KCL-PH-TH/2023-42

# Tunnelling-induced cosmic bounce in the presence of anisotropy

Jean Alexandre  Theoretical Particle Physics and Cosmology, King’s College
London, WC2R 2LS, UK  Katy Clough  Geometry, Analysis and Gravitation, School
of Mathematical Sciences, Queen Mary University of London, Mile End Road,
London E1 4NS, UK  Silvia Pla  Theoretical Particle Physics and Cosmology,
King’s College London, WC2R 2LS, UK

###### Abstract

If we imagine rewinding the universe to early times, the scale factor shrinks
and the existence of a finite spatial volume may play a role in quantum
tunnelling effects in a closed universe. It has recently been shown that such
finite volume effects dynamically generate an effective equation of state that
could support a cosmological bounce. In this work we extend the analysis to
the case in which a (homogeneous) anisotropy is present, and identify a
criteria for a successful bounce in terms of the size of the closed universe
and the properties of the quantum field.

##  I Introduction

Our universe is expanding, and on the largest scales it appears homogeneous
and isotropic with a small spatial curvature [1]. In the standard paradigm, in
which our universe emerged from a cosmological singularity, inflation provides
a dynamical mechanism to achieve the current state for a generic initial
condition [2, 3, 4]. However, the success of inflation is not entirely
independent of the initial conditions - some models require a certain level of
homogeneity in order to proceed (see [5] for a review), and all inflationary
potentials will fail to create an exponential expansion for an initially
collapsing state in the absence of a violation of the Null Energy Condition
(NEC) [6, 7, 8, 9]. An alternative paradigm, that of ekpyrosis [10, 11, 12],
commonly uses a mechanism of slow contraction to provide the smoothing of
inhomogeneities in the case of a non singular cosmic bounce, (see [13] for a
review). Such models also necessitate a violation of the NEC in order to
transition to expansion. Therefore mechanisms that violate the NEC in the
early universe are of interest for such scenarios.

Most mechanisms for NEC violation require additional exotic components or a
modification of general relativity (GR) [8]. Recently, a mechanism has been
proposed in which the NEC is violated by finite volume effects, which
necessarily occur for a scalar field with a Higgs-like potential in standard
Quantum Field Theory (QFT) on an FLRW background with a closed topology [14].
The effect arises from tunnelling between two degenerate vacua, which is
allowed if the field is confined in a finite spatial volume 111Note that this
effect is distinct to the well known Casimir effect [15]. Although both are
based on the finite-volume assumption, tunnelling is independent of the
geometry/topology of the spatial unit cell. For a comparison between the two
effects, see [16].. One nice aspect of this mechanism is that it “turns off”
in a period of expansion, meaning that after a cosmological bounce it would
quickly become suppressed - it therefore naturally favours expansion over
contraction. We note here that alternative quantum effects, involving fermion
dynamics, have been proposed to induce NEC-violation and potentially lead to a
cosmological bounce [17, 18, 19, 20].

A key question is whether the mechanism described in [14] could also provide
some kind of smoothing of inhomogeneities or anisotropies, and to what extent
it must dominate over these in order for the bounce to be successful. In this
work we will discuss the (homogeneous) anisotropic case, and explain why the
energy of the quantum fluid must already dominate over the anisotropy before
the bounce in order for it to proceed. In other bounce scenarios, a scalar
field that is dominated by its kinetic energy can play the role of a smoother
in a preceding slow contraction phase [13], so it is possible that the field
itself could be responsible for this at an earlier kinetic dominated field
(for example, as it rolls down into one of the minima of the potential)
222Also, gravitational particle creation tends to rapidly suppress
irregularities in the geometry, which can be seen with semiclassical
backreaction effects [21].. However, the description we use here is only valid
in the vicinity of the bounce, and so more work is required to quantify out-
of-equilibrium effects and the impact of inhomogeneities at an earlier stage.
These aspects are more difficult to treat and need to be explored further in
future work. What is clear is that by the time the universe is nearing the
bounce, the anisotropies must be suppressed in order for it to succeed.

