KCL-PH-TH/2023-05

aainstitutetext: Theoretical Particle Physics and Cosmology, King’s College
London, WC2R 2LS, UK

# Cosmic bounce and phantom-like equation of state from tunnelling

Jean Alexandre a and Silvia Pla<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We allow a scalar field on a flat FLRW background metric to tunnel between two
degenerate vacua. The resulting true vacuum state then violates the Null
Energy Condition, and the corresponding homogeneous fluid induces a bounce,
after which it has a phantom-like equation of state and asymptotically leads
to a de Sitter phase. The mechanism presented here requires no exotic matter
or modified gravity, it is purely generated by quantum fluctuations and is
valid for a generic double well potential.

##  1 Introduction

Unlike spontaneous symmetry breaking (SSB), which occurs in infinite volume,
tunnelling involves remarkable energetic features, among which a non-
perturbative ground state with no classical analogue. This is at the origin of
convexity of the effective potential for a scalar field Symanzik:1969ek ;
Coleman:1974jh ; Iliopoulos:1974ur ; Haymaker:1983xk ; Fujimoto:1982tc ;
Bender:1983nc ; Hindmarsh:1985nc ; Plascencia:2015pga ; Millington:2019nkw ,
and thus symmetry restoration.

The explicit calculation of the one-particle-irreducible (1PI) effective
potential, taking into account several degenerate vacua, was done in
Alexandre:2012ht ; Alexandre:2022sho  in the semi-classical approximation for
the partition function. These studies assumed an $O(4)$-symmetric Euclidean
space-time, and the corresponding work at finite but low temperature was done
in Alexandre:2022qxc  and Alexandre:2023jsl . The latter works allow for the
full tunnelling regime, involving a gas of Euclidean-time-dependent instantons
relating two degenerate vacua. It was found that the true ground state for the
scalar field is symmetric, and it violates the Null Energy Condition (NEC -
see Rubakov:2014jja ; Kontou:2020bta  for reviews), because it is non-
extensive in the thermodynamical sense. We note here that these results are
independent of symmetry-restoration by the Kibble-Zurek mechanism
Kibble:1976sj ; Zurek:1985qw , which is valid at high temperatures and does
not allow for the NEC to be violated.

The present work extends this tunnelling mechanism to a Friedmann-Lemaitre-
Robertson-Walker (FLRW) background metric, where we study the backreaction of
the fluid provided by the scalar true vacuum on the metric dynamics. Our
assumptions do not involve exotic matter or modified gravity, but a finite
volume and an adiabatic expansion instead, both to be defined in the next
section. Our results arise purely from quantum fluctuations and they have no
classical counterpart.

It is well known that in Quantum Field Theory (QFT) the energy conditions can
be violated under certain circumstances. Some examples include the Casimir
effect Casimir:1948 , radiation from moving mirrors Davies:1977yv , or black
hole evaporation Hawking:1975vcx . Another interesting example in curved
backgrounds was obtained in Onemli:2002hr . The latter work studies a self-
interacting massless field, therefore seeing only one vacuum, and not
tunnelling. Also, the background is fixed as a de Sitter metric, whereas in
our study the scale factor is determined by the backreaction of the scalar
effective vacuum. Nevertheless, it is still possible for the stress-energy
tensor to satisfy certain constraints, such as the Averaged Null Energy
Condition (ANEC), which averages the NEC over timelike or null geodesics. The
mechanism we propose here indeed does not violate the ANEC, since NEC
violation is valid temporarily only - see Sec.4. We note that an eternal
inflation scenario is described in Kontou:2020pdx , which also respects the
ANEC.

In Sec.2 we describe the semi-classical approximation in which we evaluate the
partition function, based on the different saddle points which are relevant
for two degenerate vacua: two static configurations and a gas of
instantons/anti-instantons. In the situation of non-degenerate vacua, the
relevant configurations are the Coleman bounce Coleman1 ; Coleman2  and the
shot Andreassen:2016cvx , with imaginary quantum fluctuations which arise from
a negative eigenvalue in the fluctuation determinant Ai:2019fri . In the
present case though, there are not any imaginary quantum corrections, since
the (anti-) instantons are monotonic functions of Euclidean time
Kleinert:2004ev .

The effective action is then derived in Sec.3 to the lowest order in the
field, which is enough to confirm convexity and that the ground state is
obtained for a vanishing field, unlike the situation of SSB. This calculation
is done in the adiabatic approximation, assuming that the tunnelling rate is
large compared to the space-time expansion rate. The vacuum energy induced by
tunnelling violates the NEC, which results in a cosmic bounce if we start with
a contracting universe. This is followed by an expansion with a phantom-like
equation of state $w<-1$, which asymptotically becomes de Sitter with $w=-1$.
The property $w<-1$ is usually related to a negative kinetic term in the
potential (see Ludwick:2017tox  for a review on phantom energy), but is not
the case here: the vacuum we find is homogeneous and its energetic properties
arise purely from quantum fluctuations, not from a specific bare action. We
note that the origin of energies is not relevant to this mechanism, since the
effective ground state energy is always lower than the degenerate bare
energies, and it is this negative difference which induced NEC violation.

In Sec.4 we solve numerically the Friedmann equations, where we study the
backreaction of the effective theory on gravity. As expected from NEC
violation, the solution exhibits a cosmological bounce Steinhardt:2001st ;
Khoury:2001wf ; Khoury:2001bz , known to provide an alternative to Cosmic
Inflation Guth:1980zm ; Linde:1981mu . The original idea to generate a bounce
from a tunnelling-induced scalar field true vacuum was proposed in
Alexandre:2019ygz ; Alexandre:2021imu , in the context of an $O(4)$-symmetric
Euclidean space-time though, whereas we allow here for the full tunnelling
regime, with finite volume and infinite Euclidean time.

Finally, the detailed calculations are presented in Appendix A, B and C.

##  2 Semi-classical approximation and saddle points

###  2.1 Assumptions

We consider the classical background metric

$\text{d}s^{2}=-\text{d}t^{2}+a^{2}(t)\delta_{ij}\text{d}x^{i}\text{d}x^{j}~{},$
(1)

where the scale factor $a(t)$ is kept generic. The bare matter action is

$S[\phi]=\int\text{d}^{4}x\sqrt{|g|}(L-j\phi)~{},$ (2)

where the Lagrangian $L$ involves a double-well potential, as well as a non-
minimal coupling to the scalar curvature:

$L=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\xi
R\phi^{2}-\frac{\lambda}{4!}(\phi^{2}-v^{2})^{2}-\bar{\Lambda}~{}.$ (3)

For convenience, we have also added the cosmological constant term in the
matter sector ($\bar{\Lambda}=\kappa^{-1}\Lambda$ with $\kappa=8\pi G$) to
account for vacuum energy effects after renormalisation. The important
assumptions we make are the following:

  * •

Finite volume, which allows tunnelling between the degenerate vacua. We start
from a fundamental flat spatial cell with volume $V_{0}$ and comoving volume
$a^{3}(t)V_{0}$, which can be thought of as a 3-torus, or a 3-sphere with
large enough radius to neglect curvature. Although finite, we assume the
parameters of the model to be such that quantisation of momentum can be
ignored, and the periodic boundary conditions do not play a role. Related
comments on the Casimir effect are given in Alexandre:2023jsl  for tunnelling
in flat space-time, and we focus here on the tunnelling features only;

  * •

Adiabatic approximation, where the expansion rate of the metric is assumed
small compared to the tunnelling rate for matter. According to the discussion
at the end of Sec.2.4, this is valid in the regime

$|H|\equiv\left|\frac{\dot{a}}{a}\right|\ll
v\sqrt{\frac{\lambda}{\pi}}~{}\alpha^{3/2}~{}\exp(-\alpha^{3})~{},$ (4)

where $\alpha^{3}(t)=a^{3}(t)S_{0}/\hbar$ and $S_{0}$ is the action for an
instanton interpolating the two vacua $\pm v$.

