. (Figure from Weisberg &
Taylor, 2002) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7 Orbital decay of PSR B1913+16. The data points represent mea-
sured orbital phase errors caused by assuming a fixed value of
Pb that have been translated into cumulative shift of periastron
time, in seconds. The parabola is the General Theory of Relativ-
ity prediction for the binary emitting gravitational waves. Error
bars for data points are too small to see. (Figure from Weisberg
et al., 2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 From top to bottom, aerial views of Advanced LIGO/Hanford,
Advanced LIGO/Livingston, and Advanced Virgo. (Figures from
LIGO Laboratory/Virgo) . . . . . . . . . . . . . . . . . . . . . 71
3.2 Advanced LIGO’s Michelson-Morley interferometer with the power
recycling mirror (placed between the laser and beamsplitter),
Fabry-Pérot cavities making up the 4 km arms, and signal recy-
cling mirror (placed between the beamsplitter and photodetec-
tor). The test mass setup shows the main chain side (left) and
the reaction chain side (right). (Figure from Abbott et al., 2016) 76
3.3 Broadband (orange) to narrowband (red) detector responses de-
pending on signal recycling mirror tunings, φ. Dashed/solid lines
are for lower/upper frequency sidebands. (Figure from Gabriele
Vajente, 2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
xv
3.4 Advanced LIGO’s design sensitivity curve, 1/2Sn (f). The sym-
metric binary 1.4 M neutron star and symmetric binary 30 M
black hole coalescence ranges are 173 Mpc and 1606 Mpc, re-
spectively. (Figure from Barsotti et al., 2018) . . . . . . . . . . 79
3.5 The Euler angles {ϕ, θ, ψ} convert between the detector frame
(x, y, z) and the gravitational-wave propagation frame (x′, y′,
z′). For better visual depiction, the y coordinates have their
signs inverted. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 Antenna response patterns for an interferometric detector of the
+ (left), × (center), and RMS (right) polarizations, computed
with polarization angle ψ = 0. . . . . . . . . . . . . . . . . . . 90
3.7 The RMS combination of the + and × polarization antenna re-
sponse patterns when the polarization angle, ψ = 0. The detec-
tor arms are drawn in for reference. . . . . . . . . . . . . . . . 93
3.8 A three-detector (Advanced LIGO and Virgo) BAYESTAR sky lo-
calization of GW170817 in ICRS coordinates (Aitoff projection)
overlaid on top of antenna patterns for each detector; see Chap-
ter 6. (Figures from Giuseppe Greco, 2018) . . . . . . . . . . . 94
4.1 Main analytical and numerical methods for solving the two-body
problem depends on the masses and compactness involved. Here,
m1 andm2 (m1 ≥m2) are the two individual masses and rc2/GM
is a measure of the separation distance. (Figure from Buonanno
& Sathyaprakash, 2014) . . . . . . . . . . . . . . . . . . . . . . 97
xvi
4.2 Spectrograms of an overflow glitch that requires a veto for the
corresponding GW trigger (top) and mitigation (bottom). The
bottom figure is from the Advanced LIGO/Livingston C00 “on-
line calibrated” data for GW170817 (https://dcc.ligo.org/P1700337/
public). The faint but characteristic trace of the BNS chirp can
be seen in the background; thus, the noise was modeled and
subtracted. (Figure from Abbott et al., 2017a) . . . . . . . . . 112
5.1 Timeline of LIGO/Virgo Notices and Circulars sent to our MOU
partners for GW150914 and its electromagnetic follow-up. (Fig-
ure from Abbott et al., 2016a) . . . . . . . . . . . . . . . . . . 116
5.2 Method of time slides for FAR estimation. GW triggers are de-
tected in zero-lag (top) and noise events are detected in time
shifted data with offsets greater than the GW travel time be-
tween the detectors. (Figure from Laura Nuttall, 2017) . . . . . 117
5.3 Comparison of the 90% credible regions from low-latency skymaps
produced by cWB, LIB, and BAYESTAR for GW150914 displayed in
an orthographic projection centered around the LIB localization.
In light green is the offline full parameter estimation skymap
produced by LALInference. The inset shows the distribution of
the arrival time difference, ∆tHL, across the two Advanced LIGO
detector network. (Figure from Abbott et al., 2016a) . . . . . . 120
xvii
6.1 Different regions of the ellipsoid sample component mass pa-
rameter space. Foucart’s fitting formula is applied for ellipsoid
samples in the pink and green shaded NS-BH region. In partic-
ular, the χ1-dependent green shaded regions reflect boundaries
where ellipsoid samples give non-zero remnant disk mass. Addi-
tionally, ellipsoid samples for GW170817 (red dots in the cyan
BNS/100% EMbright parameter space) and GW170618 (purple
stars in the grey BBH/0% EMbright parameter space) are shown.
(Figure from Deep Chatterjee/Abbott et al. (2019)) . . . . . . 126
6.2 BAYESTAR skymaps for GW170817 in ICRS coordinates (Moll-
weide projection) from a 1-detector network (top, Advanced LIGO/Han-
ford), 2-detector network (center, Advanced LIGO), and 3-detector
network (bottom, Advanced LIGO and Advanced Virgo). The
50% confidence region and the location of the host galaxy NGC
4993 (marked with a star) are shown. . . . . . . . . . . . . . . 131
6.3 Electromagnetic follow-up of the first observed binary neutron
star coalescence event, GW170817. This is also the first multi-
messenger event involving gravitational waves. (Figure from Ab-
bott et al., 2017b) . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.1 Types of cosmic string intersections where the intercommutation
probability, p, is assumed to be 1. From top to bottom: string-
string intersection at one point (two new long strings are formed
via partner exchange), string-string intersection at two points
(two new long strings are formed via partner exchange plus one
closed loop), and self-string intersection (one long string and a
closed loop are formed). (Figure from Sakellariadou, 2007) . . . 135
xviii
7.2 Time evolution (dotted black line) of a point along a string (red
dot) that becomes a cusp at time τ = y, starting with a string
intersecting itself at time τ = y − 2δ. (Figure from Stott et al.,
2017) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.3 An Omicron scan of the highest-ranked zero-lag cosmic string
cusp candidate event revealed it to be consistent with a tomte
blip glitch in Advanced LIGO/Livingston. Because we expect
at least one candidate event to have occurred by accident due
to noise processes, results of the search are consistent with the
hypothesis that there are no signals present. . . . . . . . . . . . 140
7.4 O2 Burst cosmic string cusp search results. . . . . . . . . . . . 141
7.5 95% confidence exclusion regions for cosmic string tension and
intercommutation probability from the LIGO and Virgo Burst
cosmic strings analysis group using O1 and O2 data for two large
loop Nambu-Goto cosmic string distribution models. The ex-
cluded regions are below the respective curves. At p = 1 for
topological strings, we cannot put a constraint on the string ten-
sion for the Blanco-Pillado et al. model (top, Blanco-Pillado et al.
(2014)). However, for the Ringeval et al. model (bottom, Lorenz
et al. (2010)), the string tension must be less than ∼4.2×10−10.
(Figures from Florent Robinet, 2018) . . . . . . . . . . . . . . . 143
8.1 From top to bottom, skymaps in ICRS coordinates (Mollweide
projection) with 90% and 50% credible regions for GW170817
(computed by BAYESTAR using Advanced LIGO data only), GRB
170817A from Fermi/GBM, and their normalized product. The
location of the apparent host galaxy NGC 4993 is marked with
a star in the joint GW-GRB skymap. . . . . . . . . . . . . . . 156
xix
List of Tables
Number Page
1.1 Supernova classification depends on the absence or presence (marked
by X) of hydrogen (H), helium (He), or silicon (Si) lines in the
peak optical spectra. . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 Intrinsic and extrinsic parameters, ϑ, of a compact binary coa-
lescence gravitational-wave waveform. . . . . . . . . . . . . . . 97
4.2 Pipeline-specific measures (marked with a X) to identify non-
Gaussian noise sources in single-detector trigger lists during the
second observing run, O2. . . . . . . . . . . . . . . . . . . . . . 100
4.3 Template bank parameters and SNR threshold for triggering
used by the low-latency CBC search pipelines during the sec-
ond observing run, O2. (Table from Abbott et al., 2019) . . . . 102
6.1 Information columns in three-dimensional BAYESTAR skymaps. . 129
7.1 O2 data were divided into 6 chunks for the Burst cosmic strings
analysis. All start and end times are in GPS time. . . . . . . . 137
7.2 The three loudest zero-lag cosmic string cusp candidate events
identified during O2. H1 and L1 stand for the Advanced LIGO/Han-
ford and Advanced LIGO/Livingston detectors respectively. . . 139
8.1 Summary of 50,000 spatial overlap integrals, IΩ, reporting the
mean and standard deviation of the mean for 100 low-latency
GW skymaps per pipeline combined with 100 Fermi/GBM GRB
skymaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.1 Expected detector sensitivities to BNS coalescences during O3. 165
xx
List of Abbreviations
1RXS first Röntgensatellit X-ray Survey; astronomical catalog
3XMM third XMM-Newton Serendipitous Source Catalogue; astro-
nomical catalog
ADVREQ advocate required
AGN active galactic nuclei
aLIGO Advanced LIGO
AXP anomalous X-ray pulsar
BAYESTAR Bayesian Triangulation And Rapid Localization; software pack-
age for gravitational-wave events
BBH binary black hole
BH black hole
BNS binary neutron star
CBC compact binary coalescence
CCO central compact object
CCSN core-collapse supernova
CDF cumulative distribution function
CMB cosmic microwave background
CSM circumstellar medium
cWB coherent WaveBurst; software package for gravitational-wave
events
DM dispersion measure
DWD double white dwarf
FAP false alarm probability
FAR false alarm rate
FGMC Farr-Gair-Mandel-Cutler
FITS flexible image transport system
xxi
GBM the Gamma-ray Burst Monitor instrument on board the Fermi
Gamma-ray Space Telescope
GCN Gamma-ray Coordinates Network
GLADE Galaxy List for the Advanced Detector Era; galaxy catalog
GPS the Global Positioning System
GraceDb gravitational-wave candidate event database
GRB gamma-ray burst
GstLAL GStreamer LAL; software package for gravitational-wave events
GUT grand unified theory
GW gravitational wave
HMNS hypermassive neutron star
ICRS international celestial reference system
IFAR inverse FAR
IIR infinite impulse response
INTEGRAL the International Gamma-Ray Astrophysics Laboratory; space
telescope
IR infrared
ISI internal seismic isolation platform
ISM interstellar medium
KAGRA Kamioka Gravitational Wave Detector
LAL LIGO algorithm library; software package for gravitational-
wave events
LIGO Laser Interferometer Gravitational-wave Observatory
LISA Laser Interferometer Space Antenna
LOOC UP Locating and Observing Optical Counterparts to Unmodeled
GW Pulses; project for gravitational-wave event follow-up
during the Initial LIGO era
LVAlert LIGO-Virgo alert system
LVT LIGO-Virgo trigger
xxii
MBTA Multi-Band Template Analysis; software package for gravitational-
wave events
MCMC Markov chain Monte Carlo
ML maximum likelihood
MOU memorandum of understanding
NGC New General Catalogue; astronomical catalog
NIR near-infrared
NS neutron star
NS-BH neutron star-black hole
oLIB Omicron LALInferenceBurst; software package for gravitational-
wave events
P_astro probability of astrophysical origin
PDF probability density function
PN post-Newtonian
PSD power spectral density
PSR pulsar
pubsub publish-subscribe
PyCBC python CBC; software package for gravitational-wave events
QED quantum electrodynamics
RAVEN rapid, on-source VOEvent coincidence monitor; software pack-
age for gravitational-wave events
RESTful API representational state transfer application program interface
RMS root mean square
RRAT rotating radio transient
RRT rapid response team
RSE resonant sideband extraction
SAA South Atlantic anomaly
SGR soft gamma-ray repeater
xxiii
sGRB short gamma-ray burst
skymap sky localization probability map
SN supernova
SNEWS SuperNova Early Warning System
SNR signal-to-noise ratio
SPI-ACS the Spectrometer and Anti-Coincidence Shield on board IN-
TEGRAL
SPIIR Summed Parallel Infinite Impulse Response; software package
for gravitational-wave events
SR signal recycling
SRC signal recycling cavity
SSB solar system barycenter
STF symmetric trace-free
TF time-frequency
TT transverse trace-free
UTC coordinated universal time
VOEvent virtual observatory event
WDM Wilson-Daubechies-Meyer
XDIN X-ray dim isolated neutron star
XMM X-ray Multi-Mirror Mission; space observatory
XMPP extensible messaging and presence protocol
XTE X-ray Timing Explorer; space telescope
1
Chapter 1
Multi-Messenger Sources and
Motivations
The state of gravitational-wave physics before the discovery of the Hulse-
Taylor binary pulsar system PSR B1913+16 (Hulse & Taylor, 1975; Taylor
et al., 1976; Taylor & Weisberg, 1982) in many ways resembled the state of
neutrino physics in the 1940’s and 1950’s. As one contemporary observer
noted, “There can be no two opinions about the practical utility of the neu-
trino hypothesis ... but ... until clear experimental evidence for the existence
of the neutrino could be obtained ... the neutrino must remain purely hypo-
thetical” (Ellis, 1937). Then, in 1959, Reines and Cowan delivered the first
crucial bit of evidence: the first direct observation of the free neutrino (Cowan
et al., 1956). Fast-forward a few decades and today neutrinos are considered
by astrophysicists to be a valuable probe of the structure of matter and per-
haps less well-known, a key ingredient to reviving stalled stellar core-collapse
explosions (Section 1.1).
The first serious attempt to detect gravitational waves began with Joseph
Weber at the University of Maryland, College Park in the 1960’s. Previous
to his efforts, gravitational waves, while deemed essential for the structural
integrity of the General Theory of Relativity (Einstein, 1916, 1918), seemed
so weak in their visible manifestations that Einstein himself believed that
they would never be detected, and perhaps even worse, that they would be of
no practical importance. Not to be deterred, Weber built his aluminum bar
detectors and his claim of detection in 1969 (Weber, 1969) propelled others to
2
attempt to do the same—and more importantly—convinced the skeptics that
detection could be possible. While results of the bar detector experiments
were not reproduced, in 1974, the discovery of the Hulse-Taylor binary pulsar
(see Section 2.4) confirmed the existence of gravitational waves.
In present-day gravitational-wave physics, theory is now confirmed by di-
rect observations with advanced ground-based interferometric detectors, lead-
ing to the advent of gravitational-wave astronomy (e.g., Abbott et al., 2016b,c,
2017c). In addition, recent discovery of gravitational waves from the first bi-
nary neutron star coalescence ever observed (Abbott et al., 2017) along with
its many electromagnetic counterparts has now signaled that multi-messenger
astronomy has officially begun (e.g., Abbott et al., 2017a; Albert et al., 2017;
Haggard et al., 2017; Savchenko et al., 2017; Troja et al., 2017; D’Avanzo et al.,
2018; Dobie et al., 2018; Margutti et al., 2018; Ruan et al., 2018).
This chapter is dedicated to exploring in depth astrophysical sources ex-
pected to be jointly observed by a network of advanced ground-based inter-
ferometric detectors and its traditional astronomy partners. Select unsolved
problems that are related to these sources are also presented since they could
be resolved by such observations.
1.1 Core-Collapse Supernovae
Core-collapse supernovae are the explosive deaths of massive stars that
require the full power of general relativity, the strong and weak interactions,
electrodynamics, and transport theory in order to be understood. The basics
of modern-day core-collapse supernova theory are summarized in this section
with a focus on the 50+ year old supernova problem: “How is the stellar core
infall eventually reversed so that the disruption of the star is triggered, along
with the ejection of the stellar mantle and stellar envelope?”
3
In the advent of a core-collapse supernova in the Milky Way (suspected to
occur once every ∼50 years), it will be possible to explore the supernova prob-
lem with advanced ground-based gravitational-wave detectors and/or their
near-term upgrades. Therefore, key gravitational-wave signatures from core-
collapse supernovae are discussed. In addition, information obtained from
gravitational-wave detections from neutron star binaries (Section 1.3) could
contribute to the effort of modeling core-collapse supernovae and are therefore
included.
The origins of the supernova problem start in 1925, when Pauli stated
in his exclusion principle (Pauli, 1925) that electrons from the same quantum
system must be in different quantum states. When applied to stars that are
counteracting their own attractive force of gravitation, the Pauli exclusion
principle prevents electrons in the stars from getting infinitely close to each
other. Instead, the electrons are forced to fill up energy levels that are available
to them, starting from the very lowest.
However, the sheer number of electrons present in a star means ulti-
mately that some of the electrons end up with high energies, and therefore
high momenta. Thus, the electrons moving outward in the star provide a kind
of ‘electron degeneracy pressure’ that can support the star against its own
gravitational implosion (e.g., Burrows(& )Ostri(ker,)2014):
π2~2 2/3 5/33 ρ
P =
5/3
5m π µemp e
where ~ is Planck’s constant divided by 2π, me is the mass of an electron, mp
is the mass of a proton, ρ is the density, and µe is the ratio of electrons to
protons.
Here, it is important to note that the same Pauli exclusion principle that
is applied for electrons applies for protons as well, although the partial pressure
4
contribution from the degenerate proton gas is much less in the non-relativistic
limit. This is simply due to the high proton-to-electron mass ratio, ∼1836,
that results in the degenerate proton gas having a kinetic energy ∼1/1836
times the energy of the degenerate electron gas (Schutz, 2004).
An important moment in the life cycle of a star is reached when hydrogen
fusion to carbon and oxygen essentially completes and the star is not massive
enough to achieve the internal pressure/temperature needed to fuse the carbon
and oxygen into heavier elements. In this case, fusion stops, leading to the
cooling down of the star at essentially fixed density (i.e., the star begins to
shrink). For stars that begin less massive than∼8−10M, electron degeneracy
pressure will halt the shrinking. This occurs when the electron Fermi energy
exceeds the electron thermal energy, kBT . For non-relativistic electrons this
occurs at ∼ 2/3T10 4ρ8 , and for relativistic electrons at ∼ 1/3T10 ρ8 with Tx ≡
T/(10x K) and ρy ≡ ρ/(10y g cm−3) (Janka, 2012). Such a star is now a white
dwarf.
Equally important as Pauli’s exclusion principle is Chandrasekhar’s dis-
covery in 1931 (Chandrasekhar, 1931) that there is an upper mass limit to stars
that can be supported via the above electron degeneracy pressure. In modern-
day core-collapse supernova theory, Chandrasekhar’s limit (e.g., Woosley et al.,
2002) is the starting point for describing the upper mass limit of metastable,
degenerate cores at the centers of massive stars near the onset of core-collapse.
These cores resemble hot white dwarfs close to their maximum effective Chan-
drasekhar mass1 given by: ( )2
≈ YeMCh, eff 1.44 M + corrections...
0.5
1The effective Chandrasekhar mass given in the equation is for cold neutron stars. The
cores of massive stars which are still hot can have a mass up to a few tenths of a solar mass
above MCh, eff without collapsing.
5
where Ye is the electron fraction, the number density of electrons divided by
the total number density of baryons, with typical values ∼0.45 (Janka et al.,
2012) and M is one solar mass. Then, because no equilibrium solutions exist
for relativistic and degenerate electron gases with M > MCh, eff, core-collapse
ensues when the stellar core mass exceeds the effective Chandrasekhar limit.
Progenitors
The lowest-mass progenitors of core-collapse supernovae have oxygen-
neon-magnesium (ONeMg) cores (Nomoto, 1984, 1987) that reach the elec-
tron degeneracy state before hydrostatic Ne-burning can be ignited. However,
ONeMg cores experience high rates of electron capture by protons and nuclei
in the cores, meaning the electron degeneracy pressure quickly lets up due to
a decrease in Ye. Therefore, these cores are the progenitors of electron cap-
ture supernovae (ECSN) which are believed to comprise up to 20−30% of all
core-collapse supernovae (Wanajo et al., 2009, 2010).
More massive progenitors are able to ignite Ne-burning and develop iron
cores which become unstable at temperatures around 1010 K (kBT ∼ 1 MeV).
Then, electron degeneracy pressure support becomes reduced due to photodis-
sociation of the iron-group nuclei (e.g., Janka, 2012; Burbidge et al., 1957) and
reactions such as
γ + 5626Fe 13α + 4n
favor the production of α-particles which proceed to capture more electrons,
speeding up the core-collapse.
In more detail, the collapsing iron core splits into two pieces, an inner core
and an outer core, because of radius-dependent local sound speeds and infall
velocities. (The outer core falls in supersonically while the inner core falls in
subsonically.) However, the collapsing inner core cannot collapse indefinitely
6
due to the repulsive nature of nucleon-nucleon interactions at extremely short
ranges (Bethe & Johnson, 1974). Thus, when the inner core reaches nuclear
densities, ρnuc ∼ 2.7 ×1014 g cm−3, the repulsive term of the strong force kicks
in, causing the nuclear equation of state there to ‘stiffen’, i.e., Γ = d lnP/d ln ρ
(or equivalently, the Γ in P ∝ ρΓ) jumps from 4/3 up to ∼2.6−2.8 as the com-
position transitions from inhomogeneous matter (nucleons, α-particles, and
nuclei) towards pure nucleons. The effect of this stiffening is that the inner
core ‘bounces’ and launches the shock wave that eventually triggers the super-
nova (e.g., Mezzacappa et al., 2014).
The shock wave however, stalls, leading to the modern-day supernova
problem. As the inner core rebounds and expands into the surrounding outer
iron core material, it loses energy due to the dissociation of iron-group nuclei
into free nucleons (∼8.8 MeV per nucleon in the post-shock matter). ∼1−2
ms after the shock formation, velocities downstream from the shock become
negative. Finally, the shock comes to a halt when the mass accretion rate from
the outer core gets low enough (Janka et al., 2012).
If the shock is not revived within ∼0.5−3 s, the shock becomes an
accretion-shock, and infalling matter from the outer core accretes onto the
central object, enabling the formation of a final black hole (Bethe, 1990).
Else, the inner core eventually becomes a neutron star or black hole. This is
where we introduce the present-day supernova problem.
There are also more energetic supernovae which require magneto-rotational
driving but these constitute only 0.1-1% of all core-collapse supernovae (Woosley
& Weaver, 1981). The observational evidence for this is indirect: as of 2016,
there were 11 long GRB—core-collapse supernovae associations. It was de-
termined that neutrino-driven core-collapse explosions (which constitute the
7
majority of core-collapse explosions) have insufficient energy to generate long
GRBs.
The stiffened nuclear equation of state describing the inner stellar core
generally has Γ > 5/3, which is the value predicted for cold neutron star inte-
riors (assuming T = 0 and that the interiors are composed of non-relativistic
Fermi neutron and proton gases) (Ott, 2014). This means that neutron stars
must be held up by more than the neutron and proton degeneracy pressures,
otherwise ones above 0.7M could not exist. Indeed, neutron stars are also
held up by the strong interactions which add another layer of difficulty to the
problem of obtaining the correct nuclear equation of state. However, figuring
out the nuclear equation of state (perhaps with the help of advanced ground-
based gravitational-wave detectors) could narrow down the number of viable
core-collapse supernovae mechanisms, a possibility that will be visited again
and explained later in this section.
The Supernova Problem
The fundamental supernova problem is that Nature is very good at pro-
ducing core-collapse supernovae, but we are not good at modeling them. Of
the ∼3× 1053 erg of gravitational binding energy released through the forma-
tion of the final neutron star or black hole, only ∼1% ends up as kinetic and
internal energy of the expanding ejecta. The remaining 99% is radiated away
as neutrinos over hundreds of seconds as the proto-neutron star (the accreting
inner core) cools (Bethe, 1990; Janka, 2001).
In 1966, Colgate and White (Colgate & White, 1966) were the first to
suggest that neutrinos could play an important role in explosion, by depositing
some of their energy into the rest of the star. In the present-day paradigm
where core-collapse supernovae are indeed neutrino-driven (albeit much dif-
8
ferently than was originally proposed by Colgate and White), the crux of
the supernova problem is that the shock from the inner core bounce becomes
stalled. In other words, neutrinos from the electron captures become trapped
by the infalling outer core which supplies a ‘ram pressure’ to the neutrinos,
preventing them from getting out. Thus, any information about the bounce is
unable to propagate out faster than it is being swept back in.
This problem was addressed by Wilson in 1985 who suggested that there
can be delayed shocks due to net neutrino heating regions that develop behind
the shock (Wilson, 1985). At matter densities near 3× 1012 g cm−3, the neu-
trino mean free paths decrease and the neutrinos become trapped due to the
shortened time between consecutive neutrino reactions. The radii at which
this occurs for each flavor of neutrino defines its respective neutrinosphere,
i.e., the approximate ‘surface’ of the proto-neutron star (Mezzacappa et al.,
2006) (Figure 1.1). Between the shock and the neutrinosphere, radially depen-
dent heating and cooling determines the ‘gain region’ where neutrino opaci-
ties, i.e., neutrino-matter interactions favor the absorption of neutrinos and
anti-neutrinos by protons and neutrons over their emissions (Bethe & Wilson,
1985):
ν −e + n→ p+ e
ν̄e + p→ n+ e+
This energy deposition onto matter can deliver delayed (however, weak) ex-
plosions Bethe & Wilson (1985).
The next two milestones in contemporary supernova theory were discov-
ered through computations. The first was the prediction that neutrino con-
vection behind the shock can produce robust explosions with 2-dimensional
(axisymmetric) models, whereas 1-dimensional (spherically symmetric) mod-
9
Heating
_νe + n → p + e-
νe + p → n + e+
_νe + n ←  p + e- Gain Radius
ν  + p ←  n + e+e
Cooling
Proto-Neutron
Star
ν-Spheres
ν-Luminosity
Matter Flow
Figure 1.1: Net heating and cooling regions between the shock and proto-
neutron star ‘surface’ which is defined by the neutrinospheres. (Figure from
Mezzacappa et al., 2006)
els cannot (Herant et al., 1994). This suggests neutrino convection can play
an important role in marginally explosive cases, and that the dimensionality
of the modeling alone can make the difference in the production (or not) of
the explosion.
The second computational prediction was of the standing accretion-shock
instability (SASI) in 2003 (Blondin et al., 2003). Previously, researchers had
studied the formation and the evolution of the shock but never its stability.
The SASI prediction shows that non-radial perturbations to the shock are
unstable and cause the shock to grow. However, radial perturbations are
stable and cause the shock to ring back to where it was.
10
Therefore, to address the supernova problem realistically requires (1) a
fully general relativistic treatment of gravity and its corrections to the neu-
trino transport equations (e.g., Bruenn et al., 2001; Liebendörfer et al., 2005),
(2) net neutrino heating in the gain regions to overcome the infalling core’s
ram pressure (e.g., Wilson, 1985; Bethe & Wilson, 1985), (3) neutrino convec-
tion to better describe neutrino transport to the gain regions (Herant et al.,
1994), (4) SASI which grows the shock upon non-radial disturbances (Blondin
et al., 2003), and (5) other corrections from stellar rotations and the pres-
ence of magnetic fields, etc. to realistically model core-collapse supernovae.
These state-of-the-art models have already been used to produce preliminary
gravitational-wave waveforms to study their detection prospects by the ad-
vanced ground-based gravitational-wave detector network.
Detection Prospects
Less than 0.01% of the available ∼3 × 1053 erg from core-collapse is ra-
diated away via gravitational waves (Ott, 2009; Kotake, 2013). This strictly
limits prospective sources for advanced ground-based gravitational-wave de-
tectors to Galactic events up to 10 kpc away (Yakunin et al., 2017). However,
much time and effort is dedicated to being ready for the next Galactic core-
collapse supernova. This is because gravitational waves directly probe the
central engine of the supernova, and particulars about the explosion dynamics
can be extracted from gravitational-wave detections (e.g., Powell et al., 2016).
Even more critically, this extracted information can also be used to validate
differing core-collapse supernovae models.
The leading order contribution to gravitational waves is due to spher-
ically asymmetric accelerations of mass and energy (Einstein, 1916, 1918).
Thus, it is fitting that observations of electromagnetic emissions from core-
11
collapse supernovae (along with theory) predict strong asymmetries (Foglizzo
et al., 2015). Then, it is possible to detect gravitational waves from neu-
trino convection and large-scale standing accretion-shock instabilities (SASI)
for neutrino-driven core-collapse supernovae. (As a reminder, these were both
historically predicted using 2-dimensional (spherically asymmetric) computer
models (Herant et al., 1994; Blondin et al., 2003)).
The core-collapse supernova rate in the Milky Way and close-by Small
and Large Magellanic Clouds is rather low, . 1 event per 30−50 years (Ott,
2009). However, the rate increases to∼1 per 20 years if the entire local group of
galaxies up to M31 at 0.8 Mpc is included (van den Bergh & Tammann, 1991).
In general, a core-collapse event at 10 kpc (which is well within the Milky Way
Galaxy), will produce gravitational-wave strains of amplitude ∼10−22−10−21
with gravitational-wave energy outputs totaling ∼10−11−10−10M 2c (Murphy
et al., 2009). This is more than enough for a detection by Advanced LIGO
which at design sensitivity, is expected to detect Galactic events with minimum
signal-to-noise ratios ∼10−20 (Harry & the LIGO Scientific Collaboration,
2010; Murphy et al., 2009).
Gravitational Wave Signatures
Gravitational wave signals were extracted from 2-dimensional and 3-
dimensional hydrodynamic simulations using a weak-field, slow-motion formal-
ism that considers strain contributions from the dominant mass-quadrupole
moment only (Murphy et al., 2009; Yakunin et al., 2015; Nakamura et al.,
2016). There are three key phases present in gravitational-wave emissions
(Figure 1.2).
The first of the three phases is the prompt convection phase which lasts
∼70−80 ms after the core bounce. This is produced by Ledoux convec-
12
tion (Keil et al., 1996) in the proto-neutron star, which is due to a nega-
tive entropy gradient, (∂ρ/∂ ln s)Y ,P , or a negative (positive) lepton gradient,l
(∂ρ/∂ lnYl)s,P for large (small) lepton fractions, Yl, below the neutrinosphere.
Figure 1.2: Sample gravitational-wave strain (h+) times the distance D vs.
time after bounce. This signal was extracted from a 2-dimensional 15 M
simulation. (Figure from Murphy et al., 2009)
The second phase is the neutrino convection/SASI phase which starts
around 120 ms for 15 M 2-dimensional and 3-dimensional models (Murphy
et al., 2009; Yakunin et al., 2017) and grows for about ∼550 ms past the
bounce (Murphy et al., 2009). This is the strongest part of the gravitational-
wave signal, as up to this point the shock is still quasi-spherical. The SASI in
particular causes dense and narrow ‘SASI plumes’ that strike the proto-neutron
star surface and correlate with large distinctive spikes in the gravitational-wave
signal (Murphy et al., 2009; Marek & Janka, 2009) (Figure 1.2). In general,
higher mass progenitors produce higher characteristic plume frequencies, fp,
which increase monotonically from ∼100 Hz at bounce to ∼300−400 Hz before
explosion. Thus, although the gravitational-wave power declines between the
13
prompt convection phase and the SASI phase, the time-frequency spectrogram
of the signal reveals the characteristic plumes and their increasing frequencies
clearly (Figure 1.3) (Murphy et al., 2009).
Figure 1.3: Comparisons of plume frequencies, fp, with the gravitational-wave
spectrogram. This signal was extracted from a 2-dimensional 15 M simula-
tion. (Figure from Murphy et al., 2009)
Comparisons of the gravitational-wave signals from the 2-dimensional
and 3-dimensional cases show that the SASI phase is strong in 2 dimensions,
although strong signal components below 250 Hz are still present in the 3-
dimensional cases (Andresen et al., 2017). On the other hand, gravitational-
wave signals from neutrino convection are stronger in 3-dimensional models
than in 2, although their overall contribution to the gravitational-wave signal
is less prominent.
The third and last phase is the explosion phase, which has a ‘memory’
feature originating from the outer exploding regions of the star, i.e., there
are distinct differences in the gravitational-wave signal that reveal the general
morphology of the explosion (i.e., prolate, oblate, and spherical) (Murphy
14
et al., 2009). As expected, when the explosion is spherical, there is very little
gravitational radiation and the strain drops to 0. However, a prolate explosion
results in a rise in strain that is specifically positive (in the reference frame of
the star where the explosion would be seen as prolate) vs. an oblate explosion
whose strain will become negative.
The explosions are also much lower in frequency, ∼tens of Hz (Fig-
ure 1.3). In many of the 3-dimensional simulations, the models did not run long
enough to produce explosions2. However, it is expected that the 3-dimensional
gravitational-wave signal will be smaller in amplitude than the 2-dimensional
case due to a smaller mass fraction contained inside the shock wave (Lentz
et al., 2015).
One critical point in this section is that 2-dimensional models by necessity
admit only one polarization state of the gravitational waves, +, due to their
axisymmetry. Therefore, realistic core-collapse supernovae which are more
likely to be similar to the 3-dimensional case will have increased chances of
detection due to both + and × polarizations being available for detection.
Electromagnetic/Neutrino Signatures
Neutrinos compose nearly 99% of the energy released in a core-collapse su-
pernova. Thus, recent 3-dimensional core-collapse simulations of non-rotating
progenitors with neutrino transport looked for correlations between the re-
sultant neutrino and gravitational-wave signals (Kuroda et al., 2017). These
models show that strong correlations between the two signals are characteristic
of vigorous SASI activity in the supernova core.
On the electromagnetic signature side, main classes and subclasses of
2Simulations take millions of processing hours to complete, which convert into months
when running on thousands of computing cores (Messer et al., 2013).
15
supernovae are determined by their spectral properties. The two main classes
of supernovae are Type I and Type II, established by the absence or presence
of hydrogen lines in the peak optical spectrum (Minkowski, 1941). Subclasses
of Type I exist to distinguish between supernovae with or without silicon or
helium lines. Of the different classes of supernovae in Table 1.1, we focus
on those with a core-collapse3 explosion mechanism only (Types Ib, Ic, and
II). In reality, there is a continuum among the core-collapse supernovae classes
depending on the amount of hydrogen and helium lost from the outer envelopes
before the explosion (due to stellar winds) and of that, how much is retained
after the explosion. This results in differences among the supernovae light
curves that leads to further refinement of classification.
H He Si Mechanism Subclassification
Type I X thermonuclear Type Ia
Type I X core-collapse Type Ib
Type I core-collapse Type Ic
Type II X core-collapse Type II
Table 1.1: Supernova classification depends on the absence or presence
(marked by X) of hydrogen (H), helium (He), or silicon (Si) lines in the peak
optical spectra.
