# Dynamical Vacuum Compressibility of Space

Yu-Cun Xie<EMAIL_ADDRESS>Department of Physics, University of
Maryland, College Park, Maryland 20742, USA  Jen-Tsung Hsiang
<EMAIL_ADDRESS>College of Electric Engineering and Computer Science,
National Taiwan University of Science and Technology, Taipei City 106, Taiwan,
R.O.C.  Center for High Energy and High Field Physics (CHiP), National Central
University, Taoyuan 320317, Taiwan, R.O.C.  Bei-Lok Hu<EMAIL_ADDRESS>Maryland
Center for Fundamental Physics and Joint Quantum Institute, University of
Maryland, College Park, Maryland 20742, USA

(Dec 26, 2023 submitted to arXiv-v2 and PRD)

###### Abstract

This paper continues the investigation initiated in Ref. Cho  _et al._ (2022)
into the quantum thermodynamic properties of space by deriving the vacuum
compressibility of a variety of dynamical spacetimes containing massive and
massless conformally coupled quantum fields. The quantum processes studied
here include particle creation, Casimir effect, and the trace anomaly. The
spaces include $S^{2},S^{3}$, and $T^{3}$ with prescribed time evolution and
$S^{1}$, where the temporal development is backreaction determined. Vacuum
compressibility belongs to the same group of quantum thermodynamic /
mechanical response functions as vacuum viscosity, a concept first proposed in
1970 by Zel’dovich Zeldovich (1970) for capturing the effects of vacuum
particle production on the dynamics of the early universe, made precise by
rigorous work of many authors in the following decade using quantum field
theory in curved spacetime methodologies and semiclassical gravity theory for
treating backreaction effects. Various subtleties in understanding the
behavior of the vacuum energies of quantum field origins, negative pressures
and novel complicated features of dynamical compressibility are discussed.

##  I Introduction

The title contains four key words in three areas of research: vacuum refers to
the ground state of a quantum field, space conjures (general) relativity, and
compressibility brings up the (nonequilibrium) thermodynamics of matter Jou
_et al._ (2010). The first two elements make up quantum field theory (QFT) in
curved spacetime (CST), a well-established field since the 1970s Birrell and
Davies (1984); Parker and Toms (2009), Hawking effect Hawking (1975) being a
well-known example. Quantum compressibility refers to the compressibility of
space filled with a quantum field at zero or finite temperature. This bears on
the emergent field of quantum thermodynamics (QTD) emphasizing how
considerations of the quantum properties of matter entice us to extend, even
revise, our understanding of classical thermodynamics and explore new laws at
low temperature for small objects. Studies on the mechanical response to
quantum fields, in particular, electromagnetic and gravitational radiation,
and the reverse process of how quantum fields respond to mechanical actions,
make up an equally exciting field of quantum optomechanics (QOM),
LIGO/VIRGO/KAGRA based on interferometry are its signature highlights.

Can one apply quantum thermodynamic concepts to understand the fundamental
properties of spacetime with the help of quantum field theory in curved
spacetimes, in more general terms than the much studied and better understood
black hole thermodynamics Bekenstein (1972, 1973); Hawking (1976); Wald (1994)
in the class of spacetimes with event horizons? This was what motivated Cho
and two of the present authors Cho  _et al._ (2022) (CHH) to investigate into
the quantum thermodynamic properties of spacetimes by deriving the quantum
heat capacity and vacuum compressibility of a variety of (static) spacetimes
containing a quantum field at zero or finite temperatures. This paper
continues that line of pursuit extending it to dynamical conditions.

#### Quantum Thermodynamics of Dynamical Spacetimes

Spacetime thermodynamics Jacobson (1995); Eling  _et al._ (2006); Padmanabhan
(2005); Chirco and Liberati (2010) is a derivative and extension of black hole
thermodynamics, also related to the view that gravity is of the nature of
thermodynamics Jacobson (1995); Padmanabhan (2010); Verlinde (2011); Hu (2011)
or hydrodynamics Hu (1996); Volovik (2003); Hu (2005); Gielen  _et al._
(2016); Oriti (2021), and should be treated on the same footing as the
collective excitations of (spacetime) atoms as in condensed matter physics Gu
and Wen (2006); Sindoni (2012); Oriti (2018); Volovik (2023). These are
examples of emergent gravity, a fundamentally different vein from quantum
gravity. (For a sample of discourses on conceptual issues of emergent gravity,
see, e.g., Hu (2009); Carlip (2014); Marolf (2015); Sindoni (2012); Barceló
_et al._ (2021)). In this broader perspective, considerations of spacetime
thermodynamics need not be limited to spacetimes with event horizons and
thermodynamics should not be restricted to equilibrium conditions. This brings
up the first key word of this paper: Dynamical.

In the context of quantum fields in curved spacetimes, a process of similar
importance as the Hawking effect in black holes is cosmological particle
creation (CPC), prominent at the Planck time. The underlying mechanism of CPC
is parametric amplification of vacuum fluctuations in a quantum field. The
backreaction effects of CPC have been shown to be strongly responsible for the
isotropization and homogenization of the early universe. Seeing the
dissipative effects of these ubiquitous quantum processes, Zel’dovich in a
short essay in 1970 Zeldovich (1970) called it “vacuum viscosity”. It was a
brilliant foresight, but showing it formally was nontrivial, as it involves
three steps: a) a well-defined vacuum for quantum fields in an evolutionary
spacetime, b) a viable procedure to regularize the UV divergences of the
stress energy tensor to arrive at a finite expression for the energy density
of particles produced, and c) calculate the backreaction of quantum processes
(involving the Casimir energy, the trace anomaly and particle creation) by
solving the semiclassical Einstein equation and the field equations self-
consistently. It wasn’t till 1982 Hu (1982) that ‘vacuum viscosity’ acquires
its full meaning by rigorous derivations in the context of QFTCST and
semiclassical gravity Hu and Verdaguer (2020). Many interesting ideas emanated
from this vein of thought on viscous cosmology, both of classical and quantum
origins, including current speculations on the nature of dark matter and dark
energy. We mention a few representative works here Grøn (1990); Mak and Harko
(1999); Zimdahl (2000); Zimdahl  _et al._ (2001); Singh and Kale (2011);
Bastero-Gil  _et al._ (2012); Chakraborty and Saha (2014); Paliathanasis  _et
al._ (2017); Da Silva and Silva (2019); Gómez  _et al._ (2023). The two 2008
papers by Klinkhamer and Volovik Klinkhamer and Volovik (2008a, b) are of
special relevance to our present work since they also invoke a time changing
vacuum compressibility in the vein of emergent gravity. While their goal is to
explain the dynamical behavior of the cosmological constant, our work at this
stage only aims at deriving the vacuum energy and pressure of a dynamical
space from the three unavoidable sources of vacuum processes, namely, the
Casimir effect, the trace anomaly and particle creation. In a way we are
applying a solid layer of prime over a rough and spotty wall, so others can
later paint on it with colors of their choice for whatever purposes without
much worry of anything underneath falling off. Only after we have done the
same for thermal quantum fields Xie  _et al._ and then for nonequilibrium
quantum fields Hsiang  _et al._ suitable for an evolutionary universe will we
feel comfortable enough to begin thinking about how these sources of vacuum
energy bear on the cosmological constant issue.

#### Goal of this work

Understanding these QFT phenomena in quantum thermodynamics terms is a primary
goal of CHH’s work: exploring whether one can see them through the quantum
capacity and vacuum compressibility of curved and dynamical spacetimes. In
this work we have the same goal in mind but a restricted scope, working with
more tangible objects which may prove more useful in analogy gravity
experiments such as a pulsating ring, a moving box or a bouncing ball. Placing
ourselves for a moment on mundane ground we can think of a simple question
like, how can you tell what a balloon is filled with? gas or liquid? What kind
of gas? The answer is known to even toddlers: bounce it on the floor. A water
balloon will probably splash rather than bounce. A lighter gas will likely
bounce higher. Compressibility often refers to the material of the ball, e.g.,
a volleyball is made of softer material and thus bouncier under the same
pressure than a basketball. Here we are not concerned with the material which
the ball is made of, rather, we consider different kinds of quantum fields
filling a variety of spaces, like, on a 2-sphere $S^{2}$, in a 3-sphere
$S^{3}$ and in a rectangular box (with topology 3-torus $T^{3}$). We are
interested in how dynamics would alter the static values of vacuum
compressibility, comparing a) nonadiabatic versus adiabatic conditions, (or
sudden versus gradual changes in the geometry of the spaces), as well as
comparing b) the contributions from the three quantum sources in various
spaces: Casimir energy, particle creation and trace anomaly, the latter
containing higher derivatives of the scale factor with time.

Nonadiabatic dynamics strongly amplifies the vacuum fluctuations
parametrically, as testified by many cosmological problems studied before:
rapid expansion of the early universe produces copious amounts of particles
and their backreaction effects are much stronger than those scantily produced
under adiabatic expansions. For the present problem we anticipate different
responses from nonadiabatic versus adiabatic processes (hitting the ball hard
versus slowly squeezing it) and a noticeable variation in its vacuum
compressibility. It would be interesting to see how such quantum effects show
up in simpler setups like a pulsating ring, a moving box or a bouncing ball.

#### Five cases studies

Indeed, these are the five cases we shall investigate: 1) a massless conformal
field in a (1+1)D ring with time-dependent radius, where the trace anomaly and
Casimir effect is the only vacuum quantum effects, 2) a massless conformal
field in a 2-sphere with changing radius, where the trace anomaly is
identically zero (for even spatial dimensions Christensen (1978a)), but with a
massive conformal field, there is particle creation. In (1+3)D we study two
situations, particle creation of a massive conformal field in 3) a 3D
conducting box with one moving side and 4) a 3-sphere in nonadiabatic
expansion, and finally, 5) trace anomaly associated with a massless conformal
field in a 3-sphere. See Table 1 for the layout.

In Case 1 we shall make use of recent results Xie  _et al._ (2023) from
backreaction studies of the effect of trace anomaly acting on the ring. This
result is derived with full-fledged QFT CST methods which involve a) deriving
a formal expression for the trace anomaly and the energy density, b) using
suitable regularization schemes to remove the ultraviolet divergences, and c)
analyzing the backreaction111Even though these quantum vacuum effects outside
of strong field conditions as in the early universe or near the black hole are
small, when future experiments reach a high enough sensitivity level the
results obtained here for these simple and easily accessible systems may prove
useful. They could, in the spirit of analog gravity, serve as a guide to
connect with the backreaction effects of conformal anomalies and cosmological
particle creation. of these effects on the rate of contraction of the ring
(over and above that due to the static Casimir effects). We shall use the
vacuum energy obtained there to calculate (numerically) the vacuum
compressibility under dynamical conditions.

For the three new cases 2, 3, 4, and 5 studied here we shall ignore
backreaction effects on the external drive and assume the dynamics is
prescribed222Notice that in all realistic experimental setups, backreaction
effects are innate, i.e., intrinsically included in the physical parameters.
Conceptually this is similar to bare versus dressed masses or charges in the
context of renormalization, namely, quantum radiative corrections are already
included in actual physical parameters or dressed quantities. To study the
backreaction effects in analog gravity experiments one needs to add the
dynamics of an external drive into the overall picture in order to study how
quantum effects such as particle creation would alter its dynamics compared to
the case where the drive undergoes a prescribed evolution. Quantum friction is
a good example where a neutral atom moving at constant speed near a dielectric
surface experiences a steady drag force from the dielectric-modulated quantum
field. Compared to the simplistic case of an atom undergoing uniform motion
driven by some external source, the drive in the case of quantum friction
certainly experiences nontrivial backreaction from the dielectric-modulated
quantum field through its action on the atom’s dynamics.