We will show that criteria for the success of the bounce can be stated in
terms of the size of the closed universe at the point at which the net energy
density is zero, and the properties of the quantum field (mainly its mass and
vacuum energy). We focus on the case of anisotropy as a second component since
it is the component that dominates the energy budget the quickest during a
collapse. Roughly speaking, at the point of zero net energy density, the size
of the universe must be comparable to the Compton wavelength of the field for
its pressure to be sufficient to turn around the collapse. We will make this
statement more precise in what follows, and give the phenomenological
consequences for the field.

The article is organised as follows: In Sec. II, we summarise the background
of tunnelling in a finite volume, in Sec. III we set out the standard
description of a homogeneous anisotropic cosmology, in Sec. IV we extend the
QFT description to the anisotropic case and in Sec. V we describe the
conditions for success in terms of the model parameters and discuss
phenomenology. We briefly provide some numerical illustrations in Sec. VI and
conclude with a brief discussion in Sec. VII.

##  II Background I: Tunnelling in an FLRW background with finite volume

Spontaneous Symmetry Breaking, where the scalar field is trapped above one
vacuum, is only strictly valid in an infinite volume, where tunnelling to
another degenerate vacuum is completely suppressed. In a finite volume,
tunnelling between two degenerate bare vacua $\phi=\pm v$ is possible, leading
to an effective potential with a lower overall minimum (see figure 1).

Figure 1: Schematic representation of the bare potential $U(\phi)$ (blue line)
and the effective potential $U_{eff}(\phi)$ obtained from tunnelling between
the vacua (orange dashed line). $U_{0}$ is given by eq.(3), from which we can
see that the effective potential curve reaches a lower value for a smaller
volume. In this way the finite volume effects act as a negative contribution
to the energy density, and can violate the NEC during a period of contraction.

Using a semi-classical approximation for the partition function, which is
dominated by a dilute gas of instantons and anti-instantons, it was shown in
[22, 16] that the resulting effective theory is such that:

  1. 1.

the true vacuum is symmetric and occurs at $\phi=0$, consistently with
convexity of the effective potential when several saddle points are taken into
account [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33];

  2. 2.

the corresponding effective action is not extensive (it is not proportional to
the volume), but has a non-trivial volume dependence;

  3. 3.

the resulting ground state fluid violates the NEC.

The above features arise from non-perturbative properties of the partition
function, since a convex effective potential cannot be obtained from a bare
double-well potential with perturbative quantum corrections only.

The (anti-)instantons considered in [22, 16] depend only on the Euclidean
time, and the tunnelling process is similar to the one described in Quantum
Mechanics, where tunnelling of a particle in a double-well potential leads to
a true ground state energy which is lower than the ground state energy in both
individual degenerate wells [34] 333Because the vacua are degenerate, the
$O(4)$-symmetric Coleman bounce [35, 36] does not play a role here, since it
would require a bubble with infinite radius. The tunnelling mechanism
described in [22, 16] is therefore not related to a first order quantum phase
transition, but rather to a second order phase transition and happens
uniformly in space..

Following the same approximation for the partition function, but in a flat and
isotropic Friedmann-Lemaitre-Robertson-Walker universe (FLRW), it was shown in
[14] that the above mechanism dynamically generates a cosmic bounce, which is
followed by an asymptotic de Sitter phase where tunnelling is suppressed
exponentially. In this context, “finite volume” is provided by a unit cell in
the form of a 3-torus of volume $V_{cell}$. Some recent works in cosmology
have considered the evidence for a closed universe, see for example [37].
However, since we do not see any periodicity in our observable universe, any
closed volume must be larger than its current size, so any finite volume
effects will now be negligible. The physical size of the closed universe can
of course be much smaller far in the past, when our comoving volume was
smaller, and thus finite volume effects could have played a role in the early
universe.