As a consequence of the second point, the scale factor $a(t)$ will be
considered constant for the calculation of the matter effective theory, and
its time dependence will be reinstated when we couple the matter effective
theory to gravity.

###  2.2 Semi-classical approximation

We work here in Euclidean signature. In the semi-classical approximation, and
focusing only on the matter sector for the reasons explained above, the
partition function takes the form

$Z[j]=\int{\cal D}[\phi]\exp(-S[\phi]/\hbar)\simeq\sum_{n}Z_{n}[j]~{},$ (5)

where

$Z_{n}=F_{n}[j]\exp(-S[\phi_{n}]/\hbar)\equiv\exp(-\Sigma_{n}[j]/\hbar)~{},$
(6)

and $\phi_{n}$ are the different dominant contributions, the saddle points,
which satisfy the equation of motion and minimise the action locally in the
space of field configurations. $F_{n}[j]$ are the fluctuation factors for
these saddle points, that we will calculate at one-loop, and $\Sigma_{n}[j]$
are the corresponding connected graphs generating functionals.

The saddle points $\phi_{n}$ satisfy then

$-\frac{1}{\sqrt{g}}\frac{\delta S}{\delta\phi}=j~{},$ (7)

and since we consider two degenerate minima, a bubble-solution cannot form,
since it would have an infinite radius Coleman1 ; Coleman2 . Hence we focus on
homogeneous saddle points only, which can depend on the Euclidean time tough.
These saddle points obey

$\ddot{\phi}+\frac{3\dot{a}}{a}\dot{\phi}-\xi
R\phi+\frac{\lambda}{6}v^{2}\phi-\frac{\lambda}{6}\phi^{3}=j~{},$ (8)

where a dot represents a (Euclidean) time derivative. In the adiabatic
approximation, the scale factor $a$ is assumed constant for the calculation of
quantum fluctuations for matter, and we will therefore take $\dot{a}=0=R$. We
discuss below the static saddle points and the instanton gas, with their
corresponding connected graphs generating functionals
$\Sigma_{1}[j],\Sigma_{2}[j]$ and $\Sigma_{gas}[j]$ respectively.

Finally, we are interested in the tunnelling-induced effective potential, such
that it is enough to consider a constant source $j$. A spacetime-dependent
source is necessary for the calculation of the derivative part of the
effective action only.

###  2.3 Static saddle points

The static saddle points satisfy

$v^{2}\phi-\phi^{3}=\frac{6j}{\lambda}~{},$ (9)

which, for $j<j_{c}\equiv\lambda v^{3}/(9\sqrt{3})$, has two real solutions

$\phi_{1}(j)=\frac{2v}{\sqrt{3}}\cos\left(\frac{\pi}{3}-\frac{1}{3}\arccos(j/j_{c})\right)~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}\phi_{2}(j)=-\phi_{1}(-j)~{},$
(10)

with the corresponding actions

$\displaystyle S_{1}[j]$ $\displaystyle\equiv$ $\displaystyle
S[\phi_{1}(j)]=\int\text{d}^{4}x\sqrt{g}\left(\bar{\Lambda}+v\,j-\frac{3}{2v^{2}\lambda}~{}j^{2}+\mathcal{O}(j^{3})\right)$
(11) $\displaystyle S_{2}[j]$ $\displaystyle\equiv$ $\displaystyle
S[\phi_{2}(j)]=S_{1}[-j]~{}.$

The one-loop fluctuation factor for a static saddle point $\phi_{n}(j)$ is
calculated in Appendix A, using the Schwinger proper time representation of
the propagator. We find for the corresponding renormalised connected graphs
generating functional

$\displaystyle\Sigma_{n}[j]$ $\displaystyle=$
$\displaystyle\int\text{d}^{4}x\sqrt{g}\Bigg{(}\bar{\Lambda}_{R}+\frac{\lambda_{R}}{4!}(\phi_{n}^{2}-v_{R}^{2})$
$\displaystyle\qquad\qquad+\frac{\hbar\lambda_{R}^{2}}{4608\pi^{2}}\Big{(}G(\phi_{n})+2(3\phi_{n}^{2}-v_{R}^{2})^{2}\ln\big{(}(3\phi_{n}^{2}/v_{R}^{2}-1)/2\big{)}\Big{)}+j\phi_{n}\Bigg{)}$

with

$G(\phi_{n})=-285v_{R}^{4}+366v_{R}^{2}\phi_{n}^{2}-81\phi_{n}^{4}\,,$ (13)

and $\lambda_{R},v_{R},\bar{\Lambda}_{R}$ are the renormalised parameters
given in Appendix A. The specific form (2.3), including the renormalised
parameters, is chosen in such a way that, in the absence of source we have

$\Sigma_{n}[0]=\int\text{d}^{4}x\sqrt{g}\,\bar{\Lambda}_{R}~{},$ (14)

which makes the discussion on vacuum energy simpler. Note that, in eq.(2.3),
the static saddle points $\phi_{n}$ can be expressed as in eq.(10), where the
parameters can be replaced by the renormalised ones, since they satisfy the
equation of motion Alexandre:2022sho .

###  2.4 Instanton gas

We describe here Euclidean time-dependent saddle points. In the absence of a
source, they obey the following equation

$\ddot{\phi}+\omega^{2}\phi-\frac{\lambda}{6}\phi^{3}=0~{},$ (15)

where $\omega=v\sqrt{\lambda/6}$, which corresponds to a problem of real-time
classical mechanics in the upside-down potential

$V(\phi)=-\frac{\lambda}{24}(\phi^{2}-v^{2})^{2}~{},$ (16)

represented in Fig. 1.

Figure 1: The upside-down potential $V(\phi)$ in which the field oscillates.
One instanton corresponds to the motion from infinitesimally close to one
hilltop to infinitesimally close to the other.