#### Approaches and Justifications

Our specific goals in the present problem is to derive the dynamical vacuum
compressibility of model spacetimes. For this we need to calculate the vacuum
energy due to all quantum field effects contributing to it, which includes the
(static) Casimir effect, the dynamical Casimir effect, particle creation and
trace anomaly. We can take advantage of the techniques developed for QFTCST
and semiclassical gravity in the decade since 1977 Hartle (1977); Hu and
Parker (1977, 1978); Fischetti  _et al._ (1979); Hartle and Hu (1979, 1980);
Hartle (1981); Anderson (1985); Calzetta and Hu (1987) and make reasonable
approximations/assumptions based on results obtained therefrom. a) Since the
backreaction effect on the drive is not of our concern here we won’t pursue
this issue in the new cases studied in Sec. III and IV. Instead, we shall
stipulate two types of temporal evolution which enable us to calculate the
effects related to particle creation in $\mathbf{R}\times S^{2}$,
$\mathbf{R}\times S^{3}$, and $\mathbf{R}\times T^{3}$ without any
approximation. b) We do so by solving the three coupled first order
differential equations governing the Bogoliubov coefficients: two complex or
four real quantities minus one Wronskian condition (e.g., Eqs. (75-77) of Hu
(1974)) numerically with the specified time-dependence of the scale factor.
This new approach is an improvement over earlier treatments. Once the vacuum
energy from these two processes in these spacetimes are obtained it is easy in
technical terms to derive the vacuum pressure and the compressibility.

Here it is perhaps appropriate to add a comment on the value of our present
work. We mention three aspects 1) An experienced researcher in cosmological
particle creation and backreaction may find there is little fundamentally new
in terms of principles. Granted, it is when these principles are applied to
corresponding analog gravity setups that calculations found in the literature
are either incomplete or incorrect333An example of the incompleteness is,
despite an abundance of work on Casimir and dynamical Casimir effects, we have
not seen much discussion of the effect of trace anomaly in the corresponding
setups, even though both effects are on the same footing. In fact the latter’s
importance takes over the former as higher time derivatives of the scale
factor contribute. An example of ‘incorrectness’ is in Ref. Nagatani and
Shigetomi (2000) where presenting the dynamics is mis-conjured as including
the backreaction. We want to employ the best methodology established in the
cosmological context and apply them to the problems under study of value to
possible future analog gravity experiments. 2) To calculate the vacuum
dynamical compressibility of a space we need to include all contributions of
vacuum energy and pressure. One cannot use one kind of partial pressure due to
one kind of vacuum process, such as Casimir or particle creation or trace
anomaly, to calculate the partial dynamical compressibility, because they
don’t add up to give the correct expression of dynamical compressibility. It
is the derivative with respect to the total pressure which counts. Because of
this, and the fact that vacuum energies have been calculated in the literature
for some but not all sources, we need to ‘fill in the gaps’ and do it in as
vigorous a way as possible. 3) Even though in this paper we have derived the
behavior of vacuum dynamical compressibility for some typical spaces, because
of its novelty, further work is needed to unlock its deeper physical meanings
in our understanding of the quantum thermodynamics of spacetime. We view this
work as only taking a first step in that direction.

#### Organization

This paper is organized as follows: In Sec. II we present a summary of the
recent results in Xie  _et al._ (2023) for the energy density of a conformal
quantum field in a 1D ring with changing radius. We then work out the
dynamical vacuum compressibility of this space, due to the trace anomaly and
Casimir effect in the ring. In Sec. III we present a calculation of the energy
density of particle creation from the nonadiabatic parametric amplification of
vacuum fluctuations of a massive conformal scalar field in a two-dimensional
sphere with time varying radius. From this we obtain the dynamical vacuum
compressibility of a two-sphere due to particle creation. In Sec. IV we do the
same for the 3-sphere, calculating the energy density first for particle
creation from a massive conformal field, then for the trace anomaly of a
massless conformal field, both under specified time variation of the radius.
After integrating over the spatial volume we obtain the energy, the pressure
and the vacuum compressibility as functions of time. In Sec. V, we present a
careful analysis of the particle creation process of a conformal massless
scalar field in a symmetric Bianchi type-I spacetime with $T^{3}$ topology, by
allowing only one-dimension to expand. We work out the energy density,
pressure and dynamical vacuum compressibility of this space, neglecting all
other less important effects. The results are shown in plots. This is a
prelude to a more complete and in-depth treatment of this problem in the more
realistic context of cosmology by the authors of Cho  _et al._ (2022) in their
second paper Hsiang  _et al._ , by way of the nonequilibrium influence action
for the free energy density. In Sec. VI we summarize the key points in the
quantum field theoretical derivations and highlight the distinct features in
the vacuum energy, pressure and compressibility in these two kinds of quantum
field processes in the different spacetimes studied.

Table 1: The red cross mark (left column) is used with the effect that is present for massless conformal field, while the blue check mark (right column) is for massive conformal field. | $\mathbf{R}\times S^{1}$ | $\mathbf{R}\times S^{2}$ | $\mathbf{R}\times S^{3}$ | $\mathbf{R}\times T^{3}$
---|---|---|---|---
Casimir energy |  $\checkmark$ $\checkmark$ |  $\times$ $\checkmark$ |  $\checkmark$ $\checkmark$ |  $\checkmark$ $\checkmark$
Trace anomaly |  $\checkmark$ $\checkmark$ |  $\times$ $\times$ |  $\checkmark$ $\checkmark$ |  $\checkmark$ $\checkmark$
Particle production |  $\times$ $\checkmark$ |  $\times$ $\checkmark$ |  $\times$ $\checkmark$ |  $\checkmark$ $\checkmark$

##  II Vacuum compressibility of $\mathbf{R}\times S^{1}$ with back reaction

In this section, we derive the dynamical vacuum compressibility from the
vacuum energy density of a conformal quantum field by considering the one-
dimensional moving mirror configuration. We impose periodic boundary
conditions on the field at the surface of the mirror. Thus the configuration
is equivalent to a one-dimensional ring with changing radius Xie  _et al._
(2023). In so doing the original problem can be recast to that of finding the
dynamical vacuum compressibility of a conformally-coupled quantum field in
$\mathbf{R}\times S^{1}$ topology. Two different vacuum quantum processes need
be considered: Casimir effect (static and dynamic) and trace anomaly. We
include the key steps in the derivation for the convenience of those readers
who may not be so familiar with the relevant quantum field theory in curved
spacetime calculations. In this case the backreaction effects of these quantum
processes on the system are included, so the scale factor $a(t)$ that account
for the mirror’s motion is dynamically determined, rather than prescribed a
priori. They will provide a benchmark for comparison with cases treated in
Sec. III, IV, and V where the backreaction effects are ignored, that is,
$a(t)$ following a given trajectory.

We shall calculate the vacuum energy, pressure, and compressibility of the
quantum field associated with the Casimir effect and the trace anomaly. In
this lower dimensional case, the expression for the trace anomaly is
sufficiently simple, enabling one to dynamically determine the evolution of
the scale factor including the backreaction effects of the quantum field.

Here we consider two mirrors in a one-dimensional space, one of which is fixed
at the spatial origin and the other is allowed to move. The spatial location
of the moving mirror varies with time $t$ as $a(t)\,L$. Here $L$ is a constant
measuring the ‘size’ and $a(t)$ is a scale factor which varies with time,
describing the motion of the second mirror. We impose a periodic condition on
the field at the surfaces of the two mirrors, so as far as the field is
concerned, the space has a $S^{1}$ topology with time-dependent scale factor
$a(t)$.

Equivalently we can consider a two-dimensional spacetime with line element

$\displaystyle ds^{2}$ $\displaystyle=dt^{2}-a(t)^{2}\,dx^{2}\,,$ (1)

Identifying $x$ and $x+L$, $L$ becomes the coordinate circumference. This is
an Einstein cylinder (See, e.g., Lin  _et al._ (2016)) with topology
$\mathbf{R}^{1}\times S^{1}$. Introducing the conformal time $\eta$ defined by
$d\eta=dt/a(t)$, the metric can be written in a conformally-flat form:

$ds^{2}=a^{2}(\eta)\quantity[d\eta^{2}-dx^{2}]\,.$ (2)

A conformally-coupled, massless scalar field $\phi$ living in this spacetime
defined by the action

$S=\frac{1}{2}\int
d^{2}x\sqrt{-g}\;\quantity[g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}]\,,$ (3)

satisfies the Klein-Gordon equation,

$\Box\phi=0\,,$ (4)

where the Beltrami-d’Alembertian operator $\Box$ on a scalar is given by

$\Box=g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}=\frac{1}{\sqrt{-g}}\,\partial_{\alpha}\Bigl{(}g^{\alpha\beta}\sqrt{-g}\,\partial_{\beta}\Bigr{)}$
(5)

and $\nabla_{\alpha}$ is the covariant derivative operator. The partial
differentiation of a tensor will be denoted by a comma in the subscript
followed by the coordinate with which the differentiation is carried out,
while the covariant derivative of a tensor is denoted by a semicolon.

Carrying out a normal mode expansion for this field, we have

$\hat{\phi}(\eta,x)=\sum_{k_{n}}\left[\hat{a}^{\vphantom{*}}_{k_{n}}f^{\vphantom{*}}_{k_{n}}(\eta,x)+\hat{a}^{\dagger}_{k_{n}}f^{*}_{k_{n}}(\eta,x)\right]\,,$
(6)

where $f^{\vphantom{*}}_{k_{n}}(\eta,x)$ is the positive-frequency solution of
(4). Under the specified spatial topology, the $k_{n}$ form a discrete set
$k_{n}=2\pi n/L$, $n\in\mathbb{Z}$. In a conformally-flat spacetime, the mode
function $f_{k_{n}}$ is exactly solvable:

$f_{k_{n}}=\frac{1}{\sqrt{2\lvert k_{n}\rvert}}\,e^{i(k_{n}x-|k_{n}|\eta)}\,,$
(7)

with the normalization condition

$i\,f^{*}_{k_{n}}(\eta,x)\,\overset{\text{\tiny$\leftrightarrow$}}{\partial}_{\eta}f^{\vphantom{*}}_{k_{n^{\prime}}}(\eta,x)=\delta_{n,n^{\prime}}\,.$
(8)

Here $\delta_{n,n^{\prime}}$ is the Kronecker delta. For positive $n$,
$f_{k_{n}}$ represent modes moving counterclockwise; when $n$ is negative,
$f_{k_{n}}$ corresponds to modes moving clockwise.

Taking the variation of the action with respect to the metric tensor, we
obtain the classical energy-momentum tensor:

$T_{\mu\nu}=\frac{2}{\sqrt{|g|}}\frac{\delta S}{\delta
g_{\mu\nu}}=(\partial_{\mu}\phi)(\partial_{\nu}\phi)-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}(\partial_{\rho}\phi)(\partial_{\sigma}\phi)\,,$
(9)

which is traceless, $T_{\mu}{}^{\mu}=0$.

With the field variable promoted to be field operator we have the energy-
momentum tensor operator. Using the mode expansion Eq. (6) for Eq. (9) the
(unrenormalized) vacuum expectation values are obtained. From there, we obtain
an expression for the energy density

$\displaystyle\langle\hat{T}_{\mu\nu}\rangle U^{\mu}U^{\nu}$
$\displaystyle=\sum_{k_{n}}\quantity[(\partial_{t}f^{\vphantom{*}}_{k_{n}})(\partial_{t}f_{k_{n}}^{*})-\frac{1}{2}g_{tt}g^{\rho\sigma}(\partial_{\rho}f^{\vphantom{*}}_{k_{n}})(\partial_{\sigma}f_{k_{n}}^{*})]=\frac{2\pi}{L^{2}}\sum_{n=0}^{\infty}n\,.$
(10)

where $U^{\mu}$ is the observer’s four-velocity. In particular, $U^{\mu}$ has
components $(1,0,\cdots,0)$ in the comoving frame.