In the isotropic case of [14], tunnelling between the minima of the double-
well potential

$U(\phi)=\kappa^{-1}\Lambda_{b}+\frac{\lambda_{b}}{24}(\phi^{2}-v_{b}^{2})^{2}~{},$
(1)

leads to the convex effective potential

$U_{eff}(\phi)=U_{0}+\frac{1}{2}M^{2}\phi^{2}+{\cal O}(\phi^{4})$ (2)

where

$\displaystyle U_{0}$ $\displaystyle=$
$\displaystyle\kappa^{-1}\Lambda\left(1-r\frac{e^{-\alpha_{iso}^{3}}}{\alpha_{iso}^{3/2}}\right)$
(3) $\displaystyle M^{2}$ $\displaystyle=$ $\displaystyle\frac{\lambda
v^{2}}{3}\left(1-\frac{27\lambda}{32\pi^{2}}\right)>0~{}.$

In the above expressions, $\lambda_{b}$, $\Lambda_{b}$ and $v_{b}$ have to be
understood as the bare parameters, while $\Lambda$, $\lambda$ and $v$ are the
renormalised parameters 444For renormalisation purposes, it is necessary to
introduce the bare vacuum energy $\Lambda_{b}$. The renormalised vacumm energy
$\Lambda$ arises from the scalar field self-interactions; it is not put by
hand but it is generated by quantum fluctuations. The latter are dominated by
ultraviolet effects and not infrared effects, such that $\Lambda$ can be
assumed to be independent of $V_{cell}$.. Also, $\kappa=8\pi G$ and we have
defined

$r=\frac{\lambda\kappa
v^{4}}{3\sqrt{3\pi}~{}\Lambda}~{}~{}~{}~{},~{}~{}~{}~{}\alpha_{iso}^{3}\equiv
a^{3}\Sigma~{},$ (4)

where $\Sigma$ is the action of one instanton relating the bare vacua and $a$
is the FLRW scale factor (we consider $\hbar=c=1)$. The potential is
illustrated in Fig. 1.

Both $r$ and $\Sigma$ depend on the field parameters, but $\Sigma$ is also
proportional to the volume $V_{cell}$ of the fundamental spatial cell, which
therefore needs to be finite for the tunnelling probability
$\propto\exp(-\alpha_{iso}^{3})$ to be finite - that is, one requires a closed
universe.

The present study extends the work of [14] to the anisotropic case, and
considers the impact of other components being present. In this work, an
adiabatic approximation is assumed, where the tunnelling rate is large
compared to the expansion rate, which allows the use of equilibrium QFT. This
approximation is very good in the vicinity of the bounce, which is the regime
on which we focus.

##  III Background II: Homogeneous anisotropic universe description

We review here features of the homogeneous but anisotropic universe relevant
to our study, and in particular discuss how the anisotropy can be treated as
an additional matter component in the Friedmann equations, assuming an
appropriate equation of state. Starting with the anisotropic Bianchi-I metric

$\text{d}s^{2}=-\text{d}t^{2}+a^{2}_{1}\,\text{d}x^{2}+a^{2}_{2}\,\text{d}y^{2}+a^{2}_{3}\,\text{d}z^{2}~{},$
(5)

the Friedmann equations read ($i,j=1,2,3$)

$\displaystyle H_{1}H_{2}+H_{1}H_{3}+H_{2}H_{3}$ $\displaystyle=$
$\displaystyle+\kappa\rho~{},$ (6) $\displaystyle(\mbox{for}~{}i\neq
j)~{}~{}~{}~{}H_{i}H_{j}+\frac{\ddot{a}_{i}}{a_{i}}+\frac{\ddot{a}_{j}}{a_{j}}$
$\displaystyle=$ $\displaystyle-\kappa p~{},$ (7)

where $H_{i}=\dot{a}_{i}/a_{i}$ and as above $\kappa=8\pi G$. Following [38,
39] we note that eq.(6) can be written

$H^{2}=\frac{\kappa}{3}\rho+\sigma^{2}~{},$ (8)

where the averaged Hubble rate $H$ and the anisotropy $\sigma^{2}$ are defined
as

$\displaystyle H$ $\displaystyle:=$
$\displaystyle\frac{1}{3}(H_{1}+H_{2}+H_{3})~{},$ (9)
$\displaystyle\sigma^{2}$ $\displaystyle:=$
$\displaystyle\frac{1}{18}\Big{[}(H_{1}-H_{2})^{2}+(H_{2}-H_{3})^{2}+(H_{1}-H_{3})^{2}\Big{]}~{}.$

Eq.(8) shows that $3\kappa^{-1}\sigma^{2}$ can be interpreted as an energy
density arising from anisotropy. Similarly, the trace of Friedmann equations
can be written