The motion starting asymptotically close to a hilltop and ending
asymptotically close to the other hilltop is given by the known solution

$\phi_{inst}(j=0)=\pm v\tanh\left(\frac{\omega}{\sqrt{2}}(t-t_{1})\right)~{},$
(17)

where $t_{1}$ corresponds to the “jump", where the instanton goes through 0,
and the corresponding action is

$S[\phi_{inst}(j=0),\bar{\Lambda}=0]=a^{3}S_{0}~{}~{}~{}~{}\mbox{with}~{}~{}~{}~{}S_{0}\equiv\frac{4\sqrt{2}}{\lambda}\omega^{3}V_{0}~{}.$
(18)

Indeed, the field spends a large (Euclidean) time close to a hilltop, with an
exponentially small contribution to both the potential and the kinetic energy,
and the main contribution to the action comes from the jump. For $p$ jumps, an
exact saddle point is a series of periodic oscillations between the two hills.
If the motion starts exponentially close to a hilltop, the distance
$|t_{i+1}-t_{i}|$ between two consecutive jumps is large compared to the width
$2\pi/\omega$ of a jump. The motion is then approximately described by

$\phi_{inst}^{(p)}(j=0)\simeq~{}\pm
v\tanh\left(\frac{\omega}{\sqrt{2}}(t-t_{1})\right)\times\tanh\left(\frac{\omega}{\sqrt{2}}(t-t_{2})\right)\times\cdots\times\tanh\left(\frac{\omega}{\sqrt{2}}(t-t_{p})\right)~{},$
(19)

where the times $t_{i}$ are regularly spread along the Euclidean time (see
Fig.2(a)), and the corresponding action is

$S[\phi_{inst}^{(p)}(j=0),\bar{\Lambda}=0]\simeq p~{}a^{3}S_{0}~{}.$ (20)

The above action remains unchanged when the jumps are shifted though, provided
the condition $|t_{i+1}-t_{i}|\gg 2\pi/\omega$ is satisfied, which is called
the dilute gas approximation (see Fig.2(b)). As a consequence, all the
corresponding configurations in the path integral $Z$ contribute as much as
the exact solution of the equation of motion. The invariance of the action
under the translation of jumps has a high degeneracy, making this dilute gas
dominant in $Z$.

(a) An exact saddle point configuration, corresponding to periodic
oscillations between the two hills provided by the upside-down potential shown
on Fig.1.

(b) An approximate saddle point configuration with the same number of
oscillations, but not periodic. The jumps are randomly distributed, but the
average distance between them is larger than their width, such that they keep
their shape and the action of the configuration is essentially the same as the
action for the exact saddle point in fig.(a).

Figure 2: Example of exact and approximate saddle points. In the dilute gas
approximation, the difference between the corresponding actions is
exponentially small, and the partition function is dominated by the whole set
of approximate saddle points.

We show in Appendix B that, in the presence of a source, the summation over
all the $p$-jump saddle points leads to

$\Sigma_{gas}[j]\simeq\frac{1}{2}\big{(}\Sigma_{1}[j]+\Sigma_{2}[j]\big{)}-\hbar\ln\big{(}\exp(\bar{N})-1\big{)}\equiv-\hbar\ln
Z_{gas}[j]~{},$ (21)

where the statistical average number of jumps between the two static saddle
points is

$\bar{N}=\sqrt{g_{00}}~{}\omega
T\sqrt{\frac{6a^{3}S_{0}}{\hbar\pi}}e^{-a^{3}S_{0}/\hbar}~{}.$ (22)

We note that the parameters in the latter equation can be understood as the
renormalised ones, since the contribution of $\bar{N}$ is at one-loop already.

The exponential of $\bar{N}$ appearing in the partition function is a known
feature in tunnelling studies, and it arises from the summation over the zero
modes of each (anti-) instanton (see Appendix B for details). Note that we are
interested here in the situation where $S_{0}$ is fixed and the total
Euclidean time $T$ goes to infinity, such that $\bar{N}$ is assumed to be
large. An alternative situation, relevant at finite temperature, consists in
fixing $T$ and taking $S_{0}\to\infty$, such that $\bar{N}\to 0$. This
corresponds to the suppression of tunnelling, and where SSB provides a better
description of the system Alexandre:2022qxc . The expression (22) can be
understood as the total Euclidean time $T$ multiplied by the tunnelling rate
$\omega\sqrt{6a^{3}S_{0}/\hbar\pi}~{}e^{-a^{3}S_{0}/\hbar}$.

##  3 Effective action

We describe here the main steps for the construction of the effective theory,
as well as its energetic properties. The details can be found in Appendix (C).

###  3.1 Symmetric ground state

From the previous section, the partition function can be expressed as

$\displaystyle Z[j]$ $\displaystyle\simeq$ $\displaystyle
Z_{1}[j]+Z_{2}[j]+Z_{gas}[j]$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{\hbar}\Sigma_{1}[j]\right)+\exp\left(-\frac{1}{\hbar}\Sigma_{2}[j]\right)+(\exp(\bar{N})-1)\exp\left(-\frac{1}{2\hbar}\Big{(}\Sigma_{2}[j]+\Sigma_{2}[j]\Big{)}\right)~{},$

from which one can derive the classical field $\phi_{c}$, which corresponds to
the vacuum expectation value in the presence of the source $j$

$\phi_{c}=\frac{-\hbar}{Z(j)\sqrt{g}}\frac{\delta Z}{\delta
j}=-M^{-2}\,j+{\cal O}(j^{3})~{}.$ (24)

In the previous expression and in the limit where $T\to\infty$, we show in
Appendix C that

$M^{-2}=\frac{3}{\lambda_{R}v_{R}^{2}}\left(1+\frac{27\hbar\lambda_{R}}{32\pi^{2}}\right)+\mathcal{O}(\hbar^{2})~{}.$
(25)

We note that $\phi_{c}$ is proportional to $j$, showing symmetry restoration:
the vacuum for $j=0$ is at $\phi_{c}=0$.
The relation $\phi_{c}[j]$ is then inverted to

$j[\phi_{c}]=-M^{2}\phi_{c}+{\cal O}(\phi_{c}^{3})~{},$ (26)

and the 1PI effective action, defined through the Legendre transform as a
functional of $\phi_{c}$, is

$\displaystyle\Gamma[\phi_{c}]$ $\displaystyle=$ $\displaystyle-\hbar\ln
Z[j\big{[}\phi_{c}]\big{]}-\int\text{d}^{4}x\sqrt{g}~{}\phi_{c}~{}j[\phi_{c}]$
$\displaystyle=$
$\displaystyle\Gamma[0]+\frac{1}{2}\int\text{d}^{4}x\sqrt{g}~{}M^{2}\phi_{c}^{2}+\mathcal{O}(\phi_{c}^{4})~{}.$

In the previous expression, the effective action for the ground state reads

$\displaystyle\Gamma[0]$ $\displaystyle=$
$\displaystyle\int\text{d}^{4}x\sqrt{g}\,\bar{\Lambda}_{R}-\hbar\ln(e^{\bar{N}}+1)$
$\displaystyle\simeq$
$\displaystyle\int\text{d}^{4}x\sqrt{g}\,\bar{\Lambda}_{R}-\hbar\bar{N}~{}.$

To summarise the essential features of the effective action (3.1):

  * •

it is convex, since $M^{2}>0$, and has its ground state at $\phi_{c}=0$;

  * •

the ground state energy has a non-trivial dependence on the comoving volume,
via $\bar{N}$, and is therefore not extensive in the usual thermodynamical
sense.