This formal expression for the energy density in Eq. (10) contains ultraviolet
divergences which need to unambiguously identified and removed, we adopt the
adiabatic regularization method, briefly explained in the following (see,
e.g., Xie  _et al._ (2023) for details).

It is more instructive to work with a massive conformally-coupled field
$\phi$. With a mode expansion in the form

$\hat{\phi}(\eta,x)=\sum_{k}\left[\hat{A}_{k}f_{k}(\eta,x)+\hat{A}^{\dagger}_{k}f^{*}_{k}(\eta,x)\right]\,,$
(11)

(we denote the annihilation operator of each mode by $\hat{A}_{k}$ to
distinguish it from its counterpart in the massless field case) in a spatially
homogeneous spacetime, the mode amplitude functions $f_{k}(\eta,x)$ are
separable into

$f_{k}(\eta,x)=(2\pi)^{-1/2}e^{ikx}\chi_{k}(\eta).$ (12)

The normalization condition for $f_{k}$ leads to
$\chi_{k}\partial_{\eta}\chi_{k}^{*}-\chi_{k}^{*}\partial_{\eta}\chi_{k}=i$.
Substituting Eq. (12) into the Klein-Gordon equation (4) yields

$\chi^{\prime\prime}_{k}(\eta)+\omega_{k}^{2}(\eta)\chi_{k}(\eta)=0,$ (13)

with $\omega_{k}=\sqrt{k^{2}+m^{2}a^{2}}$. This is the equation of motion for
the amplitude functions of the normal modes of a parametric oscillator.

To identify the UV divergences in two-dimensional spacetime considered here,
it is sufficient to find a (positive-frequency) WKB solution to Eq. (13) up to
the second adiabatic order, which should capture all the vacuum divergent
terms

$\displaystyle\chi_{k}$
$\displaystyle=(2W_{k})^{-1/2}\exp{-i\int^{\eta}\\!\differential\eta^{\prime}\;W_{k}(\eta^{\prime})}\,,$
(14) $\displaystyle W_{k}$
$\displaystyle=\omega_{k}(1+\epsilon_{2})^{1/2}\approx\omega_{k}(1+\frac{1}{2}\epsilon_{2})\,,$
$\displaystyle\epsilon_{2}$
$\displaystyle=-\omega_{k}^{-1/2}\frac{\differential^{2}}{\differential\eta^{2}}\omega_{k}^{1/2}\,.$

Substituting $f_{k}$ into Eq. (10) with $\chi_{k}$ given by (14), then taking
the massless limit, we find an expression for the energy density of the
conformally coupled field evaluated in the second-order adiabatic vacuum,
$\lvert 0_{A}\rangle$,

$\displaystyle\langle 0_{A}|\hat{T}_{\mu\nu}|0_{A}\rangle U^{\mu}U^{\nu}$
$\displaystyle=\frac{1}{2\pi a}\int_{0}^{\infty}\\!\differential
k\;\omega_{k}+\frac{1}{24\pi}\frac{\dot{a}^{2}}{a^{2}}\,,$ (15)

in the comoving frame, where the overhead dot denotes taking the derivative
with respect to the comic time $t$. The renormalized energy density of the
conformal scalar field in its conformal vacuum is given by

$\displaystyle\expectationvalue{T_{\mu\nu}}_{\text{reg}}U^{\mu}U^{\nu}=\langle\hat{T}_{\mu\nu}\rangle_{\text{c}}U^{\mu}U^{\nu}-\expectationvalue{\hat{T}_{\mu\nu}}{0_{A}}U^{\mu}U^{\nu}$
$\displaystyle=\frac{2\pi}{L^{2}}\sum_{n=0}^{\infty}n-\frac{1}{2\pi
a}\int_{0}^{\infty}\\!\differential
k\;\omega_{k}-\frac{1}{24\pi}\frac{\dot{a}^{2}}{a^{2}}$
$\displaystyle=-\frac{\pi}{6a^{2}L^{2}}-\frac{1}{24\pi}\frac{\dot{a}^{2}}{a^{2}}\,.$
(16)

In arriving at (II), we have introduced a cutoff to regularize the divergent
series and integral. Without any cutoff scale appearing in the end results,
this gives a finite well-defined expression for the energy density (see Ford
(1975, 1976) for an application to quantum fields in the Einstein universe).
We will elaborate more on this in Sec. IV.2.

The first expression on the righthand side of (II) gives the Casimir energy
density for a ring of circumference $L$. It becomes more negative when the
scale factor $a(t)$ gets smaller, independent of the derivative of $a(t)$. The
second term is due to the trace anomaly (‘anomaly’ because the classical
stress tensor is traceless). It vanishes when $a(t)$ is a constant because the
metric (2) becomes Minkowskian. Both contributions are negative definite, so
the renormalized energy density in this configuration is always negative.

We are interested in the total energy in the spacelike hypersurface orthogonal
to comoving observer’s four-velocity. To do so, we integrate the energy
density observed by the comoving observer using the induced metric on this
hypersurface. The induced metric is obtained by the projection
$\gamma_{\mu\nu}=U_{\mu}U_{\nu}-g_{\mu\nu}$, such that
$\gamma_{\mu\nu}U^{\nu}=0$. Integration gives the total energy of the field:

$\displaystyle E$
$\displaystyle=\int_{0}^{L}\\!dx\sqrt{\gamma}\;\expectationvalue{T_{\mu\nu}}_{\text{reg}}U^{\mu}U^{\nu}=-\frac{1}{24\pi}\frac{\dot{a}^{2}}{a}-\frac{\pi}{6a}\,,$
(17)

where by setting $L=1$, the physical length (perimeter) or volume of the ring
is $a(t)$ without loss of generality. This energy is negative definite, and is
unbounded from below. Since the motion of the mirror is relatively mild, the
trace anomaly, dependent on the derivative of the scale factor, plays a
subdominant role. The major contribution of vacuum energy in this case comes
from the Casimir energy.

The pressure $P$ is given by

$P=-\frac{\partial E}{\partial
V}=-\frac{\partial_{t}E}{\partial_{t}V}=-\frac{\pi}{6a^{2}}-\frac{\dot{a}^{2}}{24\pi
a^{2}}+\frac{\ddot{a}}{12\pi a}\,,$ (18)

since both $E$ and $V$ are time dependent owing to the changing scale factor
$a(t)$. The first term on the righthand side, devoid of any time derivative of
the scale factor, representing the Casimir energy is due to the topology of
the space. The remaining terms in the square brackets of (18) can be
identified as contributions from the trace anomaly. Unlike the contribution
associated with the Casimir energy, this component is not sign definite,
depending on the higher derivatives of $a(t)$.

With this, the vacuum compressibility $\kappa_{\textsc{vac}}$ is given by

$\kappa_{\textsc{vac}}=-\frac{1}{V}\frac{\partial V}{\partial
P}=\biggl{[}-\frac{\pi}{3a^{2}}-\frac{\dot{a}^{2}}{12\pi
a^{2}}+\frac{\ddot{a}}{6\pi
a}-\frac{\dddot{a}}{12\pi\dot{a}}\biggr{]}^{-1}\,.$ (19)

Again, the first term inside the brackets comes from the Casimir energy, which
is independent of $\dot{a}$, even though in arriving at this expression we
have taken the time derivative of $P(t)$. Thus this contribution has the same
form as the counterpart in the static Einstein cylinder. The trace anomaly
contribution, like that in $P$, contains higher derivatives of $a(t)$. Once we
have numerically obtained the time evolution of the scale factor $a(t)$, we
may use the above result to find the time evolution of $E$, $P$ and
$\kappa_{\textsc{vac}}$.

It is of interest to ponder over the implications of (18). Suppose we start
with a slowly varying $a(t)$ from a constant value, such that the $\ddot{a}$
term has a negligible initial contribution. The negative pressure (18)
generates a contracting stress, making the ring smaller. This trend is
independent of the sign of $\dot{a}$. A smaller $a$ makes the inward pressure
even stronger and the shrinking even faster. Once $\ddot{a}$ becomes negative,
this trend becomes unstoppable and the ring spacetime is doomed to collapse.
Positive initial $\dot{a}$ merely slight deters this from happening.

Figure 1: Here we consider a massless conformally coupled quantum field in an
Einstein cylinder of topology $\mathbf{R}^{1}\times S^{1}$ where the radius
(scale factor) is preset as $a(t)=1+e^{t^{2}/\tau^{2}}$ with $1/\tau^{2}=5$.
Plotted are the time dependence of the volume of the space, the vacuum energy
and pressure of the field due to the Casimir effect and the trace anomaly
combined, and the vacuum compressibility of the space associated with these
sources.

###  II.1 An example of prescribed $a(t)$ with positive pressure

From (18), to generate a positive pressure, we need a very large $\ddot{a}$
such that the third term in (18) becomes dominant. In other words, we would
like the factor $a\,\ddot{a}$ to be at least as large as $\dot{a}^{2}$. It
implies that $a(t)$ needs to be a rapidly increasing function of $t$,
preferably faster than exponential growing. Then we may see the pressure
taking on positive values and increase monotonically with $t$ or $a(t)$. This
is pretty easy to achieve if $a(t)$ is prescribed by, say,
$a(t)=1+e^{t^{2}/\tau^{2}}$. In this case, even though the volume is
expanding, the total energy drops to large negative values. Apparently in this
regime, the Casimir effect is negligible, so the dominant effects come from
the trace anomaly, related to the derivatives of $a(t)$. The pressure in this
case is positive, but it is not large enough to account for such a rapid
expansion, so we expect such a scenario may not be realizable self-
consistently. An external agent is needed to drive the expansion. In addition,
the pressure increases with the volume, and this is the cause of a negative
compressibility. It is in stark contrast with our experience with the ideal
gas that when the volume expands, the pressure inside the volume decreases
inversely. Even with this unusual compressibility, the spacetime seems to
respond to the vacuum energy of the field in such a way that it is quite
compressible at early times but becomes relatively less compressible at late
times due to the larger absolute values of the vacuum compressibility.

###  II.2 An example of increasing then decreasing $a(t)$

In comparison, we add one more example in Fig. 2.

Figure 2: Time dependence of the volume, energy, pressure and vacuum
compressibility in an Einstein cylinder $\mathbf{R}\times S^{1}$ with radius
(scale factor) following a preset Gaussian time-dependence given by (20), with
$a_{0}=1$, $\Delta=2$, $t_{0}=1$ and $1/\tau^{2}=5$. Same quantum field and
vacuum processes as in Fig. 1.