$\dot{H}+H^{2}=-\frac{\kappa}{6}(\rho+3p)-2\sigma^{2}~{},$ (10)

such that $3\kappa^{-1}\sigma^{2}$ can also be interpreted as a pressure
arising from anisotropy. Anisotropy therefore plays a role similar to a
homogeneous perfect fluid with equation of state $w=p/\rho=1$, and we expect
that the corresponding energy density scales as $a^{-3(1+w)}=a^{-6}$, where
$a=(a_{1}a_{2}a_{3})^{1/3}$ is the average scale factor. This is consistent
with eqs.(7) from which one can show that

$\dot{H}_{i}-\dot{H}_{j}=-3H(H_{i}-H_{j})~{},$ (11)

which implies $(H_{i}-H_{j})\propto a^{-3}$ and thus $\sigma^{2}\propto
a^{-6}$.

Finally we note that, if a bounce occurs, then at this bounce $H=0$ and
$\dot{H}>0$, such that the matter contribution should satisfy at the bounce

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{3}{\kappa}\sigma^{2}\leq 0$ (12) $\displaystyle\rho+3p$
$\displaystyle=$
$\displaystyle-\frac{6}{\kappa}\left(\dot{H}+2\sigma^{2}\right)\leq 0~{},$

as in the isotropic case.

##  IV Anisotropic quantum fluid description

As discussed above, in the isotropic case ($a_{1}=a_{2}=a_{3}\equiv a$) and
from the effective potential (2), the action for the true ground state
$\phi=0$ is

$S_{eff}^{isotropic}=\int\text{d}^{4}x\sqrt{-g}~{}\kappa^{-1}\Lambda\left(1-r~{}\frac{e^{-\alpha_{iso}^{3}}}{\alpha_{iso}^{3/2}}\right)~{}.$
(13)

In the anisotropic case (5), the only change to the instanton action $\Sigma$
is via the volume $a^{3}V_{cell}\to a_{1}a_{2}a_{3}V_{cell}$, such that the
action (13) must be modified as

$S_{eff}=\int\text{d}^{4}x\sqrt{-g}~{}\kappa^{-1}\Lambda\left(1-r~{}\frac{e^{-\alpha_{1}\alpha_{2}\alpha_{3}}}{\sqrt{\alpha_{1}\alpha_{2}\alpha_{3}}}\right)~{},$
(14)

where $\alpha_{i}\equiv a_{i}\Sigma^{1/3}$. The stress-energy tensor can be
decomposed as

$T_{\mu\nu}\equiv\frac{2}{\sqrt{-g}}\frac{\delta S_{eff}}{\delta
g^{\mu\nu}}=\textrm{diag}(\rho,a_{1}^{2}\,p,a_{2}^{2}\,p,a_{3}^{2}\,p)~{},$
(15)

and leads to the dimensionless energy density and pressure

$\displaystyle\tilde{\rho}$ $\displaystyle\equiv$
$\displaystyle\frac{\kappa\rho}{\Lambda}=1-r~{}\frac{e^{-\alpha^{3}}}{\alpha^{3/2}}~{},$
(16) $\displaystyle\tilde{p}$ $\displaystyle\equiv$ $\displaystyle\frac{\kappa
p}{\Lambda}=-1+r~{}\left(\frac{1}{2\alpha^{3/2}}-\alpha^{3/2}\right)e^{-\alpha^{3}}~{},$

where the average scale factor $\alpha$ is defined by

$\alpha^{3}\equiv a_{1}a_{2}a_{3}\Sigma~{}.$ (17)

In the previous expressions and from its definition in Eq.(4), $r$ describes
the quantum field - it is completely determined once we specify its mass,
vacuum energy and self interaction strength (via the parameters $v$, $\Lambda$
and $\lambda$). The pressure and energy density are therefore determined by
the combination of field parameters $r$ and the average size of the universe
$a\,V_{cell}^{1/3}$.