Type II supernovae are typically observed in the spiral arms of spiral
galaxies with Population I4 stars as progenitors. They have characteristic
Balmer lines in their peak optical spectra. Among Type II supernovae, those
with thicker retained hydrogen envelopes have light curves with a long plateau
phase (Type IIP) in contrast to those with thinner hydrogen envelopes retained
whose light curves decline more quickly (Type IIL). Progenitors of Type IIL
supernovae are also expected to have larger radii than Type IIP in order to
3Type Ia supernovae are produced by accreting white dwarfs in close binaries where the
silicon lines are thought to come from runaway carbon fusion in the white dwarf core.
4Population I stars are young, high-metallicity stars, formed from the gas of previous
generation stars.
16
explain their higher peak luminosities (Hicken et al., 2017).
Type IIb supernovae are a class between Type II and Type Ib, where
a few tenths of a solar mass of the original hydrogen envelope was retained.
Thus, Balmer lines that appear in the early spectra quickly disappear as the
supernova ages, and are replaced with strong helium lines. This suggests
retreat of the supernova photosphere through the hydrogen envelope and into
the helium layer. Progenitors of Type IIb supernovae are in the 10−18 M
range and are likely to be in binary systems.
Type IIn supernovae have distinct narrow hydrogen and helium emission
lines which indicate the presence of a dense circumstellar medium (CSM).
There is considerable heterogeneity in this supernova class, with some super-
novae featuring a strong Hα line5 while in others it is noticeably faint, and
some supernovae evolving into strong radio and X-ray sources while others do
not.
Type Ib and Ic supernovae have only been observed in spiral galaxies near
sites of recent star formation (i.e., H II regions6). Furthermore, Type Ib and Ic
supernovae fall into the category of stripped-envelope supernovae where Type
Ib progenitors have lost their hydrogen envelope and Type Ic progenitors have
additionally lost their helium envelope. These two clues suggest progenitors of
Type Ib/Ic supernovae are Wolf-Rayet stars (i.e., stars with mass & 25 M).
In particular, Type Ic progenitors (especially those born with sub-solar
metallicity) are interesting because they are associated with the majority of
observed long-duration gamma-ray bursts (long GRBs), which are intense
beamed signals of gamma-rays with durations lasting 2−100 s (> 10,000 s
5The Hα line is the first of the Balmer lines, meaning it occurs when a hydrogen electron
falls from its n=3 to n=2 energy level, corresponding to a wavelength of 656.28 nm.
6H II regions are regions of ionized interstellar atomic hydrogen, commonly found in
spiral galaxy disks.
17
in the case of ultra-long GRBs). It is also likely that these long GRB progen-
itors are rapidly rotating, which would consequently alter the observed GW
signal from the associated core-collapse supernova. However, it remains un-
clear whether rapidly rotating Wolf-Rayet stars exist, although some models
suggest surface rotation velocities of ∼200 km/s (Shenar et al., 2014). Thus,
there are low-latency searches in place by advanced ground-based interfero-
metric GW detectors to look for coincidences between unmodeled transient
gravitational-wave signals and GRBs/neutrinos (Section 8.1).
Lastly, supernovae leave behind spectacular supernovae remnants which
can be observed for hundreds to thousands of years after the explosion. These
remnants are created by the surrounding interstellar medium which is violently
compressed and chemically enriched by the gaseous shell ejected from the
supernova.
It is now more than 80 years since Baade and Zwicky proposed that neu-
tron stars are formed in core-collapse supernovae (Baade & Zwicky, 1934).
They were correct, although the exact internal mechanism still eludes us.
The advanced ground-based gravitational-wave detectors will allow us to di-
rectly probe the supernova central engine at design sensitivities. Thus, an
end is in sight to the 50+ year old supernova problem, likely with the help of
gravitational-wave astronomy.
1.2 Magnetars
Magnetars are a special class of neutron stars that were originally iden-
tified by their ultra-strong magnetic fields, in excess of the quantum critical
2 3
field7, B mecQED = ~ = 4.4×1013 G (Kouveliotou, 1999). In present-day magne-e
7This is the value of the magnetic field at which the cyclotron energy (the energy between
Landau levels of electrons) equals the rest-mass energy for an electron (Duncan & Thompson,
1992; Mereghetti, 2008).
18
tar physics, they are more generally defined as neutron stars with magnetically-
powered emissions, regardless of their measured or inferred surface dipole fields
(Turolla et al., 2011; Rea et al., 2012; Rea et al., 2014), and regardless of be-
ing powered in-part by rotation (Kumar & Safi-Harb, 2008). They are further
characterized by slow X-ray pulsations (periods P ∼ 2−12 s) and large spin-
down rates (period derivatives Ṗ ∼ 10−10−10−12 s s−1) (Rea et al., 2008).
Magnetars were proposed in 1992 to explain in a unified way, two obser-
vationally distinct classes of objects known by then (Duncan & Thompson,
1992): soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs),
which were discovered in 1979 (Mazets et al., 1979a,b; Mazets & Golenetskii,
1981) and 1981 (Fahlman & Gregory, 1981) respectively.
As of 2014, there were 21 confirmed magnetars and 5 magnetar can-
didates, with thousands of them expected to exist in our Galaxy (Olausen
& Kaspi, 2014; Negreiros et al., 2018). Of the confirmed magnetars, SGR
0418+5729, Swift J1822.3−1606, and 3XMM J185246.6+003317 are the low-
est magnetic field magnetars, with inferred surface dipole fields of magnitude
Bp <∼ 7.5 × 1012 G, 2.7 × 1013 G, and 4.1 × 1013 G (Turolla et al., 2011;
Rea et al., 2012; Rea et al., 2014), all less than BQED. These lower magnetic
field magnetars are aged magnetars with rapidly decaying magnetic fields.
SGR 0418+5729 and Swift J1822.3−1606 are estimated to be ∼1 Myr and
∼550 kyr old.
A central question regarding magnetars is, “What is the origin of the
high magnetic fields observed in magnetars?” In particular, the magnetic field
strength impacts the structure and evolution of neutron stars on three different
fronts: microscopically by altering the equation of state through Landau quan-
tization which leads to pressure anisotropies that affect particle composition,
19
macroscopically by requiring a full axially symmetric (rather than spherically
symmetric (Oppenheimer & Snyder, 1939)) treatment of perfect fluids within
the General Theory of Relativity framework that changes the mass/radii of
neutron stars, and temporally by altering the spin-down properties (P and
Ṗ ) which can mask the true age of the neutron star if the field exhibits non-
canonical behavior. So far, we understand that magnetar magnetospheres are
made up of twisted magnetic field lines both inside and outside of the star
as shown in Figure 1.4, which can explain their characteristic persistent emis-
sions and X-ray spectral shapes (Subsection 1.2) (Thompson & Duncan, 2001;
Thompson et al., 2002).
Furthermore, the study of magnetar subpopulations and their thermal
evolution is important to understanding physical conditions that lead to the
formation of different types of young, isolated neutron stars (Ertan et al.,
2014): AXPs, SGRs, dim isolated neutron stars (XDINs), central compact ob-
jects (CCOs), and rotating radio transients (RRATs). Past population synthe-
sis studies suggest that up to ∼10% of neutron stars are expected to have com-
panions (Iben & Tutukov, 1996), thus another lingering astrophysical question
regarding magnetars is, “Why are all the known magnetars isolated?” (Popov
& Prokhorov, 2006). In the case of AXPs, they were dubbed “anomalous” be-
cause their high X-ray luminosities could not be explained by accretion from a
binary companion or from rotational energy loss, Ė, as in the case of standard
pulsars.
Electromagnetic/Neutrino Signatures
Nearly 10% of all core-collapse supernovae explosions (Section 1.1) result
in magnetars, manifesting themselves as either SGRs or AXPs (which could
be a later phase in the evolution of SGRs) (Kouveliotou, 1999). In 1986,
20
B
ΔR0
core
z
crust
φ
Figure 1.4: Schematic diagram of magnetar magnetic field lines. Uniform
poloidal field lines thread the liquid core and the solid, outer crust of the
star while toroidal field lines are created by twisted field lines inside the core.
(Figure from Thompson & Duncan, 2001)
SGRs were recognized as a class of objects separate from GRBs, with the
most important difference being the recurrence of SGR events. This excluded
the possibility of SGRs having the same progenitors as GRBs, which were
(correctly) conjectured at the time to be caused by the catastrophic destruction
of their parent object population. Indeed, the best current model of SGR and
AXP events suggests they are glitches and flares caused by stresses built up in
the magnetar crust due to internal toroidal fields, Bφ, that are twisted up to
10 times more than the external dipole fields. This induces a rotation of the
surface neutron star layers that deforms the crust and leads to fractures (i.e.,
21
starquakes), outbursts, and flares.
There have been 3 types of magnetar bursting events detected thus far:
short bursts, intermediate bursts, and giant flares (Turolla et al., 2015). Short
bursts are the most common and observed in both SGRs and AXPs. They
typically last 0.1−1 s and have peak luminosities 1039−1041 erg/s with soft
(∼10 keV) thermal spectra. Intermediate bursts are also observed in both
SGRs and AXPs but last longer (1−40 s) and have more energetic peak lu-
minosities (1041−1043 erg/s). Finally, there are also giant flares which have
been observed only 3 times since SGRs were discovered: from SGR 0526−66
in 1979 (Mazets et al., 1979a), SGR 1900+14 in 1998 (Hurley et al., 1999), and
SGR 1806−20 in 2004 (Hurley et al., 2005; Palmer et al., 2005). These rare
events are characterized by peak luminosities in the range 1044−1047 erg/s.
As observed from Earth, giant flares could appear as short gamma-ray bursts
(sGRBs) (Section 1.3) if emitted by extragalactic SGRs (Palmer et al., 2005;
Hurley et al., 2005). Upper limits on the fraction of sGRBs due to these events
are ∼1−15% (Hurley, 2011).
Magnetars also have persistent (non-bursting), often variable X-ray lu-
minosities in the range 1033−1036 erg/s. The low-energy thermal component
of these emissions can be fit with one blackbody spectrum of temperature
∼0.3−0.6 keV and a power law with a steep photon index Γ ∼ 2−4, or by
two blackbody spectra of temperatures ∼0.3 keV and ∼0.7 keV (Rea et al.,
2008). On top of this, there are also non-thermal emissions caused by charged
particles trapped in the twisted magnetic field lines (Figure 1.4) that interact
with the surface X-ray thermal emissions through resonant cyclotron scatter-
ing (Thompson et al., 2002). While soft X-ray emissions (below 10 keV) can
be explained by resonant cyclotron scattering, hard X-ray emissions (above
∼20 keV) remain poorly understood. It is suspected that either thermal
22
bremsstrahlung from magnetar surface layers heated by returning currents or
synchrotron emissions from pair creation in the upper magnetosphere (∼100
km) are responsible for these hard emissions (Thompson & Beloborodov, 2005).
Although more than 90% of magnetar bolometric luminosities are con-
centrated in the 1−200 keV range, there are still faint (K ∼ 20) optical and/or
NIR counterparts (Israel et al., 2004) that are sometimes pulsed and transient.
As of 2003, there were four IR-identified counterparts to AXPs: 4U 0142+61
(Hulleman et al., 2000), 1E 2259+586 (Kaspi et al., 2003), 1E 1048.1−5937
(Israel et al., 2002), and 1RXS J170849−400910 (Israel et al., 2003).
Magnetar radio emissions remain controversial. In the past, it was as-
sumed they would be absent due to photon splitting, a quantum electrody-
namical effect which kicks in for B > BQED, and can overtake electron-positron
pair production deemed essential for pulsar radio emissions (Baring & Hard-
ing, 1998). However, as observed in AXP XTE J1810−197, magnetars can also
emit radio pulses after an outburst (Camilo et al., 2006; Camilo et al., 2007).
These emissions differ from standard radio pulsar emissions (Subsection 2.4)
in that their spectrum is flatter and both the flux and pulse profile show strong
variations with time. This indicates either the emission mechanism or emission
region topology differs between standard radio pulsars and magnetars.
Lastly, newborn magnetars could emit neutrinos during the ∼10−100 s
cooling epoch following core-collapse (Murase et al., 2014). These emissions
would be generated by a relativistic nucleon-rich wind where the neutrons
eventually undergo inelastic collisions, producing neutrinos in the energy range
∼0.1−1 GeV.
23
Gravitational Wave Signatures
Two types of gravitational-wave emissions are expected from magnetars.
The first are short bursts of gravitational waves which could be generated in
the event of a starquake. In the most energetic case where giant flares erupt,
there could be up to ∼1049 erg emitted in the form of gravitational waves
(Corsi & Owen, 2011). These upper limits are calculated from models where
the internal toroidal field lines of the magnetars become reconfigured during
the starquake, changing the shape of the star, causing a sudden fractional jump
in the moment of inertia, ∆I/I ∼ 10−4. A change in the moment of inertia of
this magnitude was observed in the giant flare event of SGR 1900+14 in 1998.
The second main type of gravitational-wave emission is continuous and
caused by rotation of a deformed body around one of its principal axes. Ad-
vanced ground-based interferometric detectors can detect gravitational waves
from newborn magnetars up to ∼20 Mpc away if they have internal toroidal
fields Bφ & 1016 G, especially if dipole fields Bp < a few 1014 G and initial spin
periods Pi ∼ a few ms (Kashiyama et al., 2016). Strong toroidal fields make
the star more prolate and triaxial so that in general, the principal moments of
inertia, I1 =6 I2 6= I3. Then, the deformation can be described in terms of an
ellipticity parameter, :
I1 − I2
 = ,
I3
and the gravitational-wave amplitude is proportional to I1−I2 = I3. In the
special case that I1 = I2 6= I3 is symmetric about the axis of rotation, no
gravitational waves are generated. Lastly, a population of rotating, deformed
magnetars could contribute to an overall stochastic gravitational-wave back-
ground (Marassi et al., 2011).
24
1.3 Compact Binary Coalescences
The three endpoints of stellar evolution are white dwarfs, neutron stars,
and stellar mass black holes, listed in order of increasing compactness, depen-
dent on the mass of the progenitor star. Binary systems of compact objects
(herein defined as neutron stars and stellar mass black holes) such as two
neutron stars or two stellar mass black holes orbiting around their common
center of mass have already been observed to emit gravitational waves. There
has been indirect observation in the case of binary pulsars with time-changing
orbits that are consistent with gravitational wave emission (see Section 2.4
for the Hulse-Taylor binary, the first of such systems to be discovered), and
direct observation in the case of all confident mergers listed in GWTC-1: A
Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by
LIGO and Virgo during the First and Second Observing Runs (Abbott et al.,
2018). Because orbital decay is a direct consequence of gravitational-wave
emission, compact objects in these systems eventually inspiral and merge.
During the inspiral phase, gravitational waves are emitted in a highly pre-
dictable manner that can be modeled using various techniques depending on
the masses and compactness of the system involved. Eventually, these objects
reach their final few cycles before merger, which occurs in the sensitivity band
of advanced ground-based interferometric gravitational-wave detectors. For
BNS systems (and low-mass BBH systems to a lesser extent), there can be
up to hundreds and thousands of cycles observed in the Advanced LIGO and
Virgo sensitivity band.
During these last few cycles, it becomes more and more difficult to model
the gravitational-wave emission, especially when neutron stars make up one
or both of the compact objects. Compact systems involving two neutron stars
always tidally disrupt whereas neutron star-black hole binaries might or might
25
not (depending on the mass ratio and spin of the two objects). When tidal
disruption does occur, there are opportunities for multi-messenger observa-
tions. In the case of binary stellar mass black hole systems, multi-messenger
observations are less likely, although not impossible.
Because compact binary coalescences are by far the most common class of
objects detectable by advanced ground-based interferometric detectors, there
is an entire section devoted to waveforms and searches for these events (see
Section 4.1). Here, we will explore various observing scenarios for joint grav-
itational wave, electromagnetic, and neutrino emissions from compact binary
coalescences.
Progenitors
Here, we focus on compact binary coalescences involving only neutron
stars and black holes, which are sources of gravitational waves in the Advanced
LIGO and Virgo sensitivity bands. Therefore, systems of interest are binary
neutron stars (BNS), neutron star-black holes (NS-BH), and binary black holes
(BBH).
As an aside, binary white dwarfs (known as double white dwarfs, DWD)
also emit gravitational waves, albeit in the sensitivity band of space-based in-
terferometric gravitational-wave detectors (0.1 mHz to 100 mHz) (Sathyaprakash
& Schutz, 2009). These systems involve complex mass transfer between the
donor stars and accretors that could result in the systems becoming AM
Canum Venaticorum variable stars (Nather et al., 1981; Tutukov & Yungelson,
1996) or Type Ia supernovae (e.g., Iben & Tutukov (1984); Webbink (1984)).
Thus, DWD systems are also multi-messenger sources that will detected with
the advent of LISA, the Laser Interferometer Space Antenna, the space-based
interferometric gravitational-wave detector in the works.
26
Stellar Mass Black Hole Binaries
It is possible for stellar mass binary black hole (BBH) systems to have
electromagnetic/neutrino counterparts under more exotic circumstances (Ford
et al., 2019). In particular, either disks formed around the remnant black hole
with sufficient mass accretion at late times of the merger will be required, or
dense gaseous environments in order to produce strong electromagnetic and/or
neutrino emissions. The latter case is likely to arise in Active Galactic Nuclei
(AGN), where stellar mass black holes will congregate towards the accretion
disk of the supermassive black hole and merge quickly, with mini-accretion
disks formed around the inspiraling black holes that can power relativistic
outflows.
Neutron Star Binaries
So far, there has been one confirmed observation of a BNS merger by the
Advanced LIGO and Virgo gravitational-wave detectors, GW170817 (Abbott
et al., 2017). Discovery of its electromagnetic counterparts, a short gamma-ray
burst, GRB 170817A, and optical transient, SSS17a/AT 2017gfo, has led to
separate insights about the progenitor of GW170817.
The optical transient, SSS17a/AT 2017gfo, was first identified by Swope
in a nearby (∼40 Mpc) host galaxy, NGC 4993, less than 11 hours after the
time of the neutron star merger. This identification estimated the event to
take place at a projected distance of ∼2 kpc from the galaxy’s center, giving
us the first kinematic constraints on the progenitor system’s natal (supernova)
kicks and size of the pre-supernova semi-major axes of the system (Abbott
et al., 2017b). Thus, progenitor models of the BNS system (starting at the
time of the second supernova, with various pre- and post- second supernova
properties) were evolved forward in time in the context of a model of the host
27
galaxy, and then narrowed down to those systems that became GW170817-like,
by using the physical location of the observed electromagnetic counterpart and
the gravitational-wave inspiral time/frequency as a veto. In the case of NGC
4993, an early-type spheroidal galaxy with few globular clusters (250+750−150), the
progenitor of GW170817 likely evolved as an isolated binary in the galaxy’s
field population over & 1 Gyr before merger.
Prior to observations of GW170817 and its short gamma-ray burst (sGRB)
counterpart, sGRBs (at least those produced by neutron star binaries) were
thought to be powered by collimated, relativistic jets produced during rapid
accretion onto disks formed around the remnant final object—usually a black
hole—at late times of the merger. However, GW170817 showed that these jets
are not as sharply collimated as previously assumed. In general, these electro-
magnetic counterparts are only observable by viewers within the half-opening
angle of the jets.
Observations of the short gamma-ray burst, GRB 170817A, by Fermi/GBM
and INTEGRAL/SPI-ACS (Goldstein et al., 2017; Savchenko et al., 2017),
confirm for the first time that at least of a fraction of short gamma-ray bursts
(whose origins have long been debated) are due to BNS mergers, providing
insight into the energetics and phenomenology of the progenitor system’s final
moments. Before GW170817/GRB 170817A, clues about the origin of short
gamma-ray bursts were given: the first afterglow detections in May to July
2005 of GRBs 050509b (Gehrels et al., 2004, 2005), 050709 (Villasenor et al.,
2005; Fox et al., 2005; Hjorth et al., 2005), and 050724 (Barthelmy et al., 2005;
Berger et al., 2005), revealed that short gamma-ray bursts are not associated
with core-collapse of massive stars (as in the case of long gamma-ray bursts),
occur at cosmological distances in both elliptical and star-forming galaxies,
and have afterglows with lower energy and density scales than long gamma-
28
ray bursts.
Jet−ISM Shock (Afterglow)
Optical (hours−days)
Radio (weeks−years)
Ejecta−ISM Shock
Radio (years)
θobs
GRB
(t ~ 0.1−1 s)
Kilonova
Optical (t ~ 1 day)
θj Merger Ejecta 
Tidal Tail & Disk Wind
v ~ 0.1−0.3 c
BH
Figure 1.5: Possible electromagnetic counterparts to neutron star binaries. In
blue is the cross section of the centrifugally supported disk around the final
black hole, in red is the surrounding circumburst medium, and in yellow is the
more isotropic kilonova. (Figure from Metzger & Berger, 2012)
Thus, the most probable electromagnetic counterparts to a binary neu-
tron star coalescence are the kilonova emission and the kilonova precursor.
During the neutron star merger, a small amount of neutron rich matter is
ejected both dynamically from the neutron star(s) being torn apart and from
remnant disk outflows if a disk is created. The neutrons are then captured
for r -process nucleosynthesis (including Lanthanide production) and radioac-
tively decay for days to weeks long, producing the isotropic kilonova (Li &
29
Paczyński, 1998). However, a small fraction (∼1×10−4M) escapes being cap-
tured through r -process and instead undergoes β-decay, producing the kilonova
precursor which occurs a few hours after the merger (Metzger et al., 2015).
Figure 1.6: A possible blue bump in the observed kilonova emission could be
indicative of a long-lived hypermassive neutron star phase where the electron
fraction is raised to a high enough value (Ye ∼ 0.4) that no Lanthanides are
produced. On the other hand, if the remnant black hole is formed promptly
after the merger, both the dynamical ejecta before the merger and disk outflows
after the merger will be highly neutron rich (Ye < 0.1) generating a kilonova
emission that peaks later and in the near-infrared. (Figure from Metzger &
Fernández, 2014)
Both the kilonova and the kilonova precursor are important counterparts
because they can encode information to discern the nature of the progenitor,
i.e., confirm whether it is a binary neutron star or neutron star-black hole
merger, or whether the merger went through a brief period where a hyper-
massive neutron star (HMNS) was created before collapsing into the remnant
black hole (Metzger & Fernández, 2014). For instance, the ∼week long kilo-
nova emission with a spectral peak in the near-infrared (NIR) could be indica-
30
tive of prompt (. 100 ms) remnant black hole creation and the production of
Lanthanides whereas a shorter-lived ∼day long emission with a bluer optical
peak before the late NIR peak could be indicative of a HMNS phase with
Lanthanide-free outflows (Figure 1.6).
Constraining the Nuclear Equation of State
Both core-collapse supernovae and neutron star mergers (Section 1.3) in-
volve the same rich physics: they both require general relativity, the strong
and weak interactions, fluid and magnetohydrodynamics, and transport the-
ory. One way a neutron star merger detection could aid the supernova theorists
is by narrowing down the correct nuclear equation of state from a number of
contenders.
The equation of state strongly determines the stiffness of the inner core
and the species of nuclides at the time of bounce (Togashi et al., 2017), which
ultimately affects the final mass of the proto-neutron star. This is turn de-
termines the amount and rate at which neutrinos revive the shock wave. In
general, studies have shown that increasing the proto-neutron star mass in-
creases the average neutrino luminosity and energy, as well as the evolutionary
timescale (Pons et al., 1999).
To date, the two most massive neutron stars were observed in binary
pulsar systems. They reveal masses of 1.97± 0.04 M from PSR J1614−2230
(Demorest et al., 2010) and 2.01±0.04 M from PSR J0348+0432 (Antoniadis
et al., 2013). These two masses put constraints on the viable mass-radius
curves from different equations of state in the neutron star mass vs. radius
parameter space. In particular, they rule out a large range of soft hadronic,
mixed hadronic-exotic, and strange-quark matter equations of state (Lattimer
& Prakash, 2011; Özel et al., 2010).
31
As of yet, there have not been any robust neutron star radius (or mass
plus radius) measurements to further constrain the parameter space (Lattimer,
2012; Miller, 2013; Miller & Lamb, 2016). However, radius measurements could
be determined by gravitational-wave detections. For instance, the combina-
tion of gravitational-wave and electromagnetic data from GW170817 and its
kilonova has already led to the binary’s neutron star radii estimates (Abbott
et al., 2018). It is also possible that signals from binary neutron star mergers
(Subsection 1.3), specifically the peak frequency of the post-merger emission,
could reveal the neutron star radius in the case of symmetric mass binaries
(Bauswein & Janka, 2012). For neutron star-black hole mergers, tidal defor-
mation and neutron star disruption could reveal the neutron star radius as
well (Lattimer, 2012).
32
Chapter 2
Gravitational Waves
2.1 General Theory of Relativity in a Nutshell
In the opening chapter of the book Gravitation (Misner et al., 1973),
Einstein’s geometric view of gravity is summarized in two sentences: “Space
acts on matter, telling it how to move. In turn, matter reacts back on space,
telling it how to curve.”
Mathematically, the first statement, “Space acts on matter...”, is known
as the geodesic equation and can be written as:
d2xρ dxµ dxν
+ Γρ = 0.
dλ2 µν dλ dλ
It is the statement that objects free of all forces, move in straight lines locally.
The second statement, “In turn, matter reacts back on space...” , is known
as Einstein’s field equations1 and can be written as:
− 1 8πGRµν gµνR = T4 µν .2 c
Now, there are a lot of scary looking symbols in both equations above
but most of them can ultimately be written in terms of the key players: space-
time (time as x0 or t plus three spatial coordinates x1, x2, x3, or xi) and
mass/energy. The symbols that cannot be reduced to these are Newton’s
gravitational constant (G) and the speed of light in vacuum (c).
1There are 6 truly independent equations in Einstein’s field equations: 10 independent
equations from equating symmetric rank (0,2) tensors minus 4 constraints on Rµν from the
Bianchi identity.
33
When we calculate the differential separation between two points in space-
time, we use the line element:
ds2 = g µµνdx dx
ν ,
which gives us the metric tensor2, gµν . It has a related inverse metric, gµν ,
such that 1 when µ = ρ
gµνgνρ = gρνg
νµ = δµρ =  ,0 when µ =6 ρ
where the Einstein summation convention3 is used. The inverse metric and
metric come in handy when raising and lowering indices of a given tensor.
Fundamental quantities can be computed using the metric tensor. For
example, the proper time of an∫ob√ject following a time-like path, τ , is:
dxµ dxν
∆τ = −gµν dλ,
dλ dλ
where λ is any affine parameter, i.e., it satisfies λ = aτ + b for some constants
a and b.
The Christoffel symbols4, Γρµν , which appear in the geodesic equation are
2A rank (k, l) tensor is a real-valued, linear function of k vectors and l dual vectors.
3In the Einstein summation convention, repeated upper and lower indices are summed
over. Greek letters include both spatial and temporal coordinates while Latin letters (ex-
cluding t for time) include only spatial coordinates.
4Christoffel symbols allow us to correctly switch between coordinate systems when de-
scribing physical objects and their rates of change. Fundamentally, all physical quantities
and laws must take a geometric form, i.e., one that is free of any coordinate system or basis
vectors. Thus, a vector V~ , for example, can be expressed in one coordinate system with
components V α and basis vectors êα or equivalently, in another coordinate system with
components V β and basis vectors êβ . Its derivative has the relation:
∂V~ ∂V α α ∂êα= êα + V ,
∂xβ ∂xβ ∂xβ
where in the last term, ∂êα/∂xβ is itself a vector that can be written as a linear combination
of basis vectors:
∂êα
= Γµαβ êµ.
∂xβ
Then, a little substitution and renaming indices gives us an important result, the covariant
34
defined as:
Γρ
1 ρσ
µν = g (∂µgνσ + ∂νgσµ − ∂σgµν),2
where ∂µ is shorthand for: ( )
∂ 1
∂µ = = ∂ , ∂ .
∂xµ
t i
c
Thus, we have so far understood all of the symbols that appear in the
first statement, “Space acts on matter, telling it how to move” via the geodesic
equation.
In the Einstein field equations, the two symbols Rµν and R that appear
are related to the Christoffel symbol via the Riemann tensor:
Rµνρσ = ∂ρΓ
µ
νσ − ∂σΓµνρ + Γµ ααρΓνσ − ΓµασΓανρ,
which gives a local description of space-time curvature at each point. It gives
us the change δV µ in the vector V µ, if we move it around a closed loop via
parallel transport5 first along δa êσ, then δb êρ, then −δa êσ, and −δb êρ:
δV µ = δa δb Rµ ννρσV .
Then, the Ricci tensor, Rµν , is just a contraction of the first and third indices:
R αµν = R µαν ,
derivative:
∂V~ ∂V α
= ê α
∂êα
β β α
+ V
∂x ∂x ∂xβ
∂V α
= ê + V αα Γ
µ
β αβ
êµ
∂x
∂V α
= ( ê + V µα Γαµβ êα∂xβ )
∂V α
= + V µΓαµβ êα.
∂xβ
5The parallel transport of vector V~ along a curve U~ means V~ is defined at every point
along the curve and requires V~ to be parallel and of equal length at infinitesimally close
points along the curve.
35
and the curvature scalar, R, is its trace:
R = Rµ = gµνµ Rµν .
Now, let us factor in the presence of mass and energy that curves space-
time. The symbol T µν is the energy-momentum tensor (also known as the
stress-energy tensor). It represents everything we need to know about the
energy-like aspects of the system. This includes the energy density (T 00), the
momentum densities (T 0i), and the stresses (spatial terms T ij, of which the
diagonal terms, T ii, are known as pressure).
All together, this summarizes the second statement, “In turn, matter
reacts back on space, telling it how to curve.” Thus, in the General Theory
of Relativity, gravitation is a geometric effect that arises from objects trying
to move along straight lines locally in an overall curved space-time, where
space-time is curved because of the presence of mass and energy.
One of the consequences of this theory, as shall be seen in Section 2.2, is
that changes in gravity (i.e., changes in the curvature of space-time) do not
spread instantaneously throughout the Universe. Instead, they travel at the
vacuum speed of light, c, in the form of gravitational waves6.
2.2 Gravitational Waves
We can analyze the propagation of gravitational waves, which are changes
to the curvature of space-time/gravitational field, in two spatial domains: in
the near zone and the wave zone. These are defined by the following scaling
6This statement will be clarified in Section 2.3 to distinguish between gravitational
potentials and traditional gravitational waves.
36
quantities:
tc := characteristic time scale of the source, i.e.,
:= time required for noticeable changes to
:= occur within the source,
2π
ωc := = characteristic frequency of the source,
tc
2πc
λc := = ctc = characteristic wavelength of the
ωc
:= gravitational waves,
where they are separated because (1) the difference between τ = t − r/c and
t is small in the near zone versus large in the wave zone (i.e., field retardation
is unimportant versus important) and (2) time derivatives are small compared
with spatial derivatives (multiplied by a factor of c) in the near zone versus of
order unity in the wave zone.
Then, the near zone and wave zone are defined as:
near zone,N : r  λc
wave zone,W : r  λc
where the general solution(to the wave)equation of form
∇2 − 1 ∂
2
ψ = −4πµ
c2 ∂t2
is ∫
µ(t− |x− x′|/c,x′)
ψ(t,x) = 3 ′| − d x ,x x′|
using the retarded Green’s function. Here, the full set of Einstein field equa-
tions has been converted into a wave equation plus the harmonic gauge con-
dition, i.e., the ‘relax(ed’ field equa)tions:
∇2 − 1 ∂
2
αβ 16πGh = − ταβ and
c2 ∂t2 c4
∂βh
αβ = 0,
37
where hαβ are the gravitational potentials and ταβ is the effective energy-
momentum pseudotensor. We drop the indices for simplicity to write down
the general solutions and then evaluate in the near zone and wave zone limits
to give:
ψ = ψN + ψW
for
∑∞ (− [ ∫ ]1)l 1
ψN (x) = ∂L µ(τ,x′)x′Ld3x′ ,
l! r
l=0 M
n〈L〉
[∫ R ∫ ∞ ]
ψW (x) = dsf(τ − 2s/c)A(s, r) + dsf(τ − 2s/c)B(s, r)
r 0 R
when x is in the wave z(one,)an∫d∑∞ (−1)l l∂
ψN (x) =
′
[∫ µ(t,x )|x− x
′|l−1d3x′,
l!cl ∂t
l=0 M
n〈 〉 R
∫ ∞ ]L
ψW (x) = dsf(τ − 2s/c)A(s, r) + dsf(τ − 2s/c)B(s, r)
r R−r R
when x is in the near zone.
In the solutions above, τ = t − r/c is retarded time, M is a surface of
constant time bounded by the sphere r′ = R, L is a multi-index that contains
l individual spatial indices, and µ is assumed to be of the form
1 f(τ)
µ(x) = n〈L〉
4π rm
where n is the radial unit vector x/r, and n〈L〉 is the corresponding STF
(symmetric trace-free) tensor:
∑[l/2]
n〈j j ...j 〉 − p l!(2l − 2p− 1)!!1 2 l = ( 1)
(l − 2p)!(2l − 1)!!(2p)!!
p=0
× δ(j1j2δj3j4 · · · δj2p−1j2pnj2p+1nj2p+2 · · ·njl)
38
in which [l/2] is the largest integer not larger than l/2, and the round brackets
indicate symmetrization7.
Lastly, A(s, r) and B(s, r) are:∫ r+s Pl(ξ)
A(s, r) = ∫ ′′ − dr ,r (m 1)Rr+s Pl(ξ)
B(s, r) = ′ dr
′
r (m−1)s
for ξ = (r + 2s)/r − 2s(r + s)/(rr′) where Pl are the Legendre polynomials:
1 dl
Pl(µ) = (µ
2 − 1)l.
2ll! dµl
So, a very good question one might ask at this point is, “Why did we do
all this?” The answer is, we did it for the sake of completeness. In practicality,
when solving for gravitational-wave signals that are characteristic of neutron
star and/or black hole coalescences for instance, one approach is to integrate
the wave equation in iterations, using an updated expression for the source
that must satisfy the harmonic gauge condition each iteration.
For our purposes of deriving basic properties of gravitational waves and
their effects on freely falling objects in the very far-away wave zone, we can
study plane wave solutions to the wave equation in vacuum. This neglects
corrections of order λc/|x|  1 which turn out to be very small. For example,
a binary neutron star or black hole merger with a characteristic frequency near
100 Hz (λc ∼ 3000 km) at a distance of 100 Mpc (∼3×1021 km) will have terms
we neglect of order 10−18.