In this example, the volume of $S^{1}$ or the separation of two mirrors
increases and then decreases, following a Gaussian function of time,

$a(t)=a_{0}+\Delta\,e^{-(t-t_{0})^{2}/\tau^{2}}\,,$ (20)

where without loss of generality $a_{0}$ can be identified as the scale at
$t=-\infty$, $\Delta$ the maximal amount of scale change, $\tau$ the width of
the Gaussian, and $t_{0}$ the moment of maximal separation. There are a few
unusual features. The total energy reaches the least negative value at the
moment $t=t_{0}$ where the volume is the largest. This is solely due to the
Casimir effect, the trace anomaly containing $\dot{a}$ plays no role. In
contrast, for the pressure at $t=t_{0}$, the contributions from the Casimir
effect and the trace anomaly are respectively given by

$\displaystyle P_{\textsc{ca}}(t_{0})$
$\displaystyle=-\frac{\pi}{6(a_{0}+\Delta)^{2}}\,,$ $\displaystyle
P_{\textsc{ta}}(t_{0})$
$\displaystyle=-\frac{\Delta/\tau^{2}}{6\pi(a_{0}+\Delta)}\,,$
$\displaystyle\Rightarrow$
$\displaystyle\frac{P_{\textsc{ta}}(t_{0})}{P_{\textsc{ca}}(t_{0})}$
$\displaystyle=\frac{\Delta(a_{0}+\Delta)}{\pi^{2}\tau^{2}}\,.$ (21)

Figure 3: The time dependence of the pressure for the Gaussian evolution
$a(t)=a_{0}+\Delta\,e^{-(t-t_{0})^{2}/\tau^{2}}$ is shown in the first row for
two choices of $\tau$. We decompose $P(t)$ into two contributions
$P_{\textsc{ca}}$ and $P_{\textsc{ta}}$ respectively associated with the
Casimir effect and trace anomaly. In particular, the distinct behavior of
$P_{\textsc{ta}}$ is related to the compressibility shown in the second row.
We have $1/\tau^{2}=5$ for the first column, and $1/\tau^{2}=1$ for the second
column. The other parameters chosen here are $a_{0}=1$, $\Delta=2$, and
$t_{0}=1$.

Thus their relative dominance depends on the scales we choose for the scale
factor. Generally speaking, since the trace anomaly is related to the time
variation of $a(t)$, we expect $P_{\textsc{ta}}(t_{0})$ will be more important
when the Gaussian peak is very narrow, representing rapid time variation of
the scale factor. The M shape of the curve of the pressure in Fig. 2 is
essentially caused by the trace anomaly because it is dominant for the
parameters chosen for Fig. 2. In the case when the Casimir effect dominates,
the curve monotonically increases to the least negative value, and then
monotonically decreases, simply reflecting the change of scales. The most
extraordinary feature of this example is that the compressibility is not well
defined at certain moments. From the definition of compressibility, we see
that $dV=-\kappa_{\textsc{vac}}V\,dP$, and such critical behavior tends to
happen when $P$ has a local extremum but $a$ is still changing, so that
$dP\approx 0$ but $dV\neq 0$. If the interpretation of compressibility still
holds here, then it implies that the ring is the most deformable at this
juncture. The culprit of this pathology may be traced back to the trace
anomaly, as shown in Fig. 3. In the left column, the parameter $1/\tau^{2}=5$,
while $1/\tau^{2}=1$ for the plots in the right column. It is easier to
identify the contributions from the Casimir effect and the trace anomaly in
the plots of $P(t)$ in the first row. We note by comparison that the curve for
the contribution from the trace anomaly tends to have more wiggles due to its
dependence on various derivatives of $a(t)$. In addition, as addressed
earlier, the trace-anomaly contribution in $P$ becomes more important for
large $1/\tau^{2}$. These two factor brings in the compressibility shown in
the left column of Fig. 3. In comparison, the compressibility in the right
column appears better behaved.

Figure 4: The time evolution of the volume, pressure, energy, and
compressibility of a massless conformal field in a one-dimensional moving
mirror setup, which after a periodic boundary condition is imposed, acquires a
$\mathbf{R}\times S^{1}$ topology. The evolution of the scale factor $a(t)$ is
dynamically determined by incorporating the backreactions from the vacuum
energy, see Xie  _et al._ (2023). The initial conditions of $a(t)$ are
$a(0)=1$ and $\dot{a}(0)=0$.

For the model outlined in Ref. Xie  _et al._ (2023), the behavior of $a(t)$ in
principle can be dynamically determined if we include the backreaction
considered above. The corresponding plots are shown in Fig. 4. The generic
behavior is consistent with the above analysis. Since the increasing inward
pressure causes the volume to shrink, we expect that the vacuum
compressibility takes on negative values. However, it approaches zero when
$S^{1}$ shrinks to a dot, and the space acts like a incompressible, rigid
body.

It is interesting to comment on an observation that by the numerical tests,
the results in Fig. 4 is rather generic if the initial condition
$\dot{a}(0)<\nu_{c}$, where the critical value $\nu_{c}$ is given by

$\nu_{c}=\frac{1}{\sqrt{12\pi m-1}}\,.$ (22)

When $\dot{a}(0)$ is greater than $\nu_{c}$, the dynamics of $a(t)$ has a
qualitative change. It will grow indefinitely, but the energy and the pressure
all approach zero. The consequence is that the compressibility grows rapidly
with time, as shown in Fig. 5.

Figure 5: The time evolution of the volume, pressure, energy, and
compressibility of a massless conformal field in the one-dimensional moving
mirror setup in the same configuration as in Fig. 4 except we set
$\dot{a}(0)>\mu_{c}$.

##  III Vacuum compressibility of $\mathbf{R}\times S^{2}$

Now we turn to the vacuum processes of a massive, conformally coupled quantum
field in a $\mathbf{R}\times S^{2}$ spacetime, whose spatial sector is a two-
sphere with time-dependent radius following a prescribed protocol. In this
case, the vacuum processes include the Casimir effect and spontaneous particle
creation.

The line element of $\mathbf{R}\times S^{2}$ spacetime is

$\displaystyle
ds^{2}=a^{2}(\eta)\quantity[d\eta^{2}-d\theta^{2}-\sin^{2}\theta\;d\varphi^{2}]\,,$
(23)

with $\theta\in[0,\pi]$, and $\varphi\in[0,2\pi)$. It has the same form as
that of a $(1+2)$-dimensional closed Robertson-Walker universe.

Assume that the field has mass $M$, conformally coupled to curvature, and its
action is given by

$S=\frac{1}{2}\int\\!d^{3}\mathbf{x}\sqrt{-g}\;\quantity[g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-(M^{2}+\xi
R)\phi^{2}]\,,$ (24)

where the coupling strength takes on the value $\xi=1/8$, and the Ricci scalar
$R$ is given by

$R=-\frac{2}{a^{2}}+\frac{2a^{\prime
2}}{a^{4}}-\frac{4a^{\prime\prime}}{a^{3}}\,.$ (25)

The field equation

$\quantity(\square+M^{2}+\xi R)\phi=0\,,$ (26)

permits an expansion of the form

$\phi=\int\\!\differential\tilde{u}_{\mathbf{k}}\;\quantity[\hat{a}^{\vphantom{\dagger}}_{\mathbf{k}}f_{\mathbf{k}}^{\vphantom{*}}(x)+\hat{a}^{\dagger}_{\mathbf{k}}f_{\mathbf{k}}^{*}(x)]\,,.$
(27)

where the operator $\hat{a}_{\mathbf{k}}$ annihilates the vacuum state defined
at the initial time $\eta_{0}$, and the measure
$\displaystyle\int\\!d\tilde{u}_{\mathbf{k}}$ stands for

$\displaystyle\int
d\tilde{u}_{\mathbf{k}}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\,.$ (28)

when $\mathbf{k}$ takes on discrete values permitted by the indices $l$ and
$m$ of the spherical harmonics $Y_{lm}(\theta,\varphi)$.

Because of the isotropy and homogeneity of space, the mode function
$f_{\mathbf{k}}^{\vphantom{*}}(x)$ is separable and is given by
$f_{\mathbf{k}}^{\vphantom{*}}(\eta,\theta,\varphi)=a(\eta)^{-1/2}\,Y_{lm}(\theta,\varphi)\,\chi_{l}(\eta)$
with $\chi_{l}$ satisfying

$\displaystyle\chi_{l}^{\prime\prime}(\eta)+\omega_{l}^{2}(\eta)\chi_{l}(\eta)$
$\displaystyle=0\,,$
$\displaystyle\omega_{l}^{2}=(l+1/2)^{2}+a(\eta)^{2}M^{2}\,.$ (29)

The spherical harmonics $Y_{lm}(\theta,\varphi)$ are normalized to

$\displaystyle\int\\!\differential\Omega_{2}\;Y_{lm}(\theta,\varphi)Y_{l^{\prime}m^{\prime}}^{*}(\theta,\varphi)=\delta_{ll^{\prime}}\,\delta_{mm^{\prime}}\,,$
(30)

where $d\Omega_{2}$ is the solid angle element associated with $S^{2}$.

In this case, the expectation value of the energy density of the conformal
vacuum, as measured by a comoving observer, is given by

$\displaystyle\langle\hat{T}_{\mu\nu}\rangle U^{\mu}U^{\nu}$
$\displaystyle=\frac{1}{4\pi
a^{3}}\sum_{l=0}^{\infty}(l+\frac{1}{2})\Bigl{\\{}\lvert\chi^{\prime}_{l}(\eta)\rvert^{2}+\omega_{l}^{2}(\eta)\lvert\chi_{l}(\eta)\rvert^{2}\Bigr{\\}}\,.$
(31)

Following the same procedures outlined in the previous section, we can re-
write the summand
$\lvert\chi^{\prime}_{l}(\eta)\rvert^{2}+\omega_{l}^{2}(\eta)\lvert\chi_{l}(\eta)\rvert^{2}$
in terms of two time-dependent complex functions $\alpha_{l}$ and $\beta_{l}$
by the ansatz

$\displaystyle\chi_{l}$
$\displaystyle=\frac{1}{\sqrt{2\omega_{l}(\eta)}}\,\quantity[\alpha_{l}^{\vphantom{+}}e_{l}^{(-)}+\beta_{l}^{\vphantom{+}}e_{l}^{(+)}]\,,$
$\displaystyle\chi^{\prime}_{l}$
$\displaystyle=-i\,\sqrt{\frac{\omega_{l}(\eta)}{2}}\,\quantity[\alpha_{l}^{\vphantom{+}}e_{l}^{(-)}-\beta_{l}^{\vphantom{+}}e_{l}^{(+)}].$
(32)

and arrive at three real first-order differential equations for $s_{l}$,
$p_{l}$ and $q_{l}$

$\displaystyle s^{\prime}_{l}$
$\displaystyle=\frac{\omega^{\prime}_{l}}{2\omega_{l}}\,p_{l}\,,$
$\displaystyle p^{\prime}_{l}$
$\displaystyle=\frac{\omega^{\prime}_{l}}{\omega_{l}}\bigl{(}1+2s_{l}\bigr{)}-2\omega_{l}\,q_{l}\,,$
$\displaystyle q^{\prime}_{l}$ $\displaystyle=2\omega_{l}\,p_{l}\,,$ (33)

since here $Q=0$.