###  3.2 NEC violation

For simplicity, in what follows we will drop the sub-index R and all the
parameters should be understood as the renormalised ones.

We focus here on the fluid provided by the ground state $\phi_{c}=0$. In order
to obtain the energy density and the pressure, we need to represent
$\Gamma[0]$ and thus $\bar{N}$ as the integral over a Lagrangian density,
restoring the time dependence of the scale factor. This is done in Appendix C,
where we show that the expression (3.1) can be written as

$\displaystyle\Gamma[0]=\int\text{d}^{4}x\sqrt{g}\left(\bar{\Lambda}-\rho_{0}~{}\frac{e^{-\alpha^{3}}}{\alpha^{3/2}}\right)~{},$
(29)

where

$\alpha^{3}\equiv
a^{3}\frac{S_{0}}{\hbar}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}\rho_{0}\equiv\hbar\,\frac{\omega}{V_{0}}\frac{S_{0}}{\hbar}\sqrt{\frac{6}{\pi}}=\frac{2\,\lambda~{}v^{4}}{3\sqrt{3\pi}}~{}.$
(30)

From eq.(29), the energy density and the pressure are obtained from the
components of the energy-momentum tensor

$T_{\mu\nu}=\frac{2}{\sqrt{g}}\frac{\delta\Gamma[0]}{\delta g^{\mu\nu}}~{},$
(31)

and we find

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\bar{\Lambda}-\rho_{0}~{}\frac{e^{-\alpha^{3}}}{\alpha^{3/2}}~{},$
(32) $\displaystyle p$ $\displaystyle=$
$\displaystyle-\bar{\Lambda}+\rho_{0}\left(\frac{1}{2\alpha^{3/2}}-\alpha^{3/2}\right)e^{-\alpha^{3}}~{}.$

The fluid provided by the ground state therefore features the following
properties:

  * •

It consistently satisfies the (real-time) continuity equation
$\dot{\rho}+3H(\rho+p)=0$;

  * •

It violates the NEC

$\rho+p=-\rho_{0}~{}e^{-\alpha^{3}}\left(\alpha^{3/2}+\frac{1}{2\alpha^{3/2}}\right)~{}<0~{};$
(33)

  * •

It can induce a cosmic bounce since it can simultaneously satisfy $\rho=0$ and
$\rho+3p<0$. Assuming $\bar{\Lambda}>0$, the second condition is automatically
satisfied if the NEC is violated;

  * •

Assuming $e^{-\alpha^{3}}\ll 1$, its equation of state has the phantom form

$w=\frac{p}{\rho}\simeq-1-\frac{\rho_{0}}{\bar{\Lambda}}\,e^{-\alpha^{3}}\left(\alpha^{3/2}+\frac{1}{2\alpha^{3/2}}\right)~{}<-1~{}.$
(34)

We stress here an important point: the property $w<-1$ does not arise from a
kinetic energy with the opposite sign, but is a consequence of tunnelling
between the two degenerate bare vacua, which induces a homogeneous symmetric
ground state.

##  4 Friedmann Equations

In this section we go back to Lorentzian signature. As explained in the
introduction, we study the back-reaction of the effective theory on the
metric, such that the energy-momentum tensor in the Einstein equations
$G_{\mu\nu}=\kappa T_{\mu\nu}$ contains the energy density and pressure given
by eqs. (32), and $\kappa$ is the renormalised gravity coupling. The resulting
Friedmann equations read

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{\kappa}{3}\rho$
(35) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{6}(\rho+3p)~{},$

that we study here numerically. The first equation $H^{2}\propto\rho$ gives
the initial condition $\dot{a}_{0}$ once $a_{0}$ is known, and the second
equation provides the evolution equation for $a(t)$. We then introduce the
dimensionless time

$\tau\equiv t~{}\sqrt{\frac{\Lambda}{3}}~{},$ (36)

and we use the expressions (32) to obtain from eqs.(35)

$\displaystyle\alpha^{\prime}$ $\displaystyle=$
$\displaystyle\pm\alpha\sqrt{1-r~{}\frac{e^{-\alpha^{3}}}{\alpha^{3/2}}}$ (37)
$\displaystyle\frac{\alpha^{\prime\prime}}{\alpha}$ $\displaystyle=$
$\displaystyle
1-\frac{r~{}e^{-\alpha^{3}}}{4~{}\alpha^{3/2}}(1-6\alpha^{3})~{},$

where a prime denotes a derivative with respect to $\tau$ and

$r=\kappa\frac{\rho_{0}}{\Lambda}=\frac{\rho_{0}}{\bar{\Lambda}}~{}.$ (38)

The Friedmann Equations (37) are solved numerically, and we plot in Figure 3
the solutions corresponding to a fixed value of $\alpha(0)$ and different
values of the parameter $r$. The initial condition for $\alpha^{\prime}(0)$ is
given by the negative branch $\alpha^{\prime}(0)<0$ of the first Friedmann
equation, in order to describe a cosmological bounce, which will occur if
$\rho=0$ and $\rho+3p<0$ are simultaneously satisfied at a given time $t_{b}$.
We see that such a bounce is indeed generated, after which the expansion
suppresses tunnelling: the NEC is recovered and the metric dynamics enters a
de Sitter phase, with constant $H$.

Figure 3: Time evolution of the scaled scale factor $\alpha$ (upper panel) and
the scaled Hubble rate $\mathcal{H}=\alpha^{\prime}/\alpha$ (lower panel) with
initial condition $\alpha(0)=1$, for three different values of $r$, namely
$r=2$ (solid line), $r=1$ (dashed line) and $r=0.5$ (dashed-dotted line).

##  5 Conclusions

We have described how the energetic properties arising from tunnelling could
be relevant in a cosmological context, starting from standard QFT and Einstein
gravity. To summarise the non-perturbative mechanism described in this
article:

  1. (a)

The effective theory taking into account tunnelling between two degenerate
vacua is obtained by considering the contribution of different saddle points
in the partition function;

  2. (b)

As a consequence of this interplay between the two vacua $\pm v$, the
resulting true vacuum is at $\phi_{c}=0$, with an energy which is not
proportional to the comoving volume;

  3. (c)

This non-extensive feature of the vacuum energy implies NEC violation;

  4. (d)

The NEC violation induces a cosmological bounce in the case of initial
spacetime contraction, and is valid until the resulting expansion suppresses
tunnelling, such the ANEC is satisfied.

The adiabatic approximation is well justified in the vicinity of the
cosmological bounce, but out-of-equilibrium studies would be necessary to
include the full-time dependence of the scale factor if one wishes to look at
what happens away from the bounce. A related improvement to this work would be
to derive our results in a manifestly covariant way.