7A symmetrized rank-q tensor is defined by:
C(k1k2···kq)
1
= (Ck1k2···kq + · · · ),
q!
where the remaining · · · terms consist of all possible permutations of the q indices.
39
Thus, plane wave solu(tions to the v)acuum relaxed field equations:
2
∇2 − 1 ∂ hαβ = 0
c2 ∂t2
are:
h00 = 0,
h0j = 0,
hjk
G
= Ajk| | TT(τ,x/|x|).c4 x
The TT subscript in the hjk expression means the gravitational potentials
are evaluated in the transverse trace-free gauge, which is a specialization of the
harmonic gauge condition. It indicates that the two independent components
ofAjkTT must contain all the radiative degrees of freedom regarding gravitational
fields. This can be represented inmatrix form as:
Ajk =  A+ A× 0 TT  A× −A+ 0 
0 0 0
in the case of a wave traveling in the ẑ-direction. The two degrees of free-
dom are referred to as the plus (+) and cross (×) polarizations to describe
their effect on a circular ring of freely falling particles lying in the x-y plane
(Figure 2.1).
To see this, we consider the geodesic deviation equation:
D2ξα
= −Rα ββγδu ξγuδ,
dt2
which describes the separation between two nearby freely falling masses, ξα,
each moving along their respective geodesics with four-velocities uα. Here,
D/dt is the covariant derivative along the direction of uα. Then, in the trans-
40
Figure 2.1: A circular ring of freely falling masses distorted by + (blue, top)
and × (red, bottom) polarizations of a gravitational wave propagating out of
the page. A rotation of 45◦ takes a + mode into a × mode and vice versa. For
the figure, a complete wave cycle is shown from left to right.
verse trace-free gauge for slowly moving masses, the geodesic deviation equa-
tion becomes:
d2ξj G
= | |(∂ττA
jk )ξ
dt2 2c4 x TT k
which can be integrated to give us the solution:
ξj
G
(t) = ξj(0) + | |AjkTT(t)ξk(0)2c4 x
 ∆x(t) = ∆x G0 + 4|x|(A+(t)∆x0 + A×(t)∆y0),2c
=  ∆y(t) = ∆y
G
0 + 4|x|(A×(t)∆x0 − A+(t)∆y0), 2c ∆z(t) = ∆z0.
Thus, a purely + polarization gives us a distortion of the ring into an
ellipse: ( )2 ( )2
∆x ∆y
+ − = (∆x
2 2
0) + (∆y0) ,
1 + η+(t) 1 η+(t)
and a purely (× polarizat)ion give(s us a disto)rtion of the ring into an ellipse:2 2
1 ∆x+ ∆y 1 ∆x−∆y
+ 2 2
2 1 + η×(t) 2 1−
= (∆x0) + (∆y0) ,
η×(t)
where η (t) = 1(G/c4+ |x|)A+(t) and η×(t) = 1(G/c4|x|)A×(t).2 2
41
However, the positions of freely falling masses initially at rest do not
change. To see this, we can take the geodesic equation for a freely falling mass
where dxi/dτ = 0 at τ = 0:∣ [ ]
d2xi ∣∣∣ − i dxµ dxν= Γdτ 2 µντ=0 [ (dτ d)τ ]τ=0
0 2
= − dxΓi00 .dτ
τ=0
Then, for a metric gµν = ηµν + hµν , the Christoffel symbols are:
ρ 1Γµν = η
ρσ(∂µhνσ + ∂νhσµ − ∂σhµν).
2
Thus,
∣∣∣ [∣ ( )
]
d2 i 0
2
x dx
= − Γi
dτ 2 00τ=0 [ dτ τ=0 ( ) ]
0 2
= − 1(2∂0h0i −
dx
∂ih00)
2 dτ
τ=0
0,
because h00 and h0j are both taken to be zero in the transverse trace-free
gauge. Thus, dxi/dτ must remain 0 at all times if the freely falling masses are
initially at rest (dxi/dτ |τ=0 = 0) and d2xi/dτ 2 = 0.
This means we can think of marking the coordinates of the transverse
trace-free gauge with freely falling masses. The positions of the masses (ini-
tially at rest) do not change although the coordinates are stretched and squeezed
by incident gravitational waves. This interaction will allow us to conceptually
build a gravitational-wave detector in Chapter 3.
2.3 Sources of Gravitational Waves
Although several astrophysical examples of gravitational-wave sources
were mentioned in Chapter 1, we now explore more generally how gravita-
tional waves are generated.
42
The first important result is known as Birkhoff’s Theorem in the Gen-
eral Theory of Relativity. It states that space-time outside a spherical, non-
rotating body (i.e., the spherically-symmetric solution to Einstein’s field equa-
tions in vacuum() is sta)tic and giv(en by th)e Schwarzschild external metric:−1
R R
ds2 = − 1− d(ct)2 + 1− dr2 + r2(dθ2 + sin2 θ dφ2),
r r
where R is the Schwarzschild radius, R = 2GM/c2, describing the body. This
emphatically means spherically-symmetric matter distributions do not emit
gravitational waves. Thus, for example, a perfectly spherically-symmetric core-
collapse supernova would not generate any gravitational waves, despite large
accelerations of mass and energy during the event.
This begs the question, “What kind of mass and energy motions pro-
duce gravitational waves?”. The intuitive answer is non-spherically symmetric
accelerations of mass and energy produce gravitational waves.
In the very far-away wave zone from perfect fluids moving in slow-motion
(|x|  λc), exact solut(ions to the re)laxed field equations:
1 ∂2∇2 − αβ −16πGh = ταβ
c2 ∂t2 c4
∂ hαββ = 0,
reduce to the gravitational pote[ntials: ]
00 4G 1h = | | [M + Ïjknj]nk + . . . ,c2 x 2c2
0j 4G 1h = [ Ïjk3| | nx k + . . . ,c 2c ]
jk 4G 1h = | | Ï
jk + . . .
c4 x 2
where M is the total gravitational mass of the source, nj = xj/|x|, and over-
head dots represent differentiation with respect to retarded time, t − |x|/c.
43
The lowest order time dependent parts of hαβ are dominated by changes in
the mass quadrupole moment∫, Ijk:
Ijk(t− r/c) = c−2 τ 00(t− r/c,x′)x′jx′kd3x′ +O(c−2).
Thus, at the end of Section 2.1 where I stated, “...changes in gravity
(i.e., changes in the curvature of space-time) do not spread instantaneously
throughout the Universe. Instead, they travel at the vacuum speed of light, c,
in the form of gravitational waves.”, we can see that in the case of an object
accelerating rectilinearly, the term h00 = 4GM/c2|x| is the only “gravitational
wave” that reflects the changing curvature at location x. The aforementioned
gravitational-wave polarizations and what we traditionally refer to as gravi-
tational waves are the radiative parts of the above potentials, of which the
lowest order term can be extracted in the transverse trace-free gauge, to give
the quadrupole formula:
jk 2GhTT = Ï
jk
c4R TT
,
where ∫
Ijk(t− r/c) = c−2 τ 00(t− r/c,x′)x′jx′kd3x′.
With the quadrupole formula, we can investigate gravitational waves gen-
erated by binary systems, deformed rotating neutron stars, ‘mountains’ on an
otherwise spherically symmetric neutron star, and more. We can also find a
ballpark estimate for the gravitational-wave amplitude, h0, due to a source of
mass M confined to a volume of radius rc, with changes on a characteristic
time scale tc. Then, the quadrupole moment scales as Mr2c and Ïjk is of order
M(r 2c/tc) ∼ Mv2c . This gi
GM (
ves
v )
us:
2
∼ ch0
c2R c ( )( )( )2
∼ 3.2× 10−19 M 1.5 Mpc vc ,
10M R c
44
where 10M and 1.5 Mpc (the approximate size of the Local Group of galaxies)
are used as reference. This shows that even the most violent and energetic
processes in the Universe still produce tiny gravitational waves by the time
they reach us.
2.4 The Hulse-Taylor Binary
This section presents the first indirect evidence of gravitational waves
through observation of orbital decay in the Hulse-Taylor binary pulsar system.
Readers who would like to skip ahead to learn more about advanced ground-
based interferometric gravitational-wave detectors are advised to do so (see
Chapter 3).
Pulsars were first discovered in the 1960s as point-sources emitting elec-
tromagnetic radiation in the radio band (Hewish et al., 1968). They are highly
magnetized, rotating neutron stars with periods of the order τ ∼ 10−3 to 1
seconds. Since their original detection, over two thousand pulsars have been
discovered and they are now known to emit radiation in the radio, optical,
X-ray, and/or gamma-ray wavelengths (Lorimer, 2008). The period of most
pulsars increases slowly with time and in a very regular manner. This pre-
dictability of the pulse time arrivals means pulsars can be used as clocks, on
par with the most accurate man-made clocks.
In the case of the pulsar in the Hulse-Taylor binary system, the pulsar
rotates on its axis approximately 17 times per second (Taylor & Weisberg,
1982) so the pulsation period τ is 1/17 = 0.059 s with an angular period of
2π 2π
ω = = = 106.5 s−1.
τ 0.059 s
The pulsar’s measured period derivative is
dτ
= 8.62× 10−18
dt
45
or
dω −2π dτ − 2π= = × 8.62× 10−18
dt τ 2 dt (0.059 s)2
= −1.55× 10−14 s−2.
Furthermore, the upper limit on the time-averaged X-ray and optical
flux from the pulsar region is of the order 10−10 erg s−1 cm−2 (Davidsen et al.,
1975), meaning the X-ray and optical luminosity of the pulsar region has an
upper limit of
17
L = 10−10
erg 31× 10 cm
tot × (6400 pc)2 × ( )2cm2s 1 pc
≈ 4× 1034 erg s−1,
where 6400 pc is the distance to the binary.
The source of this luminosity is mostly in the form of synchrotron radia-
tion, i.e., the radiation emitted by relativistic electrons as they spiral around
magnetic field lines.
Pulsars Are Neutron Stars
To see that pulsars are spinning neutron stars, we will consider three
possible mechanisms for producing periodicity of the observed magnitude and
regularity: binaries, stellar pulsation, and stellar rotations.
For binaries, Kepler’s law relates the angular frequency, masses, and sep-
aration distance as
G(M1 +M2)
ω2 = ,
a3
46
and therefore we get
(G(M +M ))1/31 2
a =
ω2/3
(6.7× 10−11 m3 30kg s2 × (4× 10 kg))1/3
=
(106.5 s−1)2/3
≈ 300 km,
if we assume two solar-mass objects and use the pulsar’s frequency. This gives
us a separation, a, which is much smaller than the radii of normal stars (∼105
km) or white dwarfs (∼103 km). Only a pair of neutron stars could exist in a
binary at the above scale of separation. However, if two neutron stars were or-
biting each other at such a close distance, the system would lose gravitational
binding energy via the emission of gravitational waves, causing the separation
distance to shrink and orbital frequency to grow, which contradicts our ob-
servation that pulsar frequencies decrease with time. Indeed, this is because
pulsars have other mechanisms to lose spin energy, e.g., via magnetic dipole
radiation. Thus, we can conclude that pulsars cannot be explained by or-
bital motion of stellar-mass objects, with the exception of neutron stars under
special circumstances involving large magnetic fields to account for observed
spin-downs.
Next, stars are observed to pulsate regularly in various modes, with the
pulsation period dependent on density as τ ∝ ρ−1/2. Normal stars have pul-
sation periods between hours and months and white dwarfs have pulsation
periods of 100 to 1000 s. Neutron stars are about 108 times denser than
white dwarfs, and should therefore have periods 104 times shorter than white
dwarfs, putting them in the range 0.01 to 0.1 s. However, the most common
period for pulsars is about 0.8 s, which is just outside the predicted pulsation
range. Therefore, we rule out stellar pulsations as an explanation for pulsar
periodicity.
47
And finally, we consider anisotropic emission from a rotating star as an
explanation for pulsar periodicity. First, we can work out what the maximum
mean density of the pulsar must be when it is spinning as fast as it can without
breaking apart due to centrifugal forces:
GMm 2
> mω2 ⇒ M ωr > ,
r2 r3 G
and therefore,
3M 3ω2 3(106.5 s−1)2
ρ̄ = > =
4πr3 4πG 34π × 6.7× 10−11 mkg s2
= 4× 1013 kg m−3
for the pulsar in the Hulse-Taylor binary. This means if the pulsar is a spinning
star and is not flying apart, it cannot be a white dwarf whose mean density
is four to five orders of magnitude smaller, ∼109 kg m−3. Also, pulsars with
the shortest periods of about 1 ms compared to our 59 ms, must have mean
densities about 3000 times larger to avoid breaking apart, ∼1017 kg m−3, which
is the mean density predicted for neutron stars.
Thus, it is accepted that pulsars are neutron stars. Their spin rate is what
would be expected from core-collapse of massive main-sequence stars down to
neutron star dimensions. For millisecond pulsars, it is believed they are spun
up by accretion after the neutron star formation. Also, loss of rotational energy
and decreasing pulsar frequency over time can be explained by the outgoing
radiation from the pulsar and surrounding nebula.
Pulsar Emission Mechanism and Age
Although the exact details of pulsar emission mechanism are still debated
and an active area of research, it is widely accepted that periodic emission from
48
Figure 2.2: Schematic model of a pulsar. (Figure from Lorimer & Kramer,
2004)
pulsars is due to misalignment of the magnetic field axis and the star’s rota-
tion axis by some angle θ. This misalignment has the structure of a rotating
magnetic dipole, as seen in Figure 2.2.
Then, we know that a spinning magnetic dipole emits electromagnetic
49
radiation with luminosity
L ∼ B2r6ω4 sin2 θ
where B is the magnetic field on the surface of the star, at a radius r along
the magnetic field axis. This is the basic emission mechanism that is held
responsible for the pulsar’s loss of rotational energy. Then,
dErot dω
= Iω ∝ ω4
dt dt
and
dω ∝ ω3 = Cω3
dt
for some constant C, which can be determined from present-day values of
dω/dt and ω as
ω̇0
C = .
ω30
This means that we can obtain an upper limit on the age of the pulsar
by integrating the equation as follows:( )
ω30 1 1tpulsar = − ,
2ω̇ ω2 ω20 i
where ωi is the initial angular frequency of the neutron star upon formation.
Therefore, the upper limit on the age of the Hulse-Taylor pulsar is obtained
by taking ω = ω0 and ωi =∞:
ω0 106.5 s−1
tpulsar < | | = × × − − = 3.4× 10
15 s
2 ω̇0 2 1.55 10 14 s 2
= 1× 108 years.
Now, if we call ρ the distance from the rotation axis in cylindrical dis-
tance, magnetic field lines that reach a distance ρ > c/ω must open and escape
to infinity. This is because c/ω is the maximum distance at which an object
50
can co-rotate with the star and not exceed the speed of light. Only the field
lines fully contained within that maximum distance are closed, as shown in
Figure 2.2.
Then, high-energy charged particles accelerate around the open field lines
and emit electromagnetic radiation, most easily observed in the radio wave-
lengths. Because the open field lines are concentrated near the magnetic poles,
the radiation forms two conical beams centered along the magnetic field axis,
as shown in Figure 2.2. Then, as the pulsar spins on its rotation axis, the
beams sweep out an annulus in the sky and distant observers detect a pulse
once every rotation if they happen to lie along the path of the beams.
Timing with Pulsar Profiles
Each pulse from a pulsar has a unique shape. It is only when many such
pulses are added up that the pulsar’s profile can be built. The intensity and
shape of the profile is frequency dependent. In Figure 2.3, we have the pulse
profile for the pulsar in the Hulse-Taylor binary at 430 MHz.
Given that a millisecond pulsar will rotate over one million times during
a one hour duration, an error in the rotational period will produce a shift in
the time of arrivals at the end of the hour that is a million times larger. Thus,
if we can measure the time of arrivals to a precision of 1 millisecond, then the
rotational period of the pulsar can be resolved to a precision of 10−12 seconds,
the equivalent of 9 decimal places for millisecond pulsars! This is why pulsars
can be used as very accurate clocks.
However, most pulsars do show departures from simple, uniformly slowing
rotation. One significant departure are glitches in the pulses. These glitches
are thought to be caused by ‘starquakes’, which are sudden changes in the
magnetic field configuration of the pulsar (Franco et al., 2000) due to crust
51
Figure 2.3: Pulse profile at 430 MHz for the pulsar in Hulse-Taylor binary.
Observed during July 1977 (dotted line), June 1978 (dashed line), and October
1978 (solid line). The central component has been gradually moving to the
left and becoming broader, while the third component has shifted to the right.
All profiles have been smoothed to the resolution indicated by the horizontal
bar, 400 µs. (Figure from Taylor et al., 1979)
Figure 2.4: A possible geometry to account for pulse shape changes in the
Hulse-Taylor binary. (Figure from Taylor et al., 1979)
cracking. For most pulsars, the magnetic field seems to increase after each
glitch. Ultimately, the origin is unexplained and still an active area of research.
One possible explanation for the pulse shape changes is the following.
52
The shaded regions in Figure 2.4 represent active portions of a hollow cone of
pulsar emission (the cross-section). The horizontal line in the figure represents
the loci of the line of sight through the beam as the pulsar spins. Therefore,
when the spin axis precesses, different portions of the beam move into the
line of sight and thus the pulse profile would change. After three decades
of observation at Arecibo, it is now assumed that the conal beams might be
hourglass-shaped (Weisberg & Taylor, 2002).
Pulsar Timing Formula for Isolated Pulsars
An observer on Earth records the times, τobs, of the arriving pulses from
an isolated pulsar. Our goal is to translate these times into times that are only
dependent on the intrinsic properties of the source. As recorded, these τobs’s
are affected by Earth’s motion around the Sun, Earth’s spinning on its axis,
the gravitational redshifts of the Earth and the Sun, and the dispersion of the
pulse as it travels through the interstellar medium. If we could subtract out
these effects, we would then be left with times tSSB, which are the coordinate
times at which the pulse recorded on Earth would have arrived at the Solar
System Barycenter (SSB), which is the Solar System’s center of mass. After
we figure out how to do this, we consider the case of a pulsar in a binary.
Roemer Time Delay
The first step to getting the barycentric time of arrivals is to account
for the Roemer time delay. The motion of Earth around the Sun causes a
modulation in the arrival time of the pulses. Because the orbit lies very nearly
in a plane, it can be described with Ω, the angular velocity of Earth around
the Sun, and t0, the time it takes light to travel from the Sun to Earth. And
therefore, if a pulsar is at a latitude β above the plane of the ecliptic, the
53
modulation in the arrival time of the pulses can be written as
4R, = t0 cos(Ωt− λ) cos β.
This would be precise enough classically but for pulsar timing, we need
to account for Earth spinning on its axis, the orbit of Earth being elliptical,
and the Sun moving around the SSB due to Jupiter’s influence. The most
practical way to deal with all of these effects is to account for the corrections
by referring all arrival times for the observer to the SSB. Thus, we need the
vector from the observer to the SSB:
~rob = ~roe + ~res + ~rsb
where ~roe is from the observer to the center of Earth, ~res is from the center of
Earth to the center of the Sun, ~rsb is from the center of the Sun to the SSB.
Then, the time we need to add to the times observed in the laboratory is
4R, = −~rob · n̂/c
where n̂ is the unit vector from the SSB to the pulsar. This is the Roemer
time delay in the solar system.
Shapiro Time Delay
The second step to getting the barycentric time of arrivals is to account
for the Shapiro time delay. The pulse traveling in the vicinity of the Sun will
take slightly longer to get to Earth than if the Sun were not there. To see this,
we start with the space-time interval linearly approximated as
ds2 = −(1 + 2φ(~x))c2dt2 + (1− 2φ(~x))d~x2.
And therefore photons traveling on the light-like geodesic (ds2 = 0) satisfy
to lowest order in φ:
cdt = ±(1− 2φ(~x))|d~x|.
54
This expansion is safe to do because in the solar system, |φ(~x)| = GM/rc2
is at most of order 10−6. For instance, between the Sun and Mercury we get
× 3GM 6.7 10−11 m 30− − kg s2
× 2× 10 kg
φ(~x) = =
rc2 5.8× 1010 m× (3× 108 ms )2
≈ −2.6× 10−8.
Then, we can now calculate the coordinate time difference between the
arrival time, tobs, and the emission time at the pulsar, te. If we locate the fixed
location of the pulsar with ~rp and the location of the observer at tobs as ~rp,
then we get ∫ ~rp
c(tobs − te) = |d~x|(1− 2∫φ(~x))~robs ~rp
= |~rp − ~robs| − 2 |d~x|φ(~x).
~robs
And now using the notation from above:
|~rp − ~robs| = |(~rp − ~rb) + (~rb − ~robs)|
≈ |~rp − ~rb|+ (~rb − ·
(~rp − ~rb)
~robs) |~rp − ~rb|
= |~rp − ~rb|+ (~rb − ~robs) · n̂.
Then, substituting back we get ∫ ~rp
c(tobs − te) ≈ |~rp − ~rb|+ (~rb − ~robs) · n̂− 2 |d~x|φ(~x).
~robs
and writing ~rb − ~ro(bs as ~rob and rea)rranging gives us∫
1 1 2 ~rp
tobs ≈ te + |~rp − ~rb| + ~rob · n̂− |d~x|φ(~x).
c c c ~robs
The terms in the parenthesis is the barycentric time of arrival, tSSB, which
is the time the pulse would have arrived at the SSB if there were no effects
55
from the solar system’s gravity. Therefore, w∫e can get
− 1 2
~rp
tSSB = tobs ~rob · n̂+ |d~x|φ(~x).
c c ~robs
The first correction is the Roemer delay that we found in Subsection 2.4.
The second term is (minus) the solar s∫ystem Shapiro time delay:
2 ~rp4S, = − |d~x|φ(~x)
c ~robs
so that in condensed form we have so far:
tSSB = tobs +4R, −4S,.
If we want to calculate the maximum modulation in time induced by the
Shapiro time delay, we must study the photon whose path from the pulsar to
Earth just grazes the surface of the Sun. The value for 4S, ends up being
positive when calculated which agrees physically with the pulse arriving later
since it must travel through the potential well created by the Sun.
Einstein Time Delay
Next, the Einstein time delay will account for time dilation between the
clock moving with the observer, located at ~xobs, versus the clock at the SSB.
To see this, we start with the relationship between the observer’s proper time
τ versus the coordinate time t:
−c2dτ 2 = −(1 + 2φ(~x ))c2obs dt2 + (1− 2φ(~x 2obs))d~xobs.
Then, we get ( )
dτ 2
1/2
= (1 + 2φ(~xobs))−
1 − d~x(1 2φ(~x )) obs .
dt c2
obs
dt2
56
And now using the notation ~vobs = d~xobs/dt, we have to first order in the
small parameters φ(~xobs) and ~vobs:
dτ v2≈ 1 + φ(~x obsobs)− .
dt 2c2
Integrating, we get ∫ t ( v2 (t′ ))
τ ≈ t+ dt′ φ(~x ′ obsobs(t ))−
2c2
where the lower limit of the integral corresponds to an arbitrary constant shift
in the origin of τ . Therefore, the Einstein time delay can be found from the
relation t ≈ τ +4E as ∫ t ( )v2 (t′4 ′ obs )= dt − φ(~x ′E 2 obs(t ))2c
Physically, the first term is dominant and is mostly due to the motion of
Earth around the Sun and Earth spinning on its axis. Thus, vobs ≈ v⊕. The
second term gives us the gravitational redshift of the observer. In this case,
as the observer moves away from the Sun, the rate at which time passes is
increased relative to the case when the observer is closer to the Sun.
Dispersion in the Interstellar Medium
Pulses coming from the pulsar must travel through ionized interstellar
gas before arriving at the observer. This gas effectively acts as a medium
with index of refraction, n, significantly different from 1, which is the index of
refraction of vacuum. Then, the frequency-dependent expression for the index
of refraction is
vg(ν)
2
n(ν) = ≈ − nee 11
c 2πme ν2
where vg(ν) is the group velocity of the pulse with frequency ν, e and me are
the charge and mass of the electron, and ne is the electron number density.
57
Then, the time for th∫e pulse to tra(vel a dist)ance∫L is given byL dl L e2 1 L≈ + n
2 e
dl.
0 vg c 2πmec ν 0
∫
The quantity L nedl is called the dispersion measure, DM, and is typ-0
ically given in cm−3 pc. By measuring the time of arrivals for different fre-
quency bandwidths, we can find the DM and correct for its effect. This proce-
dure is called de-dispersion and is crucial for pulsar observations. For values
of large enough DM’s, the pulses can be spread out enough that it ends up
being greater than the intrinsic period of the pulsar, making the pulsar un-
observable. In the search for pulsars, the DM is an unknown parameter and
data are de-dispersed with various possible values. In the case of the Hulse-
Taylor binary, the DM is measured to be 169 cm−3 pc (Hulse & Taylor, 1975).
The pulses were discovered near the frequency channel 430 MHz at Arecibo
Observatory, see Figure 2.3.
Thus, if we want to move the time of arrivals to the barycentric system,
we need to subtract out this di(spersion)effect. It can be summarized as
2
4 e 1 Ddisp = DM = .
2πm c ν2 ν2e
Relation to the Intrinsic Pulsar Signal
All of the delays mentioned above are small and therefore can be added
up linearly. Then, the time of arrivals in the barycentric system are related to
the time of arrivals of the observer by:
D
tSSB = τobs − +4E +4R, −42 S,.ν
These times only depend on the intrinsic properties of the pulsar because
we have accounted for the effects from the solar system’s gravitational field
58
and the pulse’s interaction with the interstellar gas. Now, we want to relate
these time of arrivals to the time of emission according to the pulse’s proper
time, T . To do this, we start by defining the accumulated phase of the spinning
pulsar as Φ.
If Φ0 is the angle at which the pulse sweeps across Earth, we will see a
pulse whenever Φ mod 2π ≡ Φ0. Now, because the pulses carry energy away
from the pulsar, the spinning frequency of the pulsar, ν, cannot be a constant.
It can be expanded around some reference value T0 as
1
ν(T ) = ν0 + ν̇ T + ν̈
2
0 0T + . . . ,
2
where ν̇0, ν̈0, etc are called spindown parameters.
Then, the accumulated ph∫ase is given byT
Φ(T ) = 2π dτν(τ)
0
1 2 1= ν0T + ν̇0T + ν̈0T
3 + . . . .
2 3!
Then, emission will take place at proper times Tn such that Φ(Tn) mod 2π ≡
Φ0, i.e., Φ(Tn) = Φ0 + 2πn. Then, the emission proper times, Tn, are given by
1 1 Φ0
ν0Tn + ν̇
2 3
0Tn + ν̈0Tn + · · · = + n.2 3! 2π
Thus, if there were no spindown parameters, the emission times would
be exactly
Φ0 n
Tn = + .
2πν0 ν0
However, spindown parameters do exist and they produce deviations from
these times. The typical dissipation mechanisms for pulsars typically behave
as ν̇ ≈ Cνn, for some constant C and n ∼ 2−3.
59
In the case of the Hulse-Taylor pulsar, ν0 ≈ 16.9 s−1 and ν̇0 ≈ −2.5 ×
10−15 s−2. Thus, from the above model, we can expect ν̈0 ≈ Cnνn−1ν̇ =
Cnνnν̇/ν = nν̇2/ν and therefore ν̈0 ≈ 3×10−31 s−3. This effect is unobservably
small and can be ignored. Then, it is sufficient to keep only up to the ν̇0 term.
The final step is to connect these pulsar proper times of emission, Tn,
with the corresponding coordinate times, tem,n. Once we find this relation, we
can relate these coordinate times of emission to the time of arrivals at the SSB
as tSSB = tem,n + d/c where d is the distance between the pulsar and the SSB.
Further Corrections for Binary Pulsars
We must also correct for the Roemer, Shapiro, and Einstein time delays
for pulsars in binaries. However, because the binary has a much stronger grav-
itational field than the Solar System, each time delay must be treated with the
General Theory of Relativity in mind and becomes technically more difficult.
We will sketch how to do this for each case and discuss other corrections that
come into play.
Einstein Time Delay
The Einstein delay in this case will relate the proper time kept on the
pulsar to the time that is kept at the center of mass system of the binary.
Here, I will sketch how to do this, still using Kepler’s laws as a description of
the pulsar’s trajectory. First, we start by writing the potential at the location
on the pulsar where the beam is emitted, ~x:
− Gmp Gmcφ(~x) = −
c2|~x− ~x | c2p |~x− ~xc|
wheremp, mc, ~xp, ~xc are the masses and locations of the pulsar and companion
star.
60
However, the first term in the potential is time independent since the
location of the emission does not change with respect to the pulsar’s center
of mass. Even though in magnitude it is significant (Gmp/c2rNS ∼ 0.2), we
can absorb it into a constant rescaling of the proper time T . Thus, the time-
dependent part of the Einstein time delay is due entirely to the second term:
Gmc
φ(~x) = − | − | .c2 ~x ~xc
Then, as in Subsection 2.4 on the Einstein time delay, we have
dT − Gm
2
c
= 1 − vp ,
dt c2|~x− ~x | 2c2c
where vp is the the pulsar’s velocity given by the relation
mc
vp = v,
mp +mc
where v is the relative velocity in the center of mass system. It can be found
from the classical relation
1 2 − G(mp +mc) −G(mp +mc)v = .
2 r 2a
Therefore, plugging this back (into our eq)uation we get2
dT − Gmc − 1 m v
2
c
= 1
dt c2|~x− ~x | ( c2c mp +)m2(c 2 )
Gmc G mc mp +mc mp +mc
= 1− − −
c2(r c2 mp +mc r ) 2a
− G mc(mp + 2m ) 1 m
2
c 1
= 1 − c .
c2 mp +mc r mp +mc 2a
And now, we need to find another expression for dT/dt that depends on
other measurable parameters of the orbit. Thus, we recall that the binary can
be described as a Keplerian orbit
u− 2πe sinu = (t− t0),
Pb
61
where t0 is a reference time of passage through the periastron and u is the
eccentric anomaly, r = a(1− e cosu). Then, differentiation gives us
du 2π
(1− e cosu) = ,
dt Pb
and thus
dT du dT 2π 1 dT
= =
dt dt du Pb 1−
.
e cosu du
Then, equating the two versio(ns of dT/dt together gives us )
2π 1 dT 2
− = 1−
G mc(mp + 2mc) 1 − mc 1
Pb 1 e cosu du c2 mp +mc r mp +mc 2a
which can be re-written(as )
2π dT G 2mcmp + 3m
2
= 1− ( cP du c2b 2a(mp +mc) )
G m2− e(cosu 1 + cc2 2a(mp +)mc)
≈ − G 2m
2
( c
mp + 3m
1 c
c2 2a((mp +mc) ))
× − Gmc(mp + 2mc)1 e cosu 1 + .
c2 a(mp +mc)
In the above equation, only the part proportional to cosu produces a
modulation and is observabl(e. And therefore, we r)escale the proper time T :
⇒ − G 2mcmp + 3m
2
T 1 c T.
c2 2a(mp +mc)
Therefore, we end up with
dT Pb
= (1− e cosu)− γ cosu
du 2π
where γ is the Einstein par(amet)er given explicitly by
Pb Gmc(mp + 2mc)
γ = e(2π) c2 a(mp +mc)1/3
P 2/3b G mc(mp + 2mc)
= e ,
2π c2 (m 4/3p +mc)
62
where we eliminated a by using the Keplerian relation G(mp + mc)/a3 =
(2π/P )2b . Then, we can find the Einstein delay by writing T as t −4E. By
using
dT d(t−4E) Pb
= = (1− e cosu)− γ cosu
du du 2π
and
dt Pb
= (1− e cosu)
du 2π
we get
d4E
= γ cosu
du
which means
4E = γ sinu.
In the case of the Hulse-Taylor binary, plugging in observed values of
Pb ≈ 27906 s and e ≈ 0.61(713 ()Ta(ylor & Weis)be(rg, 1982) g)ives us−4/3
≈ mc mp + 2mc mp +mms cγ 2.94 .
M M M
Roemer Time Delay
Because the pulsar travels in an orbit, ~x1(t), around the binary system’s
center of mass, there is a modulation in the pulsar’s location each time it
emits a pulse. Thus, there is a Roemer effect we must compute. As seen in
Subsection 2.4 for the Roemer time delay, the form for it is given by 4R =
−~x1 · n̂/c, where ~x1 is the position of the pulsar in the binary center of mass
system and n̂ is the unit vector from the SSB to the binary center of mass
(along the line of sight). If we define the following variables:
m∗~ ~x1 +m
∗~x2
X = 1 2
m∗1 +m
∗
2
where
m 2Av Gm1m2
m∗A = mA +
A − ,
2c2 2rc2
63
the 1PN8 order equations of motion become d2X~ /dt2 = 0. (We use the 1PN
corrections in this case because numerically, the Roemer time delay is quite
large when we calculate it using classical Kepler’s laws.)
Just as in the Newtonian case, for the 1PN system of equations, we end up
with conservation of total angular momentum, J~, and total energy E. These
conserved quantities allow us to find the first integrals of the equations of
motion more easily and w(e ar)rive with2
dr 2B C D
= A+ + +
dt r r2 r3
and
dψ H I
= + .
dt r2 r3
The coefficients in(the above solutio)ns are
3
A = 2ε (1 + (3ν −
ε
1) ,
2 c2
− ε
)
B = Gm(1 + (7ν 6) ),c2ε G2m2
C = −j2 1 + 2(3ν − 1) + (5ν − 10) ,
c2 c2
2
D = (8(−
GMj
3ν) ,
c2
ε )
H = j 1 + (3ν − 1) ,
c2
− GMjI = (2ν 4)
c2
where ε and ~j are the energy and angular momentum per unit value. Their
8Post-Newtonian (PN) theory is an approximation to the General Theory of Relativity
in the domain of weak fields and slow motion. It combines an expansion in powers of G
(to measure the strength of the field) with an expansion in powers of 1/c2 (to measure the
velocity of the matter distribution) (Poisson & Will, 2014). Thus, 1PN is the lowest-order
post-Newtonian correction that can be applied when the source is non-relativistic (v/c 1),
self-gravitating ((Rs/d)1/2 ∼ v/c), and weakly stressed (|T ij |/T 00 ∼ O(v2/c2)).
64
explicit solutions are given by
1 Gm 4
ε = v2 − +( 3 v(1− 3ν)2 r 8 c2 )
Gm Gm
+ (3 + ν)v2 + ν(r̂ · ~v)2 −
2rc2 r
and ( )
~ 1 v
2 Gm
j = 1 + (1− 3ν) + (3 + ν) ~r × ~v
2 c2 rc2
where ~v is the relative velocity and r̂ = ~r/r.