The Friedmann equations read, in terms of the rescaled quantities, with the
rescaled time $\tau\equiv t~{}\sqrt{\Lambda/3}$,

$\displaystyle\frac{\mathcal{H}_{1}\mathcal{H}_{2}}{3}+\frac{\mathcal{H}_{1}\mathcal{H}_{3}}{3}+\frac{\mathcal{H}_{2}\mathcal{H}_{3}}{3}$
$\displaystyle=$ $\displaystyle+\tilde{\rho}~{},$ (18)
$\displaystyle(\mbox{for}~{}i\neq
j)~{}~{}~{}~{}\frac{\mathcal{H}_{i}\mathcal{H}_{j}}{3}+\frac{\alpha_{i}^{\prime\prime}}{3\alpha_{i}}+\frac{\alpha_{j}^{\prime\prime}}{3\alpha_{j}}$
$\displaystyle=$ $\displaystyle-\tilde{p}~{},$ (19)

where a prime refers to the derivative with respect to $\tau$. We can then
define the rescaled average Hubble rate $\mathcal{H}$ and anisotropy
$\tilde{\sigma}^{2}$

$\displaystyle\mathcal{H}$ $\displaystyle:=$
$\displaystyle\frac{1}{3}(\mathcal{H}_{1}+\mathcal{H}_{2}+\mathcal{H}_{3})~{},$
(20) $\displaystyle\tilde{\sigma}^{2}$ $\displaystyle:=$
$\displaystyle\frac{1}{18}\Big{[}(\mathcal{H}_{1}-\mathcal{H}_{2})^{2}+(\mathcal{H}_{2}-\mathcal{H}_{3})^{2}+(\mathcal{H}_{1}-\mathcal{H}_{3})^{2}\Big{]}~{},$

so that eq.(18) can be simply written as

$\mathcal{H}^{2}=\tilde{\rho}+\tilde{\sigma}^{2}~{}.$ (21)

We can see from eq. (12) that $\tilde{\rho}<0$ and
$\tilde{\rho}+3\tilde{p}<0$, in order to balance out the anisotropy
contribution in the vicinity of the average bounce, defined by $\mathcal{H}=0$
and $\mathcal{H}^{\prime}>0$. In what follows, we will study under which
conditions the cosmological bounce can be induced.

##  V Critical solutions

As discussed in the previous section, in a universe with significant
anisotropy, an additional contribution to the energy density and the pressure
of the spacetime exists. Starting from some initial condition, several
scenarios are possible given the different scalings in $\alpha$. The NEC
violation from tunnelling does not necessarily win over the anisotropy during
the collapse (even where it is initially larger) - a bounce requires not only
that both contributions cancel each other such that $\mathcal{H}=0$, but also
that at this point of equality, the pressure satisfies the necessary condition
for the universe to bounce (i.e., $\mathcal{H}^{\prime}>0$). For this latter
condition to be true, the size of the universe at this point must be
sufficiently small (relative to the field parameters) for finite volume
effects to be significant, but not too small to avoid a collapse. In this
section we derive the specific requirements, and comment on the resulting
phenomenology.

###  V.1 Critical point

The critical solution of the Friedmann equations for which a bounce occurs
$(\rho_{c},p_{c},\alpha_{c},\tilde{\sigma}_{c}^{2})$ can be found by imposing
the condition $\mathcal{H}=\mathcal{H}^{\prime}=0$. This critical point is
unstable: a value of $\alpha$ that is slightly larger than $\alpha_{c}$ leads
to a bounce (the NEC violation $\propto\alpha^{-3/2}$ dominates) and a value
which is slightly smaller leads to a collapse (the anisotropy
$\propto\alpha^{-6}$ dominates).

From these conditions, the energy density and the pressure at the critical
point satisfy

$\tilde{p}_{c}=\tilde{\rho}_{c}=-\tilde{\sigma}^{2}_{c}~{}.$ (22)

Also, from eqs.(16), we find that the averaged scale factor $\alpha_{c}$ is
given by the implicit algebraic equation

$4\alpha_{c}^{3/2}+re^{-\alpha_{c}^{3}}(-3+2\alpha^{3}_{c})=0~{},$ (23)

and the anisotropy can be expressed as

$\tilde{\sigma}^{2}_{c}=\frac{1+2\alpha^{3}_{c}}{3-2\alpha^{3}_{c}}~{}.$ (24)

We can see from eq.(23) that necessarily $\alpha_{c}^{3}<3/2$, and one can
identify the two regimes

