universe

Review

Weyl Curvature Hypothesis in Light of Quantum Backreaction
at Cosmological Singularities or Bounces

Bei-Lok Hu

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Citation: Hu, B.-L. Weyl Curvature

Hypothesis in Light of Quantum

Backreaction at Cosmological

Singularities or Bounces. Universe

2021, 7, 424. https://doi.org/

10.3390/universe7110424

Academic Editor: Paulo Vargas

Moniz

Received: 12 October 2021

Accepted: 29 October 2021

Published: 7 November 2021

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Maryland Center for Fundamental Physics and Joint Quantum Institute, University of Maryland,
College Park, MD 20742-4111, USA; blhu@umd.edu

Abstract: The Weyl curvature constitutes the radiative sector of the Riemann curvature tensor and
gives a measure of the anisotropy and inhomogeneities of spacetime. Penrose’s 1979 Weyl curvature
hypothesis (WCH) assumes that the universe began at a very low gravitational entropy state, cor-
responding to zero Weyl curvature, namely, the Friedmann–Lemaître–Robertson–Walker (FLRW)
universe. This is a simple assumption with far-reaching implications. In classical general relativity,
Belinsky, Khalatnikov and Lifshitz (BKL) showed in the 70s that the most general cosmological
solutions of the Einstein equation are that of the inhomogeneous Kasner types, with intermittent
alteration of the one direction of contraction (in the cosmological expansion phase), according to the
mixmaster dynamics of Misner (M). How could WCH and BKL-M co-exist? An answer was provided
in the 80s with the consideration of quantum field processes such as vacuum particle creation, which
was copious at the Planck time (10−43 s), and their backreaction effects were shown to be so powerful
as to rapidly damp away the irregularities in the geometry. It was proposed that the vaccum viscosity
due to particle creation can act as an efficient transducer of gravitational entropy (large for BKL-M) to
matter entropy, keeping the universe at that very early time in a state commensurate with the WCH.
In this essay I expand the scope of that inquiry to a broader range, asking how the WCH would
fare with various cosmological theories, from classical to semiclassical to quantum, focusing on
their predictions near the cosmological singularities (past and future) or avoidance thereof, allowing
the Universe to encounter different scenarios, such as undergoing a phase transition or a bounce.
WCH is of special importance to cyclic cosmologies, because any slight irregularity toward the end
of one cycle will generate greater anisotropy and inhomogeneities in the next cycle. We point out
that regardless of what other processes may be present near the beginning and the end states of the
universe, the backreaction effects of quantum field processes probably serve as the best guarantor
of WCH because these vacuum processes are ubiquitous, powerful and efficient in dissipating the
irregularities to effectively nudge the Universe to a near-zero Weyl curvature condition.

Keywords: weyl curvature hypothesis; early universe cosmology; singularity and bounce; cyclic
universe; quantum fields; backreaction effects

Preface

When Professor Moniz invited me last year to write a paper for this special issue with
some review or perspective emphasis I could not identify a topic of current interest and
importance in quantum cosmology on which I have done enough work to write about. Then
in July, Professor Hendrik Ulbricht brought to my attention that this August is Professor
Penrose’s 90th Birthday (b. 8 August 1931). That got me thinking through some important
themes in cosmology from Penrose’s viewpoint—his concerns in a broad range of research
topics in gravitation, quantum physics, black holes and cosmology, as lucidly presented in
many semi-popular books (the last two being [1,2]), are all of fundamental interest. This is
how I closed in to the title theme of this paper: The Weyl Curvature Hypothesis (WCH) [3–6].

How does WCH fare when quantum backreaction is added to our considerations?
Backreaction in gravitational physics enters at four levels: classical, semiclassical, stochastic

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and quantum gravity. In classical gravity it refers to the effects of the inhomogeneous
modes on the homogeneous or long wavelength (infrared) modes; in quantum cosmology
it refers to the effects of the inhomogeneous modes truncated in a mini-superspace approx-
imation. Backreaction at the semiclassical and stochastic levels has a different meaning.
It refers to the effects of quantum field processes such as particle creation [7–9] and trace
anomaly [10–12] on the structure and dynamics of a classical background spacetime. This is
a subject which I explored between 1977–1983–1995 in my work first with Parker [13,14] and
Hartle [15,16] from 77–80 (based on [17], continued by Hartle [18,19] and Anderson [20,21]
from 1980–1984), then upgraded to a closed-time-path, in-in, or Schwinger-Keldysh treat-
ment with Calzetta in 1987 [22], including noise and fluctuations in 1994 [23], with Matacz
in 1994 [24] and with Sinha in 1995 [25]. This was continued by Campos and Verdaguer in
1994–1996 [26,27]. These works explored the backreaction effects of the trace anomaly and
particle creation in the early universe near the Planck time, at the threshold of quantum
gravity (defined as theories for the microscopic constituents of spacetime and matter). The
Weyl curvature hypothesis (WCH) [3] will be examined in the context of semiclassical and
stochastic gravity [28] in Section 5.

Taking the time to write this essay has been a good way to update my knowledge since
my 1982–1983 papers on vacuum viscosity [29] and gravitational entropy [30] in the areas
of gravitational entropy, the Weyl curvature hypothesis, the cosmological singularity or
avoidance of singularity, as predicted by different theories, and how vacuum viscosity due
to the backreaction of quantum field processes at the Planck time, such as pertaining to the
trace anomaly and vacuum particle creation, would bear on these issues. I wanted to see
how the WCH fares with other later developed models such as the cyclic cosmology of clas-
sical general relativity (GR) (Section 2.1) or particle physics (Section 2.2) origins, singularity
avoidance solutions from quantum cosmology (Sections 3.1 and 3.2), in particular, the ‘Big
Bounce’ solutions of loop quantum cosmology (Section 3.3) and quantum phase transitions
in asymptotic freedom and causal dynamical triangulation (Section 4.1)—whether there are
indications that the WCH holds in these contexts and, in some cases, whether it has proven
to be desirable or even necessary. (If the reader only wants to find out how the WCH
and gravitational entropy fare in semiclassical gravity, he/she can just selectively read
Sections 1.2, 2.3, 3.1 and 5.)

It is not my intention to cover the full range of topics mentioned above, aided with
extensive literature, like a review. Far from it. I won’t even repeat what Penrose said in
his original proposal of the WCH and how he defined gravitational entropy, or how he
argued for a conformal cyclic cosmology in classical general relativity. Instead I would urge
interested readers to read his original papers for insights which can never be reproduced.
For papers by other authors which I find relevant to our central themes here I also prefer to
keep their original words rather than add my paraphrase or re-statement. The nature of
this article is probably closer to an outline, a selective highlight, a personal guide, as the
topics chosen certainly reflect my own interests or what I view as important, which could
be very different from other practitioners’.

1. Weyl Curvature Hypothesis and Gravitational Entropy

Penrose pointed out in his 1979 essay [3] that if our Universe had begun close to
what the Friedmann–Lemaître–Robertson–Walker (FLRW) models describe, in a state of
highest degree of isotropy and homogeneity, corresponding to zero Weyl curvature, then
it is something extremely special. He reached this conclusion by comparing the entropy
of matter we see today, measured by the Hawking radiation, emitted by black holes, to
the entropy of gravity, measured by the Weyl curvature tensor squared integrated over
3-volumes. Gravitational entropy (GEnt) defined as such would increase in time from zero
in the beginning (which could be a singularity, the ‘Big Bang’, or a state of minimal volume,
a ‘bounce’, see below) and provide a cosmological arrow of time. Penrose’s proposals for
the non-activation of gravitational degrees of freedom registered in the Weyl curvature, at
the Big Bang, are referred to as the Weyl Curvature Hypothesis (WCH).



Universe 2021, 7, 424 3 of 28

Penrose’s position on this theme has not changed in 40 years, as the following state-
ment from his 2018 essay shows [6] “In such a situation (the universe in a collapsing phase),
we expect density irregularities to increase as the universe contracts, and the FLRW approx-
imation would get worse and worse. In the late stages of the collapse, these irregularities
would result in numerous black holes. Time reversing this picture (and taking into account
the invariance of Einstein’s equations under time-reversal), we find that the singular origin
that we find in the FLRW models is something extremely special.” While today we see the
Planck distribution of matter suggesting a state of maximum matter entropy, “In contrast
with the behaviour of a gas in a box, for example, where maximum entropy would be
pictured as something with great spatial uniformity, gravitating bodies, such as systems of
stars, would tend to clump more and more in their spatial distribution, as their dynamical
time evolution proceeds, representing an increase in the gravitational entropy”.

The four themes I listed above are already laden in these descriptions: WCH, GEnt,
also the Bang-vs-Bounce issue. The reason is, for people who prefer to see the universe not
going to a singularity, they have to come up with some suitable ways to avoid it, such as
through processes in particle physics or in quantum gravity. We mention four of them in
this essay: pre-Big Bang, loop quantum cosmology (LQC), causal dynamical triangulation
(CDT) and asymptotic freedom (AsyF)1.

For people who want our future universe not just to bounce once or twice, but
to continue to bounce forever without suffering much attenuation, or stumbling into a
singularity, in the so-called ‘cyclic cosmologies’, the requirements are even more stringent.
What happens to the universe at the end of the contraction phase requires extra careful
scrutiny. The highly regular FLRW type of approach to the singularity or bounce would be
preferred, because only then would the deviations be kept under control cycle after cycle.
This is in turn because gravitational forces tend to clump irregularities, so the next cycle
would be more difficult to control than the last.