Now, if we introduce other special variables, we can integrate the first-
derivative equations to find the expression for the orbit in polar coordinates
(r(u), ψ(u)), where ψ(u) is the angle the pulsar is at with respect to the peri-
astron.
Then, the Roemer delay is given by
4R = −~x1 · n̂/c = r(u) sin i sin(ω + ψ(u))
where ω and i are the angles of the periastron and inclination as shown in
Figure 2.5.
Figure 2.5: Geometry of the orbit with orbital parameters. The periastron of
a binary system has been labeled as pericenter here. (Figure from Weisstein,
2018)
Then, we can use the 1PN solutions to end up with
4R = a1 sin i((cosu− er) sinω + (1− e2)1/2θ sinu cosω)
65
where a1 is the semi-major axis of the pulsar and er = (1 + δr)e and eθ =
(1 + δθ)e. Here, δr and δθ have the values
G 3m2p + 6m
2
pmc + 2mc
δr =
c2 a(mp +mc)
G (7/2)m2 2p + 6mpmc + 2mc
δθ = .
c2 a(mp +mc)
Now, besides the Roemer time delay correction, we can also extract more
information about the Hulse-Taylor binary by studying the 1PN solutions.
The solution for ψ(u) will show that the periastron does not advance uniformly
along the orbit but at different rates according to u. Then, the derivative of
ω averaged over the orbit gives us ( )5/3
〈ω̇〉 3 2π 1= (G(m 2/3
2 p
+mc))
c Pb 1− e2
where using the measured values of e and Pb for the Hulse-Taylor binary (Tay-
lor & Weisberg, 1982) gives ( )2/3
〈 〉 mp +mcω̇ = 2.11353 deg/yr.
M
Therefore, if we measured 〈ω̇〉, we would have a way of knowing the total
mass of the binary system.
Other Corrections
Though not covered in this dissertation, there is still a Shapiro time delay
for the pulse as it feels the ‘potential well’ of the companion star. There must
also be corrections due to the loss of energy from the binary as a result of
gravitational-wave emission. This means that the orbit period Pb will ever so
slightly decrease with time. However, because the Hulse-Taylor binary is quite
relativistic, this effect can be and has been measured.
66
Two other corrections include an aberration correction and longitudinal
Doppler shift correction. The aberration correction accounts for the fact that
pulses arrive at Earth from different directions than was emitted as the pulsar
orbits the binary center of mass. The longitudinal Doppler shift correction
accounts for the proper motion of the SSB with respect to the binary center
of mass. Thus, the period of the binary that is observed is not the intrinsic
period of the binary. Then, we must study the relative acceleration of the
SSB and the binary system caused by differential rotation of the Galaxy and
correct the observed orbital period derivative for this Galactic acceleration.
Full Timing Formula and Results
In general, there are five Keplerian parameters that describe the orbital
motion of the pulsar. They are
{Pb, T0, x, e, ω}
where Pb is the orbital period, T0 is the time of passage at periastron, x =
(a/c) sin i is the projected size of the orbit, e is the eccentricity, and ω is the
longitude of periastron, as shown in Figure 2.5. A non-changing Keplerian
orbit is what is predicted by Newtonian gravity.
There are also another eight post-Keplerian parameters which character-
izes the corrections to the orbital motion of the pulsar. They are
{ω̇, γ, Ṗb, r, s, δθ, ė, ẋ}
where the ˙s signify time derivatives of mentioned Keplerian parameters, γ is
the Einstein parameter (Lorentz time dilation), −2/3 −1/3s = x(P /2π) T M2/3m−1b  2 ,
and r = Tm 32 where T ≡ GM/c = 4.925 µs. These post-Keplerian pa-
rameters are independently measurable.
67
And now, assuming that the General Theory of Relativity is correct, all
eight of the post-Keplerian parameters are predicted once we know the value of
the Keplerian parameters and the two masses of the stars in the binary. What
this means is that if we can somehow extract the five Keplerian parameters
and any two of the post-Keplerian parameters from the data, we can find the
massesmp andmc. At this point, we would be able to find all of the other post-
Keplerian parameters. Thus, an accurate fit of the observed time of arrivals
to the timing formula is extremely important.
In the case of the Hulse-Taylor binary, it was possible to extract all five
Keplerian parameters along with three post-Keplerian quantities 〈ω̇〉, γ, and
Ṗb. Then, using 〈ω̇〉 and γ, it was possible to determine mp and mc (see Fig-
ure 2.6) to check if the predicted value of Ṗb matched the corrected9 observed
value of Ṗb, which it did.
Shown in Figures 2.6 and 2.7 are two famous diagrams from nearly three
decades of observation at Arecibo. There is excellent agreement between
the observed and predicted values of binary orbital period decay, Ṗb, due to
gravitational-wave emission. Thus, the Hulse-Taylor binary pulsar is histor-
ically important as giving the first experimental evidence for the existence
of gravitational waves. For this work and “for the discovery of a new type
of pulsar, a discovery that has opened up new possibilities for the study of
gravitation”, Hulse and Taylor were awarded the Nobel Prize in 1993.
9The observed value of Ṗb must be corrected by subtracting out the Galactic acceleration
term, as mentioned previously in Subsection 2.4.
68
C
Figure 2.6: Constraints on the pulsar mass, mp, and companion mass, mc,
from extracted values of 〈ω̇〉 and γ. (Figure from Weisberg & Taylor, 2002)
69
Figure 2.7: Orbital decay of PSR B1913+16. The data points represent mea-
sured orbital phase errors caused by assuming a fixed value of Pb that have been
translated into cumulative shift of periastron time, in seconds. The parabola
is the General Theory of Relativity prediction for the binary emitting gravi-
tational waves. Error bars for data points are too small to see. (Figure from
Weisberg et al., 2010)
70
Chapter 3
Advanced Ground-Based Laser
Interferometric
Gravitational-Wave Detectors
In this chapter I present the advanced ground-based gravitational-wave
detectors, with a focus on the Advanced LIGO instruments. I capture the
basic science of how these instruments operate and how detection is made
possible. However, this requires an understanding of the fundamental and
technical noise sources that must be accounted for and minimized. Thus, I
also provide an in-depth look at the various noise sources.
3.1 A Simple Michelson interferometer
We can now capture the essential physics of how ground-based laser inter-
ferometric gravitational-wave detectors work (Figure 3.1). At the heart of one
of these detectors is a Michelson interferometer, which acts as a transducer to
convert differential displacements between freely-falling test mass mirrors into
an optical signal. We will call ωL, kL = ωL/c, and λL = 2π/kL the frequency,
wavenumber, and wavelength of the laser light1.
Consider the input light electric field in the Michelson interferometer:
E = E ei(−ωin L
t+~kL·~x)
0 . Its reflection off and transmission through a 50-50 beam-
√
splitter is described by the amplitude reflection coefficient, r = 1/ 2, and am-
√
plitude transmission coefficient, t = i/ 2. Thus, light that travels down the
√
x̂-arm of the detector has a field i(E / 2)ei(−ωLt+kLx)0 whereas light reflected
1In practice, both Advanced LIGO and Virgo use pre-stabilized 1064 nm Nd:YAG lasers.
71
Figure 3.1: From top to bottom, aerial views of Advanced LIGO/Hanford,
Advanced LIGO/Livingston, and Advanced Virgo. (Figures from LIGO Lab-
oratory/Virgo)
72
√
into the ŷ-arm has a field (E / 2)ei(−ωLt+kLy)0 . When each field reflects off
the end test mass mirrors located at (Lx, 0) and (0, Ly), the amplitudes are
multiplied by −1. Thus, the fields in the x̂- and ŷ-arms on their return path to
√ √
the beamsplitter are: −i(E / 2)ei(−ωLt+kLLx)0 and −(E0/ 2)ei(−ωLt+kLLy). At
last, when they reflect off and transmit through the beam splitter once more,
we end with a final electric field at the photodetector of:
i
E = − E ei(−ωLt+2k iLLx) − E ei(−ωLt+2kLLy)out 0 [ 02 2 ]
= − i E e−iωLt eikL((Lx+Ly()+(Lx−Ly)) + eikL((Lx+Ly)−(L02 − − − )x
−Ly))
ikL(Lx Ly) ikL(Lx Ly)
= −iE ei(−ωLt+kL(Lx+Ly)) e + e0
2
= −iE ei(−ωLt+kL(Lx+Ly))0 cos(kL(Lx − Ly)).
Thus, the power output at the photodetector is proportional to:
|E |2out = E20 cos2(kL(Lx − Ly)),
and any variation in the detector arm lengths will result in a corresponding
variation in the power output.
3.2 Interaction with Gravitational Waves in the Transverse Trace-
Free Gauge
Independent of the choice of reference frame, the physical effect of incident
gravitational waves on an interferometric detector can be seen by calculating
the (invariant) roundtrip proper times as measured at the beamsplitter of two
photons, one traveling up and down the x̂-arm versus one traveling up and
down the ŷ-arm. This calculation is simplest to do in the transverse trace-free
gauge (introduced in Section 2.2) since the waves take on a very simple form
(only hij 6= 0) and the coordinates of the freely falling mirrors and beamsplitter
do not change.
73
First, we will analyze the situation for a plus-polarized gravitational wave
propagating in the ẑ-direction:
h+(t) = h0 cos(ωgwt).
Then, the space-time interval in this reference frame is:
ds2 = −c2dt2 + (1 + h+(t))dx2 + (1− h 2 2+(t))dy + dz .
Thus, for a photon traveling up and down the x̂-arm starting at time t0, we
have:
0 = ds2 = −c2dt2 + (1 + h 2+(t))dx
dx = ± cdt (1 + h+(t))−1/2
≈ ± 1cdt (1− h+(t)),
2
which ∫leads to:L ∫x t ∫1 t1
dx = c 1− 1 1h+(∫t)dt = −c 1− h+(t)dt0 t 2 20 t2 ∫
c t1− − − − c
t2
Lx = c(t1 t0)
2 ∫ h+(t)dt = c(t2 t1) h+(t)dtt 20 t1
c t2
2Lx = c(t2 − t0)∫− h+(t)dt2 t0t2
t2 −
2Lx 1
t0 = + h+(t)dt
c 2 t0
. Thus, to order O(h0) we ha∫ve:
− 2Lx 1
t0+2Lx/c
t2 t0 = + h0 cos(ωgwt)dt
c 2 t0
2Lx h0
= + (sin(ωgw(t0 + 2Lx(/c))−(sin(ωgwt)0))c 2ωgw )
2Lx Lx sin(ωgwLx/c) Lx
= + h0 cos ωgw t0 +
c c (ωgwLx/c) c
2Lx Lx
= + h(t0 + Lx/c) sinc(ωgwLx/c),
c c
74
where we have used the identity sin(α + 2β)− sinα = 2 sin β cos(α + β).
The same calculation for the ŷ-arm gives:
2Ly Ly
t2 − t0 = − h(t0 + Ly/c) sinc(ωgwLy/c).
c c
Because sinc(x) goes to 1 as x goes to 0, in the scenario where the fre-
quency of the gravitational wave is small, i.e., the period of gravitational wave
is large compared to t1 ≈ Lx/c ≈ Ly/c, the correction to the roundtrip time
becomes adding Lxh(t1)/c or subtracting Lyh(t1)/c. Likewise, because sinc(x)
goes to 0 as x approaches ∞, if ωgwLx/c or ωgwLy/c  1, the correction to
the roundtrip time becomes suppressed.
Now, we wish to find the ideal length of the detector arms. To do this,
we turn back to the total electric field at the photodetector:
i
E = − E ei(−ωLt+2kLLx)+i∆φx(t) − i E ei(−ωLt+2kLLy)+i∆φy(t)out 0 0
2 2
i
= − E e−iωL(t−2Lx/c)+i∆φx(t)0 −
i
E e−iωL(t−2Ly/c)+i∆φy(t)0
2 2
where ∆φx(t) and ∆φy(t) are to order O(h0): ( ( ))
Lx L
∆φx(t) = ωL sinc
x
(ωgwLx/c)h0 cos
c (ωgw(t0 + c))
Lx L
= ωL sinc
x
(ωgwLx/c)h0 cos (ωgw (t−c c ))
Ly Ly
∆φy(t) = −ωL sinc(ωgwLy/c)h0 cos(ωgw t0 +c ( c))
− Ly Ly= ωL sinc(ωgwLy/c)h0 cos ωgw t− .
c c
75
By introducing L = (Lx + Ly)/2 and φ0 = kL(Lx − Ly), we get:
i
E −iωL(t−2Lx/c)+i∆φx(t)
i −iωL(t−2Ly/c)+i∆φy(t)
out = − E0e − E0e
2 2
i
= − E e−iωL(t−(2L+((Lx−Ly))/c)+i∆φx(t) − i E e−iω)L(t−(2L−(Lx−Ly))/c)+i∆φy(t)0 02 2
eiφ0+i∆φx(t)− −iω (t−2L/c) + e
−iφ0+i∆φy(t)
= iE e L0 ( 2 )
eiφ0+i∆φ(t) + e−iφ0−i∆φ(t)≈ −iE e−iωL(t−2L/c)0
2
= −iE e−iωL(t−2L/c)0 cos(φ0 + ∆φ(t))
where ( ( ))
L L
∆φ(t) = ωL sinc(ωgwL/c)h0 cos ωgw t−
c c
= |∆φ(t)| cos(ωgwt+ α).
To maximize the phase difference in the interferometer, we need to max-
imize ∆φ(t). Thus, we focus on the sin(ωgwL/c) term in the sinc(ωgwL/c)
function. Then, the smallest value that L could be is:
πc
L =
2ωgw
c
=
4fgw ( )
100 Hz
= 750 km .
fgw
Thus, detector arm lengths ∼750 km are ideal for detecting gravitational
waves with frequencies ∼100 Hz. This is achieved with the use of Fabry-
Pérot cavities to make the photons bounce back and forth many times in each
arms before recombining. Roughly this equates to O(100) bounces because
Advanced LIGO’s and Advanced Virgo’s arms are 4 km and 3 km long. For
Advanced LIGO, Fabry-Pérot cavities enhance the storage time of light and
sensitivity to a phase shift by factors of F/(2π) and 2F/π, where F = 450 is
the finesse.
76
It is important to note that for each of the reflected electric fields that
make up Eout, the effect of the gravitational wave (to order h0) is to create
sidebands of frequencies ωL ± ωgw, where the modulus of the amplitude is
|∆φ(t)|/2. For instance, if we look at the field in the x̂-arm:
E(x)
i
(t) ≈ − E e−iωL(t−2L/c)eiφ0+i∆φ(t)0
2
− i= E −iωL(t−2L/c)+iφ00e [ (1 + i|∆φ(t)| cos(ωgwt+ α))2 ]
− i i i= E eiβ e−iωLt + |∆φ(t)|eiαe−i(ωL−ωgw)t + |∆φ(t)|e−iαe−i(ωL+ωgw)t0 ,
2 2 2
where β = φ0 + ωL2L/c is an irrelevant constant phase.
3.3 Advanced Detectors
Aside from Fabry-Pérot cavities, ground-based interferometric gravitational-
wave detectors also use power recycling and signal recycling mirrors to enhance
the basic Michelson-Morley setup (Figure 3.2).
Stage 1
Metal masses
Stage 2
Steel wire
Stage 3
Fused silica 
masses
Input test mass Fused silica fibers
Stage 4
Laser Lx = 4 km
End test mass Electrostatic 
actuator Suspension 
System
Output port
Figure 3.2: Advanced LIGO’s Michelson-Morley interferometer with the power
recycling mirror (placed between the laser and beamsplitter), Fabry-Pérot cav-
ities making up the 4 km arms, and signal recycling mirror (placed between the
beamsplitter and photodetector). The test mass setup shows the main chain
side (left) and the reaction chain side (right). (Figure from Abbott et al.,
2016)
Michelson perpendicular arm
Ly = 4 km
1.6 m
77
The signal recycling mirror is placed between the beamsplitter and the
photodetector such that the signal recycling cavity is seen only by the signal
sidebands and the circulating laser power in the interferometer is not affected.
The combination of the signal recycling mirror and each input test mass mirror
can be thought of as an “equivalent input test mass mirror” for the Fabry-Pérot
cavities. Depending on slight adjustments (of order fractions of λL) to the posi-
tion of the signal-recycling mirror, the reflectivities of the equivalent input test
mass mirrors and consequently, the detector response to gravitational waves
varies. Thus, the signal recycling tuning, φ = kLlSRC+π/4, determines the de-
tector mode of operation (Figure 3.3). Here, the length of the signal recycling
cavity length2, lSRC, is the sum of the distance from the beamsplitter to the
signal recycling mirror plus the average of the distances from the beamsplitter
to each Fabry-Pérot cavity.
��
��
��
��
������
�
��
������ ��
��� ϕ� π���
� � � �
�� �� �� ��
��������������
Figure 3.3: Broadband (orange) to narrowband (red) detector responses de-
pending on signal recycling mirror tunings, φ. Dashed/solid lines are for low-
er/upper frequency sidebands. (Figure from Gabriele Vajente, 2018)
When φ = 0, the reflectivity of the equivalent input test mass mirrors is
2For Advanced LIGO, lSRC = 56.0 m (Izumi & Sigg, 2017).
√
�������������� W ���
78
the lowest possible, i.e., the signal recycling cavity is tuned to anti-resonance,
and the Fabry-Pérot cavity bandwidth is increased (the finesse decreased).
This technique, known as resonant sideband extraction (RSE), is shown in
orange in Figure 3.3. The broadband configuration is well-suited for detect-
ing signals from compact binary coalescences where the gravitational-wave
frequencies evolve and are unknown until after they happen.
When φ = π/2, the equivalent input test mass mirror reflectivity is high
and the signal recycling cavity is in resonance with a sideband for a specific
ωgw. The resulting detector bandwidth is decreased (the finesse increased).
This is simply known as the signal recycling mode (SR) and is shown in red in
Figure 3.3. The narrowband configuration is better suited for detecting peri-
odic signals from pulsars, for example, where the gravitational-wave frequency
can be derived from the angular frequency (ωgw = 2ωrot) and is already known.
Next, the power recycling mirror will be discussed in context of the shot
noise. There are two large classes of persistent noise sources for ground-based
interferometric gravitational-wave detectors (Figure 3.4): fundamental noise
(of which there are two sub-classes: displacement versus sensing) and technical
noise (of which there are hundreds). The range referred to in the figure caption
is defined as the sky-averaged distance at which the astrophysical event gives
a matched filter signal-to-noise ratio of 8 in a single detector. In the next
section, we will focus on the fundamental noise sources.
3.4 The Noise Power Spectral Density
The output of Advanced LIGO or Virgo is a time series that describes
the phase shift of laser light, s(t), composed of the gravitational-wave signal
(when present) and noise. If we think of the GW detector as a linear system
black box, an input gravitational-wave signal h(t) produces an output (in the
79
aLIGO new design curve: NSNS (1.4/1.4 M-) 173 Mpc and BHBH (30/30 M-) 1606 Mpc
Quantum
Seismic
Newtonian
Suspension Thermal
Coating Brownian
10-22 Coating Thermo-optic
Substrate Brownian
Excess Gas
Total noise
10-23
10-24
101 102 103
Frequency [Hz]
Figure 3.4: Advanced LIGO’s design sensitivity curve, 1/2Sn (f). The symmetric
binary 1.4 M neutron star and symmetric binary 30 M black hole coales-
cence ranges are 173 Mpc and 1606 Mpc, respectively. (Figure from Barsotti
et al., 2018)
absence of noise):
h̃out(f) = T (f)h̃(f)
in frequency space, where T (f) is the detector transfer function and tildes
represent Fourier transforms.
Similarly, the total noise output is related to a fictitious noise input3 via:
ñ(f) = T−1(f)ñout(f).
Because the noise input varies from one realization of the detector to
the next, n(t) is a random time series that can be characterized by its auto-
3A fictitious noise n(t) injected into the detector black box (with no other noise generated
in the detector) would produce an output noise nout(t).
Strain [1/ Hz]
80
correlation function: ∫ ∞
n ∗ n(τ) = 〈n(t)n(t+ τ)〉 = n(t)n(t+ τ)dt,
−∞
which measures the relatedness of the noise at various time offsets τ to itself.
The Fourier transform of the auto-correlation function gives us the single-sided
power spectral density(PSD), Sn(f), via: ∫2 ∞ −i2πfτ√ −∞ n ∗ n(τ)e dτ if f ≥ 0,2πSn(f) = 0 otherwise.
The interpretation of the noise power spectral density can then be found
by taking the inverse Fourier transform: ∫
1 ∞
n ∗ n(τ) = 〈n(t)n(t+ τ)〉 = S (f)ei2πfτn df
2 −∞
and evaluating at zero time offset: ∫ ∞
〈n2(t)〉 = dfSn(f).
0
Thus, the noise PSD reveals frequency-dependent contributions to the noise.
Optical Read-Out Noise
The general shape of Advanced LIGO’s design sensitivity curve, the am-
plitude spectral density 1/2Sn (f), is determined by the optical read-out noise,
i.e., the quantum noise depicted by the purple line in Figure 3.4. The optical
read-out noise is a fundamental noise source that combines radiation pres-
sure (which dominates at lower gravitational-wave frequencies and manifests
as a displacement noise) with shot noise (which is frequency-independent up
to a pole frequency and manifests as a sensing noise). Their power spectral
densities are combined as follows:
Sn(f)|opt = Sn(f)|shot + Sn(f)|rp
81
The shot noise power spectral density for a simple Michelson-Morley in-
terferometer can be derived as follows. The total power arriving at the pho-
todetector depends on the number of photons that arrive, Nγ, during an ob-
servation time T :
1
P = Nγ~ωL,
T
where ~ωL is the energy per photon and Nγ follows the Poisson distribution:
1 N
p(N ; N̄ ) = N̄ γe−N̄γγ γ γ ,
Nγ!
where N̄γ is the average value of Nγ. Because Nγ  1, the Po√isson distribution
becomes a Gaussian distribution with standard deviation Nγ. Then, the
fluctuation in the power is given by:
1
(∆P ) 1/2shot = (Nγ ~)ωLT
~ 1/2ωL
= P .
T
If there is no gravitational wa(ve, we c)an write this as:
~ 1/2ωL
(∆P )shot = P0 | cosφ0|,
T
where the output power, P , is related to the input power, P0, by P = P0 cos2 φ0
(Section 3.1). However, in the presence of a purely plus-polarized gravitational
wave, the phase is shifted by a factor 2∆φ(t) (Section 3.2), which in the limit
ωgwL/c 1, is:
4πL
2∆φ(t) = h0,
λL
giving us a fluctuation in the output power:
(∆P )gw ≈ P0 sin(2φ0)∆φ(t)
2πL
= P0 sin(2φ0) h0.
λL
82
Then, the signal-to-noise ratio (SNR) can be calculated:
(∆P )
SNR gw=
(∆P )shot ( )−1/2
2πL ~ωL
= P(0 sin()2φ0) h0 P0 | cosφ0|
−1
λL T
1/2
P0T 4πL
= h0| sinφ~ 0|,ωL λL
and written in terms of the amplitu(de spec)tral density using the relation:1/2
T
SNR = h0,
Sn(f)
which applies for optimally oriented periodic signals.
Thus, we arrive at the strain sensi(tivity)due to the shot noise:∣∣ ~ 1/21/2 λL ωL 1Sn (f) shot = ,4πL P0 | sinφ0|
and see that for a simple Michelson-Morley interferometer, indeed, it is in-
dependent of the frequency of the gravitational waves4. Thus, to lower the
shot noise, we need to increase the input laser power. This is achieved with a
power recycling mirror which is placed between the laser and the beamsplit-
ter to reflect light back towards the beamsplitter, increasing the circulating
laser power in the Fabry-Pérot cavities5. During the first Advanced LIGO Ob-
serving Run, O1, only 20 W were injected into the interferometer, which was
amplified up to 100 kW with the use of power recycling mirrors (Abbott et al.,
4In reality, the shot noise amplitude spectral density is frequency-independent up to the
pole frequency, fp = 1/4πτs (where τs is the storage time of the Fabry-Pérot cavities), and
then rises linearly. This frequency dependence arises from the interferometer with Fabry-
Pérot cavities transfer function:
TFP '
8FL√ 1 ,
λL 1 + (fgw/f )2p
which alters the SNR.
5The power recycling cavity length, lp, is the sum of the distance from the beamsplitter
to the power recycling mirror plus the average of the distances from the beamsplitter to
each Fabry-Pérot cavity. For Advanced LIGO, lp = 57.65 m (Izumi & Sigg, 2017).
83
2016). At design sensitivity, the goal is to inject 125 W into the interferometer
and increase the circulating power up to ∼750 kW, lowering the shot noise by
a factor of ∼2.7.
The other contribution to the optical read-out noise is radiation pres-
sure, a displacement noise from photons pushing on the mirrors during each
reflection. A photon of energy Eγ = |p|c changes its momentum from +p to
−p, transferring a total momentum 2|p| = 2Eγ/c to the mirrors. Thus, the
force exerted on the mirrors is F = 2P/c. However, due to fluctuations in the
number of photon arrivals in a laser beam, there is also a fluctuation in the
overall force in a time T , given by:
∆F = 2√∆P/c
~ωLP
= 2 ,
c2T
which is independent of the gravitational-wave frequency.
The power spectral density of the fluctuations in this force, S∆F , can be
calculated from the general relation: ∫
〈 1
∞
∆F (t)∆F (t′)〉 = S (f)ei2πf(t−t′)∆F df,
2 −∞
where we already know that it must be independent of the gravitational-wave
frequency. Thus, in a time T we have:
〈∆F 2(t)〉 1= S∆F ,
2T
resulting in the amplitude spectral den√sity of the fluctuations in the force:
1/2 2~ωLP
S∆F = 2 .c2
84
However, because we are ultimately interested in the displacements, x, in
the mirror, we use the relations:
F (t) = Mẍ,
F̃ (f) = −M(2πf)2x̃,
giving us: √
S1/2
2 2~ωLP
x (f) = .M(2πf)2 c2
For a simple Michelson-Morley interferometer6, the transfer function that
relates ∆L to the gravitational-wave amplitude, h, is just L, thus the amplitude
spectral density from radiation pressure is:
∣ √2 2~ωLP
S1/2 ∣n (f) rp = ,ML(2πf)2 c2
which dominates at lower frequencies. As can be seen, there is a trade-off in in-
√
creasing the laser power to lower the shot noise (∝ 1/ P ) at lower frequencies,
√
because radiation pressure noise is proportional to P .
Seismic Noise
The seismic noise is a displacement noise caused by the continuous motion
of the Earth’s ground. This could be due to natural phenomena (e.g., winds,
earthquakes, waves crashing on the shore, etc.) or human activity (e.g., nearby
traffic, trains, commercial logging, and more). To keep the test mass mirrors as
still as possible, there are both passive and active vibration isolation systems
in place.
The passive isolation system consists of suspending the test mass mirror
within a quadruple pendulum system7. The idea here is that at each stage of
6For Fabry-Pérot cavities, the equivalent length of the detector arms as a simple
Michelson-Morley interferometer is 2FL/π.
7Advanced LIGO uses 40 kg test mass mirrors suspended within a 360 kg quadruple
pendulum system using both steel and fused silica fibers (Figure 3.2).
85
suspension, the motion of the suspended mass is smaller than the motion of
the suspension point above the pendulum resonant frequency. More formally,
a mass suspended from a pendulum with a resonant frequency f0 shaking at
frequencies f  f0, experiences an attenuation in its displacement of factor
(f/f 20) , compared to if it were not suspended from the pendulum. We are only
hoping to target signals at frequencies above 10 Hz because most of the Earth’s
seismic noise occurs between 1−10 Hz. If each pendulum has a resonant fre-
quency of 1 Hz, using a quadruple pendulum system isolates the vibration in
the xy-plane of the test mass mirror ∼108 times at 10 Hz. Consequently, the
seismic noise drops off dramatically above 10 Hz (Figure 3.4).
The active isolation system consists of two parts: an internal seismic iso-
lation platform (ISI) from which the quadruple pendulum system is hung and
the reaction chain side to the test mass setup (Figure 3.2). At low frequencies,
pendulums do not help with vibration isolation and therefore, the seismic iso-
lation platform acts as the first line of defense. Vertical motion is controlled
via springs made with steel cantilevers in the seismic isolation platform. Fi-
nally, electrostatic or coil/magnet actuators on the reaction chain side push
and pull on the test mass mirrors to counteract other ground motions.
Newtonian Noise
The Newtonian noise, also known as the gravity gradient noise, domi-
nates at lower frequencies (. 30 Hz) and is caused by the coupling of Earth’s
changing gravitational field with the test mass mirrors. Motion from all nearby
objects, even atmospheric turbulence, creates a non-negligible contribution to
the Newtonian noise.
86
Suspension Thermal Noise
Suspension thermal noise is caused by thermodynamic interactions be-
tween the pendulums and their surroundings. These interactions cause ther-
mal fluctuations which induce horizontal and vertical motions in the test mass
mirrors, affecting the sensitivity of the detectors up to ∼60 Hz. There are also
the violin modes, depicted by blue spikes starting around 500 Hz in Figure 3.4.
These violin modes are due to fluctuations in the normal modes of the fused
silica fiber wires used to suspend the mirrors.
Coating Thermo-Optic and Brownian/Substrate Brownian Noise
Each test mass mirror has a multi-layered dielectric coating, alternating
between low and high refractive index materials8 to make the mirrors highly
reflective. When the temperature of the mirrors or their surrounding envi-
ronment fluctuates, the coating materials expand or contract, altering their
refractive indices. Thus, thermal noise due to thermal dissipation is known as
the coating thermo-optic noise.
Temperature fluctuations also alter the thickness of the coating material
layers and substrate9, resulting in their shear and bulk losses. Thermal noise
due to mechanical loss is called the coating and substrate Brownian noise.
Excess Gas Noise
Excess gas noise, also known as the residual gas noise, is caused by pho-
tons scattering off any residual gas molecules in the ultra-high vacuum (∼10−7
Pa) pipes that make up the detector arms. Furthermore, the residual gas must
be free of hydrocarbons in order to keep the optical surfaces clean.
8Advanced LIGO uses silica (SiO2), tantala (Ta2O5), and titania-doped tantala (Ta2O5-
TiO2), with refractive indices 1.45, 2.03, and 2.07.
9The substrate is the material beneath the coating that makes up the test mass mirrors.
Advanced LIGO mirror substrates are fused silica; KAGRA will use sapphire.
87
3.5 Interferometer Antenna Response
All gravitational-wave detectors are more like microphones rather than
telescopes, where detector sensitivities depend on propagation directions and
polarizations of incident gravitational waves. Typically, the sensitivity of a
single detector is measured by its horizon distance (i.e., the maximum distance
at which a compact binary GW source creates a maximum fiducial single-
detector SNR, ρ, of 810): √
G5/6M1/3µ1/2
∫
5 f2 f−7/3
dhorizon ≈ df,
c3/2π2/3ρ 6 f S (f)1 n
where G is Newton’s gravitational constant, c is the speed of light, M is
m1 + m2 (the sum of the component masses), µ is the reduced mass, f−7/3 is
the approximate PSD of the inspiral signal, and the integral takes place from
f1 being the low-frequency limit of the detector’s frequency band. However,
the sensitivity of a detector can also be measured by its range, (i.e., the volume
and orientation averaged distance of the source detected with SNR = ρ). It
is a factor of ∼2.26 smaller than the horizon distance due to the detector’s
directional sensitivities (Schutz, 2011) which we will now set out to derive.
In the weak-field limit where gravitational waves can be neatly separated
from the Minkowski background as metric perturbations (Section 2.2), the
waves are solutions to the w(ave equation):
2
∇2 − 1 ∂ hµν = 0,
c2 ∂t2
with hµν as a function of t− ~k′ · ~r/c representing plane waves propagating at
speed c in the k̂′-direction. In general then, we can construct two tensors,←→e +
and←→e ×, from any two unit vectors, î′ and ĵ′, that make an orthonormal triple
10The maximum single-dete√ctor SNR, ρ, is set to 8 to give a root-sum-squared 2-detector
GW network SNR of ρnet = 8 2 ≈ 11.313.
88
with k̂′:
←→e ′+ = î ⊗ î′ − ĵ′ ⊗ ĵ′
←→e = î′ ⊗ ĵ′× + ĵ′ ⊗ î′,
such that
←→
h = h ←→+ e + + h ←→× e ×.
Now, there are two coordinate systems that are ‘natural’ to describe the
situation. The first coordinate system is the gravitational-wave propagation
frame (x′, y′, z′) with k̂′ in the direction of the propagating gravitational wave
(from source towards detector), and two unit vectors, î′ and ĵ′, that make an
orthonormal triple with k̂′. The second coordinate system is in the reference
frame of the detector, (x, y, z), with two unit vectors, î and ĵ, that lie along the
directions of the detector arms with the third unit vector uniquely defined by
the cross product. Then, using the right ascension, declination, the orientation
angle to identify î′ and ĵ′, the coordinates of the detector (latitude, longitude,
elevation), and the orientation angle to identify î and ĵ, we can find the Euler
angles (Figure 3.5), (ϕ, θ, ψ), to switch between the two reference frames.
Once the Euler angles are found, the gravitational-wave strain on the
detector can be computed in a straightforward fashion, assuming that the
light travel time of the photons in the detector arms is short compared to the
period of the gravitational wave11:
←→ ←→
h = h : d
←→
= (h ←→e + h ←→+ + × e ×) : d
= h+F+ + h×F×,
11 ←→ ←→The : notation is for calculating the trace: S : T = S baabT .
89
Figure 3.5: The Euler angles {ϕ, θ, ψ} convert between the detector frame
(x, y, z) and the gravitational-wave propagation frame (x′, y′, z′). For better
visual depiction, the y coordinates have their signs inverted.
←→
where the detector tensor, d , is defined as:
←→ î⊗ î− ĵ ⊗ ĵ
d = ,
2
and F+ and F× are the antenna response patterns (Figure 3.6):
F =←→ ←→ 1 2+ e + : d = (1 + cos θ) cos 2ϕ cos 2ψ + cos θ sin 2ϕ sin 2ψ
2
and
←→ 1
F× =
←→e × : d = − (1 + cos2 θ) cos 2ϕ sin 2ψ − cos θ sin 2ϕ cos 2ψ.