If we assume that no particle is present in any mode at the initial time
$\eta_{0}$, then we have $\alpha_{l}(\eta_{0})=1$, $\beta_{l}(\eta_{0})=0$,
which in turn implies $s_{l}(\eta_{0})=p_{l}(\eta_{0})=q_{l}(\eta_{0})=0$.
Thus we can express
$\lvert\chi^{\prime}_{l}(\eta)\rvert^{2}+\omega_{l}^{2}(\eta)\lvert\chi_{l}(\eta)\rvert^{2}$
simply by $s_{l}$, i.e., $\lvert\beta_{l}\rvert^{2}$, and find

$\langle\hat{T}_{\mu\nu}\rangle U^{\mu}U^{\nu}=\frac{1}{4\pi
a^{3}}\sum_{l=0}^{\infty}\bigl{(}l+\frac{1}{2}\bigr{)}\Bigl{(}2\lvert\beta_{l}\rvert^{2}+1\Bigr{)}\,\omega_{l}\,.$
(34)

What we are interested in, particle production, is contained in the
$\lvert\beta_{l}\rvert^{2}$ term. The $\omega_{l}/2$ term contains zero-point
fluctuations, and is the main culprit of UV divergence. After regularization,
it will yield the Casimir energy density associated with $S^{2}$ topology. The
regularization is performed by subtracting out the energy density of the
corresponding continuous limit, by taking $a\rightarrow\infty$

$\displaystyle\langle\hat{T}_{\mu\nu}\rangle_{\text{phys}}U^{\mu}U^{\nu}$
$\displaystyle=\frac{1}{4\pi
a^{3}}\sum_{l=0}^{\infty}\bigl{(}l+\frac{1}{2}\bigr{)}\Bigl{(}2\lvert\beta_{l}\rvert^{2}+1\Bigr{)}\,\omega_{l}\,-\lim_{a\rightarrow\infty}\frac{1}{4\pi
a^{3}}\sum_{l=0}^{\infty}\bigl{(}l+\frac{1}{2}\bigr{)}\,\omega_{l}$
$\displaystyle=\frac{2}{4\pi
a^{3}}\sum_{l=0}^{\infty}\bigl{(}l+\frac{1}{2}\bigr{)}\,\lvert\beta_{l}\rvert^{2}\,\omega_{l}\,+\rho_{\textsc{ca}}\,,$
(35)

where

$\rho_{\textsc{ca}}=\frac{M^{3}}{2\pi}\int_{0}^{1}\\!d\xi\;\frac{\xi\sqrt{1-\xi^{2}}}{e^{2\pi
Ma\xi}+1}\,,$ (36)

is the Casimir energy density Bordag  _et al._ (2009).

Since the space is homogeneous, the total energy of the quantum field on a
spacelike hypersurface at a particular cosmic time of interest is obtained by
multiplying the energy density by the volume,

$E(\eta)=V\langle\hat{T}_{\mu\nu}\rangle_{\text{phys}}U^{\mu}U^{\nu}=4\pi
a^{2}\langle\hat{T}_{\mu\nu}\rangle_{\text{phys}}U^{\mu}U^{\nu}.$ (37)

Here we will consider two illustrative examples of prescribed $a(t)$: (1)
monotonic exponentially growth and (2) Gaussian, which contains a rise stage
and a fall stage.

Figure 6: Time evolution of the volume, pressure, energy, and compressibility
of a massive conformal field in $\mathbf{R}\times S^{2}$, with the scale
factor taking on the exponential form, (38), with $a_{0}=1$, $t_{0}=0$, and
$\tau=1/5$.

(1) Here we suppose the universe, which maintains a constant scale before
$t=t_{0}$, undergoes some phase transition such that it starts exponentially
expansion after $t=t_{0}$. The scale factor is an exponentially growing
function of cosmic time $t$,

$a(t)=\begin{cases}a_{0}\,,&t<t_{0}\,,\\\
a_{0}\,e^{\frac{t-t_{0}}{\tau}}\,,&t\geq t_{0}\,,\end{cases}$ (38)

where $\tau$ measures the rate of change of the scale factor, and $t_{0}$
represents the time theses two phases join, and, without loss of generality,
we let $a_{0}=1$, $t_{0}=0$ . We can numerically solve the set of first order
differential equations for $s_{\mathbf{k}}$, $p_{\mathbf{k}}$, and
$q_{\mathbf{k}}$, and find the energy according to (35) and (37). The pressure
and compressibility then follow suit. The relevant results are shown in Fig.
6.

The volume of $S^{2}$ is given by $V(t)=4\pi a^{2}(t)$, and the vacuum energy
scales roughly proportional to the scale factor, i.e., $E(t)\propto a(t)$.
Among various contributions of the vacuum energy, the Casimir energy is
essentially negligible for a macroscopic universe of size greater than the
Planck length, and interestingly enough, it increases to a positive constant
when $t$ is greater than of the order $\tau$. Thus at late times $t>\tau$, the
vacuum energy associated with spontaneous particle creation is dominant 444A
word of caution about the numerical calculations of the latter contribution.
It is noted that for each $t$, once we have included a sufficiently large
number of modes, the numerical result barely depends on the number of modes.
Thus, we can set a finite upper bound of the summation in (35), instead of
$\infty$, to save the computation resources. This upper bound is time
dependent, and grows very rapidly with time. If we cut the corner by including
an insufficient number of modes, the numerical results show an artefact,
demonstrating oscillator behavior. Physically, the time-dependent cutoff is
related to the fact that particles in the higher modes are created so that the
total number of created particles are expected to increase with time. Thus, we
need to raise the upper bound of the summation to accommodate the contribution
of newly created particles.. It is interesting to point out that the vacuum
energy density reaches a local maximum at $t\simeq 0.439$. Note that in
$\mathbf{R}\times S^{2}$ spacetime, the vacuum energy does not contain any
contribution from the trace anomaly because it is known that there is no trace
anomaly in odd-dimensional spacetime Birrell and Davies (1984).

The pressure $P(t)$ then takes on negative values and scales like $-a^{-1}$ at
late times. That is, the spacetime experiences a very weak inward pressure.
This pressure has a local extremum roughly at $t\simeq 0.323$, when the
Casimir energy is about to reach the saturated value. Both effects are not
causally related because, as have been argued earlier, that the Casimir energy
is typically small. At this moment, the universe is still expanding, so it
causes the vacuum compressibility $\kappa_{\textsc{vac}}$ to be ill-defined
there. This is not too unfathomable because here we stipulated the universe to
expand in a particular way. At late times, the compressbility is then negative
and decreases proportional to $-a(t)$, as a fated consequence of $E(t)$,
driven by $a(t)$.

Figure 7: Time evolution of the volume, pressure, energy, and compressibility
of a massive conformal field in $\mathbf{R}\times S^{2}$ space where the scale
factor undergoes a Gaussian expansion, (39) with $a_{0}=1$, $t_{0}=1$,
$\Delta=2$, and $\tau^{2}=1/5$

(2) The scale factor is a Gaussian

$a(t)=a_{0}+\Delta\,e^{-\frac{(t-t_{0})^{2}}{\tau^{2}}}\,.$ (39)

As seen in the $\mathbf{R}\times S^{1}$ case, it contains more scales in the
expression, so the results, shown in Fig. 7, tend to be more complicated. In
this toy model, the Universe first rapidly expands to a maximal volume at
$t=t_{0}$ and then contracts back to the original size. For the current choice
of the parameters, the curve of the vacuum energy has an inflection point at
$t=t_{0}$ and then grows to a constant at sufficiently large time. The height
of this saturated value depends on the rate of expansion and the mass of the
field .

Similar to the previous case, the Casimir energy is quite small, compared to
the energy $E_{p}$ associated with spontaneous particle production as a result
of rapid expansion. However, the shape of $E_{p}$ is sensitive to the
parameters in the configuration, in particular, the mass of the field. If we
choose the values of field mass $M=10$, as shown by the orange curve in Fig.
8, then the curve of $E_{p}$ shows a major double-peak feature, and then
quickly decays to very small values. A larger field mass in principle can
amplify the effect of the scale factor in (29), making a given mode more
energetic. Thus, the contribution to the energy attributed to particle
creation can be boosted. This sounds very paradoxical because we would expect
that particles of heavier mass are harder to produce. Indeed, as shown in Fig.
8, the density of created particles is suppressed when the mass increases from
$M=1$ to $M=10$, but the oscillatory frequency of a given mode increases in
such a way that $E_{p}$ still grows with mass during the transient moments. At
late times, when the universe asymptotically returns to its original size, the
particle density seems to saturate to a very small value, roughly proportional
to $M^{-1}$ for sufficiently large $M$. This is consistent with our
expectation that it is harder to create heavier particles.

Figure 8: Comparison of the energy $E_{p}$ associated with spontaneous
particle production and the density of created particles in the first 100
modes in $\mathbf{R}\times S^{2}$ when the scale factor is given by a Gaussian
expansion, (39) with $a_{0}=1$, $t_{0}=1$, $\Delta=2$, and $\tau^{2}=1/5$. The
orange curve corresponds to $M=10$, while the blue curve has $M=1$.

It may seem a bit strange that the positive Casimir energy acquires the
largest value when the universe is at the maximal scale. The Casimir energy
density is inversely proportional to $a(t)$, and indeed it is a minimum at
$t=t_{0}$. However, when multiplied by the volume of the universe which
increases fast the energy takes on the greatest value. Our numerical
calculations show that both $E_{p}$ and the Casimir energy are actually
proportional to $M$ even though (36) superficially has a overall factor of
$M^{3}$ outside the integral, so the contribution of the Casimir energy
remains small when the field mass takes on a larger value.

The temporal behavior of the pressure contains several local extrema, so the
compressibility becomes ill defined there.

##  IV Vacuum compressibility of conformal scalar field in $\mathbf{R}\times
S^{3}$

###  IV.1 Massive field: particle creation

We first consider spontaneous particle creation of a massive conformal field
in $(1+3)$-dimensional Robertson-Walker universe with line element

$\displaystyle
ds^{2}=a^{2}(\eta)\quantity[d\eta^{2}-\gamma_{ij}\,dx^{i}dx^{j}]\,,$ (40)

where the spatial metric $\gamma_{ij}$ is given in hyperspherical coordinates
by

$\displaystyle\gamma_{ij}\,dx^{i}dx^{j}=dX^{2}+\sin^{2}X\quantity(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\,,\qquad\qquad
X\in[0,\pi]\,.$ (41)

The wave equation governing massive conformal fields is

$\displaystyle\quantity[\square+M^{2}+\frac{1}{6}R]\phi=0\,,$ (42)

where $R=6a^{-2}(a^{-1}a^{\prime\prime}+1)$, and the curvature coupling
constant $\xi$ is $1/6$ in $S^{3}$.

The mode expansion of the field operator $\hat{\phi}(x)$ has the same form as
before:

$\displaystyle\hat{\phi}(x)=\int\\!d\tilde{u}\;\quantity[\hat{a}^{\vphantom{\dagger}}_{\mathbf{k}}f^{\vphantom{\dagger}}_{\mathbf{k}}(x)+\hat{a}^{\dagger}_{\mathbf{k}}f_{\mathbf{k}}^{*}(x)]\,,.$
(43)

with the measure $\displaystyle\int\\!d\tilde{u}$ defined by

$\displaystyle\int\\!d\tilde{u}=\sum_{k=1}^{\infty}\sum_{l=0}^{k-1}\sum_{m=-l}^{l}\,.$
(44)

The mode functions $f_{\mathbf{k}}$, in terms of the hyperspherical harmonics,
has the form

$\displaystyle f_{\mathbf{k}}(\eta,X,\theta,\varphi)$ $\displaystyle=$
$\displaystyle a(\eta)^{-1}\chi_{k}(\eta)\,Y_{klm}(X,\theta,\varphi)\,,$ (45)

with the amplitude function $\chi_{k}(\eta)$ obeying the wave equation

$\displaystyle\chi_{k}^{\prime\prime}(\eta)+\omega_{k}^{2}(\eta)\chi_{k}(\eta)=0\,,$
(46)

where $\omega_{k}^{2}=k^{2}+a(\eta)^{2}M^{2}=k^{2}+\mu(\eta)^{2}$.