Regarding the assumption of finite-volume FLRW space-time, this study has
required a toy-model geometry/topology, in the form of a 3-torus or 3-sphere,
and thus still needs to be developed for phenomenological purposes. Also,
quantum corrections in a finite volume should in principle take into account
discrete momentum, as well as periodic boundary conditions. This is done in
the framework of Casimir effect studies Bordag:2001qi , whereas the present
article focuses on the tunnel effect, with continuous momentum and effectively
Dirichlet boundary conditions. A natural step further would then consider a
discrete spectrum, which could be done numerically for example.

The situation of non-degenerate minima would avoid making the finite-volume
assumption, since the relevant instanton action (the Coleman bounce saddle
point) is independent of the volume. In this case, quantum fluctuations for
the latter saddle point would involve an imaginary part, which should be
cancelled by the imaginary part induced by other saddle points
Andreassen:2016cvx , since the effective potential is real. The whole process
is challenging to describe analytically in more than 0-dimensional space-time
though, but is a potential avenue to explore, since it could be relevant as a
component of Dark Energy (see related comment below). There are questions
which remain to be answered in the situation of non-degenerate minima though,
one of which is the possibility to obtain a NEC-violating effective ground
state. Indeed, if one considers only $O(4)$-symmetric saddle points, the NEC
should not be violated since space and (Euclidean) time are treated in the
same way. A more complex saddle point structure, to be studied in the context
of real-time tunnelling Ai:2019fri , could involve NEC violation though, but
this study goes beyond the scope of the present article. Another question in
this context is the use of a homogeneous classical field, which would be valid
at large scales only (large compared to the radius of a 4-dimensional bubble),
in order to describe an average over the gas of bubbles. But strictly
speaking, the homogenous configurations used in the present work are valid for
a second-order quantum phase transition only, whereas the presence of bubbles
is a signature for a first order phase transition.

In our model, the universe asymptotically approaches a de Sitter phase in the
late-time regime, from which one cannot escape. This phase could potentially
represent Dark Energy in the late Universe, where we would need to identify
$\Lambda$ with the current cosmological constant. Given our numerical study
where $r\sim 1$, and assuming $S_{0}\sim\hbar$ for tunnelling to be
significant, we find from the expressions (18), (30) and (38) for $S_{0}$,
$\rho_{0}$ and $r$ respectively, that

$\Lambda\sim\hbar\sqrt{\frac{\lambda}{\pi}}\frac{\kappa v}{V_{0}}~{}.$ (39)

The latter order of magnitude allows an estimate of the volume $V_{0}$ at the
bounce, which should be much smaller than the volume $V$ of the current
visible Universe, for tunnelling to be significant enough. $V_{0}$ should also
be much larger than the Plank volume $V_{P}$, for the classical gravity regime
to be valid. We obtain then the constraint

$V_{P}\ll\hbar\sqrt{\frac{\lambda}{\pi}}~{}\frac{\kappa v}{\Lambda}\ll V~{},$
(40)

which could be helpful in model building for a bouncing Cosmology scenario. In
any case, one should be careful when drawing conclusions far from the bounce,
since our approach is based on equilibrium QFT, which is valid only in the
vicinity of the bounce, i.e. when the condition (4) is satisfied. A more
complete study would involve the Keldysh formalism for example, allowing out-
of-equilibrium predictions, and which is a potential next step.

## Acknowledgements

The authors thank Katy Clough and Malcolm Fairbairn for cosmology-related
discussions, as well as Jose Navarro-Salas and Janos Polonyi for insightful
comments. This work is supported by the Leverhulme Trust (grant RPG-2021-299)
and the Science and Technology Facilities Council (grant STFC-ST/T000759/1).
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.

##  Appendix A One-loop effective action in curved space-times

In this appendix we review the main steps to obtain the one-loop effective
action in curved space-times for a real scalar field in a double-well
potential, and propagating in a curved background with Euclidean signature. We
focus here on one saddle point only.

For renormalisation purposes, we need to consider the bare action of this
model

$S[\phi,g]=\int\text{d}^{d}x\sqrt{g}\left(\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}\xi
R\phi^{2}+\frac{\lambda}{4!}(\phi^{2}-v^{2})^{2}+\bar{\Lambda}+j\phi\right)~{},$
(41)

together with the semi-classical action for gravity111 We note that the
Euclidean form of the Lagrangian differs with a minus sign with respect to its
Lorentzian form.

$S_{G}[g]=-\int\text{d}^{d}x\sqrt{g}\Big{[}(2\kappa)^{-1}R+(\epsilon_{1}R^{2}+\epsilon_{2}R^{\mu\nu}R_{\mu\nu}+\epsilon_{3}R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma})\Big{]}~{},$
(42)

in $d$ space-time dimensions, where $\kappa=8\pi G$, and
$\bar{\Lambda}=\kappa^{-1}\Lambda$. For convenience, we have included the
cosmological constant term in the matter sector. The inclusion of the higher
curvature terms is needed for the cancellation of the divergences that arise
in this context. In this setup, the Klein-Gordon equation for the scalar field
is

$(-\Box_{E}+\xi
R-\tfrac{\lambda}{6}v^{2}+\tfrac{\lambda}{3!}\phi^{2})\phi+j=0~{},$ (43)

where
$\Box_{E}=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}=\frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}g^{\mu\nu}\partial_{\mu})$,
and the scalar field can be expanded around a saddle point
$\phi=\phi_{s}+\delta\phi$. The associated Euclidean Green’s function for the
quantum fluctuation $\delta\phi$ reads

$(-\Box_{E}+Q)G_{E}(x,x^{\prime})=\frac{1}{\sqrt{g}}\delta^{(4)}(x-x^{\prime})\,,$
(44)

where

$Q=\frac{\lambda}{2}\phi_{s}^{2}-\frac{\lambda v^{2}}{6}+\xi R\,.$ (45)

The one-loop correction to the classical action can be written in terms of the
Green’s function as parker-toms

$\displaystyle\Sigma[\phi_{s},g]$ $\displaystyle=$ $\displaystyle
S_{G}[g]+S[\phi_{s},g]-\tfrac{1}{2}\hbar\,\textrm{ln}\,\textrm{Det}\,G_{E}\,$
$\displaystyle\equiv$ $\displaystyle
S_{G}[g]+S[\phi_{s},g]+\Sigma^{(1)}[\phi_{s},g]\,.$

For general background configurations, the Green’s function is unknown.
However, an approximated expression for the quantum contribution
$\Sigma^{(1)}[\phi_{s},g]$ in the case of slowly varying background fields
$\phi_{s}$ and $g$ can be computed using the proper-time formalism as follows
(see Refs. Hu-lambdaphi4 ; Rajantie18  for a detailed explanation).