Most people would agree that quantum effects are important at the beginning and
ending stages of the universe, either entering a singularity or with a minimal scale. These
developments and considerations add new impetus and interests to Penrose’s WCH-GEnt,
originally formulated without quantum considerations. To the above mentioned subjects
we shall also add semiclassical and quantum backreaction considerations. The former
refers to the backreaction of quantum field processes on the dynamics of spacetime. They
need be included because these processes were dominant and had exerted significant effects
at the Planck scale. The latter pertains to LQC: It concerns the effects of the inhomogeneous
quantum modes in the superspace of 3-geometries, so far largely ignored in the mini-
superspace formulations of LQC.

1.1. Classical GR

Before we begin our studies of WCH in a quantum setting it is important to remind
ourselves of the two very important theoretical achievements in the 60s in classical general
relativity (GR), which underlie all investigations in black holes and cosmology built upon
the GR theoretical framework: the singularity theorems from 1965–1968 of Penrose [44] ,
Hawking [45] and Geroch [46], and the general solutions of the Einstein equations near
the cosmological singularity in the work of Belinsky, Khalatnikov, Lifshitz (BLK) from the
early 60s (when the authors preferred a singularity-free scenario, later corrected, after the
Penrose-Hawking singularity theorems) to the early 1970s [47–49] with the now famous
BKL approach to cosmological singularity, and Misner’s mixmaster universe [50,51] and
minisuperspace quantum cosmology [52] from 1969 to 1972, and the ‘velocity-dominated’
approach by Eardley, Liang and Sachs [53].

1.1.1. Singularity Theorems: Penrose-Hawking-Ellis-Geroch

The Penrose 1965 singularity theorem for black holes extended by Hawking to cosmo-
logical singularities are known as the Penrose-Hawking singularity theorems in general
relativity [54]. They are described in more pedagogical terms in the book by Hawking and



Universe 2021, 7, 424 4 of 28

Ellis [55], the premium monograph on global analysis methods in general relativity. These
theorems preclude singularity-free “bounce” from occurring in the presence of any type of
matter source with stress-energy tensor satisfying a very general condition of local energy
non-negativity. For a detailed explanation of its contents with an extensive bibliography on
this fundamental topic, see the 2015 review of [56] in a mid-century celebration of Penrose’s
1965 paper.

1.1.2. Approach to Cosmological Singularity: BKL-Misner

In addition to the primary sources of the Belinsky-Khalatnikov-Lifshitz -Misner enter-
prise mentioned above, the 1975 book by Ryan and Shepley [57] has an excellent introduc-
tion to spatially-homogeneous (Bianchi) cosmologies. See also the review of Barrow and
Tipler [58]. For a discussion of the classical BKL-M mixmaster chaotic dynamics (billiard
ball in minisuperspace), see, e.g., [59–63]2. For the latest developments of this respect in
relation to indefinite Kac-Moody algebra, see [66]. The role of chaos in the decoherence of
quantum cosmology and the appearance of a classical spacetime has been pointed out by
Calzetta [67].

Since many topics we shall discuss are related to the state of the universe near the
singularity (or how it negotiates into a bounce), we need some basic knowledge of the
BKL-M behavior. The most general solutions to the Einstein equation near and toward
the cosmological singularity of the Kasner type (not the Friedmann class where matter
does matter) is the inhomogeneous Kasner universe (with two directions contracting and
one direction expanding), interspersed with the mixmaster behavior, i.e., switching of the
direction of expansion from one axis to another in very short time intervals. This so-called
“inhomogeneous mixmaster solution” represents the generic behavior of the universe
toward the cosmological singularity. This result has been reworked by many authors
employing different methods, e.g., Uggla et al. [68] constructed an invariant set which
forms the local past attractor for the evolution equations, a framework for proving rigorous
theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological
models. A more direct approach relies on methods in dynamical systems [69]. The BKL-M
behavior is also supported by Garfinkle’s numerical simulations in vacuum spacetimes
with no symmetries [70].

Three main features in the approach to cosmological singularity are, therefore: (1) “Mat-
ter doesn’t matter” in the BKL-Kasner classes of extrinsic curvature (‘velocity’)- dominated
spacetimes3 while matter does matter in the FLRW classes. (2) Extrinsic curvature (from ex-
pansion or contraction) dominates over the intrinsic (3-geometry) curvature, except in brief
intervals, which are known as the ‘mixmaster bounces’ (the universe point bouncing against
the 3-fold symmetric potential walls, receding during contraction) in mini-superspace;
(3) Inhomogeneous mixmaster behavior means that every point on the 3-geometry under-
goes an independent mixmaster dynamics.

We begin with some basics of Bianchi cosmology. Bianchi classified all spatially
homogeneous spacetimes into nine types [73] (some types are further distinguished by a
sub 0 or h, such as Type VIIh, etc.) [57]. The class of Bianchi Types I–IX spaces, especially
the Type I and Type IX spaces, encompasses many important classical cosmological models
for the early universe.

The line elements of the Bianchi universes are given by

ds2 = −dt2 +
3

∑
a,b=1

γab(t)σa(x)σb(x), (1)

where γab(t) is the metric tensor and σa(x) are the invariant basis one-forms on the homo-
geneous hypersurfaces, satisfying the structure condition dσa = 1

2 Ca
bcσb ∧ σc, where Ca

bc are
the structure constants of the underlying Lie group. For a diagonal type-IX (mixmaster)
universe, Ca

bc = εabc the totally antisymmetric tensor, and γab = l2
a δab, where la are the

three principal radii of curvature. The case when two of the three la’s are equal is called



Universe 2021, 7, 424 5 of 28

the Taub universe, the case when all three are equal gives the closed Friedmann–Lemaître–
Robertson–Walker (FLRW) universe .

The nonrotating, diagonal mixmaster universe with three principal radii of curvature
`a(=1,2,3) form a minisuperspace [50] which can be represented pictorially by the shape θ
and deformation β parameters, related to `a by

`1,2 = `0 exp[β cos(θ ∓ π/3)], `3 = `0 exp(−β cos θ)]. (2)

where `3
0 ≡ `1`2`3. With threefold symmetry, the origin of this space gives the FLRW

universe, while points along the three axes (θ = 0, 120, 240◦) constitute the family of the
Taub universes. Along any axis, say θ = 0, increasing β in the positive sense gives an
“oblate” configuration while negative β gives a “prolate” configuration. The dynamics of
the mixmaster universe is depicted by trajectories in this minisuperspace. The velocity of
the universe point is determined by the extrinsic curvature, with free motion depicting
the Kasner universe. The influence of spatial anisotropy (intrinsic curvature) is depicted
by a set of moving potential walls. The universe point bouncing off these walls signifies
a shifting in the contracting Kasner axis, while wandering into one of the three corners
signifies an oscillating and “flattening pancake” (oblate) configuration There is also one
situation when the universe point bounces off the three walls at a special incidence angle
so that it can continue to hover loosely around the origin. These characteristic regimes
of the mixmaster universe are called, respectively, (a) the “bounce” (b) the “channel run”
and (c) the “quasi-isotropic” solutions. These solutions have the approximate geometric
configurations as the three limiting cases of the Taub universe: i.e., (a) β > 0, (b) β < 0, and
(c) β = 0, respectively. For any massless field in a spacetime which is a vacuum solution to
Einstein’s equations R = 0, quantum effects of curvature are important irrespective of the
type of coupling.

The Bianchi Type I universe has zero intrinsic curvature but nonzero extrinsic curva-
ture from its expansion and contraction, whose rates differ in three directions. The line
element is given by

ds2 = −dt2 +
3

∑
i,j=1

`2
ij(t)dxidxj = a2(η)

[
−dη2 +

3

∑
i,j=1

e2β(η)ij dxidxj
]

(3)

where η =
∫

dt/a is the conformal time and βij is a symmetric, traceless 3× 3 matrix
describing the anisotropy of the geometry. For Bianchi Type I one can choose a coordinate
such that βij is diagonal4 with the three elements βi. The line element of Bianchi Type I
universe is thus

ds2 = −dt2 +
3

∑
i=1

`2
i (t)(dxi)2 (4)

where `i = a(t)eβi is the scale factor in the xi direction. This is a generalization of the spa-
tially flat isotropic FLRW universe to the case where the universe expands anisotropically. A
useful quantity for this measure is the anisotropy parameter Qβ ≡ − 1

2 ∑i>j( ˙̀ i/`i − ˙̀ j/`j)
2,

where an overdot denotes d/dt. When there is no matter present a solution to the vacuum
Einstein equation exists, called the Kasner universe:

`i(t) = tpi , where
3

∑
i=1

p2
i =

3

∑
i=1

pi = 1. (5)

We see from this relation that the universe expands when two of the pis are positive
and one negative, and contracts when two of the pis are negative and one is positive.
The Kasner solution is important because it is a generic behavior of the universe at ev-
ery point in space near the singularity where the most general solution of the Einstein
equation [47–49] is found to be an inhomogeneous Kasner solution. It is also known as a



Universe 2021, 7, 424 6 of 28

‘velocity-dominated solution’ [53] reflecting the fact that near the cosmological singularity
the extrinsic curvature (measuring the time rate of change of the scale factors) dominates
over the intrinsic curvature (in the 3-geometry), thus the saying “matter doesn’t matter”
in the BKL-M generalized Kasner class of solutions near the cosmological singularity, in
contradistinction to the FLRW class, which requires the presence of matter. Defining the
isotropic expansion rate α = ln a then we see pi = α + βi. The spatially flat FLRW universe
corresponds to the case β = 0 with scale factor a = eα.