$\displaystyle\alpha_{c}$ $\displaystyle\to$
$\displaystyle(3/2)^{1/3}~{}~{}~{}~{}\mbox{for}~{}~{}r\gg 1$ (25)
$\displaystyle\alpha_{c}$ $\displaystyle\sim$
$\displaystyle\left(\frac{3r}{4}\right)^{2/3}~{}~{}~{}~{}\mbox{for}~{}~{}r\ll
1~{}.$

One can understand the role of $\alpha_{c}$ from the point of view of the
pressure. Assume that there is a time $\tau_{1}$ where
$\mathcal{H}(\tau_{1})=0$:

  * •

A bounce requires the condition $\mathcal{H}^{\prime}(\tau_{1})>0$, and thus
$|\tilde{p}(\tau_{1})|>|\tilde{\rho}(\tau_{1})|$, such that

$4\alpha^{3/2}(\tau_{1})+re^{-\alpha^{3}(\tau_{1})}[-3+2\alpha^{3}(\tau_{1})]>0~{},$
(26)

which leads to $\alpha(\tau_{1})>\alpha_{c}$;

  * •

A collapse follows in the situation where $\mathcal{H}^{\prime}(\tau_{1})>0$,
and thus $|\tilde{p}(\tau_{1})|<|\tilde{\rho}(\tau_{1})|$, such that

$4\alpha^{3/2}(\tau_{1})+re^{-\alpha^{3}(\tau_{1})}[-3+2\alpha^{3}(\tau_{1})]<0~{},$
(27)

which leads to $\alpha(\tau_{1})<\alpha_{c}$.

###  V.2 Comparison with the isotropic case

One can also infer a maximum value for the rescaled scale factor
$\alpha_{iso}$ at the bounce, which happens when the anisotropy (and any other
components if there are) are negligible and the quantum field completely
dominates. We then have from eq.(16)

$\alpha_{iso}^{3/2}=r~{}e^{-\alpha_{iso}^{3}}~{},$ (28)

which leads to the two regimes

$\displaystyle\alpha_{iso}$ $\displaystyle\sim$ $\displaystyle(\ln
r)^{1/3}~{}~{}~{}~{}\mbox{for}~{}~{}~{}~{}r\gg 1$ (29)
$\displaystyle\alpha_{iso}$ $\displaystyle\sim$ $\displaystyle
r^{2/3}~{}~{}~{}~{}\mbox{for}~{}~{}~{}~{}r\ll 1~{},$ (30)

and we note that $\alpha_{iso}$ is not bounded when $r\to\infty$. We sketch
$\alpha_{c}$ and $\alpha_{iso}$ on Fig.2, where the region between the two
curves represents the possible range of values of the rescaled scale factor at
which a bounce can occur for a particular quantum field (as parametrised by
$r$).

Figure 2: A plot of the rescaled average scale factor at the bounce
$\alpha=a\Sigma^{1/3}$ (which is related to the size of the closed universe)
versus $r$ (which is determined by the properties of the field). We plot the
maximum $\alpha_{iso}$ (orange line) and minimum $\alpha_{c}$ (dashed blue
line) values, as a function of $r$. Although $\alpha_{c}$ asymptotically tends
to $(3/2)^{1/3}$, $\alpha_{iso}$ is not bounded and goes to infinity when
$r\to\infty$.

###  V.3 Size of the Universe at the bounce

Figure 3:  These plots illustrate consistent values of the field parameters
and the size $L_{b}$ of the closed universe at the bounce for different values
of the field parameter $r$. The upper plot shows the relation between
$\Lambda$ and $m$. The lower plot shows the relation between the maximum value
for $L_{b}$ and $m$ (the minimum of $L_{b}$ has very similar values and is not
represented for the sake of clarity - thus the range of possible values
between the limits in Fig. 2 all lie close to the lines in this plot). We note
that the function $L_{b}(m)$ is almost independent of $r$ for $r\gg 1$,
whereas it changes significantly for $r\ll 1$.