The DeWitt-Schwinger representation of the propagator $G_{E}(x,x^{\prime})$ is
given by

$G_{E}(x,x^{\prime})=\int_{0}^{\infty}\text{d}s\,H(x,x^{\prime};s)\,,$ (47)

where the kernel $H(x,x^{\prime};s)$ obeys a diffusion equation with
appropriate boundary conditions Vassilevich03 . For the one-loop connected
graph, it translates into

$\Sigma^{(1)}[\phi_{s},g]=\,\frac{\hbar}{2}\int\text{d}^{d}x\sqrt{g}\int_{0}^{\infty}\frac{\text{d}s}{s}\,H(x,x;s)\,.$
(48)

The kernel $H(x,x^{\prime};s)$ admits, in general, an asymptotic expansion in
terms of the Schwinger proper-time parameter Schwinger51 . At coincidence
$x^{\prime}\to x$ this expansion reads

$H(x,x;s)=\frac{e^{-m^{2}s}}{(4\pi
s)^{d/2}}\sum_{k=0}^{\infty}a_{k}(x)\,s^{k}\,.$ (49)

where $a_{k}(x)$ are the so-called the deWitt coefficients and $d$ is the
space-time dimension. The first few coefficients are Gilkey75 ; Vassilevich03

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle 1\,;$ (50)
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{6}R-Q\,;$ (51)
$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{180}R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}-\frac{1}{180}R^{\alpha\beta}R_{\alpha\beta}-\frac{1}{30}\Box_{E}R$
$\displaystyle+\frac{1}{6}\Box_{E}Q+\frac{1}{2}Q^{2}-\frac{1}{6}RQ+\frac{1}{72}R^{2}\,.$

This expansion captures, in its leading orders, the UV divergences ($s\to 0$)
of the theory and it is routinely used for renormalisation in the context of
QFT in curved spaces.

The expansion above (49) has an important property: it admits an exact
resummation R-summed1 ; R-summed2

$H(x,x;s)=\frac{e^{-\mathcal{M}^{2}s}}{(4\pi
s)^{d/2}}\sum_{k=0}^{\infty}b_{k}(x)\,s^{k}\,,$ (53)

with

$\mathcal{M}^{2}=Q-\frac{1}{6}R\,,$ (54)

such that, the new coefficients $b_{k}(x)$ do not contain any term that vanish
when $Q$ and $R$ are replaced by zero. For example, for the first resummed
deWitt coefficients we have

$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle 1\,;$ (55)
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle 0\,;$ (56)
$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{180}R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}-\frac{1}{180}R^{\alpha\beta}R_{\alpha\beta}-\frac{1}{30}\Box_{E}R+\frac{1}{6}\Box_{E}Q~{}.$
(57)

Therefore, the resummed expansion becomes a derivative expansion in the field
$\phi_{s}$ and the metric, physically meaningful in the case of slowly varying
background fields. Then, it is possible to truncate the expansion at a given
order $N$ \- the order of derivatives - to obtain an approximated expression
for the one-loop connected graph

$\Sigma^{(1)}[\phi_{s},g]=\,\frac{\hbar}{2}\int\text{d}^{d}x\sqrt{g}\int_{0}^{\infty}\frac{\text{d}s}{s}~{}~{}\frac{e^{-\mathcal{M}^{2}s}}{(4\pi
s)^{d/2}}\sum_{k=0}^{N}b_{k}(x)\,s^{k}\,.$ (58)

The expression above is divergent for $d=4$ and can be renormalised using
dimensional regularization. For arbitrary dimension $d$, the proper-time
integrals can be performed to give

$\Sigma^{(1)}[\phi_{s},g]=\frac{\hbar}{(4\pi)^{d/2}}\left(\frac{\mathcal{M}}{\mu_{d}}\right)^{d-4}\int\text{d}^{d}x\sqrt{g}~{}\sum_{k=0}^{N}b_{k}(x)\mathcal{M}^{d-2k}\,\Gamma\left(k-\frac{d}{2}\right)~{}.$
(59)

We have introduced a renormalisation mass parameter to proceed with
dimensional regularization in what follows. Truncating the sum at $N=2$ and
expanding around $d\to 4$ we find

$\Sigma^{(1)}=\hbar\int\text{d}^{4}x\sqrt{g}\left[\frac{\mathcal{M}^{4}}{64\pi^{2}}\Big{[}\ln\left(\frac{\mathcal{M}^{2}}{\mu^{2}}\right)-\frac{3}{2}\Big{]}+\frac{b_{2}}{32\pi^{2}}\ln\left(\frac{\mathcal{M}^{2}}{\mu^{2}}\right)\right]~{},$
(60)

where $\mathcal{M}^{2}>0$ since we quantise about stable saddle points and the
curvature effects are expected to be small. In the above expression, the
divergences have been absorbed in the scale parameter $\mu$, which is defined
by

$\ln\mu^{2}=\ln\Big{(}4\pi\mu_{d}^{2}\Big{)}-\gamma-\frac{2}{d-4}~{}~{}~{}(\mbox{finite
when}~{}d\to 4)~{}.$ (61)

From these expressions, we can directly obtain the renormalised values of the
coupling constants of the problem (see, for example Ref. Hu-book ).

In our particular problem, we are assuming an adiabatic expansion of the
universe, and that quantum processes under consideration occur at equilibrium.
Hence, we neglect the curvature of space-time. Hence we are only interested in
the couplings $\lambda,~{}v,~{}\bar{\Lambda}$. For simplicity, we will follow
Markkanen13 , and apply the renormalisation conditions at the same scale for
all bare parameters, namely,

$\displaystyle 3\frac{\partial^{2}L}{\partial\phi^{2}}\Big{|}_{\phi=\pm
v_{R},g=\eta}=\lambda_{R}\,v_{R}^{2}\,,\qquad\frac{\partial^{2}L}{\partial\phi^{4}}\Big{|}_{\phi=\pm
v_{R},g=\eta}=\lambda_{R}\,,\qquad+L\Big{|}_{\phi=\pm
v_{R},g=\eta}=\bar{\Lambda}_{R}~{},$ (62)

where $\eta$ is the Euclidean flat metric and
$\Sigma=\int\text{d}^{4}x\sqrt{g}~{}L$. From these conditions we obtain

$\displaystyle\delta\lambda$ $\displaystyle=$
$\displaystyle\frac{3\lambda_{R}^{2}}{32\pi^{2}}\left(3+\ln(\frac{v_{R}^{2}\lambda_{R}}{3\mu^{2}})\right)\,,$
(63) $\displaystyle\delta v^{2}$ $\displaystyle=$
$\displaystyle\frac{v_{R}^{2}\lambda_{R}}{16\pi^{2}}\left(10-\ln(\frac{v_{R}^{2}\lambda_{R}}{3\mu^{2}})\right)\,,$
(64) $\displaystyle\delta\bar{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{v_{R}^{4}\lambda_{R}^{2}}{1152\pi^{2}}\left(-3+2\ln(\frac{v_{R}^{2}\lambda_{R}}{3\mu^{2}})\right)~{},$
(65)

where we define $\lambda_{R}=\lambda+\hbar~{}\delta\lambda$,
$v_{R}^{2}=v^{2}+\hbar~{}\delta v^{2}$, and
$\bar{\Lambda}_{R}=\bar{\Lambda}+\hbar~{}\delta\bar{\Lambda}$. Inserting these
results in (A) and assuming $R=0$ and $\phi_{s}$ static, we obtain the final
renormalised connected graph given in Sec. 2.3.