1.1.3. Weyl Curvature

For an analysis of the Weyl curvature tensor in spatially-homogeneous cosmological
models of classical general relativity, see Barrow and Hervik 2002 [75]. For a review of
the entropy of universe with gravitational and thermal contributions, the entropy of black
holes and cosmic horizons entropy, see, Grøn 2012 [76].

1.1.4. Gravitational Entropy: Weyl and Other Measures

Assuming the interested reader would have read Penrose’s original paper on the
WCH, we will not repeat how he defines the gravitationl entropy via the Weyl curvature.
Instead, to expand the horizon somewhat we mention a few alternative definitions of
gravitational entropy.

Kinematic vs. Weyl singularity:
Lim et al. 2007 [77] and Coley et al. 2009 [78] studied what they called the ‘kinematic

singularities’, namely, the Hubble expansion scalar, the acceleration vector, the shear tensor
and the vorticity tensor in the decomposition of the covariant derivative of the 4-velocity
of a congruence of worldlines.

Bel–Robinson tensor
Clifton, Ellis and Tavakol [79] proposed a measure of gravitational entropy based on

the square-root of the Bel–Robinson tensor of free gravitational fields, which is the unique
totally symmetric traceless tensor constructed from the Weyl tensor. (For the square-root to
exist as a unique factorization of the Bel–Robinson, the spacetime is required to contain
gravitational fields that are only Coulomb-like or only wavelike.) These authors show
that their entropy measure (i) is in keeping with Penrose’s Weyl curvature hypothesis,
(ii) reduces to the usual Bekenstein–Hawking value measure for Schwarzschild black hole
and (iii) evolves like the Hubble weighted anisotropy of the free gravitational field for
scalar perturbations of a Robertson–Walker geometry thus increases as structure formation
occurs, and (iv) for the Lemaître Tolman-Bondi inhomogeneous cosmological models, they
found conditions under which the entropy increases.

Kullback-Leibler relative information entropy
While most works try to place the gravitational entropy defined by the Weyl curvature

in relation to thermodynamic entropy, Hosoya, Buchert and Morita [80] try to relate it to
information entropy. They introduced the Kullback-Leibler relative information entropy
to express the distinguishability of the local inhomogeneous mass density field from its
spatial average on arbitrary compact domains. Comparing the Kullback–Leibler entropy
to the Weyl curvature tensor invariants. Li et al. [81,82] calculated these two measures
by perturbing the standard cosmology. Up to the second order in the deviations from a
homogeneous and isotropic spacetime they find that they are correlated and can be linked
via the kinematical backreaction of a spatially averaged universe model.

1.2. Semiclassical Gravity: Entropy of Gravitons and WCH

The Weyl curvature tensor measures the anisotropy and inhomogeneities of spacetime.
Since it is the radiative sector of the Riemann tensor, it measures the curvature associated
with gravitational waves, which, of course, are the weakly inhomogeneous perturbations



Universe 2021, 7, 424 7 of 28

off a background spacetime. Quantizing the linear perturbations from a background gives
rise to gravitons. The short wavelength sector under the Brill-Hartle-Isaacson average
behaves like radiation fluid with the same equation of state as photons, namely, p = ρ/3,
with p the pressure and ρ the energy density. Gravitons, like photons, have two polariza-
tions but instead of spin-1 they are spin-2 particles. For these reasons gravitons are the
easiest entry point to see the connection between WCH and gravitational entropy.

By semiclassical we mean a quantum field, here, the graviton spin-2 field, propagating
in a classical background spacetime. (We also distinguish between quantum field theory
in curved spacetime, under the test field situation where the background is prescribed,
and semiclassical gravity where the background spacetime and the quantum field are
determined in a self-consistent manner). By quantum we mean both the matter field and
the spacetime are quantized. We reserve the term ‘quantum gravity’ to refer to theories
about the microscopic structures of spacetime at the Planck energy. We will explain what
quantum cosmology means in individual cases later, e.g., quantum cosmology of the mini-
superspace as in [52] refers to the situation where the 3-geometry is quantized, regardless
of whether the spacetime is an emergent entity or a fundamental one. However, ‘loop
quantum cosmology’ or ‘string cosmology’ is trickier, because ‘quantum’ could mean
quantizing the metric or the connection of GR, and if GR is an effective low energy theory,
the quantized collective excitations are like phonons, a far cry from the fundamental
constituents, or the ‘atoms’, of spacetime. It could also mean the cosmology (after the
Planck time) derived from a more fundamental theory of spacetime based on strings, loops,
sets or simplicies. Explaining how the spacetime manifold emerges from the interaction
of these fundamental constituents is a highly nontrivial task, probably the most severe
challenge facing the proponents of these theories of quantum gravity [83]. We shall return
to this point in the concluding remarks.

1.2.1. Graviton Backreaction in FLRW Universes

We consider this issue at two levels: (1) At the quantum field theory in curved
spacetime (QFTCST) [84–87] or test field level, we ask, under what conditions would
there be particle creation from the vacuum. It is easy to see that there is no particle
production for a massless conformally coupled field in an isotropic and homogeneous
universe, because such a spacetime is conformally-flat. But graviton production can
happen. This is because each of the two polarizations of the graviton behaves like a
massless minimally coupled scalar field [88,89]. Graviton production and its effects have
been studied by many authors [13,17,90]. As far as the matter entropy budget is concerned,
as mentioned above, with gravitons acting like photons (notwithstanding the big difference
in their scattering cross sections), their production increases the thermal entropy of the
matter content.

(2) To see how this is related to the gravitational entropy measured by the Weyl
curvature tensor, one needs to find out the backreaction of matter fields on the dynamics of
spacetime. This entails finding a self-consistent solution of the equations of motion for both
the matter fields and the spacetime they live in, namely, the semiclassical Einstein equation.
This backreaction constitutes a channel where the energy and entropy of spacetime can
be related to that of the quantum fields, with particle creation acting like a ‘transducer’
between the gravity and matter sectors. This way of thinking was first proposed by Hu in
1983 [30], having worked out the anisotropy damping scenario due to conformal massless
particle creation in Bianchi type I universes earlier with Hartle and Parker. We shall discuss
these results in Section 4.

Here, with the FLRW universe, the problem is simpler, since the background spacetime
is not only homogenous but also isotropic. In the beginning the Weyl curvature tensor is
nonzero because of the perturbations, the gravitational waves or the gravitons, imparting
the universe with some gravitational entropy. At the end, after graviton production ceases,
the spacetime returns to the FLRW universe with zero Weyl curvature tensor. One can



Universe 2021, 7, 424 8 of 28

see that the increase of matter entropy from the production process is accompanied by a
decrease of gravitational entropy. This is discussed in [91,92].

Now, what causes the graviton production to cease? The answer comes from solving
the semiclassical Einstein equations. Doing so, Hu and Parker 77 [13] show that the backre-
action of created gravitons leads to a change in the power expansion law a = tq towards
q = 1/2, i.e., radiation dominated. And a radiation-dominated RW universe admits no par-
ticle creation because the time-dependent frequency of the normal modes of the quantum
field contains a term d2a/dη2 where η is the conformal time. A radiation-dominated RW
universe behaves like a = η, rendering d2a/dη2 = 0. Thus, after a certain relaxation time
the behavior of the gravitons created corresponds to the evolution of classical radiation in
an isotropic universe. With backreaction we see that as the gravitational entropy density
goes to zero, graviton production will cease and no further backreaction will take place.

The fact that the backrection of graviton production can alter the equation of state of
matter was pointed out by Grishchuk in 1975 [90] and shown with field theory calculations
by Hu and Parker [13] and Hartle [17]. We will continue this description of the backreaction
of particle creation on background spacetime dynamics in Sec. V and discuss how the
damping of anisotropy and inhomogeneities of spacetime bears on gravitational entropy
and the WCH.

1.2.2. Entropy of Gravitons in the Gowdy Universe

Husain [93] paraphrased Penrose’s ideas of gravitational entropy based on the WCH
into three conditions: “If the Weyl curvature is to be identified with entropy, it should
satisfy the following criteria that are normally associated with entropy. (l) It should increase
monotonically with respect to a time coordinate from the initial singularity where it is finite
or zero. (2) Its value should increase with the number of quanta of the gravitational field.
(3) Clumped configurations of matter or gravitational quanta should correspond to higher
Weyl curvature values than unclumped configurations.”

Conditions (1) and (2) are illustrated in the example above, regarding graviton pro-
duction and its backreaction on the spacetime in a FLRW universe. For conditions (2) and
(3), we can see an example for the Gowdy universe [94,95]. The Gowdy three-torus (T3)
universe is an exact solution to the vacuum Einstein equations interpreted to be a single
polarization of gravitational waves propagating in an anisotropic, spatially inhomogeneous
background.

Appying Hamiltonian quantization via the Arnowitt-Deser-Misner (ADM) method
Berger [96,97] studied the production of gravitons from empty space. At large times,
graviton number is well defined since the solution is in WKB form. The creation process
produces the anisotropic collisionless classical radiation. Near the singularity, the model
behaves like an empty Bianchi Type I universe at each point in space (locally Kasner).

Beginning with the square of the curvature as an operator, Hussain calculated its
expectation values in states of clumped and unclumped gravitons. These results led him to
conclude that the curvature contains information about the entropy of the gravitational
field in this class of quantum cosmology models.