The procedure henceforth is similar to what was presented in the previous
section for $\mathbf{R}\times S^{2}$ spacetime, except for the difference in
the prefactors of the mode sums

$\displaystyle\expectationvalue{T_{\mu\nu}}U^{\mu}U^{\nu}=\sum_{k=1}^{\infty}\sum_{l=0}^{k-1}\sum_{m=-l}^{l}\;T_{\mu\nu}[f_{\mathbf{k}}^{\vphantom{*}},\,f_{\mathbf{k}}^{*}]U^{\mu}U^{\nu}=\frac{1}{4\pi^{2}a^{4}}\sum_{k=1}^{\infty}k^{2}\quantity[|\chi^{\prime}_{k}|^{2}+\omega_{k}^{2}(\eta)|\chi_{k}|^{2}]\,.$
(47)

The form of $\chi_{k}$ is the same as in the $\mathbf{R}\times S^{2}$ case, so
is the expression of the energy density

$\displaystyle\expectationvalue{T_{\mu\nu}}U^{\mu}U^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}a^{4}}\sum_{k=1}^{\infty}k^{2}\quantity[\omega_{k}(2|\beta_{k}|^{2}+1)]\,.$
(48)

The above expression is formally UV divergent, and subtraction of terms of the
corresponding adiabatic orders is needed to obtain a finite expression. The
latter have been calculated in Bunch (1980). The finite physical energy
density is given by

$\displaystyle\expectationvalue{T_{\mu\nu}}_{\text{phys}}U^{\mu}U^{\nu}$
$\displaystyle=\frac{1}{4\pi^{2}a^{4}}\sum_{k=1}^{\infty}k^{2}\quantity[\omega_{k}(2|\beta_{k}|^{2}+1)]$
$\displaystyle-\frac{1}{4\pi^{2}a^{4}}\int_{0}^{\infty}\\!dk\;k^{2}\Biggl{\\{}\Biggr{.}\omega_{k}+\frac{M^{4}a^{4}}{8\omega_{k}^{5}}\frac{a^{\prime
2}}{a^{2}}-\frac{M^{4}a^{4}}{32\omega_{k}^{7}}\quantity[2\frac{a^{\prime\prime\prime}a^{\prime}}{a^{2}}-\frac{a^{\prime\prime
2}}{a^{2}}+\frac{4a^{\prime\prime}a^{\prime 2}}{a^{3}}-\frac{a^{\prime
4}}{a^{4}}]$
$\displaystyle+\frac{7M^{6}a^{6}}{16\omega_{k}^{9}}\quantity[\frac{a^{\prime\prime}a^{\prime
2}}{a^{3}}+\frac{a^{\prime
4}}{a^{4}}]-\frac{105M^{8}a^{8}}{128\omega_{k}^{11}}\frac{a^{\prime
4}}{a^{4}}\Biggl{.}\Biggr{\\}}\,.$ (49)

The Casimir energy on the other hand is given by

$\rho_{\textsc{ca}}=\frac{1}{4\pi^{2}a^{4}}\sum_{k=1}^{\infty}k^{2}\omega_{k}-\frac{1}{4\pi^{2}a^{4}}\int_{0}^{\infty}\\!dk\;k^{2}\omega_{k}\,.$
(50)

After performing the integration, we obtain

$\displaystyle\expectationvalue{T_{\mu\nu}}_{\text{phys}}U^{\mu}U^{\nu}$
$\displaystyle=\frac{1}{4\pi^{2}a^{4}}\sum_{k=1}^{\infty}\quantity[2|\beta_{k}|^{2}\,k^{2}\omega_{k}]-\frac{M^{2}a^{\prime
2}}{48\pi^{2}a^{4}}+\rho_{\textsc{ca}}$
$\displaystyle\qquad\qquad+\frac{6}{2880\pi^{2}a^{4}}\quantity{\frac{a^{\prime\prime\prime}a^{\prime}}{a^{2}}-\frac{1}{2}\frac{a^{\prime\prime
2}}{a^{2}}-\frac{2a^{\prime\prime}a^{\prime
2}}{a^{3}}+\frac{1}{2}\frac{a^{\prime 4}}{a^{4}}}\,.$ (51)

Expressing the result in cosmic time

Figure 9: The time evolution of the volume, pressure, energy, and
compressibility for a massive scalar field in $\mathbf{R}\times S^{3}$ with
$a(t)$ following an exponential expansion. The parameters are the same as
before, but here we choose a shorter time scale. The pressure will tend to a
negative constant, not shown in the plot.

$\displaystyle\expectationvalue{T_{\mu\nu}}_{\text{phys}}U^{\mu}U^{\nu}$
$\displaystyle=\frac{1}{4\pi^{2}a^{4}}\sum_{k=1}^{\infty}\quantity[2|\beta_{k}|^{2}\,k^{2}\omega_{k}]-\frac{M^{2}\dot{a}^{2}}{48\pi^{2}a^{2}}+\rho_{\textsc{ca}}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{480\pi^{2}a^{2}}\quantity{\frac{\dot{a}^{2}\ddot{a}}{a}-\frac{\dot{a}^{4}}{a^{2}}-\frac{\ddot{a}^{2}}{2}+\dot{a}\dddot{a}}$
(52)

gives the total energy. The third term is the Casimir energy and the
expressions in the second line (shown in the next section) come from the trace
anomaly. The second term is related to the nonzero trace of the classical
stress tensor of the massive field. The numerical results for a) an
exponential expansion and b) a Gaussian evolution are respectively shown in
Fig. 9 and Fig. 11, with the volume given by $V(t)=2\pi^{2}a^{3}(t)$.

For the case of exponential expansion, the vacuum energy contains four
components. The contribution from the Casimir energy, like before, is
negligibly small for a macroscopic universe. The vacuum energy $E$ in this
$\mathbf{R}\times S^{3}$ case scales like $a^{3}(t)$. At first glance it
appears plausible to relate it to the fact that the number of allowed modes
increases exponentially with the spatial dimensions. It turns out this is not
entirely true. As will be explained later, this scaling mainly results from
the trace-related contribution of the vacuum energy in a rapidly expanding
universe. There are two types of contributions associated with the trace of
the renormalized expectation value of the energy momentum tensor of the
massive field: One is related to the nonzero trace of the classical stress
tensor, and the other, independent of mass, results from the trace anomaly.
For a rapidly expanding case like the current one, their contributions tend to
dominate over the one due to spontaneous particle creation, because these
contributions contain higher-order time derivatives of the scale factor. In
particular, the trace anomaly contribution in a fast expanding universe
typically is more important than the one associated with the classical trace
as long as $m$ is smaller than the inverse of the expansion time scale. On the
other hand, for a slowly (but not adiabatically slowly) expanding universe,
say $\tau=1$, the contribution associated with the trace can be comparable or
even smaller than the counterpart by spontaneous particle creation.

Figure 10: Comparison of the time evolution of the energy densities associated
with spontaneous particle creation and the trace for a massive scalar field in
an exponentially expanding 3-sphere $\mathbf{R}\times S^{3}$. We use the same
set of parameters as before.

In this case the time evolution of the vacuum may reveal additional complexity
due to the competition of different time scales such as the inverse mass,
$\tau$, or their combinations. From the time dependence of the vacuum energy,
we may roughly deduce the time evolution of the pressure and the vacuum
compressibility accordingly. At late times $t\gg\tau$, $P(t)$ scales like
$a^{0}(t)$ and is thus a negative constant. Again the pressure has a local
extremum at $t\simeq 0.225$, its presence causes the compressibility to be ill
defined there.

It is interesting to compare this with the above time scale $t\simeq 0.344$
when both the vacuum energy density $E/V$ and the ratio $E_{p}/E_{t}$ reach
the local maximum, in which $E_{t}$ is the contribution associated with the
trace. At first glance it is not clear why both ratios reach the local maximum
at the same moment. A closer numerical inspection shows that the energy
density associated with the trace quickly reaches a saturated constant for the
exponential expansion case. This can be readily seen from (52)

$-\frac{M^{2}\dot{a}^{2}}{48\pi^{2}a^{2}}+\frac{1}{480\pi^{2}a^{2}}\quantity{\frac{\dot{a}^{2}\ddot{a}}{a}-\frac{\dot{a}^{4}}{a^{2}}-\frac{\ddot{a}^{2}}{2}+\dot{a}\dddot{a}}=-\frac{M^{2}}{48\pi^{2}\tau^{2}}\biggl{[}1-\frac{1}{20M^{2}\tau^{2}}\biggr{]}\,.$
(53)

This further reveals how the magnitude and the sign of the trace-related
contribution in vacuum energy can depend on the mass $M$ of the field and the
expansion rate $\tau^{-1}$, and allow us to assess the importance of this
contribution.

Thus, the time variation of the vacuum energy density $\rho_{\textsc{vac}}$ is
mostly governed by the contribution due to spontaneous particle creation, as
shown in Fig. 10.

Figure 11: The time evolution of the volume, pressure, energy, and
compressibility for a massive scalar field in $\mathbf{R}\times S^{3}$ with a
Gaussian $a(t)$. We use the same set of parameters as before.

Compared with the $\mathbf{R}\times S^{2}$ case, there are two major
distinctions: 1) The contribution of the vacuum energy scales differently
among the two cases. Thus, this variation will cause the pressure and
compressbility to scale in a different way. 2) The vacuum energy in
$\mathbf{R}\times S^{2}$ spacetime is always dominated by the process of
vacuum spontaneous particle production for a macroscopic universe, but in
$\mathbf{R}\times S^{3}$ spacetime, the most important contribution depends on
the expansion rate and the mass scale. For a fast expanding universe, the
trace-related contribution dominates.

When the scale factor assumes the Gaussian form, (39), the time evolution of
the vacuum energy gets increasingly complicated because the trace related
contribution depends on various time derivatives of the scale factor,
introducing multiple scales into the evolution. The behavior of the energy
$E_{p}$ associated with spontaneous particle production is very similar to
that in $\mathbf{R}\times S^{2}$, but the curve of vacuum energy $E$ looks
somewhat different from the counterpart in the $\mathbf{R}\times S^{2}$ case.
It can be negative before the universe reaches the maximum size. This is
related to the fact that the trace related contribution $E_{t}$ is negative
about $t=t_{0}$, and its absolute value is slightly larger than $E_{p}$ for
the current choice of parameters. It is worth mentioning that the time
evolution of $E_{t}$ is symmetric about $t=t_{0}$ because each term contains
an even number of derivatives of the scale factor of the Gaussian form. Thus,
the contribution $E_{t}$ becomes negligible at late times.

It is not too surprising that the compressibility has pretty wild behavior,
but it approaches zero at late times. This suggests that the space manifest
itself as a rigid body. That is, the volume barely changes when the outward
pressure is slightly increased.

###  IV.2 Massless field: Trace anomaly

As a final example we return to the effects due to the trace anomaly. The
action of a massless conformally couplws scalar field in a closed Robertson-
Walker spacetime with scalar curvature $R$ is

$S=\frac{1}{2}\int\\!d^{3}x\sqrt{-g}\;\quantity[g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-\xi
R\,\phi^{2}]\,.$ (54)

It is invariant under conformal transformations, i.e., $g_{\mu\nu}\rightarrow
a^{2}g_{\mu\nu}$ if we set $\xi=1/6$.