For completeness, we also give the renormalised values of $\kappa^{-1}$ and
$\xi$. The renormalisation conditons we impose are

$-2\frac{\partial L}{\partial R}-\xi_{R}\phi^{2}\Big{|}_{\phi=\pm
v_{R},g=\eta}=\kappa^{-1}_{R}\,,\qquad\frac{\partial^{3}L}{\partial
R\partial\phi^{2}}\Big{|}_{\phi=\pm v_{R},g=\eta}=\xi_{R}\,,$ (66)

that lead to

$\displaystyle\delta\xi$ $\displaystyle=$
$\displaystyle\frac{\lambda_{R}(6\xi_{R}-1)}{192\pi^{2}}\left(3+\ln(\frac{v_{R}^{2}\lambda_{R}}{3\mu^{2}})\right)\,,$
(67) $\displaystyle\delta(\kappa^{-1})$ $\displaystyle=$
$\displaystyle\frac{v_{R}^{2}\lambda_{R}(6\xi_{R}-1)}{2304\pi^{2}}\left(11+\ln(\frac{v_{R}^{2}\lambda_{R}}{3\mu^{2}})\right)\,.$
(68)

##  Appendix B Quantisation over instanton configurations

In Section 2.4 we describe few features of the gas of instantons for a
vanishing source. In the presence of an infinitesimal source $j\ll j_{c}$, the
jump is not modified, and what changes is the position of the asymptotically
"flat" parts of the instantons, which now go from one saddle point
$\phi_{i}(j)$ to the other, instead of going from one vacumm $\pm v$ to the
other $\mp v$. We have then, instead of eq.(18),

$S[\phi_{inst}(j)]\simeq
a^{3}S_{0}+\frac{1}{2}\big{(}S_{1}[j]+S_{2}[j]\big{)}~{},$ (69)

since on average the configuration spends half the Euclidean time
exponentially close to $\phi_{1}(j)$ and the other half close to
$\phi_{2}(j)$. The contribution of one instanton
$F_{inst}\exp(-S[\phi_{inst}]/\hbar)$ to the partition function is the product
of the following contributions

  * •

The "flat" part close to each static saddle point, leading to the fluctuation
factor $F_{i}$ about each of the static saddle points, for half of the total
Euclidean time

$\sqrt{F_{1}F_{2}}e^{-(S_{1}+S_{2})/(2\hbar)}=\exp\left(-\frac{1}{2\hbar}\big{(}\Sigma_{1}[j]+\Sigma_{2}[j]\big{)}\right)~{},$
(70)

where $\Sigma_{n}[j]$ is given in eq.(2.3).

  * •

Fluctuations above one jump which, discounting the zero mode corresponding to
the translational invariance of the jump, lead to the factor (see
Kleinert:2004ev ; Dunne2009 )

$\sqrt{\frac{6a^{3}S_{0}}{\hbar\pi}}~{};$ (71)

  * •

The zero mode corresponding to the position of the jump, which can happen at
any Euclidean time between 0 and $T$, and thus gives the extra factor

$\omega\int_{0}^{T}\sqrt{g_{00}}~{}\text{d}t=\sqrt{g_{00}}~{}\omega T~{}.$
(72)

Note that the summation over the different positions of the jump is done with
the comoving proper time, since the jump is observed by the comoving observer.
Here, $S_{0}$ and $\omega$ are defined with the renormalised parameters.

All together, the contribution of one instanton to the partition function is

$F_{inst}\exp\left(-\frac{S[\phi_{inst}]}{\hbar}\right)=\sqrt{g_{00}}~{}\omega
T\sqrt{\frac{6a^{3}S_{0}}{\hbar\pi}}\exp\left(-a^{3}\frac{S_{0}}{\hbar}-\frac{1}{2\hbar}\big{(}\Sigma_{1}[j]+\Sigma_{2}[j]\big{)}\right)~{}.$
(73)

For a $p$-jump saddle point in the dilute gas approximation, and where the
width of an instanton is negligible compared to the total Euclidean time $T$,
the classical action is

$S[\phi_{inst}^{p}(j)]\simeq
pa^{3}S_{0}+\frac{1}{2}\big{(}S_{1}[j]+S_{2}[j]\big{)}~{}.$ (74)

Also, whereas the first jump can happen at any time $t_{1}\in[0,T]$, the jump
$i$ can happen at a time $t_{i}\in[t_{i-1},T]$ only, such that the degeneracy
of a $p$-jump configuration leads to the factor Kleinert:2004ev

$\prod_{i=1}^{p}\left(\omega\int_{t_{i-1}}^{T}\sqrt{g_{00}}~{}\text{d}t_{i}\right)=\frac{1}{p!}(\sqrt{g_{00}}~{}\omega
T)^{p}~{}~{}~{}~{}~{}~{}~{}(\text{with}~{}t_{0}=0)~{}.$ (75)

Summing over all the possibilities for $p$, we obtain the final expression for
the dilute gas contribution to the partition function

$\displaystyle\exp\left(-\frac{1}{\hbar}\Sigma_{gas}[j]\right)$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{\infty}\frac{1}{p!}(\sqrt{g_{00}}~{}\omega
T)^{p}\left(\frac{6a^{3}S_{0}}{\hbar\pi}\right)^{p/2}\exp\left(-pa^{3}\frac{S_{0}}{\hbar}-\frac{1}{2\hbar}\big{(}\Sigma_{1}[j]+\Sigma_{2}[j]\big{)}\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2\hbar}\big{(}\Sigma_{1}[j]+\Sigma_{2}[j]\big{)}\right)\left[\exp\left(\sqrt{g_{00}}~{}\omega
T\sqrt{\frac{6a^{3}S_{0}}{\hbar\pi}}e^{-a^{3}S_{0}/\hbar}\right)-1\right]~{}.$

##  Appendix C Effective action, energy density and pressure

We give here details on the derivation of the one-loop effective action. We
start from the partition function

$\displaystyle Z[j]$ $\displaystyle=$ $\displaystyle
Z_{1}[j]+Z_{2}[j]+Z_{gas}[j]$ $\displaystyle=$ $\displaystyle
e^{-\Sigma_{1}/\hbar}+e^{-\Sigma_{2}/\hbar}+(e^{\bar{N}}-1)e^{-(\Sigma_{1}+\Sigma_{2})/2\hbar}~{},$

where $\Sigma_{2}[j]=\Sigma_{1}[-j]$ which, for small source, can be expanded
as

$\Sigma_{1,2}[j]=\int\text{d}^{4}x\sqrt{g}\left(\bar{\Lambda}_{R}\pm\sigma_{(1)}\,j+\frac{1}{2}\sigma_{(2)}\,j^{2}+\mathcal{O}(j^{3})\right)~{},$
(77)

with

$\sigma_{(1)}=v_{R}-\hbar\,\frac{9\lambda_{R}v_{R}}{32\pi^{2}}\,,\qquad\sigma_{(2)}=-\frac{3}{v_{R}^{2}\lambda_{R}}-\hbar\,\frac{81}{32\pi^{2}v_{R}^{2}}~{}.$
(78)