Finally, in the context of quantum cosmology based on the Wheeler-DeWitt (WdW)
equation we mention the work of Grøn and Hervik [98] These authors introduced a measure
of ‘gravitational entropy’ behaving in accordance with the second law of thermodynamics
and investigated its evolution in Bianchi type I and Lemaître–Tolman universe models.
They work with the quantity Π ≡ P

√
h =
√

h(C2

R2 )
1
2 which is the ratio between the Weyl

tensor squared C2 and the Ricci tensor squared R2, integrated over dV =
√

hd3x in a
comoving coordinate xi, where h is the 3-volume element. They also considered the
expectation value of the Weyl curvature squared and the Ricci curvature squared, taken
with respect to the wave functions of the universe satisfying the Wheeler-DeWitt equation.
They then investigated whether a quantum calculation of initial conditions for the universe
based upon the Wheeler–DeWitt equation supports Penrose’s Weyl curvature conjecture or
not for the Bianchi type I universe models with dust and a cosmological constant and the



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Lemaître–Tolman universe models. They investigated two versions of the hypothesis. The
local version based on P2 fails to support the conjecture whereas a non-local entity based
on Π2 showed more promise concerning the conjecture. Grøn and Hervik commented that
their findings seem to be corroborated by that of Pelvas and Lake [99] on the one hand,
who showed that local entities like P2 cannot be a measure of gravitational entropy, and on
the other hand, by that of Rothman and Anninos [100,101], who showed that a nonlocal
quantity they constructed using the volume of the phase space could. The investigations of
these authors seem to suggest that while one cannot say that the Weyl tensor is directly
linked to the gravitational entropy, a certain non-local entity which is constructed from it
has an entropic behavior, and reflects the tendency of the gravitational field to produce
inhomogeneities.

We shall say some more about ADM and WdW, and about loop quantum cosmology
in Section 3. But we will not discuss quantum geometry and gravitational entropy (see,
e.g., [102]) applied to black holes or in the much studied holographic entanglement entropy
vein [103] for spacetimes with boundaries.

2. Cyclic Cosmology and Trace-Anomaly Induced Bounces
2.1. Penrose’s Conformal Cyclic Cosmology (CCC)

This theory is Penrose’s recent favorite in terms of cosmological models–favorite in
that after his 2005 proposal he has gathered his thoughts on entropy, black holes, time and
cosmology in a 300 page 2010 book [1] which dwells on these foundational issues of Nature
in a lucid and inspiring way. In his more recent 2017 book [2] of over 500 pages, “Fashion,
Faith, and Fantasy in the New Physics of the Universe” Penrose refers (in Section 4.3) to
his “conformal cyclic cosmology” as an idea so fantastic that it could be called “conformal
crazy cosmology”5.

To see what CCC is, the reader may benefit from reading just 4 pages of exposition
in [4]. For the background it is useful to know something about singularities in GR, e.g.,
in [104,105], and in particular, the class of isotropic cosmological singularities, where the
Weyl tensor is finite or preferably zero in the WCH. At the isotropic singularity even
though the Ricci tensor related to matter may approach infinity, the Weyl curvature remains
finite. Goode and Wainwright [106–108] proposed a definition of isotropic cosmological
singularities which is adopted by Tod [109,110] whose work Penrose refers to. The essential
idea is that the physical metric which is singular at cosmic time T = 0, is conformally
related to a metric which is regular at T = 0, i.e., the singularity arises solely due to the
vanishing of the conformal factor. The conformal transformation which is used to define
an isotropic singularity is an essential ingredient in Penrose’s conformal cyclic cosmology.
Namely, the Big Bang can be conformally represented as a smooth spacelike 3-surface,
across which the space-time is, in principle, extendable into the past in a conformally
smooth way.

According to Penrose [6], the Big Bang (BB) of our ‘aeon’ is the conformal continuation
of the remote exponential expansion of a previous aeon. There is a pre-BB aeon, and after
our future Big Crunch (BC)—actually Penrose prefers to see a long exponential expansion
(EE) to our future—there is a post-BC or post-EE aeon. In this sense a cyclic cosmology
scenario naturally emerges, where the aeons are linked by conformal transformations.

2.2. Cyclic Cosmology from Particle/ String Models. Big Crunch

No classical matter can exist with a pressure p to energy density ρ ratio w ≡ p/ρ > 1.
The stiffest Zel’dovich equation of state with w = 1 is the upper limit of classical matter
because that is when the speed of sound equals the speed of light. In the recent two decades
cosmological models with a scalar field providing an effective equation of state w > 1 have
drawn some interest, motivated by the possibility of cyclic cosmologies based on string theory-
related models, notably the Pre-Big Bang model of Gasperini and Veneziano [111–113] and
the ekpyrotic cosmologies of Steinhardt and Turok [114,115]. The designers of these theories
want to see the universe contract towards a big crunch singularity in a smooth homogeneous



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and isotropic (FLRW) manner, so it can be reborn in an equally smooth manner to be able to
sustain a large number of cycles, ideally ad infinitum.

2.2.1. Cyclic and Ekpyrotic Cosmologies

Ekpyrotic cosmologies solve the flatness problem in the contraction phase in a way
similar to how inflation solves it in the expansion phase. Both are looking for the conditions
where the time rate of change of the scale factor (Hubble) becomes close to a constant for
a period of time (around 65 e-folding time to be compatible with the entropy contents
of the observed universe). It is easy to see this from two simple equations, the Einstein
G00 equation for the dynamics of spacetime and the continuity equation (divergence-free
condition) for matter. In a FLRW universe with scale factor a and Hubble expansion
rate H = ȧ/a, where an overdot denotes taking the derivative with respect to cosmic
time, the Einstein G00 = 8πGρ equation reads: H2 = −k/a2 + 8πG

3 ρ, where k = 0, 1,−1
correspond to the flat, closed and open universes and ρ, the total energy density, contains
different kinds of classical matter and quantum fields, represented by equations of state
with different values for w ≡ p/ρ. The familiar cases of w = 0, 1/3 in classical cosmology
are for dust and radiation, respectively. We shall first show (i) why w = −1 for inflaton
field, and then explain (ii) how a special kind of matter with an equation of state w > 1
affects the Hubble rate, and what it does for the ekpyrotic cosmologies.

It is well-known how inflation solves the flatness problem, namely, by assuming that
the potential V(φ) of the inflaton field φ is sufficiently flat so that it rolls down the slope very
slowly. That imparts a constant energy density in the Einstein equation which dominates
over the other sources, which all drop as power laws of the scale factor during an expansion

phase. The pressure-energy density ratio for a quantum field is w ≡ p/ρ =
1
2 φ̇2−V
1
2 φ̇2+V

. When
1
2 φ̇2 � V as in slow-roll inflation, w→ −1.

From the continuity equation we easily get ρ ∝ a−3(1+w), namely, for non-relativistic
matter ρnrel ∝ a−3, for radiation ρrad ∝ a−4, for curvature anisotropy ρanis ∝ a−6, and for
some ‘ultra-stiff’ nonclassical matter with w > 1, ρust ∝ a−r where r > 6. Now it is clear
why during contraction, as a gets smaller, contributions from this ultra-stiff matter with
w > 1 dominate over other forms in governing the expansion rate.

This is why the cyclic and ekpyrotic cosmological models focus on the Big Crunch
singularity and why scalar fields with equations of state w > 1 are of special interest. This
‘ultra-stiff’ matter in a homogeneous and isotropic universe can be realized by a scalar field
φ with a steep negative potential energy V(φ) < 0. The wits and skills in designing the
potential of a quantum field to fit one’s special needs– these activities which dominate
the theoretical stage in inflationary cosmology–will also be active for this newer vein of
cosmology. We refer interested readers to the nice review of Lehners [116] from which they
can trace back to the original papers.

Of special interest to our concern is whether the mixmaster behavior might be altered
by the effect of a scalar field with equation of state w > 1. The authors of Ref. [117] show
that if w > 1, the chaotic mixmaster oscillations due to anisotropy (extrinsic curvature) and
(intrinsic) curvature are suppressed, and the contraction is described by a homogeneous
and isotropic Friedmann equation. Referring to string theory related models, they show
that chaotic oscillations are suppressed in Z2 orbifold compactification and contracting
heterotic M-theory models if w > 1 at the crunch.

However, as pointed out by Barrow and Yamamoto [118] this result stands only when
the ultrastiff pressures are isotropic. The inclusion of simple anisotropic pressures when
pressures can exceed the energy density stops the isotropic Friedmann universe from being
a stable attractor as an initial or final singularity is approached. Thus the situation with
isotropic pressures, studied earlier in the context of cyclic and ekpyrotic cosmologies, is not
generic, and Kasner-like behavior occurs when simple pressure anisotropies are present.

Including the consideration of stochastic effects, as invoked in Linde’s ‘chaotic’ in-
flation or Starobinsky’s stochastic inflation scenarios [37,39], the effective scalar field can
climb up the potential in some regions of space, which leads to an increase in the energy



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density. However, as Brandenberger et al. [119] show, the backreaction of fluctuations
that have already exited the Hubble radius will lead to a decrease in the effective energy
density, which is strong enough to prevent eternal expansion in the late-time dark-energy
phase of ekpyrotic cosmology.

2.2.2. Big Crunch with Dark Energy in Semiclassical Cosmology

As was mentioned above, how the universe evolves in the future– whether it ap-
proaches a singularity (Big Crunch), or it can avoid it (a Bounce), and how it approaches the
singularity or the minimal radius, are important factors to consider in any cyclic cosmology
proposal. Without assuming any exotic matter, Kennedy et al. [120] studied the influence of
the vacuum expectation value of the energy-momentum tensor of a conformally-invariant
free quantized field without particle creation on the spherically symmetric gravitational
collapse of a dust cloud. Using the same method as used by Fischetti et al. [15] and
Anderson [20] for the early universe, they found qualitatively similar results, namely, for
certain values of an arbitrary parameter the collapse proceeds to singularity, although
in a manner which differs from the classical (Oppenheimer-Snyder) behaviour; for other
values of the parameter the collapse terminates before singularity and thereafter the dust
cloud expands.