2
To see why this is true, we calculate the time it takes for a photon to travel
(in the detector reference frame) from the beam splitter located spatially at
(0, 0, 0) to a freely-falling mirror at (L0, 0, 0) and back. This requires solving
the equation:
ds2 = −c2dt2 + (1 + h11)(dx1)2 = 0,
90
Figure 3.6: Antenna response patterns for an interferometric detector of the +
(left), × (center), and RMS (right) polarizations, computed with polarization
angle ψ = 0.
which in the limit that the light travel time of the photons is short compared
to the gravitational-wave period (i.e., L0  λGW), gives us:
√
( 1 + h11
∣
dt = ∣ ∣)dx1∣c
≈ 1 |dx
1|
1 + h11
2 c
such that ( )
1 2L0
Troundtrip = 1 + h11
2 c
2L1
= .
c
In other words: ( )
1
L1 = L0 1 + h11 ,
2
or written more generally: ( )
1
Lû = L0 1 + î ·
←→
h · î ,
2
91
which allows us to calculate the difference in roundtrip times down the two
arms. Then, the gravitational-wave strain, h, can be defined as:
Lî − Lĵ
h = (L01 ←→ ←→ )
= ( î · h · î− ĵ · h · ĵ2 ( )1 )a b − a b iaib − jajb= i habi j habj = hab
2 2
←→ î⊗ î− ĵ ⊗ ĵ
= h :
2
←→ ←→
= h : d ,
as shown above.
The derivation of the antenna response patterns, F+ and F×, requires
use of the rotation matrix constructed from the Euler angles that transform
between the two reference frames(Figure 3.5): 
′ 
Ra =  A B C a  D E F  =
 G H I  cosψ cosϕ− cos θ sinϕ sinψ cosψ sinϕ+ cos θ cosϕ sinψ sinψ sin θ  − sinψ cosϕ− cos θ sinφ cosψ − sinψ sinϕ+ cos θ cosϕ cosψ cosψ sin θ  .
sin θ sinφ − sin θ cosϕ cos θ
92
Then, the antenna response patterns are:
F =←→ ←→+ e + : d = e+abdab
= Ra
′Rb′e aba b +a′b′d    

1 
= Tr Rᵀ
1 0 0   1 0 0 
0 −1 0 R 0 −1 02 
0 0 0 0 0 0
A2 −D2 −B2 + E2
=
2
1
= (1 + cos2 θ) cos 2ϕ cos 2ψ + cos θ sin 2ϕ sin 2ψ
2
and
F =←→ ←→e : d = e dab× × ×ab
= Ra
′Rb′e ′ ′daba b ×a b    
0 1 0
1  
 
1 0 0 
= Tr Rᵀ 1 0 0 R 0 −1 0 
2   
0 0 0 0 0 0
= AD − EB
−1= (1 + cos2 θ) cos 2ϕ sin 2ψ − cos θ sin 2ϕ cos 2ψ.
2
In Figure 3.7, the root mean square (RMS) combination of F+ and F×
when ψ = 0 is shown enlarged with the L-shaped arms of the detector drawn
in for reference. Thus, the detectors are most sensitive to gravitational waves
coming from directly above and below, forming the two antipodal regions of
the peanut-shaped antenna pattern. In Figure 3.8, this explains the two bright
yellow regions in each sky localization probability map (skymap).
Next, the detectors are least sensitive to gravitational waves coming from
directions within the plane of the detector arms, especially along the diagonals.
93
Figure 3.7: The RMS combination of the + and × polarization antenna re-
sponse patterns when the polarization angle, ψ = 0. The detector arms are
drawn in for reference.
This is because the arms would be stretched and squeezed equally by passing
gravitational waves from these directions, creating zero strain. This explains
the four indentations (i.e., nodes) in the RMS antenna pattern and the four
black islands of ∼0 probability in each skymap of Figure 3.8.
94
Figure 3.8: A three-detector (Advanced LIGO and Virgo) BAYESTAR sky local-
ization of GW170817 in ICRS coordinates (Aitoff projection) overlaid on top
of antenna patterns for each detector; see Chapter 6. (Figures from Giuseppe
Greco, 2018)
95
Chapter 4
Low-Latency Searches for
Gravitational-Wave Candidate
Events
Of the many different kinds of gravitational-wave signals Advanced LIGO
and Virgo can detect (bursts, periodic signals, coalescing binaries, and stochas-
tic backgrounds), we focus on low-latency searches for gravitational-wave tran-
sients (i.e., bursts and coalescing binaries). We will introduce the two types of
searches in place (modeled and unmodeled) which produce gravitational-wave
triggers that enter the online Gravitational Wave Candidate Event Database
(GraceDb). Then, we will walk through the automated and human data-
quality vetting procedures to select the gravitational-wave trigger (now ele-
vated to gravitational-wave candidate event status) to report for electromag-
netic/neutrino follow-up.
Portions of this chapter will resemble the section on online gravitational-
wave analysis I wrote for the LIGO Scientific Collaboration and Virgo Col-
laboration paper, Low-Latency Gravitational Wave Alerts for Multi-Messenger
Astronomy During the Second Advanced LIGO and Virgo Observing Run (Ab-
bott et al., 2019).
4.1 Compact Binary Coalescence Searches
Compact binary systems involving neutron stars and stellar mass black
holes, (i.e., binary neutron star, neutron star-black hole, and binary black hole
systems) merge in the sensitive frequency band of advanced ground-based in-
96
terferometric GW detectors. The modeled compact binary coalescence (CBC)
searches specifically look for signals from these systems.
The waveform, h(t), is the collected history of the stretching and squeez-
ing of space by the coalescing binary. It requires solving the two-body problem
with different tools of attack depending on the masses and compactness that
are involved (Figure 4.1). Because the General Theory of Relativity dictates
that the binary loses energy and angular momentum via gravitational radia-
tion, the waveform features three general phases: an inspiral phase while the
binary shrinks, a merger phase while the remnant object is formed, and a ring-
down phase if the remnant object is a perturbed black hole. More specifically,
during the inspiral, the binary ‘chirps’ (i.e., the gravitational-wave frequency
and amplitude both increase) and during the merger, the peak frequency and
amplitude are reached. Lastly, during the ringdown, the perturbed remnant
black hole quickly settles into its final state by emitting gravitational waves
with characteristic quasinormal modes.
The waveforms vary depending on the binary’s intrinsic parameters such
as the masses and spins that are involved, and also on extrinsic parameters such
as orientation and distance with respect to the detectors (Table 4.1). A specific
combination of the two individual masses (m1 and m2, m1 ≥ m2) known as
the chirp mass, Mc, largely determines the binary’s observed frequency (f)
and frequency evolution (ḟ) at lower freq(uencies before th)e merger:3/5
M (m1m
3/5
2) c
3 5
= = π−8/3f−11/3c ḟ ,
(m 1/51 +m2) G 96
where c is the vacuum speed of light and G is Newton’s gravitational con-
stant. Unfortunately, the chirp mass alone does not reveal the two individual
masses or their spins. Instead, we need more information such as the mass
ratio, m1/m2, which can be found by knowing the binary’s waveform to higher
97
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��� ���������������������
���
������������������������
��� ��������������������
�������������������� ������������������������
���
��� ��� ��� ��� ��� ���
�����
Figure 4.1: Main analytical and numerical methods for solving the two-body
problem depends on the masses and compactness involved. Here, m1 and m2
(m1 ≥ m2) are the two individual masses and rc2/GM is a measure of the
separation distance. (Figure from Buonanno & Sathyaprakash, 2014)
precision. Thus, the modeled CBC searches require knowing the waveforms
from the last few orbits to very high precision.
Intrinsic Parameters ϑin Extrinsic Parameters ϑex
m1 mass of primary object α right ascension
m2 mass of secondary object δ declination
S~1 spin of primary object r distance
S~2 spin of secondary object t⊕ arrival time at detector
ι inclination angle
ψ polarization angle
φc coalescence phase
Table 4.1: Intrinsic and extrinsic parameters, ϑ, of a compact binary coales-
cence gravitational-wave waveform.
We can then solve for thousands upon thousands of these waveforms
to construct a waveform template bank that suitably covers the parameter
�������
98
space of compact binary systems. To determine the best-fit waveform (i.e.,
the matched template) that describes the gravitational-wave signal buried in
the noisy detector output, we use a process known as matched filtering.
First, imagine a filter function k(t) applied to our noisy detector output
s(t) = h(t) + n(t): ∫ ∞
ŝ = s(t)k(t)dt.
−∞
The signal (S) is the expected value of ŝ when the gravitational-wave signal,
h(t), is present. The noise (N) is the RMS value of ŝ when h(t) is absent. Then,
the matched template is determined by the filter function that optimizes the
signal-to-noise ratio (SNR, S/N).
The exact solution is as follows. If we already know the waveform h(t)
of the gravitational-wave signal, we can construct the Wiener filter (i.e., the
matched filter/template):
h̃(f)
k̃(f) = ,
Sn(f)
to optimize the signal-to(-nois)e ratio∫(i.e., the template’s auto-correlation):2
S ∞ h̃∗(f)h̃(f)
= 4 df.
N 0 Sn(f)
However, in practice, we don’t know h(t), and instead have in place dis-
crete waveform template banks with templates hi(t) that cover the target
parameter space of compact binary systems (Table 4.3). Then, the SNR time
series is determined by using the Wiener filter to compute the inner product
of the whitened templates and the whitened detector output (i.e., the cross-
correlation sequenc(e bet)ween t∫he data output and template):2
S ∞ h̃∗i (f) s̃(f)= 4 ei2πftdf.
N 1/2 1/2i 0 Sn (f) Sn (f)
When the SNR peaks above a predetermined threshold set by the low-latency
search pipeline, a single-detector gravitational-wave trigger with time of arrival
99
and coalescence phase information is produced for that detector’s output data
stream.
Sufficiently high SNR thresholds prevent Gaussian noise from being iden-
tified as a trigger because its Gaussian probability distribution drops off very
quickly away from the mean. However, non-Gaussian noise is handled dif-
ferently, by performing χ2 waveform consistency tests and computing sine-
Gaussian noise correlation statistics (using signal-based vetoes, (Nitz, 2018)),
explicitly zeroing out loud and short instrumental noise transients (gating the
data), and vetoing triggers occurring during times flagged as contaminated by
known noise sources (applying the low-latency data quality vector).
Once pipeline-specific vetoes have been applied to the single-detector trig-
ger lists (Table 4.2), the lists are then combined to search for coincident trig-
gers across the detector network. Consistencies in the SNR, coalescence phase,
trigger time, and template parameters determine the ranking statistic for the
coincident trigger which can then be converted into a false alarm rate.
Triggers of high significance (low false alarm rates) are then elevated to
gravitational-wave candidate event status if they pass scrutiny under auto-
mated and human data-quality checks. By this point, the BAYESTAR algorithm
has already generated and uploaded a sky localization probability map for the
candidate. Two things happen in parallel next. One, we do a more thorough
job determining the source parameters, relaxing assumptions made by the
low-latency pipelines: all low-latency CBC pipelines assume that the compact
objects are aligned or anti-aligned with the orbital angular momentum and
that the orbital eccentricity is negligible. Two, we report the candidate event
to our observing partners for electromagnetic/neutrino follow-up.
Currently, with a few months before the advent of the third observing
100
run, O3, we have 4 low-latency modeled CBC searches in place: GstLAL,
MBTAOnline, PyCBC Live (which all produced triggers during the first and
second observing runs), and a new low-latency search pipeline called SPIIR.
Pipeline-specific matched filtering technique and background estimation are
outlined next.
PyCBC Live GstLAL MBTAOnline SPIIR*
Signal-based vetoes X X X X
Low-latency data
quality vector X X X
Gated data X X
Table 4.2: Pipeline-specific measures (marked with a X) to identify
non-Gaussian noise sources in single-detector trigger lists during the
second observing run, O2.
* Alerts for triggers from SPIIR were not sent to observing part-
ners during O2. However, this is subject to change for the third
observing run, O3.
GstLAL
The GStreamer LIGO Algorithm Library (GstLAL) low-latency search
pipeline performs matched filtering in the time domain with real template
waveforms (Messick et al., 2017). Trigger significance is calculated by con-
structing a likelihood-ratio ranking statistic that models the distribution of
trigger properties for noise and GW events (Cannon et al., 2015). The back-
ground is computed by synthesizing likelihood ratios from a random sampling
of a probability density that is estimated using non-coincident triggers accu-
mulated over the course of an observing run, which are taken to be noise.
MBTAOnline
The Multi-Band Template Analysis (MBTAOnline) low-latency search pipeline
performs matched filtering in the frequency-domain with complex template
101
waveforms. It uses several matched filters to cover the detector bandwidth,
i.e., the matched filter is split across multiple frequency bands (Adams et al.,
2016). This allows for shorter templates to be used in each frequency band
and GW candidate events to be identified with sub-minute latencies. The
SNR is defined to be the modulus of the complex matched filter response and
is evaluated at its maximum value to extract the signal time of arrival and
coalescence phase.
The background distribution of the ranking statistic is constructed by
making every possible coincidence from single-detector triggers over few hours
of recent data. It then folds in the probability of a pair of triggers passing the
temporal coincidence test.
PyCBC Live
The Python CBC (PyCBC Live) low-latency search pipeline performs
matched filtering in the frequency-domain with complex template waveforms
(Nitz et al., 2018). It estimates the background of accidental coincidences by
using time slides between triggers from different detectors generated within
the 5 most recent hours of live time data. This choice limits the online inverse
false alarm rate to ∼100 yr.
SPIIR
The Summed Parallel Infinite Impulse Response (SPIIR) low-latency search
pipeline performs matched filtering in the frequency-domain using a set of in-
finite impulse response (IIR) filters to approximate the template waveforms
(Chu, 2017). It produces gravitational-wave candidate events with sub-minute
latencies where the significance is evaluated by the distribution of background
events. To estimate this background, SPIIR uses time slides of live time data
from the past week.
102
PyCBC Live GstLAL MBTAOnline
Total mass m1 +m2 2−500aM 2−150aM 2−100M
Mass ratio m1/m2 1−98 1−98 1−99
Minimum component mass m2 1M 1M 1M
Spin magnitudeb (m < 2M) 0−0.05 0−0.05 0−0.05
Spin magnitudeb (m > 2M) 0−0.998 0−0.999 0−0.9899
SNR threshold for triggering 5.5 4 (L), 3 (V)c 5.5d
Table 4.3: Template bank parameters and SNR threshold for triggering used
by the low-latency CBC search pipelines during the second observing run,
O2. (Table from Abbott et al., 2019)
a The maximum total mass for PyCBC Live and GstLAL is in fact a function
of mass ratio and component spins (Dal Canton & Harry, 2017; Mukherjee
et al., 2017). In this table, we indicate the highest total mass limit over
all mass ratios and spins.
b For detection, the component spins, S~1 and S~2, are often assumed to be
non-precessing and aligned with the binary’s total angular momentum.
c GstLAL requires an SNR threshold of 4 for triggers from Advanced LIGO
versus 3 for triggers from Virgo.
d MBTAOnline uses a higher SNR threshold of 6 for triggers from Advanced
LIGO to form coincidences with triggers from Virgo.
4.2 Burst Searches
There are also two unmodeled “Burst” searches, cWB and oLIB, that are
capable of detecting gravitational waves from a wide variety of astrophysical
sources in addition to compact binary coalescences. This includes (but is not
limited to) core-collapse supernovae, magnetar starquakes, and more specula-
tive sources such as intersecting cosmic strings.
The Burst searches work by looking for excess power in the time-frequency
(TF) domain of the GW strain data (Klimenko et al., 2016; Lynch et al., 2017).
To understand the time-frequency domain, first let’s take the Fourier transform
of a function s(t) defined on the real axis −∞ < t <∞. The power spectrum
of the transform, |s̃(f)|2, reveals the dominant Fourier components but tells
us nothing about the phase relations between the components (i.e., we do not
know when things happened). Thus, the simplest way to recover temporal
103
information is to transform segments of length δt on the real axis.
The power spectrum of the Fourier transform of s(t) on the interval
0 < t < δt reveals the dominant Fourier components for that interval with
resolution 1/δt. We can repeat this process for δt < t < 2δt, and so on. Thus,
working in the TF plane we can determine a reasonable estimate of the total
duration of the signal along with its dominant frequency range.
The excess power method works as follows. Advanced LIGO and Virgo’s
strain data are available using a sampling rate of either 16384 Hz or 4096 Hz.
Thus, a detector output data segment of duration δt = N∆t (∆t equal to
1/16384 s or 1/4096 s) consists of a discrete set of N + 1 values:
sj = s[tstart + j∆t],
where tstart is the start time of the segment and j = 0, 1, . . . , N .
In Fourier space, these values become:
N∑−1
s̃k = s e
−i2πjk/N
j ,
j=0
which can be re-written as:
N∑−1
s̃ = s[t ]e−i2π(tj−tstart)fkk j ,
j=0
with tj = tstart + j∆t and fk = k/δt. Thus, the frequency resolution is 1/δt
and the maximum frequency is N/δt since s̃k = s̃k+N .
However, the detector output is the sum of the noise and a potential
signal, s̃k = ñk + h̃k. Thus,
N∑−1
ñ = n[t ]e−i2π(tj−tstart)fkk j and
∑j=0N−1
h̃ = h[t ]e−i2π(tj−tstart)fkk j .
j=0
104
If we assume the different Fourier components of noise are uncorrelated (i.e.,
the noise is stationary and the detector is stable), we also have:
〈ñ∗ñ′ 1k k〉 = δkk′ Sn[fk],2
where Sn(fk) is the single-sided noise power spectral density and δkk′ is the
Kronecker delta function.
Then, minimal assumptions about the gravitational-wave signal allow us
to calculate its excess power statistic. For simplicity, let’s say that the signal is
of duration δt with its power concentrated in the frequency band f1 < f < f2,
where f1 = k1/δt and f2 = k2/δt. The excess power statistic is defined as:
∑k2 2
E |s̃k|= 4 ,
Sn[fk]
k=k1
thus we compute values of E over all possible start times. Segments with excess
power above a predetermined threshold indicate potential triggers.
cWB
The coherent WaveBurst (cWB) low-latency search algorithm uses Wilson-
Daubechies-Meyer (WDM) transform to convert detector output from a net-
work of M GW detectors into the TF plane at different frequency resolutions.
These transforms are generalizations of the Fourier transform:
N∑−1
Sm;k,l = fk,l[tj]sm[tj],
j=0
where f are WDM filter functions and s is output from the mthk,l m detector
(Necula et al., 2012; Klimenko et al., 2016). Then, cWB identifies clusters of
TF pixels with excess power above the baseline detector noise. Clusters that
overlap in time and frequency at different frequency resolutions indicate the
presence of a GW trigger.
105
Using a maximum likelihood approach, principal components of theWDM
transform from different resolutions are maximized over all possible time-of-
flight delays in the network of GW detectors for the purpose of selecting pixels
from the cluster that describe the GW signal completely and without overlap-
ping information. The contribution from the trigger energy which is coherent
among the involved detectors is used to calculate the detection statistic, which
is compared to the background to estimate the trigger’s significance. Trig-
gers of high significance are followed-up with waveform reconstruction and the
creation of sky localization probability maps (skymaps) since they could be
elevated to candidate event status.
oLIB
For the Omicron LALInferenceBurst (oLIB) low-latency search algorithm,
we define a central time (tc), central frequency (fc), duration (σt), and band-
width (σf) for gravitational-wave transients that are well localized in both time
and frequency: ∫ ∞ |h(t)|2
tc = ∫ t ||h|| dt,2−∞∞ |h̃(f)|2
fc = ∫2 f ||h|| df,20∞ |h(t)|2
σ2 2t = ∫ (t− tc)−∞ ||h|| dt,2∞
2 2 |h̃(f)|2σf = 2 (f − fc)
0 ||
df,
h||2
where ∫ ∞ ∫ ∞
||h||2 = |h(t)|2dt = |h̃(f)|2df.
−∞ −∞
The dimensionless quality factor Q = fc/σf is a measure of this transient’s
aspect ratio in the TF plane.
106
Then, oLIB uses Q transform to decompose single detector data output
into several TF planes of constant quality factors Q ∼ τf0, where τ and f0
are the time resolution and central frequency of the transform’s filter/wavelet
(Chatterji et al., 2004; Lynch et al., 2017):
N∑−1
sf0 [tm] = s[tj]wf0 [tj − tm]e−i2πjf0/N .
j=0
Above, wf0 [tj − tm] is the time-domain window function centered around tm
with a duration that is inversely proportional to the frequency under consid-
eration.
Then, the Omicron software in oLIB clusters data segments of excess
power with identical f0 and Q spaced within 100 ms of each other. Searches
for coincidences are performed in two stages: first with ± 50 ms and second
with ± ∼10 ms. Finally, Omicron clusters the coincident triggers such that
only one trigger for each analysis window of 100 ms is analyzed.
The LALInferenceBurst software in oLIB then coherently analyzes the
coincident triggers to produce two Bayes factors: BSN and BCI. BSN compares
a signal model (S) to a Gaussian noise model (N) to roughly measure the
loudness of the signals and BCI compares a ‘coherent’, i.e., correlated signal
model (C) to an ‘incoherent’, i.e., uncorrelated signal model (I) to roughly
measure how similar the signals look in each detector. A likelihood ratio test
uses both of these statistics to compute the significance of the gravitational-
wave trigger which is then submitted into GraceDb if above a set threshold.
4.3 GraceDb
All triggers produced by the low-latency search pipelines enter an interac-
tive database known as GraceDb1 (the Gravitational Wave Candidate Event
1https://gracedb.ligo.org/
107
Database), the centralized hub for aggregating and disseminating informa-
tion about GW candidate events. It features a human-friendly web interface
for displaying information as well as a RESTful API (representational state
transfer application program interface) for programmatic interaction with the
service. Tools provided by GraceDb’s Python-based client package allow users
to add new events to the database, annotate existing events, perform searches,
upload files, and more. GraceDb also alerts follow-up advocates (humans on
duty who analyze and decide whether or not a GW trigger is appropriate for
electromagnetic/neutrino follow-up) via text messaging and phone calls using
Twilio.
Automated follow-up processes of GW candidate events that perform
tasks such as parameter estimation or detector characterization are alerted
of new event creation and updates to events in GraceDb via push notifica-
tions using the LIGO-Virgo Alert System (LVAlert). Users and listeners of
the LVAlert notification service based on the extensible messaging and pres-
ence protocol (XMPP) create and subscribe to specific messaging nodes to
receive target notifications (e.g., from a specific low-latency search pipeline or
heartbeat process for debugging).
4.4 Supervised Electromagnetic/Neutrino Follow-Up Process
For O1 and O2, several follow-up processes responded to the entry of a
GW trigger into GraceDb, notified by the receipt of an LVAlert message. Three
of these processes were of immediate relevance to the electromagnetic/neutrino
follow-up effort: the low-latency skymap generator for CBC candidate events
(BAYESTAR, the BAYESian TriAngulation and Rapid localization pipeline for
O1 and O2), the tracker of candidate event status/updates and alert genera-
tor/sender (approval_processor for O1, upgraded to approval_processorMP
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for O2), and the tracker of other follow-up processes starting and ending on
time (eventSupervisor for O2 only).
In particular for O1 and O2, I wrote the majority of the software respon-
sible for making the decision to send alerts based on incoming state informa-
tion about the GW triggers, approval_processor/approval_processorMP
(Chapter 5).
4.5 Online Automated Data Vetting
Low-latency gravitational-wave search pipelines receive detector state in-
formation indicating when the detectors’ data are suitable for use in astro-
physical analysis. This includes times when the detectors are operating in a
nominal state and data calibration is accurate. However, additional informa-
tion is required to deal with non-Gaussian transient noise artifacts known as
glitches, which can mimic true gravitational-wave signals and often plague the
detectors’ data.
In Table 4.2, we see that some known forms of instrumental noise are
passed to the low-latency search pipelines through the ‘low-latency data qual-
ity vector’. This vector produces data quality vetoes which flag times of known
data quality issues with 1/16 s resolution. They are generated in real time
using sensors that measure the behavior of the instruments and their environ-
ment. If a GW trigger occurs during a time that is vetoed, it is not reported
for electromagnetic/neutrino follow-up. Because of this feature, data quality
vetoes are reserved for severe noise sources.
There are also low-latency glitch detection algorithms which search de-
tector data for correlations between witness sensors and GW strain data. In
the case of the streaming machine learning pipeline, iDQ, which houses several
of these algorithms, i.e., ‘classifiers’, a glitch false alarm probability (FAP) is
109
reported around the time of a GW trigger which measures the probability that
the detector was in a nominal state based on the presence of glitches in witness
sensors (Essick et al., 2013; Biswas et al., 2013).
For O1 and O2, approval_processor and approval_processorMP col-
lected iDQ glitch FAP information produced by the ovl classifer to define an
iDQ-based veto:
∏ N veto if joint glitch FAP ≤ 0.01joint glitch FAP = glitch FAPi ⇒
i=1 pass if joint glitch FAP > 0.01,
where i = 1, . . . , N are all relevant detectors in a GW trigger detection. This
veto was applied to triggers from Burst low-latency search pipelines only. Dur-
ing O1, I found that CBC triggers did not benefit from using glitch FAP in-
formation generated using time windows comparable to waveform template
durations because of the large deadtimes that were introduced.
4.6 Human Vetting
During O1 and O2, GW triggers required human approval to be elevated
to candidate event status, i.e., to officially be released to our observing partners
via the LIGO/Virgo Initial Notice (a machine-readable alert containing the
GW candidate event’s basic properties plus a sky localization probability map
(skymap)). For O3, we will enter the era of public alerts, where LIGO/Virgo
Preliminary Notices will be released to the public without human vetting. (By
definition, Preliminary Notices contain preliminary information with zero hu-
man vetting. In O1 and O2, we kept these alerts internal to the LIGO Scientific
Collaboration and Virgo Collaboration.) However, in O3, LIGO/Virgo Initial
Notices will still require human vetting and the process will be as follows.
For humans to become involved in the mix, GW triggers must pass all
the previous major hurdles: have a low enough false alarm rate, not occur
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during a data quality vetoed time, not be a hardware injection, and have a
high enough iDQ joint glitch false alarm probability (for Burst triggers only).
At this point for O1 and O2, approval_processor/approval_processorMP
requested sign-offs from the appropriate mix of LIGO/Hanford, LIGO/Liv-
ingston, and Virgo (near the end of O22) detector operators by triggering an
alarm in the observatory control rooms. Essentially, detector operators were
to qualitatively answer the question, “At the time of the trigger, were things
running normally?”, as OK or NOT OK. This way, we could ensure unusual
events such as thunderstorms, trucks driving close the buildings, etc. did not
occur at the time of the GW trigger and that the interferometers were running
optimally.
The OK response from operators generated two actions. One, the Rapid
Response Teams (RRTs) consisting of commissioning, computing, and cali-
bration experts at each detector site documented the state of the detectors
and two, approval_processor/approval_processorMP applied the special
GraceDb label, ADVREQ (advocate required), to generate texts, emails, and
phone calls from GraceDb to the electromagnetic/neutrino (EM) follow-up
advocates.
Thus, many different groups of persons from the collaborations were in-
volved in the decision-making process: detector operators, RRTs, low-latency
search pipeline experts, detector characterization (data quality) experts, and
EM follow-up advocates. We all met at our designated TeamSpeak channel
for the on-call validation process within minutes of being notified of the GW
trigger.
First, the EM follow-up advocates on duty perform non-stationary noise
2Advanced Virgo joined the Advanced LIGO GW detector network in August 2017 for
the last month of O2 data acquisition.
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checks and daily/weekly event rate checks. For the non-stationary noise check,
if there are 2 or more CBC GW triggers more than 1 second apart from
the same low-latency search pipeline in a given time frame, the pipeline is
likely responding to non-stationary noise which affected the false alarm rate
estimates and we reject all triggers. Likewise, if there are two or more Burst
triggers more than 5 seconds apart from the same Burst pipeline, we reject
all triggers. However, it is of course okay to have different search pipelines
produce triggers close in time; this is what we want! For the event rate check,
if too many viable candidate events appear to be occurring too close together,
this suggests a problem with the background estimation. In this case, the
advocate on duty would double check with the search pipeline expert to see if
anything could be wrong with the analysis.
Next, all previously mentioned data vetting products (Section 4.5) and
additional data quality information (e.g., Omega and Omicron scans) not acces-
sible at low-latency timescales are considered. The Omega and Omicron scans
are spectrograms, i.e., power scans, that help us visualize witness sensor data
in the TF domain around the time of the GW trigger (Chatterji et al., 2004;
Robinet, 2016) when we ask two essential questions regarding data quality:
1. Could transient noise account for the trigger that we see? If yes, veto.
If no, proceed to next question.
2. Could transient noise bias the estimate of the source parameter and
properties? If yes, mitigate. If no, proceed with the vetting process.
This is because the same type of noise might in one case be vetoed, and in
another case, be mitigated. For example, in Figure 4.2, spectrograms featuring
similar type overflows but requiring different types of response actions are
presented.
112
Figure 4.2: Spectrograms of an overflow glitch that requires a veto for the
corresponding GW trigger (top) and mitigation (bottom). The bottom fig-
ure is from the Advanced LIGO/Livingston C00 “online calibrated” data for
GW170817 (https://dcc.ligo.org/P1700337/public). The faint but character-
istic trace of the BNS chirp can be seen in the background; thus, the noise was
modeled and subtracted. (Figure from Abbott et al., 2017a)
113
The low-latency search pipeline experts also have to defend or clarify
the significance of their triggers. In the past, when we had more than one
viable GW trigger ((declared as neighbors if their event times were contained
within ±5 seconds of each other), the EM follow-up advocates selected the
most promising candidate based on pre-established criteria (e.g., lowest FAR,
choosing CBC over Burst triggers for compact binary events). For O3, the
grouping of neighbor triggers is handled automatically through the creation of
‘Superevents’ in GraceDb.
Finally, the EM follow-up advocates select the skymap to include in the
Initial Notice depending on the validation done at different instrument sites.
For O2, a priority was given to the two Advanced LIGO detectors since they
were more sensitive. At this point, we are ready to send the Initial Notice for
the GW trigger (for O1 and O2 to our MOU observing partners) or Superevent
(for O3 to the public), and compose to corresponding Initial Circular.
When necessary, search pipeline experts and the data quality team with
the help of the RRTs recommended LIGO/Virgo Retraction Notices after days
or weeks, using extended data investigation and/or updated FAR calculation
based on additional background data. For example, during O2, GW candi-
date event G275404 originally detected by PyCBC Live and GstLAL did not
appear in the offline analysis by either search pipeline and its 50%/90% cred-
ible regions in the skymaps increased from 460/2100 deg2 for BAYESTAR to
2000/17000 deg2 for LALInference. Thus, a LIGO/Virgo Retraction Circular
was sent stating: “Neither search produced a significant trigger at the time of
G275404. We conclude that G275404 is not a trigger of interest and does not
warrant further follow-up.” (LIGO Scientific Collaboration & Virgo Collabo-
ration, 2017).
114
Chapter 5
O1: The First Observing Run
The first observing run (O1) from September 12, 2015 to January 19,
2016 involved only the two Advanced LIGO detectors in the United States.
During the course of O1, there were 2 confident gravitational-wave events and
1 LIGO-Virgo trigger (LVT), later named GW150914 (“The First Monday
Event”), GW151226 (“The Boxing Day Event”), and LVT151012 (“The Second
Monday Event”).
The confident GW detections were both stellar mass binary black hole
mergers, and they revealed that there exists a class of heavier stellar mass
black holes than those deduced previously from electromagnetic (X-ray binary)
observations. LVT151012, was also recently declared a true gravitational-wave
event (GW151012) as stated in the post-O2 CBC “catalog” paper, GWTC-1:
A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed
by LIGO and Virgo during the First and Second Observing Runs (Abbott et al.,
2018).
In this chapter, I detail work I began before O1 started. Namely, I
wrote a Python-based information-checking and decision-making software pro-
gram named approval_processor to prepare for the event we might have a
serendipitous gravitational-wave candidate requiring electromagnetic/neutrino
follow-up.
115
5.1 approval_processor: The First of the Advanced Detector Era
Gravitational-Wave Candidate Event Annotators
The first low-latency search for EM counterparts was conducted by the
Locating and Observing Optical Counterparts to Unmodeled GWPulses (LOOC
UP) Project between December 2009 and October 2010, during the Initial
LIGO era (2005 to 2010). LOOC UP provided sky localization estimates for
potential GW events with latencies on the order of thirty minutes to radio,
optical, and X-ray telescopes (Kanner et al., 2008; Kanner, 2011).
For the Advanced Detector era, the EM follow-up program was largely
orchestrated by the approval_processor software program. The primary
role of approval_processor was to internally alert humans within the LIGO
Scientific Collaboration and Virgo Collaboration of significant GW triggers
entering GraceDb that required more thorough (i.e., human) vetting, before
taking on its secondary role of sending alerts for vetted candidates to our
observing partners via the Gamma-ray Coordinates Network1 (GCN). Basic
trigger properties from the pipelines (false alarm rate, trigger time-of-arrival
on Earth, detectors involved with the trigger), data quality and data prod-
ucts, detector operator and advocate signoffs determining the result of human
vetting, and other labels identifying temporally correlated external triggers or
signal injections performed in hardware at the sites2 were considered.
Although updates to approval_processor were made during the course
of the first observing run, it successfully sent LIGO/Virgo Notices to our ob-
serving partners to generate EM/neutrino follow-up for the first binary black
hole merger detections and candidate events in history (see Figure 5.1 to view
the alerts/follow-up timeline for GW150914, the first ever observed binary
1https://gcn.gsfc.nasa.gov
2Hardware signal injections are simulated GW signals created by physically displacing
the detectors’ test masses (Biwer et al., 2017)
116
black hole merger).
Below are the information included by approval_processor in its alerts.
Initial GW Initial Updated GCN Circular Final
Burst Recovery GCN Circular (identified as BBH candidate) sky map
Fermi GBM, LAT, MAXI, Swift Swift Fermi LAT,
IPN, INTEGRAL (archival) XRT XRT MAXI
BOOTES-3 MASTER Swift UVOT, SkyMapper, MASTER, TOROS, TAROT, VST, iPTF, Keck, Pan-STARRS1Pan-STARRS1, KWFC, QUEST, DECam, LT, P200, Pi of the Sky, PESSTO, UH VST TOROS
VISTA
MWA ASKAP, ASKAP, VLA, VLA,LOFAR MWA LOFAR LOFARVLA
100 101 102
t− tmerger (days)
Figure 5.1: Timeline of LIGO/Virgo Notices and Circulars sent to our MOU
partners for GW150914 and its electromagnetic follow-up. (Figure from Ab-
bott et al., 2016a)
5.2 Information Sent to MOU Partners
False Alarm Rate
The measure of significance for a gravitational-wave trigger is the esti-
mated false alarm rate (FAR), which quantifies the rate at which noise events
with as high a ranking as the foreground trigger are generated. For each of the
observing runs, there was a predetermined FAR threshold for sending alerts
to our observing partners. For the majority of the first and second observ-
ing runs, this FAR threshold was one CBC and one Burst candidate event
per month of live-time for O1 (3.8×10−7 Hz) and one CBC and one Burst
candidate event per two months of live-time for O2 (1.9×10−7 Hz).