The classical field $\phi_{c}$ is

$\phi_{c}=\frac{-\hbar}{Z(j)\sqrt{g}}\frac{\delta Z}{\delta
j}=-M^{-2}\,j+{\cal O}(j^{3})~{},$ (79)

with

$M^{-2}=-\sigma_{(2)}+\frac{V_{4}}{\hbar}\frac{2}{(e^{\bar{N}}+1)}\sigma_{(1)}^{2}=\frac{3}{\lambda_{R}v_{R}^{2}}\left(1+\frac{2A}{3}+\hbar\lambda_{R}\,\frac{27}{2\pi^{2}}\left(\frac{1}{16}-A\right)\right)+\mathcal{O}(\hbar^{2})\,,$
(80)

and

$V_{4}=\int\text{d}^{4}x\sqrt{g}~{}~{}~{}~{},~{}~{}~{}~{}A=\frac{V_{4}\,\omega^{4}_{R}}{\hbar\lambda_{R}(e^{\bar{N}}+1)}~{}.$
(81)

The relation $\phi_{c}[j]$ is then inverted to $j[\phi_{c}]$, in order to
define the 1PI effective action as the Legendre transform

$\Gamma[\phi_{c}]=-\hbar\ln
Z[j\big{[}\phi_{c}]\big{]}-\int\text{d}^{4}x\sqrt{g}~{}\phi_{c}~{}j[\phi_{c}]~{}.$
(82)

An expansion in the classical field finally gives

$\Gamma[\phi_{c}]=\Gamma[0]+\int\text{d}^{4}x\sqrt{g}~{}\frac{M^{2}}{2}\phi_{c}^{2}+\mathcal{O}(\phi_{c}^{4})~{},$
(83)

with

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\left(-\sigma_{(2)}+\frac{V_{4}}{\hbar}\frac{2}{e^{\bar{N}}-1}\sigma_{(1)}^{2}\right)^{-1}$
$\displaystyle=$
$\displaystyle\frac{\lambda_{R}v_{R}^{2}}{3}\left(\frac{1}{1+24A}-\hbar\lambda_{R}\frac{27}{32\pi^{2}}\frac{1-16A}{(1+24A)^{2}}\right)+\mathcal{O}(\hbar^{2})~{},$

and

$\Gamma[0]=\int\text{d}^{4}x\sqrt{g}\,\bar{\Lambda}_{R}-\hbar\ln(e^{\bar{N}}+1)\simeq\int\text{d}^{4}x\sqrt{g}\,\bar{\Lambda}_{R}-\hbar\bar{N}~{}.$
(85)

In the limit $T\to\infty$ we obtain then

$M^{2}=\frac{\lambda_{R}v_{R}^{2}}{3}\left(1-\hbar\lambda_{R}\frac{27}{32\pi^{2}}\right)+\mathcal{O}(\hbar^{2})~{}.$
(86)

The next step is the analysis of the energy density and pressure for the
ground state. The stress-energy tensor can be obtained from the definition

$T^{E}_{\mu\nu}=\frac{2}{\sqrt{g}}\frac{\delta\Gamma(0)}{\delta
g^{\mu\nu}}~{}.$ (87)

where we have explicitly written the super-index E as a reminder that we are
working in Euclidean signature. Because of homogeneity and isotropy, the
stress-energy tensor can be decomposed as

$T^{E}_{\mu\nu}=\text{diag}(-\rho,a^{2}p,a^{2}p,a^{2}p)~{},$ (88)

so that we directly obtain

$\rho=-T^{E}_{00}=-\frac{2}{\sqrt{g}}\frac{\delta\Gamma(0)}{\delta
g^{00}}\Big{|}_{g^{00}=1}\,\,,\qquad
p=g^{11}T_{11}^{E}=\frac{2}{a^{2}\sqrt{g}}\frac{\delta\Gamma(0)}{\delta
g^{11}}\Big{|}_{g^{11}=a^{-2}}\,.$ (89)

In order to express $\bar{N}$ as a Lagrangian density we restore the time
dependence of the scale factor with the replacement

$\sqrt{g_{00}}~{}T~{}f(a)\to\int_{0}^{T}\text{d}t~{}\sqrt{g_{00}}~{}f(a)$ (90)

and we express the cell 3-volume at $t=t_{0}$ as

$V_{0}=\int\text{d}^{3}x~{}a^{3}(t_{0})=\int\text{d}^{3}x~{}~{}~{}~{}\mbox{if
we choose}~{}~{}~{}~{}a(t_{0})=1~{}.$ (91)

The effective action for the ground state for $\omega_{R}T\gg 1$ [see Eq.
(85)] can then be expressed as

$\displaystyle\Gamma[0]$ $\displaystyle\simeq$
$\displaystyle\int\text{d}^{4}x\sqrt{g}~{}\bar{\Lambda}_{R}-\hbar\omega_{R}\sqrt{\frac{6S_{0}}{\hbar\pi}}\int_{0}^{T}\text{d}t\sqrt{g_{00}}\int\frac{\text{d}^{3}x}{V_{0}}a^{3/2}~{}e^{-a^{3}S_{0}/\hbar}$
$\displaystyle=$
$\displaystyle\int\text{d}^{4}x\sqrt{g}\left(\bar{\Lambda}_{R}-\rho_{0}~{}\frac{e^{-a^{3}S_{0}/\hbar}}{\sqrt{a^{3}S_{0}/\hbar}}\right)~{},$

where

$\rho_{0}\equiv\frac{\omega_{R}S_{0}}{V_{0}}\sqrt{\frac{6}{\pi}}=\frac{2\lambda_{R}v_{R}^{4}}{3\sqrt{3\pi}}~{},$
(93)

and where $S_{0}$ is defined with the renormalised parameters.

From Eqs. (89) and (C) we can easily obtain the energy density and the
pressure, namely

$\displaystyle\rho=-T^{E}_{00}=-\left.\frac{2}{\sqrt{g}}\frac{\delta\Gamma(0)}{\delta
g^{00}}\right|_{g_{00}=1}$ $\displaystyle=$
$\displaystyle+\bar{\Lambda}_{R}-\rho_{0}~{}\frac{e^{-a^{3}S_{0}/\hbar}}{\sqrt{a^{3}S_{0}/\hbar}}\,,$
(94) $\displaystyle
p=g^{11}T^{E}_{11}=\left.\frac{2}{a^{2}\sqrt{g}}\frac{\delta\Gamma(0)}{\delta
g^{11}}\right|_{g_{11}=a^{2}}$ $\displaystyle=$
$\displaystyle-\bar{\Lambda}_{R}+\rho_{0}\left(\frac{1}{2\sqrt{a^{3}S_{0}/\hbar}}-\sqrt{a^{3}S_{0}/\hbar}\right)e^{-a^{3}S_{0}/\hbar}~{}.$

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