Ever since the discovery of a sizable fraction (0.7) of dark energy in the cosmological
matter-energy contents, theories about the future of our universe need to take into account
the bounds provided by stringent cosmological data. The combined 2018 Planck data
together with other major observations give the dark-energy equation of state parameter
wde = −1.03± 0.03, consistent with a cosmological constant (w = −1). One can regard
this as weakly favoring ‘phantom’ dark energy (w < −1), which can lead to a future
singularity called ‘Big Rip’ [121]. (A rather complex classification of different possible
future singularities is given in, e.g., Ref. [122] Sec. IA. ) Assuming that the dark energy
which causes these singularities to behave like a perfect fluid, Carlson et al. [122] studied
the effects which quantum fields and an α0R2 term in the gravitational Lagrangian have
on future singularities. Special emphasis was placed on those values of α0 which are
compatible with the universe having undergone Starobinsky inflation [123] in the past, the
same values which allow for stable solutions to the semiclassical backreaction equations in
the present universe.

2.3. Semiclassical Gravity: Trace Anomaly Induced Bounces

For conformally-invariant fields in conformally-flat spacetimes in the conformal vac-
uum state, the regularized stress energy tensor is given by [20]

< 0|T̂ab|0 > =
α

3
(gabR;c

;c − R;ab + RRab −
1
4

gabR2) (6)

+ β(
2
3

RRab − Ra
cRbc +

1
2

gabRcdRcd − 1
4

gabR2),

where Rab is the Ricci tensor, R is the scalar curvature, and α and β are constants which
depend on the number and types of fields present. Dimensional regularization gives

α =
1

2880π2 (NS + 6Nν + 12NV); β =
1

2880π2 (NS + 11Nν + 62NV) (7)

where NS, Nν, NV are respectively the number of scalar fields, four-component neutrino
fields, and Maxwell fields included in one’s theory.

Taking the trace of (6) we see that it is non-vanishing even though for conformal fields
the classical stress energy tensor is traceless. This is why it is called the trace anomaly.
The semiclassical Einstein equations have been solved for the effects of backreaction in the
70s in various cases by the following authors: Ruzmaikina & Ruzmaikin [124] were the
first to work on this problem, they, and later, Gurovich & Starobinsky [125] studied the
case β = 0, Wald [11,12] investigated the case α = 0, and Fischetti, Hartle, and Hu [15] the



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cases α ≤ 0, all values of β and α > 0, β < 3α. Each of these was done for spatially-flat
FLRW spacetimes containing classical radiation and having no cosmological constant.
Starobinsky [123] investigated the case α < 0, β > 0 for FLRW spacetimes with no classical
radiation or matter and no cosmological constant.

The most complete investigation of the back-reaction problem for conformally-invariant
free quantum fields in spatially-flat homogeneous and isotropic spacetimes containing
classical radiation was carried out by Anderson [20]. He considered only solutions which
at late times approach the appropriate solution to the field equations of general relativity,
the so-called asymptotically classical solutions, with the following results: For all α, β with
α > 0 there are many such solutions, while for all α, β with α < 0 there is only one such
solution. For β > 3α > 0, there is always one solution which undergoes a “time-symmetric
bounce” and which contains no singularities or particle horizons. For α > 0 there is
always at least one solution which begins with an initial singularity and which has no
particle horizons. For all α, β there is always at least one solution which begins with an
initial singularity.

In a sequel paper [21], Anderson added the consideration of nonzero spatial curvatures
and/or a nonzero cosmological constant. The qualitative behaviors are the same as for
spatially-flat spacetimes with zero cosmological constant. Instead, for β > 3α > 0, there
is always one solution which undergoes a “time-symmetric bounce” and which contains
no singularities or particle horizons. The differences caused by the spatial curvature and
cosmological constant include the initial behavior of the time-symmetric bounce solution
and, if the spatial curvature is nonzero, the initial behavior of many solutions for the cases
β ≈ 3α > 0 and β ≈ 3α < 0. For further work on semiclassical backreaction of conformal
quantum fields in a universe with a cosmological constant, see, e.g., Ref. [126].

3. Singularity Avoidance in Quantum Cosmology

Readers interested in having a broader perspective of this issue might want to first
learn of the major milestones in the developments of quantum gravity. A summary is given
by C. Rovelli [127].

The first stage of quantum gravity began with a series of papers by Bergmann [128]
pursuing a program concerned with the quantization of field theories which are covariant
with respect to general coordinate transformations, like the general theory of relativity. All
these theories share the property that the existence and form of the equations of motion is
a direct consequence of the covariant character of the equations. Dirac’s work [129,130] on
the quantization of constrained systems is a cornerstone in this pursuit. A more sophisti-
cated and systematic treatment can be found in the book by Henneaux and Teitelboim [131].

ADM Quantization.
As most researchers would agree, the canon of canonical quantization of general

relativity is the ADM formalism for quantizing 3-geometries in a 3+1 (space-time) decom-
position, developed by Arnowitt, Deser and Misner in a series of papers 1959-1961. See
their summary in [132]. Read also Kuchař [133], Ashtekar [134], the monographs of Rovelli
[135] and Thiemann [136].

Wheeler-DeWitt equation.
The Wheeler-DeWitt equation [137,138] is a Hamilton-Jacobi equation formulation of

the ADM quantization on the superspace, the space of all 3-geometries. (For a short intro-
duction with some historical tidbits, read [139].) Minisuperspace refers to the truncation of
the infinite dimensional superspace to a few dimensions, such as the scale factors of the
Bianchi universes. Quantum cosmology was studied by Misner [50,52] by applying the
ADM quantization to the mini-superspaces of Bianchi Type I and IX universes. Read also
Ryan [74].

The second wave of quantum cosmology arrived in the mid-80s with the considera-
tions of the initial conditions for the wave functions of the universe, notably, Hartle and



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Hawking’s no boundary condition [140] and Vilenkin’s tunneling wave function [141–143].
That was also the time when the two major programs of quantum gravity took off, namely,
superstring theory [144,145] and loop QG [146,147] (and its modern version of spin foam
network [148–150] which has a more obvious microscopic appeal). Supersymmetric quan-
tum cosmology is well represented by the monographs of D’Eath [151] and of Moniz [152].
Other major approaches to quantum gravity are nicely represented in Oriti’s edited book
[153]. A state-of-the-art overview of multiple approaches from the words of 37 prominent
practitioners is recorded in this most recent book [154]. Please refer to the original papers,
reviews and monographs on this subject.

3.1. Quantum Cosmology: Bounce in Gowdy T3 Universe

Here we summarize Berger’s work [96,97]. The Gowdy T3 universe [94,95] can be inter-
preted as consisting of a single polarization of gravitational waves whose amplitudes satisfy
a linear wave equation propagating in a non-linear background spacetime. One can then
view these wave amplitudes acting quadratically as a source on the first-order equations for
the non-linear background part of the gravitational field as a backreaction problem. Classi-
cally the model has an initial “big bang” singularity near which it is velocity-dominated
behaving as a different Kasner solution at each value of the spatial coordinate orthogonal
to the symmetry plane. Far from the singularity, the method of Isaacson [155,156] may be
used to identify the gravitational waves. Their effective stress-energy tensor is that for a
p = ρ fluid in two dimensions propagating in a spatially-homogeneous background space-
time. At the quantum level, Berger performed a canonical quantization of the dynamical
degrees of freedom. An adiabatic vacuum state was introduced and adiabatic regulariza-
tion used to obtain non-divergent stress-energy tensor vacuum expectation values. The
regularized expectation value was used as a source for the classical background spacetime
in the spirit of semi-classical gravity. Berger found that “The effect of the regularization
of the stress-energy tensor expectation value is to replace the classical singularity with a
symmetric bounce. The semi-classical spatially-averaged Gowdy T3 cosmological model
thus collapses from a state of large volume to a minimum volume near r = 0 and then
re-expands. The classical singularity is replaced by a symmetric bounce”.

3.2. Quantum BKL-Mixmaster Scenarios

Earlier we mentioned applications of dynamical system methods and chaos theories
to cosmology, in particular, to the analysis of the BKL-mixmaster behavior [157–159]. For
quantum chaos applied to the BKL-M models, we mention the numerical solutions of
the Wheeler-DeWitt equation by Furusawa [160,161] and Monte Carlo simulations of the
vacuum Bianchi IX universe by Berger [162]. For more recent work we mention two
groups of authors on this subject. The first group’s authors are Bergeron, Czuchry, Gazeau,
Małkiewicz and Piechocki. Of particular interest to our theme here is their results on
singularity avoidance [163,164]. How to get the universe to enter an inflationary stage
without soliciting the aid of an inflaton field (cf. Starobinsky inflation [123] invokes a
scalar field, at least the trace anomaly of it, as was also studied in [15]) is certainly of
interest [165]. (See [166,167] for an earlier work on mixmaster inflation via an interacting
quantum field). These authors constructed the quantum mixmaster solutions using affine
and Weyl–Heisenberg covariant integral quantizations. According to these authors these
quantization methods can regularize classical singularities which, they claim, the commonly
implemented canonical quantization cannot. From this they show some promising physical
features such as the singularity resolution, smooth bouncing, the excitation of anisotropic
oscillations and a substantial amount of post-bounce inflation as the backreaction to the
latter. For a recent summary of these authors’ work on this subject, see [168].