The FAR estimation method is specific to the low-latency search pipeline
that triggers the candidate event (Section 4), although the most common
method uses time slides (Figure 5.2). With the method of time slides, data
sets from the various GW detectors are time-shifted by a time offset greater
than the GW travel time between the detector sites to generate years of ef-
117
Figure 5.2: Method of time slides for FAR estimation. GW triggers are de-
tected in zero-lag (top) and noise events are detected in time shifted data with
offsets greater than the GW travel time between the detectors. (Figure from
Laura Nuttall, 2017)
fective background. Then, the search pipeline analyzes this data (which must
de facto be void of any real coincident gravitational waves) and the rate of
noise events arising from uncorrelated noise sources occurring simultaneously
is calculated.
Two-Dimensional Sky Localization Probability Maps
For O1, we wanted to provide our observing partners with prompt and
accurate sky localization probability maps (skymaps) in order to increase their
chances of finding electromagnetic/neutrino counterparts where time is of the
essence (e.g., a rapidly fading X-ray/optical afterglow occurring within min-
utes to hours of a neutron star merger). Thus, low-latency source localization
for CBC candidate events was provided by BAYESTAR, the BAYESian TriAngu-
lation and Rapid localization pipeline (a pun on the powerful Cylon Basestar
from Battlestar Galactica), which produced skymaps for follow-up on the order
of ∼30 seconds to minutes (Singer & Price, 2016).
118
As we can see in Table 4.1, there are 11 total parameters, ϑ, that describe
the physical properties of a compact binary coalescence. Given that Advanced
LIGO and/or Virgo detects a gravitational-wave candidate in a data set DGW,
we can write the posterior distribution, P (ϑ|DGW), which describes the prob-
ability of the parameters given the data:
P (ϑ| P (DGW|ϑ)P (ϑ)DGW) = ,
P (DGW)
using Bayes’ theorem. For purposes of localization, the only two parameters of
interest are the right ascension and declination (α and δ), where all the other
parameters are nuisance parameters (λ) that can be marginalized over, i.e.,
integrated away. Thus, we∫can write the marg∫inal posterior:
| | P (DGW|ϑ)P (ϑ)P (α, δ DGW) = P (ϑ DGW) dλ = dλ,
P (DGW)
which involves complicated multi-dimensional integrals that are in practice
performed with the Markov chain Monte Carlo (MCMC) integration tech-
nique. However, Advanced LIGO and Virgo’s source localization pipeline
LALInference that utilizes a rigorous MCMC analysis takes on the order of
hours to days and weeks, rendering it unhelpful for low-latency electromag-
netic/neutrino follow-up. Thus, we utilize BAYESTAR which works with the
maximum likelihood (ML) parameter estimates of the GW amplitude, coales-
cence phase, and arrival time at each detector as determined by the triggered
template of the low-latency CBC search pipeline when producing GW trig-
gers3. In this manner, the dimensionality of the original marginal posterior
problem is greatly reduced and marginalization can be carried out using various
methods of quadratures in a matter of seconds to minutes (e.g., Newton-Cotes
for integrating over the polarization angle, Legendre-Gauss for cosine of the
inclination angle, etc.) (Singer, 2015).
3In other words, for O1, BAYESTAR computed the triggered template’s auto-correlation
sequence to calculate the marginal posterior.
119
For O1, BAYESTAR produced two-dimensional (direction information only)
skymaps packaged into convenient Flexible Image Transport System (FITS)
files for CBC candidate events. These skymaps were created via GW trig-
gers produced from the two Advanced LIGO detector network and therefore
typically constrained the localization to ∼100 to 1000 deg2 consisting of two
long, thin sections of a great circle, one in the Northern hemisphere and one
in the Southern hemisphere. In the event that the orbital plane of the binary
is nearly face-on towards the Earth and the GW phases on arrival can be ex-
plained by two different binary inclination angles of opposite handedness, each
arc-shaped mode features two narrow stripes at the ends resembling a snake’s
forked-tongue, aptly naming this feature the forked-tongue morphology.
We also sent low-latency two-dimensional (directional information only)
skymaps for Burst GW candidate events (Essick et al., 2015). The latency for
producing cWB skymaps is on the order of ∼minutes. Because the cWB low-
latency search detection statistic is sensitive to the arrival times of the GW
signal at different detector sites, it allows us to compute a constrained likeli-
hood functional dependent on the source sky position (i.e., localization comes
from a combination of time-delay information (triangulation) and amplitude
coupling through the antenna patterns). Thus, the skymap is constructed by
maximizing the constrained likelihood functional for all possible GW signals
for the candidate event at each point in the sky.
The other low-latency Burst search oLIB uses LALInferenceBurst (LIB)
to compute its skymaps. The latency for oLIB skymaps is higher, on the order
of ∼hours to days, because LIB is an MCMC parameter estimation algorithm
which uses sine-Gaussian templates for filtering. It reports a posterior in
nine parameters, of which all parameters apart from sky position must be
marginalized away to produce the skymap.
120
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Figure 5.3: Comparison of the 90% credible regions from low-latency skymaps
produced by cWB, LIB, and BAYESTAR for GW150914 displayed in an ortho-
graphic projection centered around the LIB localization. In light green is the
offline full parameter estimation skymap produced by LALInference. The in-
set shows the distribution of the arrival time difference, ∆tHL, across the two
Advanced LIGO detector network. (Figure from Abbott et al., 2016a)
In the case of short-duration compact binary coalescences (i.e., binary
black hole mergers like GW150914) where the candidate event is detect by
both CBC and Burst low-latency searches, we can compare the low-latency
Burst skymaps to BAYESTAR (Figure 5.3). In this case, the CBC skymaps
have much better localization (smaller credible regions) although most of the
probability from both Burst and CBC skymaps will still contain the region
around the true GW source location.
121
Chapter 6
O2: The Second Observing Run
The second observing run (O2) from November 30, 2016 to August 25,
2017 involved the two Advanced LIGO detectors in the United States joined
by the Advanced Virgo detector in Italy for the last month of data-taking
(starting August 1, 2017). During the course of O2, there were 8 confident
gravitational-wave events: 7 of them stellar mass binary black hole mergers and
1 binary neutron star coalescence (Abbott et al., 2018). Of the 8 confident GW
events, only 6 were found by the low-latency search pipelines and reported for
EM/neutrino follow-up. The remaining 2 events (GW170729 and GW170818)
were recovered by the offline search analyses.
Much of the work done to enable successful follow-up of low-latency GW
candidates during O2 was documented in a LIGO Scientific Collaboration and
Virgo Collaboration paper that I co-chaired with Sarah Antier, Low-Latency
Gravitational Wave Alerts for Multi-Messenger Astronomy During the Second
Advanced LIGO and Virgo Observing Run (Abbott et al., 2019). Our paper
writing team included (in alphabetical order) Sarah Antier, Deep Chatterjee,
myself, Shaon Ghosh, Giuseppe Greco, Barbara Patricelli, Karelle Siellez, and
Koh Ueno. In this chapter, you will find more details on various sections from
this paper.
In preparation for O2, I made upgrades (working with Reed Essick) to
turn approval_processor into a multi-processing information-tracking soft-
ware, approval_processorMP. This upgrade was inspired by the fact that
during O1, each incoming LIGO-Virgo Alert (LVAlert) from approved low-
122
latency search pipelines triggered approval_processor to run, but it would
begin its vetting process each time from scratch by querying GraceDb for GW
trigger information. We wanted to cut down the latency of this process by
preserving state information about the GW triggers and candidate events in
local memory.
Other updates to approval_processorMP were planned simultaneously.
Again, working with Reed Essick, I wrote additional software to “throttle”
low-latency search pipelines if they overproduced GW triggers during a set
time duration (as determined by Poisson statistics), signaling misbehavior in
the streaming analysis and the need for approval_processorMP to ignore that
pipeline’s incoming GW triggers. This pipeline throttle feature was complete
with an option to send commands to the live approval_processorMP process
to un-throttle misbehaving pipelines, or to just let it run and wait until the
pipeline settled down again. We debuted this upgrade during the middle of
O2 and it was helpful in a handful of cases.
It was also during O2, I began working on the idea of a “grouper” (with
help from Deep Chatterjee, who implemented the feature for O3) that would
select the “superevent” out of a group of neighboring GW triggers for per-
forming EM/neutrino follow-up. Although a beta version of this software ran
during O2, it ran into problems regarding backlogging for loud GW triggers
as described next.
All of these upgrades rendered the former approval_processor obsolete,
although this came with its own drawbacks. We saw during the course of O2
that there were issues related to memory leakage (holding onto too many GW
and external triggers’ state information after many days to weeks of continuous
live time, even with built-in functionality to expire old triggers), becoming a
123
zombie process that would stop responding to incoming LVAlerts, and (in the
event of an interesting GW candidate event) getting backlogged by the unten-
able number of LVAlerts being generated for the group of GW triggers from
different low-latency search pipelines (most of them GraceDb log comments
that were not useful information to approval_processorMP for performing its
checks).
Still, with active monitoring of approval_processorMP, we were able to
successfully send the Initial LIGO/Virgo Notices with tremendously reduced
(∼30 minutes to hours) latencies for O2 compared to hours (sometimes days)
we saw in O1. Our efforts were well repaid with the first ever detection of
a binary neutron star coalescence, GW170817, observed on August 17, 2017
(Abbott et al., 2017a), followed by its short gamma-ray burst counterpart,
GRB 170817A, ∼1.7 s later detected by Fermi/GBM (the Gamma-ray Burst
Monitor) (Goldstein et al., 2017) and INTEGRAL/SPI-ACS (the spectrometer
anti-coincidence shield) (SPI-ACS) (Savchenko et al., 2017). There was also
an unprecedented amount of EM/neutrino follow-up generated for GW170817,
that led to the discovery of its kilonova, X-ray, and radio counterparts (Fig-
ure 6.3).
Thus, this chapter details additional information approval_processorMP
sent to our observing partners during O2, to aid EM/neutrino follow-up efforts.
6.1 Information Sent to MOU Partners
EM-Bright Source Classification
For O2, a new low-latency source classification pipeline, EM-Bright, was
created for CBC triggers. The purpose of EM-Bright was to provide our ob-
serving partners with two probabilities: one regarding the presence of at least
one neutron star in the binary system (ProbHasNS), and the other regarding
124
the presence of remnant disk mass outside the final black hole (ProbHasRem-
nant). The motivation for creating EM-Bright was to generate more follow-up
for compact binaries with neutron stars as they are more likely to be accom-
panied by an electromagnetic/neutrino counterpart. For instance, neutron
star-black hole (NS-BH) mergers and binary neutron star (BNS) mergers are
likely progenitors of short gamma-ray bursts if the neutron star tidally dis-
rupts and a hot, massive (∼few percents of 1 M) accretion disk is formed
around the remnant black hole.
The earliest CBC trigger information available for producing source clas-
sification information are the point estimates of the masses (m1, m2) and
aligned spin components (χ1, χ2) of the two objects in the binary, with m1 ≥
m2. The point estimates are provided by the low-latency search pipelines via
the waveform template that triggered to give the lowest false alarm rate dur-
ing the search. However, point estimates have associated uncertainties and are
expected to be offset from the true component values. Thus, the EM-Bright
pipeline constructs an ambiguity ellipsoid around the triggered point estimate
using an effective Fisher formalism (Cho et al., 2013). Each ambiguity ellip-
soid is constructed in the three-dimensional (Mc, η, χ1) parameter space where
Mc = (m m )3/51 2 /(m 1/51 + m2) is the chirp mass and η = m m /(m + m )21 2 1 2
is the symmetric mass ratio (Pannarale & Ohme, 2014). Additionally, the
ambiguity ellipsoids are populated with 1000 points, i.e., ellipsoid samples, to
enclose a region of 90% match within its boundary for a total of 1001 ellipsoid
samples including the original point estimate. The twofold source classification
probabilities are computed for each ellipsoid sample.
For the first classifer, ProbHasNS, only the mass of the secondary object,
m2, is required. Simply, EM-Bright checks whether m2 < 2.83 M where 2.83
M is the maximum neutron star mass allowed by the very stiff equation of
125
state, 2H EOS (Kyutoku et al., 2010, 2011), to err on the side of having more
counterparts than not. ProbHasNS is the fraction of ellipsoid samples with
this property.
The second classifier, ProbHasRemnant, is model-dependent and poten-
tially requires more parameters than just the secondary mass. For instance, if
the secondary mass indicates that the system is a BBH, i.e., m2 > 2.83 M,
the system and all its ellipsoid samples are immediately classified as EMdark,
i.e., ProbHasRemnant is set to 0%. On the other hand, if the primary mass
indicates that the system is a BNS, i.e., m1 ≤ 2.83 M, the system and all
its ellipsoid samples are classified as EMbright, i.e., ProbHasRemnant is set to
100%, and we would highly recommend EM follow-up if the GW trigger is
promoted to GW candidate event status.
In the case of NS-BH binaries, the EM-Bright pipeline adopted Foucart’s
fitting formula1 (Foucart, 20(12) to)calcu(late the r)emnant disk mass, mrem:1/3
mrem ≈ 3m1 2m2 risco0.228 1− − 0.148
m2,b m2 r2 r2
wherem2,b is the baryon mass of the neutron star, m1 andm2 are the masses of
the black hole and neutron star, r2 is the radius of the neutron star calculated
using the 2H EOS, and risco is the innermost stable circular orbit of the black
hole which depends on the black hole’s dimensionless spin parameter, χ 2bh :
Z1 = √ [ ]1 + (1− χ2bh)1/3 (1 + χ )1/3bh + (1− χ )1/3bh ,
Z2 = 3χ2bh + Z
2
1),
risco √
= 3 + Z2 − sign(χbh) (3− Z1)(3 + Z1 + 2Z2).
m1
1In Foucart’s fitting formula, Newton’s gravitational constant G, and the speed of light
c, are both equal to 1.
2In our implementation of the EM-Bright pipeline, we assume that the spin of the
black hole, χbh, is parallel to the orbital angular momentum. Similarly, our waveform
models assume that the spins of the compact objects are aligned with the orbital angular
momentum. Thus, χbh = χ1.
126
Thus, for each ellipsoid sample, mrem is computed and ProbHasRemnant
is the fraction of ellipsoid samples for which mrem is greater than zero. This is
because although remnant disk masses ∼0.03M could launch a GRB engine,
masses as low as ∼0.01 M are thought to be enough to generate a kilonova.
Thus, we take a conservative lower limit of 0 M.
10.0
EMbright; BNS region
EMbright; χBH ≤ 0.00
EMbright; χBH ≤ 0.50
EMbright; χBH ≤ 0.80
EMbright; χBH ≤ 0.90
EMdark; χBH ≤ 0.90
EMdark; BBH region
GW170817 ellip. sam.
GW170608 ellip. sam.
2.83
χ χ
B B
H H
= =
0 0.0 .5
1.0
1.0 2.83 10.0 100.0
m1[M�]
Figure 6.1: Different regions of the ellipsoid sample component mass param-
eter space. Foucart’s fitting formula is applied for ellipsoid samples in the
pink and green shaded NS-BH region. In particular, the χ1-dependent green
shaded regions reflect boundaries where ellipsoid samples give non-zero rem-
nant disk mass. Additionally, ellipsoid samples for GW170817 (red dots in the
cyan BNS/100% EMbright parameter space) and GW170618 (purple stars in
the grey BBH/0% EMbright parameter space) are shown. (Figure from Deep
Chatterjee/Abbott et al. (2019))
Qualitatively speaking, Foucart’s fitting formula shows that the more
symmetric the masses are, and the more the black hole spin component (which
affects risco) is aligned with the orbital angular momentum (i.e., the higher the
m2[M�]
9
0.
=
H
χB
0.
8
=
H
χB
�
6.0
M
=
Mc
M�
3.5
=
Mc
�
1.5
M
=
Mc
127
value of χ1), the more likely it is that the neutron star will be sufficiently
tidally disrupted (Figure 6.1). For instance, an ellipsoid sample with masses
(7, 2) M will give non-zero remnant disk mass according to Foucart’s fitting
formula if the value of χ1 is slightly greater than 0.5, but no remnant disk
mass below this value.
During O2, EM-Bright provided ProbHasNS and ProbHasRemnant source
classification with a latency of a few minutes for CBC GW candidate events.
This information was included in the machine-readable LIGO/Virgo Notices
and in the human-prose LIGO/Virgo Circulars with a link to the EM-Bright
technical document3.
Three-Dimensional Sky Localization Probability Maps
For O2, we also sent our observing partners distance-resolved three-
dimensional BAYESTAR skymaps for CBC gravitational-wave candidate events.
This requires calculating th∫e marginal posterio∫r:
P (DGW|ϑ)P (ϑ)
P (r, α, δ|DGW) = P (ϑ|DGW) dλ = dλ,
P (DGW)
where λ are all the other parameters, i.e., “nuisance parameters”, excluding r,
α, and δ in the table of waveform parameters, ϑ (Table 4.1). This marginal
posterior distribution can be written as the product of the two-dimensional
(direction information only) skymap and the conditional distance distribution:
P (r, α, δ|DGW) = P (r|α, δ,DGW)P (α, δ|DGW),
which in O2 is computed using the matched-filter SNR time series (the cross-
correlation sequence between the detector output and the template) versus the
triggered template’s auto-correlation sequence used in O1.
3https://dcc.ligo.org/public/0139/T1600571/010/Description_Document.pdf
128
However, the conditional distance distribution can be approximated using
our intuition that along a given line of sight, the probability per unit distance
(assuming that the GW source is in this direction) will be unimodal4 and
well-fit by a Gaussian ansatz (Singer et al., 2016):
(r−µ̂(α,δ))2
P (r|α, δ,D ) ≈ √N̂(α, δ) −e 2σ̂2GW (α,δ) r2,
2πσ̂(α, δ)
where N̂(α, δ), µ̂(α, δ), and σ̂(α, δ) are the direction-dependent normalization
coefficient, location parameter, and scale parameter, respectively for r ≥ 0.
(The r2 at the end ensures that the probability density per unit volume van-
ishes at the origin.) The location parameter and its scale (i.e., standard de-
viation) of the ansatz distribution are calculated by fitting the ansatz to the
true marginal posterior distribution using the method of moments.
Thus, similarly to O1, the three-dimensional (direction plus distance in-
formation) BAYESTAR skymaps are packaged into Flexible Image Transport
System (FITS) files which are backwards compatible with software that read
O1 style two-dimensional skymaps. Each skymap contains four columns in
total (Table 6.1) such that the two-dimensional (direction information only)
column, ρi, and the probab∑ilities along all given lines of sight are normalized:N−1
ρi = 1, and
∑i=0N−1∫ ∞
P (r,ni) dr = 1,
i=0 0
where N is the total number of pixels, n is the direction of the ithi pixel,
and P (r,ni) is the approximate marginal posterior probability distribution in
spherical polar coordinates:
(r−µ̂ )2N̂i − i2
P (r,n 2σ̂ 2i) = ρi√ e i r .
2πσ̂i
4The SNR of the GW candidate event is a degenerate combination of luminosity distance
plus binary inclination angle. The unimodality of the distance comes the broad, universal
distribution of the binary inclination angle arising from the Malmquist bias.
129
Column Information
1. ρi probability contained in pixel i (i.e., the two-dimensional skymap)
2. µ̂i mean of location distance in direction of pixel i
3. σ̂i standard deviation of location distance in direction of pixel i
4. N̂i normalization coefficient for pixel i
Table 6.1: Information columns in three-dimensional BAYESTAR skymaps.
When the three-dimensional skymaps are plotted in totality, they have
a non-trivial geometry for candidate events from the two Advanced LIGO
detector network. The two long, thin sections of a great circle that are typical
of two-dimensional skymaps now become two thin, rounded, slightly oblique
petals in three dimensions. The forked-tongue morphology that arises from
binary inclination angle degeneracy when the binary’s orbital plane is nearly
face-on towards the Earth now becomes narrow crevices that run along the
outer edges of the petals. All in all, these two-detector three-dimensional
BAYESTAR skymaps end up looking like jacaranda tree seeds or space potato
chips. In the three-detector Advanced LIGO plus Advanced Virgo network,
one can imagine the two-dimensional islands of directional probabilities turning
into spindles when including distance information.
When these three-dimensional skymaps are used to follow-up on binary
neutron star candidate events, they are helpful in multiple ways. First, we can
reduce the area to be searched over because the nearness of the event to the
local Universe allows us to combine the skymap with a galaxy catalog (e.g.,
the Galaxy List for the Advanced Detector Era5 (Dálya et al., 2018)). This
way, nearby galaxies with consistent redshifts can be targeted for follow-up.
Second, expanding on the first idea, we can also minimize the total exposure
time required to observe every galaxy in the 90% credible volume by applying a
flux limit and taking into consideration the type of instrument used for follow-
5http://aquarius.elte.hu/glade
130
up (e.g., large versus small field of view, etc.) (Singer et al., 2016). Lastly,
distance estimates allow us to exclude false positive transient electromagnetic
counterpart candidates.
6.2 The Advanced Virgo Detector
Starting August 1, 2017, Advanced Virgo, the European ground-based
interferometric GW detector in Cascina, Italy with 3 km-long arms (Acernese
et al., 2015), joined the Advanced LIGO detector network for the last month
of data acquisition. Because Advanced Virgo was still being commissioned
during most of its run for O2, the low-latency search pipelines set lower SNR
thresholds for triggering using its data (Table 4.3), or did not use it at all in the
case of PyCBC Live. However, once a GW trigger made it to the candidate
event stage, Advanced Virgo data were used in the post-processing skymap
generation. For the real events GW170814 and GW170817, Advanced Virgo
played a critical role in improving the localization down to ∼tens of square
degrees for the 50% confidence regions. In the case of GW170817, the binary
neutron star coalescence came from a direction near a node of the Advanced
Virgo detector. The weak signal there was still enough to help constrain the
localization and break down the degeneracy using data from the two Advanced
LIGO detectors only (Figure 6.2).
131
Figure 6.2: BAYESTAR skymaps for GW170817 in ICRS coordinates (Moll-
weide projection) from a 1-detector network (top, Advanced LIGO/Hanford),
2-detector network (center, Advanced LIGO), and 3-detector network (bot-
tom, Advanced LIGO and Advanced Virgo). The 50% confidence region and
the location of the host galaxy NGC 4993 (marked with a star) are shown.
132
500
400 LIGO - Virgo Fermi/GBM t-tc SALT(days) ESO-NTT
300 1.2 SOAR
ESO-VLT
200 7000o
1.4
INTEGRAL/SPI-ACS
100
2.4
50 4000
o
 -12 -10 -8 -6 -4 -2 0 2 4 6 4000 6000 10000 20000
t-tc (s) wavelength (A
o )
GW
LIGO, Virgo
γ-ray
Fermi, INTEGRAL, Astrosat, IPN, Insight-HXMT, Swift, AGILE, CALET, H.E.S.S., HAWC, Konus-Wind
X-ray
Swift, MAXI/GSC, NuSTAR, Chandra, INTEGRAL
UV
Swift, HST
Optical
Swope, DECam, DLT40, REM-ROS2, HST, Las Cumbres, SkyMapper, VISTA, MASTER, Magellan, Subaru, Pan-STARRS1, 
HCT, TZAC, LSGT, T17, Gemini-South, NTT, GROND, SOAR, ESO-VLT, KMTNet, ESO-VST, VIRT, SALT, CHILESCOPE, TOROS, 
BOOTES-5, Zadko, iTelescope.Net, AAT, Pi of the Sky, AST3-2, ATLAS, Danish Tel, DFN, T80S, EABA
IR
REM-ROS2, VISTA, Gemini-South, 2MASS,Spitzer, NTT, GROND, SOAR, NOT, ESO-VLT, Kanata Telescope, HST
Radio
ATCA, VLA, ASKAP, VLBA, GMRT, MWA, LOFAR, LWA, ALMA, OVRO, EVN, e-MERLIN, MeerKAT, Parkes, SRT, Effelsberg
-100 -50 0 50 10-2 10-1 100 101
t-tc (s) t-tc (days)
1M2H Swope DLT40 VISTA Chandra 
10.86h i 11.08h h 11.24h YJKs 9d X-ray
MASTER DECam Las Cumbres J VLA
11.31h W 11.40h iz 11.57h w 16.4d Radio
Figure 6.3: Electromagnetic follow-up of the first observed binary neutron star
coalescence event, GW170817. This is also the first multi-messenger event
involving gravitational waves. (Figure from Abbott et al., 2017b)
frequency (Hz)
counts/s (arb. scale)
normalized Fλ
133
Chapter 7
Cosmic Strings
Between the second and third observing runs, I joined LIGO and Virgo’s
Burst cosmic strings analysis group, consisting of myself, Imène Belahcene,
Kipp Cannon, Florent Robinet, and Daichi Tsuna. We performed a matched
filter based search looking for GW bursts from cosmic string cusps. In this
chapter I present our goals and search methods, and include our results from
analyzing O2 data, which are officially included in the “O2 Burst All-Sky
Paper” titled All-sky search for short gravitational-wave bursts in the second
Advanced LIGO and Virgo run (Abbott et al., 2018a).
7.1 Basic Properties
The Universe has gone through several phase transitions since the Big
Bang, leaving behind clues such as the Cosmic Microwave Background (CMB),
created by photons scattering off hot, dense, ionized matter (i.e., the last
scattering surface) before the Universe cooled down sufficiently enough for
electrons to recombine into atoms, and for photons to cease scattering and
propagate freely. Thus, while the CMB provides a snapshot of the Universe
∼377,000 years after the Big Bang, it is the limit of an electromagnetic probe
of the history of the Universe. Gravitational waves, on the other hand, may
penetrate through this last scattering surface, to provide clues about large-
scale mass and energy transitions and distributions in the early Universe. In
this respect, we are motivated to study and search for gravitational waves from
a more speculative source known as cosmic strings.
134
Strings vs. Superstrings
There are two viable contexts in which cosmic strings arise. In the
framework of Grand Unification Theories (GUTs), they are linear topologi-
cal defects (similar to vortex filaments in superfluid helium) formed during
the grand unification epoch (or any axial or cylindrical symmetry-breaking
phase transition of the early Universe) (Sakellariadou, 2007). In the frame-
work of String Theory, they are called cosmic superstrings, which are coherent
macroscopic states of fundamental F-strings and Dirichlet D-strings (Copeland
et al., 2004). Strings and superstrings have different intercommutation proba-
bilities, p, which is the probability that a string/superstring will swap partners
or chop itself off/form a loop when intersecting with itself or another string/-
superstring. Cosmic strings from GUTs have p = 1 while cosmic superstrings
have p < 1.
If they exist, cosmic strings and superstrings are topologically stable ob-
jects with finite widths that are less then a trillion times smaller than the radius
of a hydrogen atom. Thus, their large-scale dynamics can be described using a
zero-width approximation known as the Nambu-Goto action in a low-density
(or otherwise empty) Universe (Copeland & Kibble, 2009). However, strings
also have a string tension, Gµ (c = 1, where µ is the mass per unit length),
and due to their cosmological sizes, emit gravitational waves in a number of
different ways when they intersect/interact with one another and themselves.
135
Figure 7.1: Types of cosmic string intersections where the intercommutation
probability, p, is assumed to be 1. From top to bottom: string-string inter-
section at one point (two new long strings are formed via partner exchange),
string-string intersection at two points (two new long strings are formed via
partner exchange plus one closed loop), and self-string intersection (one long
string and a closed loop are formed). (Figure from Sakellariadou, 2007)
Loops, Cusps, and Kinks
Loops are created when strings interact with themselves or each other
(Figure 7.1), providing several gravitational-wave signatures (and a way for
strings to lose energy/not dominate the energy density of the Universe). Dur-
ing the “looping off” process, discontinuities appear along the tangent vector of
the original string(s), which are called kinks. The loops that are created also
oscillate periodically under their own tension, typically creating cusps, which
are points along the loop that accelerate to momentarily reach the speed of
light (Figure 7.2). The density of cusps in a network of strings depends on the
strings’ intercommutation probability, p. We consider for the O2 Burst cosmic
strings analysis, gravitational waves emitted from individual cusps, which are
136
well-modeled and can be searched for with matched filter templates (Damour
& Vilenkin, 2000).
����������
���������
�������
���������
����������
�
�
�
�
Figure 7.2: Time evolution (dotted black line) of a point along a string (red
dot) that becomes a cusp at time τ = y, starting with a string intersecting
itself at time τ = y − 2δ. (Figure from Stott et al., 2017)
7.2 Cosmic String Cusps Search Algorithm
Gravitational waves from cosmic string cusps follow a f−4/3 power law at
high frequencies:
h(f) = A(z, l, Gµ)f−4/3Θ(fh − f)Θ(f − fl)
in the frequency domain, where Θ is the Heaviside function, and fl/fh are
the low/high frequency cutoffs of the gravitational-wave signal (Stott et al.,
2017). In practice, the low frequency cutoff is determined by the sensitivity
of the gravitational-wave detectors, ∼10 Hz due to seismic noise, and the
high frequency cutoff is determined by the angle between the line of sight of
the observer and the direction of the beamed gravitational-wave signal. The
�
�
137
amplitude, A, is a function of the redshift-dependent distance to the source
r(z), string length l, and string tension Gµ.
To perform the analysis, we divided O2 data into 6 separate chunks, each
one of duration 1−2 months, where chunk boundaries were positioned to co-
incide with significant detector maintenance breaks (Table 7.1). 31 templates
with varying high frequency cutoffs, 30 Hz < fh < 4096 Hz, were used for the
matched filtering. The lowest high frequency cutoff of 30 Hz corresponds to the
maximum angle between the line of sight and the beamed gravitational-wave
signal.
Chunk Start End
28 1164499217 1166486417
29 1167523218 1170547218
30 1170547218 1178323218
31 1179792018 1183420818
32 1183420818 1185580818
33 1185580818 1187740818
Table 7.1: O2 data were divided into 6 chunks for the Burst cosmic strings
analysis. All start and end times are in GPS time.
The search detection pipeline works as follows. Single-detector triggers
are recovered from each detector data set and then combined when performing
the coincidence search. In zero-lag, detector data are shifted temporally within
the bounds of the gravitational-wave travel time between detectors, allowing
us to find candidate events. When time slides offset the detector data with
times greater than the GW travel time between detectors, accidental noise
coincidences are found. A multivariate log-likelihood ratio is then computed
for each coincident trigger (candidate events and noise events) to be used for
ranking.
We ran two parallel analyses twice using O1 reviewed software and O2
138
updated software. For the first parallel analyses, C00 “raw” data (including
data from Virgo in Chunk 33) were used after CAT 11 data quality vetoes
were removed. The search determined that all 5 loudest events originated from
Chunk 33/Virgo, and found four useful data quality flags: two from the UPV
(Used Percentage Veto) data quality pipeline which uses statistical correlations
between witness sensor channels and the GW strain channel (similar to iDQ)
to determine noisy times, and one data quality flag each for control system
glitches and photodiode glitches.
The parallel analyses also showed slight discrepancies in the loudest re-
covered events from the two pipelines which allowed us to track down bugs
that had been fixed by hand during O1, and bugs introduced during updates
to the parent software repository LALsuite (LIGO Algorithm Library Suite).
Once these were fixed, for the second parallel analyses, we used C02 “clean”
data (excluding Virgo data) after CAT 1 and CAT 42 data quality vetoes were
removed.
1CAT 1 data quality flags determine times of critical issues with a key detector compo-
nent not operating in its nominal configuration.
2CAT 4 data quality flags exclusively remove transient hardware injections.
139
7.3 Search Results
From the second parallel analyses, we found a total of 69,554 zero-lag
cosmic string cusp candidate events (i.e., coincident triggers) from ∼107 s
of coincident time between the two Advanced LIGO detector data sets. The
highest ranked candidate event had a log-likelihood ratio of ∼9 (λmax ∼ 8178),
within 1σ of the expected background distribution (Figure 7.4a). To construct
the 5.8 ×1010 s ≈ 1, 846.7 yr of effective background, we used 6000 time slides
with ∼3.5 s offsets. We also investigated the three loudest triggers (Table 7.2)
with Omicron power scans and discovered they were all consistent with tomte
blip glitches3 (Figure 7.3).
Rank Detector Peak Time SNR Log-Likelihood
1 H1 1185584772.3114 3.82L1 1185584772.31958 9.43 9.01
2 H1 1175720634.21753 3.92L1 117572063.21716 9.01 8.03
3 H1 1175010906.68774 4.79L1 1175010906.68872 7.72 7.48
Table 7.2: The three loudest zero-lag cosmic string cusp candidate events
identified during O2. H1 and L1 stand for the Advanced LIGO/Hanford and
Advanced LIGO/Livingston detectors respectively.
We also estimate the sensitivity of our search pipeline by injecting a
few million random simulated signals of known amplitude A into the data.
The detection efficiency, e(A), is defined as the fraction of simulated signals
recovered with log-likelihood greater than ln λmax (Figure 7.4b).
7.4 Constraints
Lastly, even without a confirmed GW detection, we can constrain the
(Gµ, p) parameter space for two Nambu-Goto large loop distribution models
3Tomte blip glitches are a class of glitches with peak frequencies less than 100 Hz, named
after the Scandinavian mythological creatures, tomte, whose tall skinny hats resemble the
morphology of this glitch class in the TF domain.
140
Figure 7.3: An Omicron scan of the highest-ranked zero-lag cosmic string
cusp candidate event revealed it to be consistent with a tomte blip glitch in
Advanced LIGO/Livingston. Because we expect at least one candidate event
to have occurred by accident due to noise processes, results of the search are
consistent with the hypothesis that there are no signals present.
developed for topological strings (i.e., p = 1): the Blanco-Pillado et al. model
(Blanco-Pillado et al., 2014) and the Ringeval et al. model (Lorenz et al., 2010),
which were studied during the first observing run. Because it is unknown how
the loop densities scale with p for cosmic superstrings, we assume for our
purposes that they scale as 1/p.
Then, for each mo∫del, M, we can ∫write the effective detection rate:∞ ∞ d2R(M)
R(M)(Gµ, p) = e(A) dA× (A, z, f ∗;Gµ, p) dz,
0 0 dzdA
where e(A) is the detection efficiency curve using the log-likelihood of the loud-
est candidate event in the search, d2R(M)/dz dA (or equivalently, d2R(M)/dz dh,
h = Af−4/3), is the rate at which cosmic string cusps create GW bursts in a
loop distribution model M and f ∗ is the lowest high-frequency cutoff used in
the template searches (i.e., 30 Hz) (Abbott et al., 2018b). We can now con-
strain the parameter space of model M by excluding regions where cosmic
string cusps of a given string tension Gµ and intercommutation probability p
141
(a) Cumulative event rate as a function of the log-likelihood ratio ranking statistic.