The frequencies $\omega_{k}$ of the normal modes in Eq. (46) become time
independent and are given by $\lvert k\rvert$,

$\chi_{k}^{\prime\prime}(\eta)+\omega_{k}^{2}\chi_{k}(\eta)=0\,.$ (55)

The energy density is formally divergent at the UV end:

$\displaystyle\langle\hat{T}_{\mu\nu}\rangle U^{\mu}U^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}a^{4}}\sum_{k=1}^{\infty}k^{3}\,.$ (56)

Before we carry out the renormalization of the energy density expression, we
note that although the trace of the classical energy momentum tensor vanishes,
the trace of the renormalized expectation values of the energy momentum tensor
operator is nonzero, this anomalous trace, known as the trace anomaly, is a
purely quantum effect. The renormalization procedure breaks conformal
symmetry. This nonzero trace anomaly has been calculated by various
approaches, such as dimensional regularization, zeta function regularization,
point-splitting regularization, and so on. Here we will take the adiabatic
regularization, and the corresponding expressions for the trace anomaly for a
massless conformally coupled scalar field in the Bianchi type I and the
Robertson-Walker universes have been derived in Hu (1978, 1979) by exploiting
the difference between a massless conformal quantum field theory and a massive
conformal quantum field theory after renormalization in the mass approaching
zero limit. The observation was that finite terms appear in the reduction of
integrals for the energy density and the pressure containing higher powers of
mass to integrals containing a lower power of mass when one sets the mass of a
massive field to zero. These finite terms from the integrals containing higher
powers of mass add up to produce the trace anomaly which would otherwise not
be there if one works ab initio with a massless field throughout.

With this in mind, we obtain the energy density in a comoving frame:

$\displaystyle\expectationvalue{T_{\mu\nu}}_{\text{phys}}U^{\mu}U^{\mu}$
$\displaystyle=\frac{6}{2880\pi^{2}a^{4}}\quantity[1+\frac{a^{\prime\prime\prime}a^{\prime}}{a^{2}}-\frac{a^{\prime\prime
2}}{2a^{2}}-\frac{2a^{\prime\prime}a^{\prime 2}}{a^{3}}+\frac{a^{\prime
4}}{2a^{4}}]$
$\displaystyle=\frac{1}{480\pi^{2}a^{2}}\quantity{\frac{1}{a^{2}}+\frac{\dot{a}^{2}\ddot{a}}{a}-\frac{\dot{a}^{4}}{a^{2}}-\frac{\ddot{a}^{2}}{2}+\dot{a}\dddot{a}}\,.$
(57)

In the second line, we express the result in terms of cosmic time. The first
term in the curly brackets is the Casimir energy density, and the remaining
expressions is associated with the trace anomaly of a massless conformal field
in the Robertson-Walker universe Fischetti  _et al._ (1979); Anderson (1983).
It is interesting to note that the same expression for the contribution
associated with trace anomaly in (57) also appears in the second line of (52)
in the massive field case.

Figure 12: The time evolution of the volume, pressure, energy, and vacuum
compressibility for a massless scalar field in $\mathbf{R}\times S^{3}$ when
an exponentially expanding scale factor $a(t)$.

From this expression of the energy density for the massless conformal field,
retracing the steps in the massive field case, we obtain the results for the
pressure, and the vacuum compressibility for a massless conformal scalar field
due to the Casimir effect and the trace anomaly. There is no spontaneous
particle production for a conformally coupled, massless field in a conformally
static spacetime.

Their time evolution due to these two quantum sources are shown in Fig. 12 for
an exponential expansion and Fig. 13 for a Gaussian evolution. It is seen that
the vacuum energy due to the Casimir effect is too small to be of our concern.
The contribution from the trace anomaly of a massless field is smaller than
that of a massive field because the mass dependent term is absent. As a
consequence, we do not see a local extremum in the time evolution of the
pressure and thus the vacuum compressibility is well defined. However, since
the pressure saturates rapidly to a negative value, the compressibility
diverges accordingly.

This is a good place to comment on the comparison with the results for the
$\mathbf{R}\times S^{1}$ case. a) Recall that in the $\mathbf{R}\times S^{1}$
case the scale factor $a(t)$ is dynamically determined, with the backreaction
effects. That is, we solve for the dynamics of $a(t)$ with the Casimir effect
and the trace anomaly incorporated, by the self-consistent interaction between
the dynamics of the quantum field and the geometry. Here, the scale factor is
prescribed a priori, not determined dynamically. b) In the $\mathbf{R}\times
S^{1}$ case, the presence of the trace anomaly accelerates the collapse of the
$S^{1}$ space, over and above that already present due to the Casimir effect.
c) The expressions for the trace anomaly in two dimensions and four dimensions
are quite different: the four-dimensional case contains higher-order
derivative of $a$. Therefore it is not surprising to see many qualitative
differences between these two cases.

Figure 13: Time evolution of the volume, pressure, energy, and vacuum
compressibility for a massless scalar field in $\mathbf{R}\times S^{3}$ for a
scale factor $a(t)$ taking on a Gaussian dynamics

If the universe expands and contracts according to the Gaussian profile given
by (20), then, as analyzed earlier, the vacuum energy is essentially given by
the contribution of the trace anomaly only. This is because the Casimir energy
is negligible when $a(t)$ is large, and the contribution from the nonzero
classical trace vanishes. Hence the features related to the trace-anomaly
contribution we highlighted earlier can be directly applied here. Since there
is no particle production, the vacuum energy is very small at late times.
However, at $t\gg t_{0}$ the contribution from the trace anomaly is
exponentially small, while that from the Casimir energy dominates.

##  V Vacuum compressibility of $\mathbf{R}\times T^{3}$

We now turn our attention to the quantum effect due to anisotropic expansion.
Creation of field quanta occurs quite generically for the quantum field in a
time-dependent background. The conformally-coupled, massless scalar field in a
conformally-flat spacetime is a well known exception. For the effects of
particle creation of conformally-coupled massless scalar fields we need to
consider anisotropic expansion, such as was done for the Bianchi Type I
universe Hu and Parker (1978); Hartle and Hu (1979).

Here we consider the effects of vacuum particle creation in a three-
dimensional rectangular box. We allow the longitudinal length to change in
time, but set the two transverse lengths equal and fixed. Inside the box is
filled a massless, conformally-coupled quantum scalar field in its vacuum
state. We assume that the field takes on periodic boundary conditions on the
surfaces of the box, so for this quantum field, the box configuration is
analogous to a space with three-torus $T^{3}$ topology. Thus, we have cast the
dynamical Casimir effect of a massless, conformally-coupled scalar field into
an anisotropically expanding spacetime. We are interested in the vacuum energy
of created particles in this $\mathbf{R}\times T^{3}$ spacetime.

This spacetime has a line element of the same form as that of an
anisotropically expanding, spatially-flat (Bianchi type-I) universe:

$ds^{2}=dt^{2}-\quantity[a_{1}^{2}(t)\,dx^{2}+a_{2}^{2}(t)\,dy^{2}+a_{3}^{2}(t)\,dz^{2}]\,,$
(58)

where $a_{i}$ are the scale factors in the three Cartesian directions. We let
$a_{2}=a_{3}=1$, but $a_{1}=a(t)$, i.e., allowing the scale factor in the $x$
direction to change in time.

A conformally-coupled, massless scalar field $\phi$ in this anisotropically
expanding space obeys the Klein-Gordon equation:

$\displaystyle\partial_{t}^{2}\phi+\frac{\dot{a}}{a}\dot{\phi}-\sum_{i=1}^{3}\frac{\partial_{i}^{2}\phi}{a_{i}^{2}}+\frac{1}{6}R\phi=0\,,$
(59)

where $\xi=\dfrac{n-2}{4(n-1)}=\dfrac{1}{6}$ is the conformal coupling
strength, and $R=-\dfrac{2\ddot{a}}{a}$ is the scalar curvature of spacetime.

The classical energy-momentum tensor of this scalar field is given by

$\displaystyle T_{\mu\nu}$
$\displaystyle=(\partial_{\mu}\phi)(\partial_{\nu}\phi)-\frac{1}{2}g_{\mu\nu}g^{\lambda\sigma}(\partial_{\lambda}\phi)(\partial_{\sigma}\phi)-\frac{1}{6}\left[\nabla_{\mu}\partial_{\nu}(\phi^{2})-g_{\mu\nu}g^{\lambda\sigma}\nabla_{\lambda}\partial_{\sigma}(\phi^{2})+\phi^{2}G_{\mu\nu}\right]\,,$
(60)

where $G_{\mu\nu}$ is the Einstein tensor and $\nabla_{\mu}$ is the covariant
derivative with respect to the metric tensor $g_{\mu\nu}$. The corresponding
energy density is

$T_{\mu\nu}U^{\mu}U^{\nu}=\frac{1}{2}\,\dot{\phi}^{2}+\frac{1}{3}\frac{\dot{a}}{a}\,\phi\dot{\phi}+\frac{1}{6}\sum_{i=1}^{3}\left(\frac{\partial_{i}\phi}{a_{i}}\right)^{2}-\frac{1}{3}\sum_{i=1}^{3}\frac{\phi\,\partial_{i}^{2}\phi}{a_{i}^{2}}-\frac{1}{6}\,G_{tt}\phi^{2},$
(61)

where an overdot indicates differentiation with respect to $t$.

It is convenient to introduce a new field variable $\chi=a^{1/3}\phi$ and a
conformal time variable $\eta$ according to
$\differential\eta=a^{-1/3}{\differential t}$ with a prime indicating
differentiation respect to $\eta$, and write (61) in terms of conformal time

$T_{\mu\nu}U^{\mu}U^{\nu}=a^{-2/3}\quantity[\frac{1}{2}\,a^{-2/3}\chi^{\prime
2}+\frac{1}{6}\sum_{i=1}^{3}\quantity(\frac{\partial_{i}\chi}{a_{i}})^{2}-\frac{1}{3}\sum_{i=1}^{3}\frac{\chi\,\partial_{i}^{2}\chi}{a_{i}^{2}}-\frac{1}{2}a^{-2/3}Q\chi^{2}]\,.$
(62)

The positive-definite quantity $Q$ defined by

$Q=\frac{1}{18}\sum_{i<j=1}^{3}\quantity(\frac{a_{i}^{\prime}}{a_{i}}-\frac{a_{j}^{\prime}}{a_{j}})^{2}=\frac{1}{9}\quantity(\frac{a^{\prime}}{a})^{2}>0\,,$
(63)

measures the severity of anisotropic expansion. The new field variable $\chi$
satisfies the equation of motion

$\chi^{\prime\prime}-2a^{2/3}\sum_{i=1}^{3}\frac{\partial_{i}^{2}\chi}{a_{i}^{2}}+Q\chi=0\,.$
(64)

Under the periodic condition imposed at the boundaries, the corresponding
field operator $\hat{\chi}$ has mode expansion of the form:

$\hat{\chi}=l^{-3/2}\sum_{\mathbf{k}}\quantity[\hat{A}_{\mathbf{k}}\chi_{\mathbf{k}}(\eta)e^{+i\mathbf{k}\cdot\mathbf{x}}+\hat{A}_{\mathbf{k}}^{\dagger}\chi_{\mathbf{k}}^{*}(\eta)e^{-i\mathbf{k}\cdot\mathbf{x}}]\,,$
(65)

with the compact notation

$\sum_{\mathbf{k}}=\prod_{i=1}^{3}\sum_{n_{i}}$

where $k_{i}=2\pi n_{i}/l$, $n_{i}\in\mathbb{Z}$, and $l$ is the coordinate
length of each side of the box. The Fourier modes $\chi_{\mathbf{k}}$ satisfy
the parametric oscillator equation:

$\displaystyle\chi^{\prime\prime}_{\mathbf{k}}+[\Omega_{\mathbf{k}}^{2}(\eta)+Q(\eta)]\chi_{\mathbf{k}}=0\,,$
(66)

with $\Omega_{\mathbf{k}}(\eta)$ given by

$\Omega_{\mathbf{k}}^{2}=a^{2/3}(\eta)\quantity[\frac{k_{x}^{2}}{a^{2}(\eta)}+k_{y}^{2}+k_{z}^{2}]\,,$
(67)

and the normalization condition
$\chi^{\prime*}_{\mathbf{k}}\chi^{\vphantom{*}}_{\mathbf{k}}-\chi_{\mathbf{k}}^{*}\chi^{\prime}_{\mathbf{k}}=i$.