The other group’s authors are Góźdź, Kiefer, Kwidzinski, Piechocki, Piontek and G.
Plewa. Kiefer et al. [169,170] formulated the criteria of singularity avoidance for general
Bianchi class A models in the framework of quantum geometrodynamics based on the
Wheeler–DeWitt equation and give explicit and detailed results for Bianchi I models with



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and without matter. The singularities in these cases are big bang and big rip. They find
that the classical singularities can generally be avoided in these models.

Góźdź et al. [171] use affine coherent states quantization in the physical phase space
and show that during quantum evolution the expectation values of all considered observ-
ables are finite. The classical singularity of the BKL scenario is replaced by the quantum
bounce that presents a unitary evolution of the considered gravitational system. In a
later paper [172] they asked whether their claims may change because the affine coherent
states quantization method depends on the choice of the group parametrization. Using
the two simplest parameterizations of the affine group, they show that qualitative features
of their quantum system do not depend on the choice. They maintain that the quantum
bounce replacing a singular classical scenario is expected to be a generic feature of the
considered system.

3.3. Loop Quantum Cosmology (LQC): ‘Big Bounce’?

We devote this space to LQC because singularity avoidance is supposed to be a major
accomplishment of LQG6. This is referred to as the ‘Big Bounce’, in capital letters. There
are many reviews of LQC, e.g., Ashtekar et al. [178–182], Bojowald [183,184], and a recent
book [185]. (Note the loop QG community does not sing in unison in regard to many major
claims in LQC. Read Ashtekhar and Singh’s 2011 status report [182] and the critiques of
Bojowald [186,187]).

Without getting into the detailed claims and critiques we ask two key physical ques-
tions: (1) What role would genuine quantum degrees of freedom play in the cosmological
singularity proven to exist in classical general relativity? (2) Does quantization of grav-
ity alter the Big Bang? If it does, will there be a minimum finite scale in the quantum
geometry—the quantum bounce? (Here, the canonical formulation of cosmic ‘time’ is often
provided by a relational variable such as the scale factor in the ‘3-volume’ or a scalar field.)

There seems to be some consensus in the LQC community on the existence of a
quantum bounce. On the first issue Bojowald asked the question [188]: How Quantum is
the Big Bang? Studying isotropic models with an interacting scalar field he concluded that
quantum fluctuations do not affect the bounce much. Quantum correlations, however, do
play an important role and could even eliminate the bounce.

In [189] a new model is studied which describes the quantum behavior of transitions
through an isotropic quantum cosmological bounce in loop quantum cosmology sourced
by a free and massless scalar field. Using dynamical coherent states Bojowald provided
a demonstration that in general quantum fluctuations before and after the bounce are
unrelated. “Thus, even within this solvable model the condition of classicality at late times
does not imply classicality at early times before the bounce without further assumptions.
Nevertheless, the quantum state does evolve deterministically through the bounce. These
analytical bouncing solutions corroborate results from numerical simulations attesting to
robustly smooth bounces under the assumption of semiclassicality”.

There seems to be no controversy in this last statement within the LQC community.
However, how much of the state of the universe before the bounce is retained, is a subject
of debates.

Cosmic memory over bounces
For theorists favoring the bounce scenario, especially the cyclic cosmologies, an

important question to ask is, does the universe retain, after the bounce, its memory about
the previous phase.

Bojowald is of the opinion of ‘cosmic forgetfulness’ [190], meaning, not all the fluc-
tuations (and higher moments) of a state before the bounce can be recovered after the
bounce, and values depend very sensitively on the late time state. In [191] he showed that
quantum fluctuations before the big bang are generically unrelated to those after the big
bang. A reliable determination of pre-big bang quantum fluctuations of geometry would
thus require exceedingly precise observations.



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Countering this claim, Corichi and Singh [192,193], in one exactly solvable model,
found that a semiclassical state at late times on one side of the bounce, peaked on a pair
of canonically conjugate variables, strongly bounds the fluctuations on the other side,
implying semiclassicality. From this result these authors assert that cosmic recall is almost
perfect. See Bojowald’s Comments [194] and Corichi & Singh’s Reply [195].

Bianchi I model as a prototype for a cyclical Universe
For more recent work on this topic, we mention two. Montani, Marchi and Moriconi

[196] investigated the classical and quantum behavior of a Bianchi I model with stiff matter,
an ultra-relativistic component, and a small negative cosmological constant (at the turnover
point). They apply the Vilenkin wave function, using the volume of the universe, a quasi-
classical variable, as the dynamical clock for the pure quantum degrees of freedom. They
found that in a model where the isotropic variable is considered to be on a lattice, the
big-bang singularity is removed at a semiclassical level in favor of a big bounce. The mean
value of the Universe anisotropy variables remains finite during the whole evolution across
the big bounce, assuming a value that depends on the initial conditions fixed far from the
turning point.

An even simpler picture is obtained by Wilson-Ewing [197–199]: “In the loop quantum
cosmology effective dynamics for the vacuum Bianchi type I and type IX space-times, a
non-singular bounce replaces the classical singularity. The bounce can be approximated
as an instantaneous transition between two classical vacuum Bianchi I solutions, with
simple transition rules relating the solutions before and after the bounce. These transition
rules are especially simple when expressed in terms of the Misner variables: the evolution
of the mean logarithmic scale factor Ω is reversed, while the shape parameters β± are
unaffected. As a result, the loop quantum cosmology effective dynamics for the vacuum
Bianchi IX space-time can be approximated by a sequence of classical vacuum Bianchi
I solutions, following the usual Mixmaster transition maps in the classical regime, and
undergoing a bounce with this new transition rule in the Planck regime.” At the level of
effective dynamics in loop quantum cosmology, for this type of bounce solutions, Gupt and
Singh [200] found the selection rules and underlying conditions for all allowed and forbid-
den transitions.

Open Issues of LQC
Bojowald raised a few cautionary points about the bounce solutions in LQC [201]: (a) A

conceptual gap exists in the form of several unquestioned links between bounded densities,
bouncing volume expectation values, and singularity avoidance, made commonly in
arguments in favor of a generic bounce in LQC. (b) While bouncing solutions exist and may
even be generic within a given quantum representation, they are not generic if quantization
ambiguities such as choices of representations are taken into account. Small-volume BKL
behavior in a collapsing universe is shown to be crucially different from the large-volume
behavior exclusively studied so far in LQC.

4. Quantum Phase Transition and Quantum Backreaction

In addition to the popularly known string and loop theories, there is a handful of
proposals of quantum gravity theories describing sub-Planckian (length scale) physics [153].
Our present aim is to see how the WCH fares in these theories, i.e., whether some of these
theories predict that the Weyl curvature diminishes by comparison with other components
of the Riemann tensor near the singularity or bounce. Note that this question may not be
answerable by all of them because many of these theories have not yet presented a full
account of how their basic constituents interact, and at the Planck length and above, give
rise to the familiar manifold structure of spacetime described by GR. What makes this non-
trivial and non-straightforward is the possibility of the emergence of mesoscopic structures
between the micro (QG) and the macro (GR), or the occurrence of phase transitions which,
in our opinion, is very likely. Quantum gravity not being discussed here since it is a topic



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covered by many books and reviews, we will limit our attention to identifying those micro-
theories which predict a robust and smooth (near-isotropic and homogeneous) geometry at
the Planck scale rising from sub-Planckian scale by phase transitions or other processes or
mechanisms (e.g., polymerization, crystalization).

4.1. Quantum Phase Transition: Continuum to Discrete

Asymptotic Safety
First proposed by Weinberg [202], the asymptotic safety program in quantum

gravity [203–211] asserts that the short-distance behavior of gravity is governed by a
nontrivial renormalization group fixed point. As describd by Bonano and Saueressig [212]
It provides an elegant mechanism for completing the gravitational force at sub-Planckian
scales. At high energies, the fixed point controls the scaling of couplings such that un-
physical divergences are absent, while the emergence of classical low-energy physics is
linked to a crossover between two renormalization group fixed points. These features make
asymptotic safety an attractive framework for the building of a cosmological model.

D’Odorico and Saueressig [213] studied quantum corrections to the classical Bianchi
I and Bianchi IX universes. The correction terms induce a phase transition in the dy-
namics of the model, changing the classical, chaotic Kasner oscillations into a uniform
approach to a point singularity. This seems to be consistent with the results obtained
from the backreaction of particle creation from quantum fields, namely, the isotropization
of anisotropy, which we mentioned in Section 2.3 and which will be described in Section 5.1.

Causal Dynamical Triangulation CDT
A systematic and vigorous treatment of discrete quantum geometry is the Causal

Dynamical Triangulations (CDT) program pursued by Ambjorn, Jurkiewicz, Loll et al.
extensively for decades. CDT is a lattice model of gravity that has been used to study
non-perturbative aspects of quantum gravity. Read the review of Loll [214], and for the
most recent development, Ambjorn et al. [215]: “CDT has a built-in time foliation but is
coordinate-independent in the spatial directions. The higher-order phase transitions observed
in the model may be used to define a continuum limit of the lattice theory. Some aspects
of the transitions are better studied when the topology of space is toroidal rather than
spherical. In addition, a toroidal spatial topology allows us to understand more easily the
nature of typical quantum fluctuations of the geometry. In particular, this topology makes
it possible to use massless scalar fields that are solutions to Laplace’s equation with special
boundary conditions as coordinates that capture the fractal structure of the quantum geometry.
When such scalar fields are included as dynamical fields in the path integral, they can have
a dramatic effect on the geometry.” One can view this as a discrete modeling of the Gowdy
T3 universe we described in Section 3.1. The quantum phase transition is of course the
interestingly new and exciting development.