The black line and shaded region are the expected background distribution (with
±1σ statistical error). The upper corners of the steps in the red line are the 69,554
zero-lag candidate events, all consistent with background.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 Injections recovered with logΛ > 9.01
0
10−21 10−20 10−19
1
Injection Amplitude (s− 3 )
(b) Search detection efficiency as a function of the cosmic string cusp signal ampli-
tude. The detection efficiency is defined as the fraction of simulated cusp events
recovered with log-likelihood greater than ∼9.
Figure 7.4: O2 Burst cosmic string cusp search results.
would have registered with our analysis pipeline with log-likelihoods as loud
as that of our most significant candidate event. The 95% confidence exclusion
Detection Efficiency
142
regions are shown in Figure 7.5. It is important to stress that the models
studied were derived for topological strings (p = 1) and therefore, only the
string tension of the Ringeval et al. model was constrained to be less than
∼4.2×10−10.
In the parameter space where we have chosen to report results (the
large loop regime), the Stochastic cosmic strings search, which looks for the
gravitational-wave background created by the superposition of multiple, un-
resolved cusps and kinks, is more sensitive and therefore places tighter con-
straints on the intercommutation probability and string tension.
143
1
O1+O2
O2
−1 O110
10
10−3
10−12 10−11 10−10 10−9 10 8 10−7 10 6
String Tension, Gµ
1
O1+O2
O2
O1
10−1
10−2
10−3
10−12 10−11 10−10 10−9 10−8 10−7 10−6
String Tension, Gµ
Figure 7.5: 95% confidence exclusion regions for cosmic string tension and
intercommutation probability from the LIGO and Virgo Burst cosmic strings
analysis group using O1 and O2 data for two large loop Nambu-Goto cos-
mic string distribution models. The excluded regions are below the respective
curves. At p = 1 for topological strings, we cannot put a constraint on the
string tension for the Blanco-Pillado et al. model (top, Blanco-Pillado et al.
(2014)). However, for the Ringeval et al. model (bottom, Lorenz et al. (2010)),
the string tension must be less than ∼4.2×10−10. (Figures from Florent Robi-
net, 2018)
Intercommutation probability, p Intercommutation probability, p
144
Chapter 8
En Route to O3: The Third
Observing Run
8.1 Low-Latency Gravitational Wave-Electromagnetic and Neutrino
Counterpart Coincidence Searches
For the third observing run, O3, I updated and built upon an analysis
pipeline known as RAVEN (the Rapid, on-source VOEvent Coincidence Moni-
tor), created by Alex Urban during the first and second observing runs, O1 and
O2. RAVEN’s responsibility is to perform low-latency searches for temporally
coincident external triggers and gravitational-wave candidate events. When
the coincidence involves a gamma-ray burst (GRB), RAVEN also computes the
false alarm rate (FAR) that this could be due to noise in the gravitational-
wave detectors being temporally and/or spatio-temporally coincident with a
real GRB.
Temporal Coincidence Searches
During O1 and O2, for each external trigger entering GraceDb, RAVEN
looked for gravitational waves in two time windows, [−5, +1] s and [−600,
+60] s, around the event time of the external trigger. The time windows
were selected based off models of compact binary mergers (Metzger & Berger,
2012) and supernova emissions, where time delays could be due to differences in
emission times and/or propagation speeds of the gravitational waves and elec-
tromagnetic/neutrino counterparts. Reciprocal searches were also performed1
1Reciprocal searches are necessary because of latencies in reporting either the external
trigger or the gravitational-wave candidate event to GraceDb.
145
for each new gravitational-wave candidate event entering GraceDb. These
searches looked for external triggers in time windows, [−1, +5] s and [−60,
+600] s, around the gravitational-wave event time.
For O3, I made RAVEN discern between the two types of external triggers
(GRBs versus supernova (SN) neutrinos) and two types of gravitational-wave
events (modeled/compact binary coalescences versus unmodeled/generic tran-
sient bursts) entering GraceDb. In general, we expect short GRBs to be coin-
cident with neutron star binary coalescences and SN neutrinos to be coincident
with SN gravitational-wave bursts. However, complications arise because long
GRBs could be associated with supernovae with rotating progenitors. Thus,
the searches are performed as follows: for a GRB external trigger reported by
Swift or Fermi, we look for modeled compact binary coalescences within [−5,
+1] s and for unmodeled bursts within [−600, +60] s of the external trigger
event time. For a neutrino reported by SNEWS (Supernova Early Warning
System), we look for unmodeled bursts within [−600, +60] s of the external
trigger event time. The reciprocal searches are as follows: for modeled com-
pact binary coalescences, we look for external triggers within [−1, +5] s of the
gravitational-wave event time, and for unmodeled burst transients, we look for
external triggers within [−60, +600] of the gravitational-wave event time.
Gravitational Wave-Gamma-Ray Burst Coincidences
In the event that RAVEN finds a coincidence involving a GRB, RAVEN can
compute a corresponding false alarm rate that the coincidence is in fact due
to noise in the gravitational-wave (GW) detectors being coincident with a real
GRB. The derivation for the FAR computation was first detailed and outlined
by Michał Was within a Bayesian framework (à la Ashton et al. (2018)) and
is re-derived here.
146
Suppose Fermi or Swift detects a GRB candidate in data set DGRB and
Advanced LIGO and/or Virgo detects a gravitational-wave (GW) candidate
in data set DGW. We want to compute the Bayes factor that compares the
common-source hypothesis HC (both GRB and GW detections are real and of
a common-source origin) to the signal/noise hypothesis HSN (GRB is real but
GW is noise):
C
B P (DGRB, DGW|H )C/SN(DGRB, DGW) = .
P (D SNGRB, DGW|H )
In the common-source hypothesis, the GRB and GW share common-
source parameters such as source direction/orientation, luminosity distance,
characteristic time of the event (emission times related by compact binary
merger models), etc. Thus, the∫likelihood must be computed as follows:
P (DGRB, DGW|HC) = ∫ P (D CGRB, DGW, θ|H )dθΘ
= P (DGRB, D |θ,HCGW )P (θ|HC)dθ,
ΘS
where ΘS is the region of parameter space where P (θ|HC) > 0.
The first part of the integrand, P (DGRB, DGW|θ,HC), can be rearranged
as follows:
P (DGRB, DGW|θ,HC)
= P (DGRB|DGW, θ,HC)P (D CGW|θ,H )
= P (DGRB|θ,HC)P (DGW|θ,HC)
P (D C C CGRB|H )P (θ|DGRB,H ) P (DGW|H )P (θ|D CGW,H )
=
P (θ|HC ,) P (θ|HC)
where we have just used Bayes’ Theorem2 in the second to third line. Substi-
2Bayes’ Theorem is:
| P (A ∩B) P (B|A)P (A)P (A B) = = .
P (B) P (B)
147
tuting this into the integral gives us:
P (D CGRB,DG∫W|H )
= ∫ P (DGRB, DGW|θ,HC)P (θ|HC)dθΘS
P (DGRB|HC)P (θ|D CGRB,H )P (DGW|HC)P (θ|D CGW,H )
= ∫ |HC dθΘS P (θ )
|HC |HC P (θ|D
C
GRB,H )P (θ|D CGW,H )
= P (DGRB )P (DGW ) dθ
ΘS P (θ|HC)
= P (D CGRB|H )P (DGW|HC)Iθ(DGRB, DGW),
where the integral in the last line is the posterior overlap integral which quan-
tifies the agreement in the posterior parameter distributions derived indepen-
dently from each data set.
Then, the Bayes’ factor comparing the common-source hypothesis to the
signal/noise hypothesis becomes:
C
B P (DGRB, DGW|H )C/SN(DGRB, DGW) =
P (D SNGRB, DGW|H )
P (D CGRB|H )P (D CGW|H )
= |HS |HN Iθ(DGRB, DGW).P (DGRB GRB)P (DGW GW)
However, in the special case of a true common-source astrophysical event,
we expect:
P (θ|HC) = P (θ|HSGRB) = P (θ|HSGW)
to be true, which leads to:
S
P (D CGRB/GW|H ) P (DGRB/GW|HGRB/GW)
= , i.e.,
P (D N NGRB/GW|HGRB/GW) P (DGRB/GW|HGRB/GW)
BC/N(DGRB/GW) = BS/N(DGRB/GW)
and
|HC P (D |HSP (DGRB/GW ) GRB/GW GRB/GW)
|HS = = 1 , i.e.,P (DGRB/GW GRB/GW) P (D SGRB/GW|HGRB/GW)
BC/S(DGRB/GW) = BS/S(DGRB/GW) = 1.
148
Thus, the Bayes’ factor comparing the common-source hypothesis to the
signal/noise hypothesis is:
B P (DGW|H
S
GW)
C/SN(DGRB, DGW) = Iθ(DGRB, DGW)
P (D NGW|HGW)
= BS/N(DGW)Iθ(DGRB, DGW).
The first part of the right hand side, BS/N(DGW), is proportional to
1/FARGW (the inverse FAR or IFAR) of the gravitational-wave signal because
P (D |HSGW ) is the rate at which true gravitational-wave astrophysical events
occur that are as loud (as determined by the analysis detection statistics) as
observed data DGW times some observing time ∆t while P (DGW|HN) is the
rate at which noise events are detected as GW candidate events (as loud as
the observed data DGW) times the same observing time ∆t.
The IFARGW takes the particular form that its probability density func-
tion (PDF) and cumulative distribution function (CDF) take the particular
forms:
k
P (IFARGW) = ∫IFAR2GW∞ k
P (IFARGW > IFAR∗GW) = dx
IFA∣∣R∗ x2
= −k ∣ GW∣∞ k=x ∗ ,IFAR∗ IFARGW GW
for some constant k3. This IFARGW PDF has the important property that it
is invariant under multiplication by a real, non-negative random variable, S,
3By definition, the expected number of triggers below FAR∗GW due to background is
FAR∗GW × the foreground time analyzed. Thus, the expected number of triggers above
IFAR∗GW due to background is k/IFAR
∗
GW where k is the foreground time analyzed.
149
whose expectation value, 〈S〉, is 1: ∫ ∞
R × S S 1P ( ) = P (IFARGW ) = ∫ P ( )P (IFARGW)−∞ |S|dS∞
S k 1= ∫ P ( ) dS−∞ IFAR2GW S∞ k
=
−∞∫P (S)(IFARGW × S SdS)2
k ∞ k
= R SP (S)dS = R .2 2−∞
This gives us the important result that if the expectation value of the
overlap integrals, 〈Iθ〉, is 1, the Bayes’ factor itself can be thought of as an
inverse FAR for the coincidence:
BC/SN(DGRB, DGW) ∝ IFARGWIθ(DGRB, DGW)
∝ IFARcoinc.
Thus, RAVEN can report a FAR for each coincidence it finds, which can be
interpreted as the rate at which noise in the GW detectors is coincident with
a real GRB:
FAR
FAR GWcoinc = I .θ(DGRB, DGW)
Intuitively, this agrees with common sense. The coincidence becomes
more significant (i.e., the value of the coincidence FAR decreases) if parame-
ters describing the astrophysical event from the GRB and GW data streams
independently agree (i.e., the overlap integral is large). The following subsec-
tions detail exactly how the overlap integral is computed and double-check the
condition that its expectation value is 1.
Temporal Coincidence False Alarm Rates
The RAVEN temporal coincidence inverse false alarm rate takes the form:
IFARcoinc = IFARGWIt(DGRB, DGW)
150
which depends o∫n the temporal posterior overlap integral:
I P (t
C C
GW|DGRB,H )P (tGW|DGW,H )
t = ∫ |HC dtGWΘS P (tGW )
P (t |D S SGW GRB,HGRB)P (tGW|DGW,H= GW)dtGW.
ΘS P (t
S
GW|HGRB/GW)
The temporal posterior overlap integral can be simplified because all
gravitational-wave candidate events entering GraceDb have an observed event
time, t̂GW, which gives:
P (t SGW|DGW,HGW) = δ(tGW − t̂GW)
and
S
I P (tGW = t̂GW|DGRB,HGRB)t = .
P (tGW = t̂ SGW|HGRB/GW)
However, the time of the GRB observation, t̂GRB, does not directly infer
t̂GW, the time of the gravitational-wave candidate event. Instead, the two
observation times are related by an astrophysical model which gives:
∆t = tGRB − tGW
and ∫
P (t S SGW|DGRB,HGRB) = P (tGW + ∆t|DGRB,HGRB)P (∆t)d∆t.
One of the built-in assumptions for RAVEN is that the GRB will come
in some time within [∆tmin, ∆tmax] s ([−1, +5] s for modeled compact bi-
nary coalescences and [−60, +600] s for unmodeled burst transients) of the
gravitational-wave event time. Thus, P (∆t) is the uniform distribution:
∆tmaxP (∆t) = U∆tmin (∆t).
151
Because the PDF for the GRB event time is also a delta function, we
have: ∫
max
P (tGW|D SGRB,HGRB) = δ(∆t+ tGW − t̂ ∆tGRB)U∆tmin (∆t)d∆t
max
= U∆t∆tmin (t̂GRB − tGW).
This gives us:
max
I U
∆t
∆tmin (t̂GRB − t̂GW)
t =
P (tGW = t̂GW|H
,
S
GRB/GW)
where the denominator is the uniform distribution:
P (tGW|HS
GRB
GRB) = U
∆t
0 (tGW),
where ∆tGRB is the time between independent GRB discoveries by Swift and
Fermi4.
Thus, the temporal posterior overlap integral is:
U∆t
max
I ∆tmin (t̂GRB − t̂GW)t = ∆tGRBU0 (tGW)
∆tGRB ∆tGRB
=
∆tmax −∆tmin = ∆tsearch
1
= .
RGRB∆tsearch
If the expectation value of the temporal coincidence statistic is 1, i.e.,
〈S〉 = 〈1/(RGRB∆tsearch)〉 = 1, we can safely multiply It with IFARGW and in-
terpret the product as the temporal coincidence IFAR. And indeed it is! There
are only two values of S (0 when there is no coincidence and 1/(RGRB∆tsearch)
when there is, with a coincidence ocurring with probability RGRB∆tsearch),
giving us:
〈S〉 1= 0 · (1− RGRB∆tsearch) + GRB search
RGRB
· R ∆t = 1.
∆tsearch
4In practice, ∆tGRB is computed as 1/RGRB, where RGRB = 0.807/(60 × 60 × 24 s) is
the empirical combined rate of independent GRB discoveries by Swift and Fermi.
152
Thus, the temporal coincidence FAR is:
FAR = FAR RGRB∆tsearchcoinc GW .
If the O3 RAVEN pipeline had been available and running during O2, a co-
incidence would have been found between the single-detector modeled compact
binary coalescence candidate event G298048 (which later became GW170817)
and GRB 170817A. The search window would have been 6 seconds. Thus, the
reported temporal coincidence FAR would have been:
× −12 × 0.807FARcoinc = 3.478 10 Hz
60× × × 6 s60 24 s
≈ 1.95× 10−16 Hz = 1 per 1.63× 108 yr.
Spatio-Temporal Coincidence False Alarm Rates
When sky localization probability maps (skymaps) are available from both GW
and GRB data sets, RAVEN can also compute the spatio-temporal coincidence
FAR. We start with the inverse false alarm rate which takes the form:
IFARcoinc = IFARGWIt,Ω(DGRB, DGW)
= IFARGWIt(DGRB, DGW)IΩ(DGRB, DGW),
where the posterior overlap integral cleanly factorizes into a temporal part
and a spatial part. The temporal posterior overlap integral was calculated in
Subsection 8.1 and t∫he spatial posterior overlap integral is:
I P (Ω|D ,H
C
GRB )P (Ω|D CGW,H )
Ω = ∫ C dΩΘS P (Ω|H )
P (Ω|D ,HSGRB GRB)P (Ω|DGW,HS= GW)
ΘS P (Ω|HS
dΩ.
GRB/GW)
We assume for the sake of simplicity a uniform all-sky prior, although
in reality, this is not strictly true. For instance, instruments aboard Fermi
153
do not take data while transiting the South Atlantic Anomaly (SAA) and the
advanced ground-based interferometric GW detectors are sensitive to direction
(Section 3.5). Then, the spatial overlap integral takes the continuous and
discrete forms: ∫
IΩ = 4π P (Ω|D S SGRB,HGRB)P (Ω|DGW,HGW)dΩ
Θ∑SNpix
= Npix skymapGRB[i]× skymapGW[i],
i=1
where the GRB and GW skymaps are assumed to have the same pixel resolu-
tion5.
Now, the value of IΩ can range from 0 to infinitely high depending on
how well the GRB and GW spatial posteriors overlap and are localized. It is
not obvious a priori that the expectation value of IΩ will be 1, and until this
is checked, we cannot report the spatio-temporal coincidence FAR. Thus, for
each low-latency GW search pipeline (GstLAL, PyCBC Live, MBTAOnline, cWB,
and oLIB), 100 low-latency GW skymaps produced from its candidate events
were combined with 100 Fermi Gamma-ray Burst Monitor (GBM) skymaps
to produce 10,000 overlap integrals (Table 8.1).
As can be seen, all pipelines are within ∼1σ of a mean value approxi-
mately equal to 1, except for GstLAL which is ∼3.5σ away. However, even this
systematic shift in the spatial overlap integrals using GstLAL GW candidate
event skymaps is contained at ∼8.5%.
Thus, because 〈IΩ〉 ≈ 1, the spatio-temporal coincidence FAR is:
FAR RGRB∆tsearch
FAR GWcoinc = I .Ω
5Npix is the total number of pixels in a skymap. When the pixel resolutions differ, we
match the GRB skymap resoluti∑on to the GW skymap and normalize its probabilities:Npix
I i=1 sk∑ymapGRB[i]× skymapGW[i]Ω = Npix N .pix
i=1 skymapGRB[i]
154
√
Pipeline IΩ σ(IΩ)/ 104
GstLAL 1.085 0.024
PyCBC Live 1.017 0.033
MBTAOnline 0.995 0.025
cWB 1.019 0.027
oLIB 0.949 0.049
Table 8.1: Summary of 50,000 spatial overlap integrals, IΩ, reporting the
mean and standard deviation of the mean for 100 low-latency GW skymaps
per pipeline combined with 100 Fermi/GBM GRB skymaps.
To continue our example of the single-detector GW candidate event G298048
coincidence with GRB 170817A, we compute the spatial overlap integral. Us-
ing the skymap from Fermi/GBM6 and the low-latency skymap produced by
BAYESTAR with only LIGO/Hanford data, we compute:
× −12 × 0.807FARcoinc = 3.478 10 Hz × × ×
1
6 s×
60 60 24 s 2.243
≈ 8.69× 10−17 Hz = 1 per 3.66× 108 yr.
Joint GW-GRB Sky Localization Probability Maps
We calculate the spatial overlap integral, IΩ, in Subsection 8.1 using two
skymaps: one from Fermi/GBM for the GRB localization and one from Ad-
vanced LIGO and Advanced Virgo for the GW candidate event localization.
For O3, these same two skymaps are used to provide our astronomer partners
with a combined skymap.
As an example, in Figure 8.1, the low-latency localization pipeline BAYESTAR
used data from both Advanced LIGO detectors to localize GW1708177. Then,
Fermi/GBM provided localization for GRB 170817A. In this case, IΩ = 10.39,
which is greater than one, but not significantly large due to uncertainties in
6https://gammaray.nsstc.nasa.gov/gbm/science/grbs/grb170817a/gbuts_healpix_
systematic.fit
7https://dcc.ligo.org/public/0146/G1701985/001/BAYESTAR_no_virgo.fits.gz
155
both skymaps. The normalized product of these two skymaps reveals that
although the GW170817 BAYESTAR skymap is bimodal with two long, thin is-
lands of probability, the overlap involves only the island from the Northern
antenna pattern.
156
Figure 8.1: From top to bottom, skymaps in ICRS coordinates (Mollweide
projection) with 90% and 50% credible regions for GW170817 (computed by
BAYESTAR using Advanced LIGO data only), GRB 170817A from Fermi/GBM,
and their normalized product. The location of the apparent host galaxy NGC
4993 is marked with a star in the joint GW-GRB skymap.
157
8.2 P_astro: The Probability of Astrophysical Origin
Currently, there is an effort to provide our observing partners with a new
data product from the P_astro pipeline, which computes the probability that
a CBC GW candidate event is of astrophysical origin, accounting for both
foreground GW trigger rate distribution and background trigger rate distribu-
tion. It uses a multicomponent/extended FGMC (Farr-Gair-Mandel-Cutler)
method (Kapadia et al., 2018), explained below. Specifically, the pipeline
looks at four astrophysical regions, α, of a low-latency search pipeline’s tem-
plate bank parameter space: regions of binary neutron stars (BNS), neutron
star-black holes (NS-BH), binary black holes (BBH), and mass-gap binaries
(binaries involving at least one compact object of masses between 3M and
5M).
The original FGMC method applies Poisson counting statistics to con-
struct a two-component posterior on expected trigger counts from astrophys-
ical (Λ1) and terrestrial (Λ0) sources during a set observing time (Farr et al.,
2015). There is a built-in assumption that the search pipeline’s ranking statis-
tic/SNR threshold for triggering is set low such that the number of background
triggers vastly exceeds the number of astrophysical triggers, and the expected
trigger counts, Λ0,1, for each type of trigger follows counting statistics:
P (k|Λ ) ∝ Λk e−Λ0,10,1 0,1 .
Then, the two-component posterior can be written as:
∏N
P (Λ0,Λ1|{x1, x2, . . . , xN}) ∝ P (Λ0,Λ1) (Λ0b(xj) + Λ f(x ))e−Λ0−Λ11 j ,
j=1
where N is the observed number of candidate events above threshold, xi are
the ranking statistic/SNR of the candidate events, P (Λ0,Λ1) is the prior on
the expected counts, and b(xi) and f(xi) are the background and foreground
158
probability density functions (i.e., models) evaluated at the observed xi (i.e.,
b(xi) = P (xi|noise) and f(xi) = P (xi|signal)).
In the multicomponent/extended FGMC method, the foreground triggers
are split into their α categories such that there are multiple, source-specific
foreground trigger distributions, f~(x) = fα(x), where x is generalized to con-
tain more information about the trigger’s properties than just the ranking
statistic/SNR (now denoted as L). The corresponding astrophysical origin
expected trigger count is also split into source-specific categories, Λ~ 1 = Λα.
Then, the multicomponent posterior can be written as:
∏N
~
P (Λ ~0,Λ1|{x1, x2, . . . , xN}) ∝ P (Λ0,Λ~ 1) (Λ b(x ) + Λ~ · f~(x ))e−Λ0−Λ1·~u0 j 1 j ,
j=1
where ~u is the corresponding unit vector for each source-specific class we are
interested in (i.e., for 4 astrophysical categories of interest, the multiplicative
factor at the end must be e−Λ0−Λ1−...−Λ4).
In practice, for the GstLAL low-latency search pipeline where P_astro
has been implemented, the foreground trigger distributions, f~α(x), can be ap-
proximated using conditional probability and by dividing the search pipeline’s
template bank parameter space into multiple bins (denoted as m):
f~(x) = fα(x) = P (L,m|α) = P (L|m,α)P (m|α) ≈ P (L|m, signal)P (m|α),
where L is GstLAL’s ranking statistic (the likelihood ratio), m is the template
bin number, P (m|α) are the template weights (i.e.,Wα(m)), and P (L|m, signal)
are the bin-dependent foreground trigger probabilities. Likewise, the back-
ground trigger distribution also has a bin-dependent form:
b(x) = P (L,m|noise) = P (L|m, noise)P (m|noise),
where P (m|noise) are the noise template weights (i.e., W0(m)).
159
This allows us to re-write the multicomponent posterior more compactly,
using source-specific Bayes factors for a trigger that registers with x’s ranking
statistic value L:
~ ~
K~
f(x) P (L|m, signal)W1(m)
1(x) = = ,
b(x) P (L|m, noise) W0(m)
P (Λ ,Λ~0 1)|~x) = P (Λ0,Λ~ 1|{∏x1, x2, and . . . , xN})N
∝ P (Λ ,Λ~ ) (Λ + Λ~ ·K~ (x ))e−Λ0−Λ~ 1·~u0 1 0 1 1 j .
j=1
The conditional probability that an event with properties x comes from
the αth astrophysical source category is given by:
P (Λ , x|Λ ,Λ~ Λαfα(x) ΛαKα(x)α 0 1) = = .
Λ0b(x) + Λ~ 1 · f~1(x) Λ0 + Λ~ 1 ·K~ 1(x)
Then, marginalizin∫g over the posterior for the expected counts gives us:∞
P (Λα, x|~x) = ∫ P (Λα, x|Λ ,Λ~0 1)P (Λ0,Λ~ 1|~x) dΛ dΛ~0 10∞ Λαfα(x)
= ∫ P (Λ ,Λ~ |~x) dΛ dΛ~~ ~ 0 1 0 10 Λ0b(x) + Λ1 · f1(x)∞ ΛαKα(x)
= P (Λ0,Λ~ |~x) dΛ dΛ~~ · ~ 1 0 1
.
0 Λ0 + Λ1 K1(x)
With low-latencies in mind, the sub-second P_astro pipeline in fact com-
putes the astrophysical probability that the N+1th candidate event (i.e., the
new candidate event) is in the αth astrophysical source category by working
with mean values of the background and source-specific astrophysical Pois-
son expected counts, 〈Λ0〉N and 〈Λ~ 1〉N , which are pre-computed on a weekly
cadence during maintenance8: ∫ ∞
〈Λ 〉 = Λ P (Λ ,Λ~ |~x) dΛ dΛ~α α 0 1 0 1, and
0
| 〈Λα〉NKα(xN+1)P (Λα, xN+1 ~xN+1) = .〈Λ0〉N + 〈Λ~ 〉 ·K~1 N 1(xN+1)
8The computation of the mean values themselves during maintenance is on the order of
∼minutes.
160
To continuously update the template weights, injection campaigns are
conducted on a weekly basis to see which parts of the template banks are
triggered, i.e., activated. Lastly, as of yet, the probabilities produced by the
P_astro pipeline are not used for candidate event vetting. They will, however,
be included in the LIGO/Virgo Circulars if available.
8.3 Public Alerts
For Advanced LIGO and Virgo’s upcoming third observing run, O3, we
enter the era of public alerts, where LIGO/Virgo Preliminary Notices will be
sent fully autonomously within ∼1 to 10 minutes of a promising GW trigger
entering GraceDb. Much of the low-latency follow-up processes for annotating
and orchestrating LVAlerts will be triggered under GWCelery9, an umbrella
Python-based package based on the asynchronous task queue, Celery. The
most up-to-date information regarding alert content, including instructions
for signing up to receive these alerts, can be found at the LIGO/Virgo Public
Alerts User Guide: https://emfollow.docs.ligo.org/userguide/.
During the few minutes’ latency for sending the Preliminary Notice, sev-
eral automated processes will occur for incoming GW triggers. Most impor-
tantly, there is a new notion of a “Superevent”, which unifies a group of GW
triggers from multiple low-latency search pipelines that correspond to the same
physical event. Each Superevent has a so-called preferred_event (whose FAR
and trigger properties are reported in the outgoing Notices and Circulars) and
a GW_events list (which includes all the other related GW triggers). The
Superevents follow a naming convention, Syymmdd{abc}, where S stands for
Superevent, yymmdd is the UTC date, and letters at the end increment as a,
b, . . . , aa, . . . , to allow for the possibility of many time-separated GW triggers
9https://gwcelery.readthedocs.io/
161
being recovered on a given day.
The Superevent’s preferred_event is selected to preference GW triggers
recovered with data from multiple detectors versus a single detector for im-
proved localization. For compact binary coalescence events, a CBC low-latency
search trigger is preferred over a Burst low-latency search trigger for improved
waveform reconstruction and parameter estimation. If at this point, there
still hasn’t been a down-selection to one GW trigger, the Superevent algo-
rithm selects for its preferred_event the GW trigger with the lowest reported
FAR for Burst triggers, but highest SNR for CBC triggers. There is also a
preference for CBC triggers over Burst, and multi-interferometer triggers over
single-detector triggers.
Once the preferred_event has been set, we check its FAR to see if it falls
below the threshold for reporting (∼1/2 months for CBC Superevents using
a trials factor of 5 and ∼1/yr for Burst Superevents using a trials factor of
4)10. In this scenario, the first LIGO/Virgo Preliminary Notice is sent (with
reference to a skymap if available), with the possibility of a second Preliminary
Notice if a skymap becomes available before the Superevent has been vetted
for its data quality. All Superevents that have been released to the public will
have GraceDb pages that are viewable by persons outside the LIGO Scientific
Collaboration and Virgo Collaboration.
The turnaround for sending the LIGO/Virgo Initial Notice and Circu-
lar is within 24 hours, (with the goal of ∼30 minutes for BNS and NS-BH
Superevents). During this time, human vetting procedures similar to those
that occurred during O1 and O2 will be performed, and all Initial Notices and
Circulars will be distributed with an update for the sky localization.
10The trials factors correspond to the number of pipelines for CBC and Burst low-latency
searches. The CBC pipelines are GstLAL, PyCBC Live, MBTAOnline, SPIIR-HighMass, and
SPIIR-LowMass. The Burst searches are cWB-AllSky, cWB-BBH, cWB-IMBH and oLIB-AllSky.
162
In the case of CBC Superevents, we will also release EM-Bright source
classification and P_astro data products through the Initial Notice and Circu-
lar, although quantitative estimates of the masses and spins, GW strain data,
and the waveform regressed from the data will be kept inside the Collabora-
tions.
For publications related to LIGO/Virgo Superevents that become con-
firmed events, the LIGO/Virgo Initial Circular should be cited as the first
formal publication of the candidate event. If data quality inspections deter-
mine the Superevent to be a noise event, a LIGO/Virgo Retraction Notice will
be sent, indicating that the Superevent is no longer a GW candidate event.
163
Chapter 9
Conclusions and Outlook
The science of gravitational-wave (GW) astronomy is maximized when we
have joint detections by instruments of traditional astronomy. A joint detec-
tion can serve to confirm the astrophysical origin of the signal, to determine
the host galaxy (and therefore distance to the event), to discern viable models
concerning the central engine and event environment, and much more. Thus,
the majority of my time as a graduate student was devoted to enabling elec-
tromagnetic/neutrino follow-up of Advanced LIGO and Virgo’s GW candidate
events.
A collaboration-wide goal for the first observing run (O1) was to enable
joint detections in the event that we saw something interesting—a possible
GW signal. Thus, for O1 I created the first GW candidate event annotator
during the Advanced Detector era, approval_processor. The software that I
wrote served to (1) select the candidate events for follow-up and (2) send the
alerts out to the traditional astronomy community. Looking back, I remember
the excitement and almost-palpable tension in the air when we had our first
viable GW trigger (the one we would later name GW150914, the first ever
observed binary black hole merger). The rest (to be honest) is a blur; we were
all so busy with the prospect of our first discovery.
In the months leading up to the second observing run (O2), we determined
within the LIGO/Virgo follow-up group that approval_processor would re-
quire an upgrade to deal with increased rates of GW triggers expected during
O2. We needed to design the software to avoid race conditions and backlogging
164
in how it processed incoming streams of information from various pipelines
(e.g., data quality products, sky localization probability maps, labels, etc.).
Therefore, I worked with Reed Essick to create the second GW candidate
event annotator during the Advanced Detector era, approval_processorMP.
Our responsibilities were divided so that he created the programming objects
for the annotator’s multi-processing infrastructure, and I created the canvas
that utilized these objects to make logical decisions in selecting candidate
events and sending alerts for follow-up.
As a result of these efforts, we successfully sent alerts out to the wider
astronomy community during both O1 and O2. We even had a few confirmed
GW detections along the way (11 of them to date!), and at least one highly-
confident joint detection, GW170817, the first ever observed binary neutron
star merger. The other less-confident (possible) joint detection occurred with
a Fermi/GBM weak transient at the time of GW150914 (Connaughton et al.,
2016).
To summarize, when I first started my graduate research in GW astron-
omy, we had not yet detected any GW events. It had still been an open
question whether or not we would see anything during O1. Fast-forward a
couple of years and the current landscape is very different. The question has
evolved from, “Will we see anything?” to now, “Will we have enough follow-up
advocates on duty? And what will we see? What counterparts will be found?
What will joint detections tell us about the event?"
Advanced LIGO and Virgo’s third observing run (O3) has already offi-
cially started as of April 1, 2019. KAGRA, the cryogenic underground GW
detector in Japan, will also join for O3. In less than 4 years since the detection
of GW150914 (which admittedly had a more improvised vetting and EM/neu-
165
trino follow-up alert process), we have streamlined the follow-up procedure to
best facilitate our observing partners and increase our science returns.
There will be an uptick in the number of alerts sent during O3 for two
reasons. One, the sensitivities of our detectors improve with each observing
run (Table 9.1) and two, we enter the era of public alerts.
We expect numerous compact binary coalescence (CBC) GW events, with
BBH candidates occurring ∼1 per week and BNS candidates occurring up to
∼1 per month (for a total of 1 to 10 for the totality of the observing run). The
NS-BH coalescence event rate remains uncertain (Pankow, 2018). For each
CBC candidate event, low-latency sky localization probability maps that are
both rapid and accurate are generated, source classification and astrophysical
origin probabilities for four source categories (BNS, NS-BH, BBH, and mass-
gap binaries) are provided, and human-vetted alerts are sent/composed within
hours of a promising candidate event.
Detector BNS Range (Mpc)
Advanced LIGO 120 to 170
Advanced Virgo 65 to 85
KAGRA 8 to 25
Table 9.1: Expected detector sensitivities to BNS coalescences during O3.
For Burst candidate events, we let our imagination wander. What unex-
pected GW sources await our detection? What does the Universe have in store
for us? Perhaps cosmic strings exist and are intersecting, emitting bursts of
GW radiation. Even an “expected” GW source such as a Galactic supernova
or magnetar starquake will be a boon for new scientific discovery and inquiry.
If anything, history has shown unforeseen fundamental discoveries accom-
pany each new window of observation. Radio astronomy, for instance, led to
the discovery of the cosmic microwave background (the earliest electromagnetic
166
relic of the Big Bang) and also quasi-stellar objects (i.e., quasars—accretion
disks surrounding supermassive black holes at cosmological distances). Us-
ing Type Ia supernovae as standard candles, we also learned to our surprise
that the Universe is expanding and accelerating, powered by dark energy, the
dominant energy/mass component of the Universe... Thus, we have much to
look forward to with the relatively new fields of GW and multi-messenger
astronomy.