From the previous results one can put bounds on the typical physical size of
the universe at the bounce, for a given field. The instanton action is of the
order [16]

$\Sigma\sim\frac{m^{3}}{\lambda}V_{cell}~{},$ (31)

where $m=v\sqrt{\lambda/3}$. The physical length $L$ is given by
$L=a(t)V_{cell}^{1/3}$ and its value at the bounce then satisfies

$\lambda^{1/3}~{}\frac{\alpha_{c}}{m}~{}\lesssim~{}L_{b}~{}\lesssim~{}\lambda^{1/3}~{}\frac{\alpha_{iso}}{m}~{}.$
(32)

To get a feel for the phenomenological consequences of the model, we can
relate the field quantities $m$ and $\Lambda$ with the size of the closed
universe at the bounce $L_{b}$, parameterised by the ratio $r$. For simplicity
we assume here that $\lambda^{1/3}$ is of order 1 555This assumption is not
necessarily justified for an axion-like particle, with a potential of the form
$M^{4}\cos(\phi/f)$. Indeed, a small mass $\propto M^{2}/f$ compared to 1eV
[40] implies an extremely small self-coupling constant $\propto M^{4}/f^{4}$.
However, there are axion models not requiring such a small self coupling, as
for example in the string-inspired model presented in [41]. Our results can
easily be adapted to other values for $\lambda$ depending on the model..

The current size of the visible universe is $\sim 10^{26}$ meters, and we do
not see any evidence of periodicity in it [42]. Any bounce must have happened
before the electroweak phase transition, at which point the the size of our
observable universe was about $10^{11}$ meters. This therefore imposes a
minimum on the size of the closed universe at the bounce - any smaller and we
would see evidence for periodicity now. However, a bounce could also have
occurred much earlier than this and so the universe could have been smaller.
If instead we take the bounce to occur at the era of grand unification, the
size of the closed universe would be of order $1$ meter or larger.

As can be seen from Fig.3, for $r=1$, if the bounce occurred when $L\sim 1$ m,
the scalar field would need a bare mass of $\sim 10^{-7}$ eV and a vacuum
energy $\Lambda$ of order $10^{-140}~{}\ell_{p}^{-2}$, therefore much smaller
than the current cosmological constant. The plot illustrates how the values
change for different values of $r$, but in general one requires a larger mass
to be consistent with a smaller $L$, and small values for the vacuum energy
are required.

##  VI Numerical solutions

In this section we numerically integrate the Friedmann equations (18), both in
the bouncing case (where NEC violation dominates) and in the collapsing case
(where the anisotropy dominates), to illustrate the possible outcomes.

We choose the initial anisotropy $\tilde{\sigma}^{2}(\tau_{0})$ as one of the
parameters. For simplicity, we will consider
$\alpha_{2}(\tau_{0})=\alpha(\tau_{0})$ and
$\mathcal{H}_{2}(\tau_{0})=\mathcal{H}(\tau_{0})$. Hence from (20) we find for
all times

$\displaystyle\mathcal{H}_{1}$ $\displaystyle=$
$\displaystyle\mathcal{H}\pm\sqrt{3\tilde{\sigma}^{2}}~{},$ (33)
$\displaystyle\mathcal{H}_{2}$ $\displaystyle=$ $\displaystyle\mathcal{H}~{},$
(34) $\displaystyle\mathcal{H}_{3}$ $\displaystyle=$
$\displaystyle\mathcal{H}\mp\sqrt{3\tilde{\sigma}^{2}}~{}.$ (35)

The initial value of $\mathcal{H}$ can then be determined by equations (21)
and (16), namely

$\mathcal{H}^{2}(\tau_{0})=\tilde{\sigma}^{2}(\tau_{0})+1-r\frac{e^{-\alpha^{3}(\tau_{0})}}{\alpha^{3/2}(\tau_{0})}~{}.$
(36)

We take the negative root $\mathcal{H}(\tau_{0})<0$ since we want to start
from a contracting phase. As a consequence, the initial Hubble rates are
entirely determined by $\tilde{\sigma}^{2}(\tau_{0})$, and for the numerical
analysis the quantities we fix are $\tilde{\sigma}^{2}(\tau_{0})$,
$\alpha(\tau_{0})$ and $r$.