Causal Set: Continuum to crystalline phase transition
Causal set theory, first proposed by Sorkin [216] and developed extensively by

Dowker [217,218] and their associates, has seen significant developments both in the
number of adherents and in scope of its research topics. How the classical continuum
spacetime as we know it evolved from discrete causal sets is one of the key issues.
Brightwell et al. [219] noted that non-perturbative theories of quantum gravity inevitably
include configurations that fail to resemble physically reasonable spacetimes at large scales.
Yet these configurations are entropically dominant and pose an obstacle to obtaining the
desired classical limit. These authors examine this ‘entropy problem’ in a model of causal
set quantum gravity corresponding to a discretization of 2D spacetimes. Using results from
the theory of partial orders they show that, in the large volume or continuum limit, its par-
tition function is dominated by causal sets which approximate a region of 2D Minkowski
space.



Universe 2021, 7, 424 17 of 28

Surya [220] continued this investigationand presented evidence for a continuum phase
in a theory of 2D causal set quantum gravity, which contains a dimensionless non-locality
parameter ε ∈ (0, 1]. She also found a phase transition between this continuum phase and
a new crystalline phase which is characterized by a set of covariant observables. For more
recent developments, see her review [221].

Berezenskii-Kosterlitz-Thouless (BKT) transition
Antoniadis, Mazur, and Mottola [222] presented a simple argument which determines

the critical value of the anomaly coefficient in four dimensional conformal factor quantum
gravity, at which a phase transition between a smooth and elongated phase should occur.
The argument is based on the contribution of singular configurations (“spikes”) which
dominate the partition function in the infrared. The critical value is the analog of c = 1
in the theory of random surfaces, and the phase transition is similar to the Berezenskii-
Kosterlitz-Thouless transition. The critical value they obtain is in agreement with the
previous canonical analysis of physical states of the conformal factor and may explain why
a smooth phase of quantum gravity has not yet been observed in simplicial simulations.

2D Quantum Gravity and Random Geometry
This was an active topic of research in the second half of the 80s. As a representative

work, Gross and Migdal [223] proposed a nonperturbative definition of two-dimensional
quantum gravity, based on a double scaling limit of the random matrix model which they
solve to all orders in the genus expansion. They derived an exact differential equation
for the partition function of two-dimensional gravity as a function of the string coupling
constant that governs the genus expansion of two-dimensional surfaces. The works of the
principal contributors to this subject are collected in a book [224].

4.2. Quantum Backreaction of Inhomogeneous Superspace Modes

We have mentioned the backreaction problem earlier. It may refer to several differ-
ent effects. We use different symbols to denote them: (C) the backreaction of inhomo-
geneous modes of classical gravitational perturbations on the homogeneous modes in
classical general relativity, as in the works of Buchert, Clifton, Ellis et al. versus Green &
Wald. (Q) The backreaction of the inhomogeneous modes of quantum geometry (in the
full superspace of 3-geometries) on the mini-superspace quantum cosmology as studied
in [225,226]. In between the classical and quantum levels lie the semiclassical and stochastic.
(S) The backreaction in semiclassical and stochastic gravity refers to the effects of quantum
matter field processes, such as the trace anomaly or particle creation, back-reacting on a
classical background geometry: (SC) Semiclassical gravity when the mean value of the
stress energy tensor of matter fields act as the source in the semiclassical Einstein equation;
(ST) stochastic gravity when the fluctuations of the quantum fields are included as sources
of the Einstein-Langevin equation. The ‘quantum backreaction’ in the title of this paper
refers to the middle two levels between the classical and the quantum.

These 4 levels (C-SC-ST-Q) of backreaction are summarized in a recent review with an
extensive bibliography [225]. These authors start by assessing the question of backreaction,
i.e., whether cosmological inhomogeneities have an effect on the large scale evolution of
the Universe, focusing on the purely quantum mechanical backreaction. They present
one recent approach based on mathematical tools inspired by the Born–Oppenheimer
approximation to include backreaction in quantum cosmology.

We mention three groups of representative work on this issue in the 80s to 90s , then
leave this topic for the readers to explore with the help of this excellent review.

(1) Using path-integral quantization, Hawking and Halliwell [227] assume that the
Universe is in the quantum state defined by a path integral over compact four-metrics. This
can be regarded as a boundary condition for the wave function of the Universe on super-
space, the space of all three-metrics and matter field configurations on a three-surface, the
same as was proposed by Hartle and Hawking earlier. They treated the homogeneous and



Universe 2021, 7, 424 18 of 28

isotropic degrees of freedom of the Friedmann Universe exactly and the inhomogeneous
and anisotropic degrees of freedom in superspace to second order in the perturbations. For
the same model Kiefer [228] calculated explicitly the wave functions for all multipoles of
the matter field and for the tensor modes of the metric. D’Eath and Halliwell [229] have
considered the backreaction of the fermionic perturbations on the homogeneous modes.
Origin of structure in supersymmetric quantum cosmology was studied by Moniz [230].
By employing the supersymmetry and Lorentz constraint equations a set of quantum
states was obtained and a particular quantum state which has properties typical of the
conventional no-boundary Hartle-Hawking solution was identified.

(2) In canonical quantum gravity Kuchař and Ryan [231] questioned the validity of
physical predictions based on minisuperspace quantization of Einstein’s theory of gravita-
tion. They proposed to investigate a hierarchy of models with higher symmetry embedded
in models of lesser symmetry, to spell out the criteria under which minisuperspace quan-
tum results can be expected to make meaningful predictions about full quantum gravity.
As a concrete example they studied a homogeneous, anisotropic cosmological model of
higher symmetry (the Taub model) embedded in one of lesser symmetry (the mixmaster
model) and showed that the respective behavior is widely different.

(3) With quantum field theory placed in an open systems setting, Sinha and Hu [232,233]
used the example of an interacting λΦ4 scalar quantum field in a closed Robertson-Walker
universe, where the scale factor and the homogeneous mode of the scalar field model the
minisuperspace degrees of freedom. They explicitly computed the back-reaction of the
inhomogeneous modes on the minisuperspace sector using a coarse-grained effective action
and show that the minisuperspace approximation is valid only when this backreaction
is small.

For backreaction due to quantized matter fields, its decoherence effects, and how the
semiclassical limits are reached in quantum cosmology, see, e.g., [67,234–237].

Concerning quantum backeaction, specifically the question of how the inhomogeneous
quantum modes backreact on the lower dimensional superspace modes, and whether the
results from mini- or midi-superspace can give a fair representation of the full picture,
a recent paper by Bojowald [238] also expressed such a concern: “Even though the BKL
scenario allows us to use the classical dynamics of homogeneous models to understand
space-time near a spacelike singularity, it is a poor justification of minisuperspace models in
quantum cosmology. Using a minisuperspace model to evolve from a nearly homogeneous
geometry at late times to a BKL-like geometry at early times means that we begin with
a well-justified, approximate infrared contribution of the full theory, but then push the
infrared scale all the way into the ultraviolet.”

Meanwhile, we shall wait to see further research results from the practitioners of loop
and spin-foam quantum cosmology to see how the singularity avoidance in LQG might be
altered by including the backreaction of inhomgeneous quantum modes.

In the next section we shall discuss the backreaction of quantum field processes and
how that affects the dynamics of classical background spacetimes. Before ending, we men-
tion two backreaction effects, one in inflation, the other in classical GR. Finelli et al. [239]
studied the backreaction of scalar fluctuations on the space-time dynamics in the long
wavelength limit using a set of gauge invariant variables. Below, we add a short interlude
in the classical backreaction research programs for completeness, as it bears on gravitational
entropy, albeit less so on the WCH at the cosmological singularity or bounce.

Backreaction of inhomogeneous mode in classical GR
There are two schools of thought on classical backreaction: This research began around

2000, with the work of Buchert [240] and Ellis with collaborators [241]. The central themes
are summarized in this letter [242] where these authors make the observation that “A
large-scale smoothed-out model of the universe ignores small-scale inhomogeneities, but
the averaged effects of those inhomogeneities may alter both observational and dynamical
relations at the larger scale”. It also comments briefly on the relation to gravitational entropy.



Universe 2021, 7, 424 19 of 28

A broader representation of their work can be found in these two reviews [243,244] See
also [245,246].

Green and Wald, on the other hand, showed in a series of papers from 2011–2014 [247]
that the backreaction effects of classical inhomogenous modes are insignificant. These issues
are related to how one should treat the inhomogeneous modes of shorter wavelengths, such
as using the Brill-Hartle-Isaacson average [155,156,248,249], and how much they impact on
the behavior of the theory if only the lowest modes are retained. Buchert, Ellis et al. [250]
countered the Green-Wald arguments, reasserting their claims. In turn Green and Wald
replied to this [251,252] with a simple derivation of their no backreaction results.

5. Semiclassical Backreaction Supports WCH

In this last section we examine to what degree is WCH consistent or contradictory
with the results of semiclassical backreaction, namely, the backreaction of quantum field
processes involving vacuum polarization or fluctuations on the geometrodynamics near
the Planck time, like those due to the trace anomaly or cosmological particle creation.
These processes have been studied rather thoroughly in the twenty years 1977–1996 as
we mentioned before. Earlier, in Sections 1.2, 2.3 and 3.1, we focused on the avoidance
of singularity due to the backreaction effects of the trace anomaly and particle creation.
Here we complete our narrative by adding the considerations of the damping of anisotropy
and inhomogeneity.

5.1. Damping of Irregularities Ensures a Smooth Transition, Critical for Cyclic Cosmologies

Sparing the reader the details of the derivations of regularized stress energy tensors
for particle creation and the calculations of the backreaction effects from the solutions of
the semiclassical Einstein equation (which can be found in the papers referred to in earlier
sections or in a recent book [28]), we shall just highlight the key features with relevance to
WCH in the following. More comments about gravitational entropy in this context can be
found in [30].