167
Appendix A
Appendix A: LIGO/Virgo Notices
for GW150914
The following are the machine-readable LIGO/Virgo Notices (i.e., VOEvents)
sent to GCN for the gravitational-wave candidate event G184098, later known
as GW150914, the first ever observed binary black hole merger (Abbott et al.,
2016b). VOEvent information is repackaged by GCN before being sent to our
observing partners. The skymap sent with the LIGO/Virgo Update Notice
was computed with the LIB parameter estimation algorithm.
1 <?xml version="1.0" ?>
2 <voe:VOEvent xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
3 xmlns:voe="http://www.ivoa.net/xml/VOEvent/v2.0"
4 xsi:schemaLocation="http://www.ivoa.net/xml/VOEvent/v2.0 http://www.ivoa.
net/xml/VOEvent/VOEvent-v2.0.xsd"
5 version="2.0" role="observation" ivorn="ivo://gwnet/gcn_sender#G184098-7-Initial
">
6 <Who>
7 <Date>2015−09−16T03:11:58</Date>
8 <Author>
9 <contactName>LIGO Scientific Collaboration and Virgo Collaboration<
/contactName>
10 </Author>
11 </Who>
12 <What>
13 <Param name="internal" dataType="string" value="0">
14 <Description>Indicates that this event should be distributed to
LSC/Virgo members only</Description>
15 </Param>
16 <Param name="Pkt_Ser_Num" dataType="string" value="7"/>
17 <Param name="GraceID" dataType="string" value="G184098" ucd="meta.id">
18 <Description>Identifier in GraceDB</Description>
19 </Param>
20 <Param name="AlertType" dataType="string" value="Initial" ucd="meta.
version" unit="">
21 <Description>VOEvent alert type</Description>
22 </Param>
23 <Param name="EventPage" dataType="string" value="https://gracedb.ligo.
org/events/G184098" ucd="meta.ref.url">
24 <Description>Web page for evolving status of this candidate event<
/Description>
25 </Param>
168
26 <Param name="Instruments" dataType="string" value="H1,L1" ucd="meta.
code">
27 <Description>List of instruments used in analysis to identify this event
</Description>
28 </Param>
29 <Param name="FAR" dataType="float" value="1.17786e-08" ucd="arith.rate;
stat.falsealarm" unit="Hz">
30 <Description>False alarm rate for GW candidates with this strength or
greater</Description>
31 </Param>
32 <Param name="Group" dataType="string" value="Burst" ucd="meta.code"
unit="">
33 <Description>Data analysis working group</Description>
34 </Param>
35 <Param name="Pipeline" dataType="string" value="CWB" ucd="meta.code"
unit="">
36 <Description>Low−latency data analysis pipeline</Description>
37 </Param>
38 <Param name="Search" dataType="string" value="AllSky" ucd="meta.code"
unit="">
39 <Description>Specific low−latency search</Description>
40 </Param>
41 <Param name="CentralFreq" dataType="float" value="123.828491" ucd="gw.
frequency" unit="Hz">
42 <Description>Central frequency of GW burst signal</Description>
43 </Param>
44 <Param name="Duration" dataType="float" value="0.024773" ucd="time.
duration" unit="s">
45 <Description>Measured duration of GW burst signal</Description>
46 </Param>
47 <Param name="Fluence" dataType="float" value="122.840988205" ucd="gw.
fluence" unit="erg/cm^2">
48 <Description>Estimated fluence of GW burst signal</Description>
49 </Param>
50 <Group type="GW_SKYMAP" name="skyprobcc_cWB_complete-BurstAllSky">
51 <Param name="skymap_fits_shib" dataType="string" value="
https://gracedb.ligo.
org/events/G184098/files/skyprobcc_cWB_complete.fits" ucd="meta.
ref.url" unit="">
52 <Description>Sky Map FITS Shibboleth protected</Description>
53 </Param>
54 <Param name="skymap_fits_x509" dataType="string" value="
https://gracedb.ligo.
org/api/events/G184098/files/skyprobcc_cWB_complete.fits" ucd="
meta.ref.url" unit="">
55 <Description>Sky Map FITS X509 protected</Description>
56 </Param>
57 <Param name="skymap_fits_basic" dataType="string" value="
https://gracedb.ligo.
org/apibasic/events/G184098/files/skyprobcc_cWB_complete.fits"
ucd="meta.ref.url" unit="">
58 <Description>Sky Map FITS basic auth protected</Description>
59 </Param>
60 <Param name="skymap_png_shib" dataType="string" value="
https://gracedb.ligo.
org/events/G184098/files/skyprobcc_cWB_complete.png" ucd="meta.
ref.url" unit="">
61 <Description>Sky Map image Shibboleth protected</Description>
62 </Param>
63 <Param name="skymap_png_x509" dataType="string" value="
https://gracedb.ligo.
169
org/api/events/G184098/files/skyprobcc_cWB_complete.png" ucd="
meta.ref.url" unit="">
64 <Description>Sky Map image X509 protected</Description>
65 </Param>
66 <Param name="skymap_png_basic" dataType="string" value="
https://gracedb.ligo.
org/apibasic/events/G184098/files/skyprobcc_cWB_complete.png" ucd
="meta.ref.url" unit="">
67 <Description>Sky Map image basic auth protected</Description>
68 </Param>
69 </Group>
70 </What>
71 <WhereWhen>
72 <ObsDataLocation>
73 <ObservatoryLocation id="LIGO Virgo"/>
74 <ObservationLocation>
75 <AstroCoordSystem id="UTC-FK5-GEO"/>
76 <AstroCoords coord_system_id="UTC-FK5-GEO">
77 <Time>
78 <TimeInstant>
79 <ISOTime>2015−09−14T09:50:45.391000</ISOTime>
80 </TimeInstant>
81 </Time>
82 </AstroCoords>
83 </ObservationLocation>
84 </ObsDataLocation>
85 </WhereWhen>
86 <How>
87 <Description>Candidate gravitational wave event identified by low−latency
analysis</Description>
88 <Description>H1: LIGO Hanford 4 km gravitational wave detector</Description>
89 <Description>L1: LIGO Livingston 4 km gravitational wave detector<
/Description>
90 </How>
91 <Citations>
92 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−6−
Preliminary</EventIVORN>
93 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−5−Update
</EventIVORN>
94 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−4−
Initial</EventIVORN>
95 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−3−
Preliminary</EventIVORN>
96 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−2−
Retraction</EventIVORN>
97 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−1−
Preliminary</EventIVORN>
98 <Description>Initial localization is now available</Description>
99 </Citations>
100 <Description>Report of a candidate gravitational wave event</Description>
101 </voe:VOEvent>
170
1 <?xml version="1.0" ?>
2 <voe:VOEvent xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
3 xmlns:voe="http://www.ivoa.net/xml/VOEvent/v2.0"
4 xsi:schemaLocation="http://www.ivoa.net/xml/VOEvent/v2.0 http://www.ivoa.
net/xml/VOEvent/VOEvent-v2.0.xsd"
5 version="2.0" role="observation" ivorn="ivo://gwnet/gcn_sender#G184098-8-Update"
>
6 <Who>
7 <Date>2015−09−16T03:14:30</Date>
8 <Author>
9 <contactName>LIGO Scientific Collaboration and Virgo Collaboration<
/contactName>
10 </Author>
11 </Who>
12 <What>
13 <Param name="internal" dataType="string" value="0">
14 <Description>Indicates that this event should be distributed to
LSC/Virgo members only</Description>
15 </Param>
16 <Param name="Pkt_Ser_Num" dataType="string" value="8"/>
17 <Param name="GraceID" dataType="string" value="G184098" ucd="meta.id">
18 <Description>Identifier in GraceDB</Description>
19 </Param>
20 <Param name="AlertType" dataType="string" value="Update" ucd="meta.
version" unit="">
21 <Description>VOEvent alert type</Description>
22 </Param>
23 <Param name="EventPage" dataType="string" value="https://gracedb.ligo.
org/events/G184098" ucd="meta.ref.url">
24 <Description>Web page for evolving status of this candidate event<
/Description>
25 </Param>
26 <Param name="Instruments" dataType="string" value="H1,L1" ucd="meta.
code">
27 <Description>List of instruments used in analysis to identify this event
</Description>
28 </Param>
29 <Param name="FAR" dataType="float" value="1.17786e-08" ucd="arith.rate;
stat.falsealarm" unit="Hz">
30 <Description>False alarm rate for GW candidates with this strength or
greater</Description>
31 </Param>
32 <Param name="Group" dataType="string" value="Burst" ucd="meta.code"
unit="">
33 <Description>Data analysis working group</Description>
34 </Param>
35 <Param name="Pipeline" dataType="string" value="CWB" ucd="meta.code"
unit="">
36 <Description>Low−latency data analysis pipeline</Description>
37 </Param>
38 <Param name="Search" dataType="string" value="AllSky" ucd="meta.code"
unit="">
39 <Description>Specific low−latency search</Description>
40 </Param>
41 <Param name="CentralFreq" dataType="float" value="123.828491" ucd="gw.
frequency" unit="Hz">
42 <Description>Central frequency of GW burst signal</Description>
43 </Param>
44 <Param name="Duration" dataType="float" value="0.024773" ucd="time.
duration" unit="s">
45 <Description>Measured duration of GW burst signal</Description>
171
46 </Param>
47 <Param name="Fluence" dataType="float" value="122.840988205" ucd="gw.
fluence" unit="erg/cm^2">
48 <Description>Estimated fluence of GW burst signal</Description>
49 </Param>
50 <Group type="GW_SKYMAP" name="LIB_skymap-BurstAllSky">
51 <Param name="skymap_fits_shib" dataType="string" value="
https://gracedb.ligo.org/events/G184098/files/LIB_skymap.fits.gz"
ucd="meta.ref.url" unit="">
52 <Description>Sky Map FITS Shibboleth protected</Description>
53 </Param>
54 <Param name="skymap_fits_x509" dataType="string" value="
https://gracedb.ligo.org/api/events/G184098/files/LIB_skymap.fits
.gz" ucd="meta.ref.url" unit="">
55 <Description>Sky Map FITS X509 protected</Description>
56 </Param>
57 <Param name="skymap_fits_basic" dataType="string" value="
https://gracedb.ligo.org/apibasic/events/G184098/files/LIB_skymap
.fits.gz" ucd="meta.ref.url" unit="">
58 <Description>Sky Map FITS basic auth protected</Description>
59 </Param>
60 <Param name="skymap_png_shib" dataType="string" value="
https://gracedb.ligo.org/events/G184098/files/LIB_skymap.png" ucd
="meta.ref.url" unit="">
61 <Description>Sky Map image Shibboleth protected</Description>
62 </Param>
63 <Param name="skymap_png_x509" dataType="string" value="
https://gracedb.ligo.org/api/events/G184098/files/LIB_skymap.png"
ucd="meta.ref.url" unit="">
64 <Description>Sky Map image X509 protected</Description>
65 </Param>
66 <Param name="skymap_png_basic" dataType="string" value="
https://gracedb.ligo.org/apibasic/events/G184098/files/LIB_skymap
.png" ucd="meta.ref.url" unit="">
67 <Description>Sky Map image basic auth protected</Description>
68 </Param>
69 </Group>
70 </What>
71 <WhereWhen>
72 <ObsDataLocation>
73 <ObservatoryLocation id="LIGO Virgo"/>
74 <ObservationLocation>
75 <AstroCoordSystem id="UTC-FK5-GEO"/>
76 <AstroCoords coord_system_id="UTC-FK5-GEO">
77 <Time>
78 <TimeInstant>
79 <ISOTime>2015−09−14T09:50:45.391000</ISOTime>
80 </TimeInstant>
81 </Time>
82 </AstroCoords>
83 </ObservationLocation>
84 </ObsDataLocation>
85 </WhereWhen>
86 <How>
87 <Description>Candidate gravitational wave event identified by low−latency
analysis</Description>
88 <Description>H1: LIGO Hanford 4 km gravitational wave detector</Description>
89 <Description>L1: LIGO Livingston 4 km gravitational wave detector<
/Description>
90 </How>
91 <Citations>
172
92 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−7−
Initial</EventIVORN>
93 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−6−
Preliminary</EventIVORN>
94 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−5−Update
</EventIVORN>
95 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−4−
Initial</EventIVORN>
96 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−3−
Preliminary</EventIVORN>
97 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−2−
Retraction</EventIVORN>
98 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G184098−1−
Preliminary</EventIVORN>
99 <Description>Updated localization is now available</Description>
100 </Citations>
101 <Description>Report of a candidate gravitational wave event</Description>
102 </voe:VOEvent>
173
Appendix B
Appendix B: LIGO/Virgo Notices
for GW170817
The following are the machine-readable LIGO/Virgo Notices (i.e., VOEvents)
sent to GCN for the gravitational-wave candidate event G298048, later known
as GW170817, the first ever observed binary neutron star coalescence (Abbott
et al., 2017a). VOEvent information is repackaged by GCN before being sent
to our observing partners. As can be seen, for O2, we provided our observ-
ing partners with EM-Bright information, in this case giving both ProbHasNS
and ProbHasRemnant equal to 1 for GW170817. The three-detector skymap
available with the LIGO/Virgo Update Notice sent nearly 5 hours later aided
the follow-up efforts that led to the detection of the afterglow counterparts.
1 <?xml version="1.0" ?>
2 <voe:VOEvent xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
3 xmlns:voe="http://www.ivoa.net/xml/VOEvent/v2.0"
4 xsi:schemaLocation="http://www.ivoa.net/xml/VOEvent/v2.0 http://www.ivoa.
net/xml/VOEvent/VOEvent-v2.0.xsd"
5 version="2.0" role="observation" ivorn="ivo://gwnet/gcn_sender#G298048-1-Initial
">
6 <Who>
7 <Date>2017−08−17T13:08:15</Date>
8 <Author>
9 <contactName>LIGO Scientific Collaboration and Virgo Collaboration<
/contactName>
10 </Author>
11 </Who>
12 <What>
13 <Param name="internal" dataType="int" value="0">
14 <Description>Indicates whether this event should be distributed to
LSC/Virgo members only</Description>
15 </Param>
16 <Param name="Pkt_Ser_Num" dataType="string" value="1"/>
17 <Param name="GraceID" dataType="string" value="G298048" ucd="meta.id">
18 <Description>Identifier in GraceDB</Description>
19 </Param>
20 <Param name="AlertType" dataType="string" value="Initial" ucd="meta.
version" unit="">
174
21 <Description>VOEvent alert type</Description>
22 </Param>
23 <Param name="Retraction" dataType="string" value="false" ucd="meta.code
" unit="">
24 <Description>Set to true if the event is retracted.</Description>
25 </Param>
26 <Param name="HardwareInj" dataType="int" value="0" ucd="meta.number"
unit="">
27 <Description>Indicates that this event is a hardware injection if 1, no
if 0</Description>
28 </Param>
29 <Param name="Vetted" dataType="int" value="1" ucd="meta.number" unit=""
>
30 <Description>Indicates whether this candidate has undergone basic
vetting by humans</Description>
31 </Param>
32 <Param name="OpenAlert" dataType="int" value="0" ucd="meta.number" unit
="">
33 <Description>Indicates that this event is an open alert if 1, no if 0<
/Description>
34 </Param>
35 <Param name="EventPage" dataType="string" value="https://gracedb.ligo.
org/events/G298048" ucd="meta.ref.url">
36 <Description>Web page for evolving status of this candidate event<
/Description>
37 </Param>
38 <Param name="Instruments" dataType="string" value="H1" ucd="meta.code"
>
39 <Description>List of instruments used in analysis to identify this event
</Description>
40 </Param>
41 <Param name="FAR" dataType="float" value="3.47773444765e-12" ucd="arith
.rate;stat.falsealarm" unit="Hz">
42 <Description>False alarm rate for GW candidates with this strength or
greater</Description>
43 </Param>
44 <Param name="Group" dataType="string" value="CBC" ucd="meta.code" unit=
"">
45 <Description>Data analysis working group</Description>
46 </Param>
47 <Param name="Pipeline" dataType="string" value="gstlal" ucd="meta.code"
unit="">
48 <Description>Low−latency data analysis pipeline</Description>
49 </Param>
50 <Param name="Search" dataType="string" value="O2VirgoTest" ucd="meta.
code" unit="">
51 <Description>Specific low−latency search</Description>
52 </Param>
53 <Param name="ProbHasNS" dataType="float" value="1.0" ucd="stat.
probability" unit="">
54 <Description>Probability that at least one object in the binary is less
than 3 solar masses</Description>
55 </Param>
56 <Param name="ProbHasRemnant" dataType="float" value="1.0" ucd="stat.
probability" unit="">
57 <Description>Probability that there is matter in the surroundings of the
central object</Description>
58 </Param>
59 <Group type="GW_SKYMAP" name="bayestar-CBCO2VirgoTest">
60 <Param name="skymap_fits_shib" dataType="string" value="
https://gracedb.ligo.org/events/G298048/files/bayestar.fits.gz"
175
ucd="meta.ref.url" unit="">
61 <Description>Sky Map FITS Shibboleth protected</Description>
62 </Param>
63 <Param name="skymap_fits_x509" dataType="string" value="
https://gracedb.ligo.org/api/events/G298048/files/bayestar.fits.
gz" ucd="meta.ref.url" unit="">
64 <Description>Sky Map FITS X509 protected</Description>
65 </Param>
66 <Param name="skymap_fits_basic" dataType="string" value="
https://gracedb.ligo.org/apibasic/events/G298048/files/bayestar.
fits.gz" ucd="meta.ref.url" unit="">
67 <Description>Sky Map FITS basic auth protected</Description>
68 </Param>
69 <Param name="skymap_png_shib" dataType="string" value="
https://gracedb.ligo.org/events/G298048/files/bayestar.png" ucd="
meta.ref.url" unit="">
70 <Description>Sky Map image Shibboleth protected</Description>
71 </Param>
72 <Param name="skymap_png_x509" dataType="string" value="
https://gracedb.ligo.org/api/events/G298048/files/bayestar.png"
ucd="meta.ref.url" unit="">
73 <Description>Sky Map image X509 protected</Description>
74 </Param>
75 <Param name="skymap_png_basic" dataType="string" value="
https://gracedb.ligo.org/apibasic/events/G298048/files/bayestar.
png" ucd="meta.ref.url" unit="">
76 <Description>Sky Map image basic auth protected</Description>
77 </Param>
78 </Group>
79 </What>
80 <WhereWhen>
81 <ObsDataLocation>
82 <ObservatoryLocation id="LIGO Virgo"/>
83 <ObservationLocation>
84 <AstroCoordSystem id="UTC-FK5-GEO"/>
85 <AstroCoords coord_system_id="UTC-FK5-GEO">
86 <Time>
87 <TimeInstant>
88 <ISOTime>2017−08−17T12:41:04.445710</ISOTime>
89 </TimeInstant>
90 </Time>
91 </AstroCoords>
92 </ObservationLocation>
93 </ObsDataLocation>
94 </WhereWhen>
95 <How>
96 <Description>Candidate gravitational wave event identified by low−latency
analysis</Description>
97 <Description>H1: LIGO Hanford 4 km gravitational wave detector</Description>
98 </How>
99 <Description>Report of a candidate gravitational wave event</Description>
100 </voe:VOEvent>
176
1 <?xml version="1.0" ?>
2 <voe:VOEvent xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
3 xmlns:voe="http://www.ivoa.net/xml/VOEvent/v2.0"
4 xsi:schemaLocation="http://www.ivoa.net/xml/VOEvent/v2.0 http://www.ivoa.
net/xml/VOEvent/VOEvent-v2.0.xsd"
5 version="2.0" role="observation" ivorn="ivo://gwnet/gcn_sender#G298048-2-Update"
>
6 <Who>
7 <Date>2017−08−17T17:49:40</Date>
8 <Author>
9 <contactName>LIGO Scientific Collaboration and Virgo Collaboration<
/contactName>
10 </Author>
11 </Who>
12 <What>
13 <Param name="internal" dataType="int" value="0">
14 <Description>Indicates whether this event should be distributed to
LSC/Virgo members only</Description>
15 </Param>
16 <Param name="Pkt_Ser_Num" dataType="string" value="2"/>
17 <Param name="GraceID" dataType="string" value="G298048" ucd="meta.id">
18 <Description>Identifier in GraceDB</Description>
19 </Param>
20 <Param name="AlertType" dataType="string" value="Update" ucd="meta.
version" unit="">
21 <Description>VOEvent alert type</Description>
22 </Param>
23 <Param name="Retraction" dataType="string" value="false" ucd="meta.code
" unit="">
24 <Description>Set to true if the event is retracted.</Description>
25 </Param>
26 <Param name="HardwareInj" dataType="int" value="0" ucd="meta.number"
unit="">
27 <Description>Indicates that this event is a hardware injection if 1, no
if 0</Description>
28 </Param>
29 <Param name="Vetted" dataType="int" value="1" ucd="meta.number" unit=""
>
30 <Description>Indicates whether this candidate has undergone basic
vetting by humans</Description>
31 </Param>
32 <Param name="OpenAlert" dataType="int" value="0" ucd="meta.number" unit
="">
33 <Description>Indicates that this event is an open alert if 1, no if 0<
/Description>
34 </Param>
35 <Param name="EventPage" dataType="string" value="https://gracedb.ligo.
org/events/G298048" ucd="meta.ref.url">
36 <Description>Web page for evolving status of this candidate event<
/Description>
37 </Param>
38 <Param name="Instruments" dataType="string" value="H1" ucd="meta.code"
>
39 <Description>List of instruments used in analysis to identify this event
</Description>
40 </Param>
41 <Param name="FAR" dataType="float" value="3.47773444765e-12" ucd="arith
.rate;stat.falsealarm" unit="Hz">
42 <Description>False alarm rate for GW candidates with this strength or
greater</Description>
43 </Param>
177
44 <Param name="Group" dataType="string" value="CBC" ucd="meta.code" unit=
"">
45 <Description>Data analysis working group</Description>
46 </Param>
47 <Param name="Pipeline" dataType="string" value="gstlal" ucd="meta.code"
unit="">
48 <Description>Low−latency data analysis pipeline</Description>
49 </Param>
50 <Param name="Search" dataType="string" value="O2VirgoTest" ucd="meta.
code" unit="">
51 <Description>Specific low−latency search</Description>
52 </Param>
53 <Param name="ProbHasNS" dataType="float" value="1.0" ucd="stat.
probability" unit="">
54 <Description>Probability that at least one object in the binary is less
than 3 solar masses</Description>
55 </Param>
56 <Param name="ProbHasRemnant" dataType="float" value="1.0" ucd="stat.
probability" unit="">
57 <Description>Probability that there is matter in the surroundings of the
central object</Description>
58 </Param>
59 <Group type="GW_SKYMAP" name="bayestar-HLV-CBCO2VirgoTest">
60 <Param name="skymap_fits_shib" dataType="string" value="
https://gracedb.ligo.org/events/G298048/files/bayestar-HLV.fits.
gz" ucd="meta.ref.url" unit="">
61 <Description>Sky Map FITS Shibboleth protected</Description>
62 </Param>
63 <Param name="skymap_fits_x509" dataType="string" value="
https://gracedb.ligo.org/api/events/G298048/files/bayestar-HLV.
fits.gz" ucd="meta.ref.url" unit="">
64 <Description>Sky Map FITS X509 protected</Description>
65 </Param>
66 <Param name="skymap_fits_basic" dataType="string" value="
https://gracedb.ligo.org/apibasic/events/G298048/files/bayestar-
HLV.fits.gz" ucd="meta.ref.url" unit="">
67 <Description>Sky Map FITS basic auth protected</Description>
68 </Param>
69 <Param name="skymap_png_shib" dataType="string" value="
https://gracedb.ligo.org/events/G298048/files/bayestar-HLV.png"
ucd="meta.ref.url" unit="">
70 <Description>Sky Map image Shibboleth protected</Description>
71 </Param>
72 <Param name="skymap_png_x509" dataType="string" value="
https://gracedb.ligo.org/api/events/G298048/files/bayestar-HLV.
png" ucd="meta.ref.url" unit="">
73 <Description>Sky Map image X509 protected</Description>
74 </Param>
75 <Param name="skymap_png_basic" dataType="string" value="
https://gracedb.ligo.org/apibasic/events/G298048/files/bayestar-
HLV.png" ucd="meta.ref.url" unit="">
76 <Description>Sky Map image basic auth protected</Description>
77 </Param>
78 </Group>
79 </What>
80 <WhereWhen>
81 <ObsDataLocation>
82 <ObservatoryLocation id="LIGO Virgo"/>
83 <ObservationLocation>
84 <AstroCoordSystem id="UTC-FK5-GEO"/>
85 <AstroCoords coord_system_id="UTC-FK5-GEO">
178
86 <Time>
87 <TimeInstant>
88 <ISOTime>2017−08−17T12:41:04.445710</ISOTime>
89 </TimeInstant>
90 </Time>
91 </AstroCoords>
92 </ObservationLocation>
93 </ObsDataLocation>
94 </WhereWhen>
95 <How>
96 <Description>Candidate gravitational wave event identified by low−latency
analysis</Description>
97 <Description>H1: LIGO Hanford 4 km gravitational wave detector</Description>
98 <Description>A gravitational wave trigger identified a possible counterpart
GRB</Description>
99 </How>
100 <Citations>
101 <EventIVORN cite="supersedes">ivo://gwnet/gcn_sender#G298048−1−
Initial</EventIVORN>
102 <Description>Updated localization is now available</Description>
103 </Citations>
104 <Description>Report of a candidate gravitational wave event</Description>
105 </voe:VOEvent>
Index
1E 1048.1−5937, 22
1E 2259+586, 22
1RXS J170849−400910, 22
3XMM J185246.6+003317, 18
4U 0142+61, 22
Advanced LIGO, 2, 70
amplitude spectral density, 79
arm length, 75
coating, 86
data/candidate event vetting
approval_processor, see approval_processor
approval_processorMP, see approval_processorMP
detector operators, 110
electromagnetic/neutrino follow-up advocates, 110
event rate check, 111
false alarm rate, 116
iDQ, 108
low-latency data quality vector, 108
low-latency detector state information, 108
low-latency search pipeline experts, 110
non-stationary noise check, 111
omega scan, 111
Omicron scan, 111
rapid response teams, 110
Fabry-Pérot cavities
179
180
finesse, 75
noise sources
coating Brownian noise, 86
coating thermo-optic noise, 86
excess gas noise, 86
Newtonian noise, 85
optical read-out noise, 80
radiation pressure, 80, 83
seismic noise, 84
shot noise, 80
substrate Brownian noise, 86
suspension thermal noise, 86
violin modes, 86
observing run
O1, 82, 114
O2, 121
O3, 144
power recycling, 76, 82
power recycling cavity length, 82
power spectral density, 78
definition, 80
sampling rate, 103
search, see low-latency search
sensitivity
horizon distance, 87
range, 78
signal recycling, 76
181
substrate, 86
test mass mirror mass, 84
vibration isolation, 84
Advanced Virgo, 2, 70, 130
arm length, 75
sampling rate, 103
affine parameter, 33
anomalous X-ray pulsar, 18, 19
1E 1048.1−5937, see 1E 1048.1−5937
1E 2259+586, see 1E 2259+586
1RXS J170849−400910, see 1RXS J170849−400910
4U 0142+61, see 4U 0142+61
XTE J1810−197, see XTE J1810−197
approval_processor, 107, 114, 115, 163
approval_processorMP, 107, 121, 164
AXP, see anomalous X-ray pulsar
Bayes’ theorem, 146
Bianchi identity, 32
black hole, 6
binaries, 24, 38
GW150914, see GW150914
Burst
excess power, 102
search, see low-latency search
CBC, see compact binary coalescence
Christoffel symbols
second kind, 33
182
transverse trace-free gauge, 41
compact binary coalescence
chirp mass, 96
horizon distance, 87
matched filtering, 98
P_astro, 157
range, 78
search, see low-latency search
symmetric mass ratio, 124
waveform parameters, 97
cosmic microwave background, 133, 166
cosmic string
basic properties, 133
cusps, 135
constraints, 139
search algorithm, 136
search results, 139
intercommutation probability, 134
kinks, 135
superstrings, 134
tension, 134
covariant derivative, 34, 39
curvature, 35
cWB, 104
WDM transform, 104
dispersion
interstellar medium, 56
183
measure, 57
Einstein
field equations, 32, 34
relaxed, 36, 39, 42
general theory of relativity, see general theory of relativity
index notation, 33
summation convention, 33
time delay, 55, 59
EM-Bright source classification, 123, 162
Foucart’s fitting formula, 125
energy-momentum tensor, 35
post-Newtonian theory, 63
Fabry-Pérot cavities, 75
gamma-ray burst, 20
general theory of relativity, 32, 35
Birkhoff’s theorem, 42
post-Newtonian theory, see post-Newtonian theory
geodesic
deviation, 39
equation, 32, 33, 41
GraceDb, 106
gravitational wave
basic properties, 35
detections
GW150914, see GW150914
GW170817, see GW170817
184
detector
Advanced LIGO, see Advanced LIGO
Advanced Virgo, see Advanced Virgo
amplitude spectral density, 79
antenna response pattern, 88
Fabry-Pérot cavities, see Fabry-Pérot cavities
Initial LIGO, see Initial LIGO
KAGRA, see KAGRA
Michelson interferometer, see Michelson interferometer
noise sources, 76
power recycling, 76, 82
power spectral density, 78
signal recycling, 76
localization, see sky localization probability map
polarizations, 39, 43, 87
quadrupole formula, 43
sidebands, 76
sources, 41
strain, 88
GRB, see gamma-ray burst
GRB 050509b, 27
GRB 050709, 27
GRB 050724, 27
GRB 170817A, 123
Green’s function, 36
GstLAL, 100
GW150914, 163, 164
185
EM follow-up timeline, 116
LIGO/Virgo initial notice, 167
LIGO/Virgo update notice, 170
GW170618
EM-Bright source classification, 126
GW170729, 121
GW170817, 123, 130, 164
EM follow-up timeline, 132
EM-Bright source classification, 126
LIGO/Virgo initial notice, 173
LIGO/Virgo update notice, 176
localization
BAYESTAR, 94, 131, 154
neutron star radius, 31
GW170818, 121
GWCelery, 160
harmonic gauge, 36
Hulse-Taylor binary pulsar, 1, 44
Initial LIGO, 115
KAGRA, 164
substrate, 86
kilonova, 28
precursor, 28
Legendre polynomials, 38
line element, 33
LISA, 25
186
long gamma-ray burst, 6, 16
LOOC UP, 115
low-latency search
coincident gravitational wave-electromagnetic counterpart
RAVEN, see RAVEN
modeled gravitational wave/compact binary coalescence
GstLAL, see GstLAL
MBTAOnline, see MBTAOnline
PyCBC Live, see PyCBC Live
SPIIR, see SPIIR
unmodeled gravitational wave/generic transient burst
cWB, see cWB
oLIB, see oLIB
LVAlert, 107
magnetar, 17
3XMM J185246.6+003317, see 3XMM J185246.6+003317
electromagnetic/neutrino signatures
AXP, see anomalous X-ray pulsar
giant flare, 21
intermediate burst, 21
near-infrared, 22
neutrinos, 22
non-bursting, 21
optical, 22
SGR, see soft gamma-ray repeater
short burst, 21
gravitational waves, 23
187
resonant cyclotron scattering, 21
starquakes, 21, 23, 50
Swift J1822.3−1606, see Swift J1822.3−1606
mass quadrupole moment, 43
MBTAOnline, 100
metric tensor, 33
inverse, 33
Michelson interferometer, 70
neutron star, 6
binaries, 2, 24, 31, 38
electromagnetic/neutrino signatures, 26
GW170817, see GW170817
PSR B1913+16, see Hulse-Taylor binary pulsar
equation of state, 30
hypermassive, 29
isolated
central compact object, 19
magnetar, see magnetar
rotating radio transient, 19
X-ray dim isolated neutron star, 19
mass, 30
pulsar, see pulsar
radius, 31
oLIB, 105
Q transform, 106
P_astro, 157, 162
188
parallel transport, 34
Poisson distribution, 81
post-Newtonian theory, 63
power spectral density, 83
proper time, 33
PSR B1913+16, see Hulse-Taylor binary pulsar
public alerts, 109, 160, 165
pulsar, 44
age, 49
mechanism, 47
PyCBC Live, 101
RAVEN
spatial overlap integral, 153
spatio-temporal coincidence false alarm rate, 152
temporal coincidence false alarm rate, 149
temporal coincidence search, 144
temporal overlap integral, 151
Ricci tensor, 34
Riemann tensor, 34
Roemer time delay, 52, 62
scalar
curvature, see curvature
Schwarzschild
external metric, 42
radius, 42
SGR, see soft gamma-ray repeater
SGR 0418+5729, 18
189
SGR 0526−66, 21
SGR 1806−20, 21
SGR 1900+14, 21, 23
sGRB, see short gamma-ray burst
Shapiro time delay, 53, 65
short gamma-ray burst, 21, 124
GRB 050509b, see GRB 050509b
GRB 050709, see GRB 050709
GRB 050724, see GRB 050724
sky localization probability map, 92
BAYESTAR
forked-tongue morphology, 119
three-dimensional, 127
two-dimensional, 117
cWB, 119
joint GRB-GW, 154
LALInferenceBurst, 119
skymap, see sky localization probability map
soft gamma-ray repeater, 18, 19
SGR 0418+5729, see SGR 0418+5729
SGR 0526−66, see SGR 0526−66
SGR 1806−20, see SGR 1806−20
SGR 1900+14, see SGR 1900+14, see SGR 1900+14
SPIIR, 101
stress-energy tensor, see energy-momentum tensor
superevent, 160
supernova
190
core-collapse, 2
electromagnetic/neutrino signatures, 14
electron capture, 5
gravitational waves, 10, 11
magneto-rotational, 6
neutrino-driven, 7
progenitor, 5
rate, 11
spherically-symmetric, 42
standing accretion-shock instability, 9
thermonuclear, 15
Swift J1822.3−1606, 18
symmetric tensor, 38
symmetric trace-free tensor, 37
tensor
definition, 33
metric, see metric tensor
Ricci, see Ricci tensor
Riemann, see Riemann tensor
stress-energy, see energy-momentum tensor
symmetric, see symmetric tensor
symmetric trace-free, see symmetric trace-free tensor
transverse trace-free gauge, 39, 43, 72
white dwarf, 46
binaries, 25
XTE J1810−197, 22
191
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