We take $r\leq e$, so that we are before the point of equality in the energy
densities in the anisotropy and the quantum field. For these values of $r$, we
choose the initial average scale factor such that
$\alpha^{3}(\tau_{0})=\alpha_{1}(\tau_{0})\alpha_{2}(\tau_{0})\alpha_{3}(\tau_{0})=1$.
This allows the bounce to happen soon after the initial time, compared to the
typical time scale of the whole process. A larger initial scale factor would
shift the time when the bounce occurs.

Fig.4 shows an example of a bouncing solution. We include the time evolution
of the scale factors $\alpha_{i}$, and Hubble rates $\mathcal{H}_{i}$, the
anisotropy $\tilde{\sigma}^{2}$, and the energy density $\tilde{\rho}$. We
choose $r=2$, and initial conditions at $\tau_{0}=0$
$\alpha_{i}(\tau_{0})=\\{2/3,1,3/2\\}$, and
$\tilde{\sigma}^{2}(\tau_{0})=0.05$.

Fig.5 shows an example of a case where the bounce is not reached, due to the
anisotropy dominating, but that is close to the critical case. The initial
conditions are $r=2$, $\alpha_{i}(\tau_{0})=\\{2/3,1,3/2\\}$ and
$\tilde{\sigma}^{2}(\tau_{0})=0.18247$.

##  VII Discussion

In this work we have studied the possibility of a cosmic bounce occurring in a
universe in which there is a significant (but not dominant) anisotropy, in
addition to the presence of a scalar field which is subject to finite volume
effects in a closed universe. We have shown that criteria for the success of
the bounce can be stated in terms of the size of the closed universe at the
point at which the net energy density is zero, and have studied the properties
of the quantum field (mass and vacuum energy) that permit a bounce of a size
consistent with our own cosmological history.

At the point of zero net energy density, the size of the universe must be
roughly comparable to the Compton wavelength of the field for its pressure to
be sufficient to turn around the collapse, so values smaller than $\sim
10^{-5}$ eV are needed for the mass. We also find that the vacuum energy of
the field must be extremely small, even in comparison to the current day
cosmological constant. After the bounce, the universe transitions to expansion
and the contribution of the quantum field to the energy density reduces to its
vacuum energy, with finite volume effects completely suppressed. The fact that
this vacuum energy is smaller than the current value is therefore consistent
with what we observe (it could be a small contribution to its value), but the
smallness of the value seems to require some further explanation – although
this is of course true of the cosmological constant itself.

Examples of further consequences of tunnelling effect in finite volume are as
follows. One could involve an out-of-equilibrium QFT description of tunnelling
in the background of a time-varying metric, which would allow a more accurate
study away from the bounce. Then, the inclusion of the Casimir effect due to
the finite volume $V_{cell}$ could give rise to new effects, with possible
cosmological relevance. Also, the effect of spatial curvature should be
included, in the situation where the space fundamental cell is a 3-sphere
instead of a 3-torus, and these studies are left for future works.

###### Acknowledgements.

We thank Tim Clifton, Malcolm Fairbairn and David J. Marsh for helpful
conversations. This work is supported by the Leverhulme Trust (grant
RPG-2021-299). JA is also supported by the Science and Technology Facilities
Council (grant STFC-ST/T000759/1). KC is supported by an STFC Ernest
Rutherford fellowship, project reference ST/V003240/1. For the purpose of Open
Access, the authors have applied a CC BY public copyright licence to any
Author Accepted Manuscript version arising from this submission.

Figure 4: Bouncing case: In the two upper panels we show the time evolution of
the rescaled scale factors $\alpha_{i}$ and the rescaled Hubble rates
$\mathcal{H}_{i}$ with the initial conditions given in the main text. The
lower panel shows the time evolution of the anisotropy $\tilde{\sigma}^{2}$
and energy density $\tilde{\rho}$. Figure 5: Near critical no-bounce case: In
the two upper panels we show the time evolution of the rescaled scale factors
$\alpha_{i}$ and the rescaled Hubble rates $\mathcal{H}_{i}$ with the initial
conditions given in the main text. The lower panel shows the time evolution of
the anisotropy $\tilde{\sigma}^{2}$ and energy density $\tilde{\rho}$. The
initial conditions are close to the critical point and for a time the value of
the rescaled average Hubble rate $\mathcal{H}$ remains near zero. However,
eventually the average scale factor grows, leading to another collapsing phase
in which the anisotropy diverges.

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