(1) Vacuum fluctuations of a quantum field are ubiquitous, one need not add them
in by hand, nor can one wish them away. Their effects on the background spacetime
have to be included in considerations of whether the universe contracts to a singularity or
undergoes a bounce.

(2) Many vigorous quantum field- theoretical studies of particle creation in (weakly)
anisotropic and inhomogeneous spacetimes show that vacuum particle production is
abundant near the Planck time, their backreaction effects are strong and the dissipation
of these irregularities happens swiftly. (Note the significant qualitative differences be-
tween semiclassical backreaction of quantum field processes and classical backreaction of
inhomogeneous modes in GR mentioned earlier).

(3) The damping of irregularities drives down the Weyl curvature with their strong
and swift actions. This seems to suggest that quantum field processes act as an effective
protector, maybe even a guarantor, of WCH.

(4) These quantum field processes act in such a way as to draw the spacetime into a
fixed point in superspace, a three-geometry of high isotropy and homogeneity, the special
status bestowed by the WCH.

(5) The overall backreaction effects of quantum field processes can be summarized by
something resembling a Lenz law: the production of particles acts in such a way as to dimin-
ish their production. This is because, when the universe is isotropized and homogenized,
it becomes conformally-flat, whence no more conformal particles are produced. (There is
ground to believe that in the very early universe massless particles are overwhelmingly
more abundant.)

(6) For any aficionado of cyclic cosmology, in addition to showing that a bounce is
possible, the ‘reining-in’ of the irregularities near the bounce–in fact, near every bounce– is
particularly important [118]: If the isotropic expansion were unstable as the scale of the
universe approaches zero, then huge irregularities and anisotropies could accumulate and



Universe 2021, 7, 424 20 of 28

the successive cycles would be very different and increasingly anisotropic. Some powerful
and ever-ready damping mechanisms need be there to prevent this from happening.

In summary, if we ponder upon the ubiquitous nature of these quantum field processes,
their strongly dissipative effects on the spacetime irregularities, and the isotropized and
homogenized outcome ‘locked-in’ by the Lenz law, we can see that WCH is more likely to
be validated when these factors are included in our considerations. Add to this the fact
that these processes apply both in the expansion and contraction phases of the universe,
rendering both the beginning and ending of a cycle to be smooth, we can see that quantum
field processes are good facilitators of cyclic cosmology.

5.2. Quantum Gravity: Macro Spacetime Manifold from the Interaction of Micro Constituents

Going from the Planck scale down toward the singularity or bounce, even from our
rather skimpy description earlier, it may appear to the reader that some quantum gravity
theories, e.g, those based on the WdW equation, Hamiltonian quantum gravity, and loop
quantum cosmology are more capable of providing a detailed description of quantum
spacetime than others, e.g., strings, spin foams, causal sets. This is not the case. One needs
to first scrutinize whether some tacit, commonplace, yet unproven assumption has crept in,
that the macro variables are the same as the micro variables. One should ask, “Are they?”. This
is a pivotal question, because by assuming there is no change of variables from micro to
macro, one only needs to deal with the quantum to classical transition issue. In my opinion,
a more important and challenging question is that of micro to macro transition.

Theoretically, this cuts into the key issue of whether the dynamical variables in GR are
‘fundamental’ all the way from the scale of our universe now, captured amazingly well by
Einstein’s theory, to the sub-Planckian scales, and, for the former group of theories, even
all the way down toward the singularity. What if the variables are collective variables, that
the macroscopic spacetime is emergent, and that general relativity is an effective theory
valid only at large scales and low energies? We know Nature has many, possibly an infinite
number of, levels of structure, separated by characteristic scales, and each level can be
described by a suitably constructed effective theory. If so, the theories we now know of the
large scale structures of spacetime will very likely become inapplicable at scales below the
Planck length.

Are we to believe that once a macro-variable is quantized, it becomes a micro-variable?
This is a big leap of faith. In Nature, a quantized macro-variable can be a very different
physical entity from a micro-variable, which can be of a quantum nature. E.g., sound is a
collective excitation of atoms. Quantizing them yields phonons. But quantization of sound
will never lead to the microvariables we want–atoms. In fact, these collective variables
lose their meanings once the atomic scale is reached. If the metric or connection forms
are collective variables, quantizing them is the wrong way to discover the sub-Planckian
microscopic constituents. In contrast, those theories which propose certain entities as the
microscopic constituents of spacetime, such as strings or causets, need to show that their
interactions can indeed produce the macroscopic spacetime we know, thus is the difference
between quantum and emergent gravity [83].

5.3. Quantum Cosmology: Singularity or Bounce Ruled by the Interaction of Micro-Constituents

Varied as the many schools of thought and practice are, there is agreement that quan-
tum gravity refers to theories of the microscopic structures of spacetime. Quantum cosmology
is supposed to be the application of quantum gravity to the description of the universe
from the Planck scale down to the singularity or the bounce. Thus, some knowledge of the
underlying theory of quantum gravity is required. Many theories of quantum cosmology
assume the existence of a metric or connection, and many papers, such as in string cosmol-
ogy, start with a statement like, “Let us consider strings in the FLRW or de Sitter universe”.
Right there, I must say, an ontological issue already shows up. Where does your spacetime
manifold come from? If we believe that strings or loops are the fundamental microscopic
constituents of spacetime at scales below the Planck length, then, before one talks about



Universe 2021, 7, 424 21 of 28

string or loop cosmology in terms of the dynamics of the metric or connection, one needs to
explain how the spacetime manifold, this macroscopic structure of our universe, befittingly
described by general relativity, comes into being.

It appears that a clear demonstration of how the spacetime manifold emerges from
string or loop interactions is still largely lacking. If one takes the dynamical variables of
a macroscopic theory of spacetime and quantizes them, thinking that one would get the
microscopic degrees of freedom in this way, while disregarding the possibly huge structural
and behavioral differences between the micro and the meso- and macro-scopic emergent
structures, the claims made pertaining to the nature of a singularity or a bounce in the
sub-Planckian length regions may not make much sense or even be outright inconsistent.

Therefore, unless one can provide a proof that the micro-variables describing sub-
Planckian physics are the same as the macro variables of the large scale spacetime we are
familiar with, in our opinion, all predictions and claims about bounces made in quantum
cosmology based on quantizing the macro geometric variables in general relativity need
be reassessed.

It is with this broader perspective that we should look at the meaning and applicability
of WCH. It may very well be that it is enough to examine the validity and implications of
WCH at the Planck scale and not extend it to the sub-Planckian scale, because, after all, the
Weyl curvature, like all the geometric objects, is well defined only when the spacetime it
lives in has a manifold structure, which could be an emergent entity of the underlying QG
theories for the microscopic constituents of spacetime.

Funding: This research received no external funding.

Acknowledgments: I would like to thank Paul Anderson and Paulo Moniz for a careful proof-
reading of the completed manuscript with helpful suggestions, and for providing the references
for the approach to the Big Crunch and for supersymmetic quantum cosmology, respectively. The
assistance of Cici Xia at MDPI Wuhan to put the references in the MDPI format is much appreciated.

Conflicts of Interest: The author declares no conflict of interest.

Notes
1 This list is far from exhaustive. We are not treating inflationary cosmology [31–34] here, since it already has a wide coverage

and how the WCH fits in with inflation is rarely discussed. On the relation of inflation and singularity, Borde and Vilenkin [35]
made a categorical statement that a physically reasonable spacetime that is eternally inflating to the future must possess an
initial singularity. Vilenkin [36] also questioned the necessity of quantum cosmology in the face of eternal inflation. As for the
speculations into our universe’s future, there is considerable amount of work based on two types of inflationary models, one is
the ‘stochastic inflation’ of Starobinsky [37,38] which allows the noise associated with the fluctuations of a quantum field to drive
the universe to inflation. The other is the ‘eternal inflation’ proposed and developed by Linde, Vilenkin, Guth and others [39,40],
where a universe can live forever with the continual births of baby universes in the sequential and parallel generation of multiple
branches. For a recent assessment of the likelihood for eternal inflation in a variety of popular models, including the swampland
of string theory, see, e.g., [41–43].

2 Chaotic dynamics of minisuperspace cosmology is not limited to the mixmaster type. As Calzetta and Hasi show [64,65] it can
also occur in the dynamics of a spatially-closed FLRW universe conformally coupled to a real, free, massive scalar field for large
enough field amplitudes.

3 An important exception is the case when a massless scalar field is present, which obeys an equation of state p = ρ like a stiff
fluid, and, in Friedmann models, has the same dependence of the density on the scale factor as anisotropies (ρ ≈ a−6). As is
rigorously shown in [71], such a scalar field will suppress the BKL oscillations during the evolution towards the singularity. The
relevance of stiff matter in the early universe is noted by Barrow [72].

4 This diagonal form has full generality because there is no spatial curvature. In a Bianchi Type IX universe where the spatial
curvature is present, spacetimes represented by the full matrix βij are more general than that of the diagonal metric which is
Misner’s mixmaster universe [50]; the off-diagonal components signify rotation [74].

5 An element of craziness needs to creep in before an idea ascends to the order of the three Fs: Fashion, Faith, and Fantasy. I read
this as a warning of an enlightened guru speaking to his present adherents and future believers: Safeguard your independent
thinking before you become completely converted.



Universe 2021, 7, 424 22 of 28

6 We could include group field theory (GFT) [173–176] in the present considerations (only) to the extent that (partial) equivalence
of models employed or results reported can be shown to exist between these two theories [177]. Of course we need to hear the
views of the GFT community on this.

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