Let the initial state $\ket{0_{A}}$ be the vacuum state annihilated by
$\hat{A}_{\mathbf{k}}$. Then after transforming back to the cosmic time, the
vacuum expectation value of the $tt$ component of the stress-energy tensor

$\expectationvalue{\hat{T}_{\mu\nu}}{0_{A}}U^{\mu}U^{\nu}={\frac{a^{-4/3}}{2l^{3}}}\sum_{\mathbf{k}}\bqty{\vqty{\chi_{\mathbf{k}}^{\prime}}^{2}+(\Omega_{\mathbf{k}}^{2}-Q)\vqty{\chi_{\mathbf{k}}}^{2}}\,,$
(68)

gives the formal expression of the energy density. In order to obtain the
explicit expressions for $\vqty{\chi_{\mathbf{k}}}$ and
$\vqty{\chi_{\mathbf{k}}^{\prime}}$, we solve the equation of motion (66) by
the ansatz

$\displaystyle\chi_{\mathbf{k}}$
$\displaystyle=\frac{1}{\sqrt{2\Omega_{\mathbf{k}}(\eta)}}\,\quantity[\alpha_{\mathbf{k}}^{\vphantom{+}}e_{\mathbf{k}}^{(-)}+\beta_{\mathbf{k}}^{\vphantom{+}}e_{\mathbf{k}}^{(+)}]\,,$
$\displaystyle\chi^{\prime}_{\mathbf{k}}$
$\displaystyle=-i\,\sqrt{\frac{\Omega_{\mathbf{k}}(\eta)}{2}}\,\quantity[\alpha_{\mathbf{k}}^{\vphantom{+}}e_{\mathbf{k}}^{(-)}-\beta_{\mathbf{k}}^{\vphantom{+}}e_{\mathbf{k}}^{(+)}],$
(69)

for two new time-dependent functions $\alpha_{k}(t)$ and $\beta_{k}(t)$, with

$e_{\mathbf{k}}^{(\pm)}\equiv\exp\quantity[\pm
i\int_{\eta_{0}}^{\eta}\\!\differential\eta^{\prime}\;\Omega_{\mathbf{k}}(\eta^{\prime})]\,.$
(70)

The normalization condition requires
$\quantity|\alpha_{k}(\eta)|^{2}-\quantity|\beta_{k}(\eta)|^{2}=1$. Combining
Eqs. (66) and (69), we obtain a simultaneous set of first-order differential
equations for $\alpha_{\mathbf{k}}$ and $\beta_{\mathbf{k}}$,

$\displaystyle\alpha^{\prime}_{\mathbf{k}}$
$\displaystyle=\frac{1}{2}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega_{\mathbf{k}}}-i\,\frac{Q}{\Omega_{\mathbf{k}}})(e_{\mathbf{k}}^{+})^{2}\,\beta_{\mathbf{k}}-i\frac{Q}{2\Omega_{\mathbf{k}}}\,\alpha_{\mathbf{k}}\,,$
(71) $\displaystyle\beta^{\prime}_{\mathbf{k}}$
$\displaystyle=\frac{1}{2}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega_{\mathbf{k}}}+i\,\frac{Q}{\Omega_{\mathbf{k}}})(e_{\mathbf{k}}^{-})^{2}\,\alpha_{\mathbf{k}}+i\,\frac{Q}{2\Omega_{\mathbf{k}}}\,\beta_{\mathbf{k}}\,.$
(72)

These are constrained complex-valued equations, so they can be re-written into
three real-valued differential equations by introducing three real variables

$\displaystyle s_{\mathbf{k}}$ $\displaystyle=|\beta_{\mathbf{k}}|^{2}\,,$
$\displaystyle p_{\mathbf{k}}$
$\displaystyle=\alpha_{\mathbf{k}}^{\vphantom{*}}\beta_{\mathbf{k}}^{*}e_{-}^{2}+\alpha_{\mathbf{k}}^{*}\beta_{\mathbf{k}}^{\vphantom{*}}e_{+}^{2}\,,$
$\displaystyle q_{\mathbf{k}}$
$\displaystyle=i\,(\alpha_{\mathbf{k}}^{\vphantom{*}}\beta_{\mathbf{k}}^{*}e_{-}^{2}-\alpha_{\mathbf{k}}^{*}\beta_{\mathbf{k}}^{\vphantom{*}}e_{+}^{2})\,.$
(73)

Thus we obtain a simultaneous set of first-order real-valued ordinary
differential equations Hu (1974)

$\displaystyle s^{\prime}_{\mathbf{k}}$
$\displaystyle=\frac{1}{2}\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega_{\mathbf{k}}}p_{\mathbf{k}}+\frac{1}{2}\frac{Q}{\Omega_{\mathbf{k}}}q_{\mathbf{k}}\,,$
(74) $\displaystyle p^{\prime}_{\mathbf{k}}$
$\displaystyle=\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega_{\mathbf{k}}}(1+2s_{\mathbf{k}})-\quantity(\frac{Q}{\Omega_{\mathbf{k}}}+2\Omega_{\mathbf{k}})q_{\mathbf{k}}\,,$
(75) $\displaystyle q^{\prime}_{\mathbf{k}}$
$\displaystyle=\frac{Q}{\Omega_{\mathbf{k}}}(1+2s_{\mathbf{k}})+\quantity(\frac{Q}{\Omega_{\mathbf{k}}}+2\Omega_{\mathbf{k}})p_{\mathbf{k}}\,.$
(76)

Expressing the energy density in term of $s_{\mathbf{k}}$, $p_{\mathbf{k}}$,
and $q_{\mathbf{k}}$, we find

$\displaystyle\expectationvalue{\hat{T}_{\mu\nu\;\text{reg}}}{0_{A}}U^{\mu}U^{\nu}$
$\displaystyle={\frac{a^{-4/3}}{2l^{3}}}\sum_{\mathbf{k}}\bqty{\Omega_{\mathbf{k}}(2s_{\mathbf{k}}+1)-\frac{Q}{2\Omega_{\mathbf{k}}}(1+2s_{\mathbf{k}}+p_{\mathbf{k}})}.$
(77)

With a suitably defined adiabatic vacuum state the parameter $s_{\mathbf{k}}$
can be used as a measure of particle number density at time $\eta$due to the
moving boundary.

Eq. (77) contains ultraviolet divergences which we remove by adiabatic
regularization

$\displaystyle\expectationvalue{\hat{T}_{\mu\nu\;\text{reg}}}{0_{A}}U^{\mu}U^{\nu}$
$\displaystyle={\frac{a^{-4/3}}{2l^{3}}}\sum_{\mathbf{k}}\bqty{\Omega_{\mathbf{k}}(2s_{\mathbf{k}}+1)-\frac{Q}{2\Omega_{\mathbf{k}}}(1+2s_{\mathbf{k}}+p_{\mathbf{k}})}-\rho_{(0)}-\rho_{(2)}-\rho_{(4)}\,.$
(78)

We take the zeroth- and second-order adiabatic expansion of the energy density
$\rho_{(0)}$, $\rho_{(2)}$ from Fulling  _et al._ (1974), and subtract them
from the corresponding formally divergent terms

$\displaystyle\expectationvalue{\hat{T}_{\mu\nu\;\text{reg}}}{0_{A}}U^{\mu}U^{\nu}$
$\displaystyle={\frac{a^{-\frac{4}{3}}}{2l^{3}}}\sum_{\mathbf{k}}2s_{\mathbf{k}}\Omega_{\mathbf{k}}+\quantity{{\frac{a^{-\frac{4}{3}}}{2l^{3}}}\sum_{\mathbf{k}}\Omega_{\mathbf{k}}-\frac{a^{-\frac{4}{3}}}{2}\int\\!\frac{d^{3}k}{(2\pi)^{3}}\;\Omega_{\mathbf{k}}}$
$\displaystyle\qquad+\quantity{{\frac{a^{-\frac{4}{3}}}{2l^{3}}}\sum_{\mathbf{k}}\quantity[-\frac{Q}{2\Omega_{\mathbf{k}}}]-\frac{a^{-\frac{4}{3}}}{2}\int\\!\frac{d^{3}k}{(2\pi)^{3}}\;\quantity[-\frac{Q}{2\Omega_{\mathbf{k}}}]}$
(79)
$\displaystyle\qquad+\quantity{{\frac{a^{-\frac{4}{3}}}{2l^{3}}}\sum_{\mathbf{k}}\quantity[-\frac{Q}{2\Omega_{\mathbf{k}}}(2s_{\mathbf{k}}+p_{\mathbf{k}})]}-{\frac{a^{-\frac{4}{3}}}{2}\int\\!\frac{d^{3}k}{(2\pi)^{3}}\;\quantity[\frac{1}{8}\frac{\Omega^{\prime}{}_{\mathbf{k}}^{2}}{\Omega_{\mathbf{k}}^{3}}]}-\rho_{(4)}\,.$

The first term in the first line on the righthand side is the energy density
of the created particles, the second term, denoted by $\rho_{\textsc{ca}}$, is
the Casimir energy density due to the compactness of the spatial section.

The Casimir energy in $T^{3}$ topology has been calculated in e.g. Mamaev and
Trunov (1979), and is given by

$\rho_{\textsc{ca}}(\ell_{1},\ell_{2},\ell_{3})=-\frac{\pi^{2}}{90\ell_{1}^{4}}-\frac{\pi^{2}}{6\ell_{1}\ell_{2}\ell_{3}^{2}}-\frac{\zeta_{R}(3)}{2\pi\ell_{1}\ell_{2}^{3}}+\frac{8\pi}{\ell_{1}\ell_{2}^{2}\ell_{3}}G\quantity(\frac{\ell_{3}}{\ell_{2}})-\frac{16}{\ell_{1}^{2}\ell_{2}\ell_{3}}R\quantity(\frac{\ell_{2}}{\ell_{1}},\frac{\ell_{3}}{\ell_{1}}),$
(80)

where $\ell_{1}$, $\ell_{2}$, $\ell_{3}$ are the side lengths of the
rectangular box, the modified Bessel function of the second kind is given by

$\displaystyle K_{\nu}(z)$
$\displaystyle=\frac{(z/2)^{\nu}\Gamma(1/2)}{\Gamma(\nu+1/2)}\int_{1}^{\infty}\\!dt\;e^{-zt}(t^{2}-1)^{(2\nu-1)/2}\,,$
(81) and $\displaystyle G(z)$
$\displaystyle=-\frac{1}{2\pi}\sum_{n=1}^{\infty}\sum_{l=1}^{\infty}\frac{n}{l}\,K_{1}(2\pi
nlz)\,,$ (82) $\displaystyle R(z_{1},z_{2})$
$\displaystyle=\frac{z_{1}z_{2}}{8}\sum_{l,p=-\infty}^{\infty}(1-\delta_{l0}\delta_{p0})\sum_{j=1}^{\infty}\quantity(\frac{j}{\sqrt{l^{2}z_{1}^{2}+p^{2}z_{2}^{2}}})^{3/2}K_{3/2}\quantity(2\pi
j\sqrt{l^{2}z_{1}^{2}+p^{2}z_{2}^{2}})\,.$ (83)

For a cubic box
$\rho_{\textsc{ca}}(\ell_{1},\ell_{1},\ell_{1})=-0.838/\ell_{1}^{4}$, and for
the geometry where the length of one side goes to infinity,
$\rho_{\textsc{ca}}(\ell_{1},\ell_{1},\infty)=-0.301/\ell_{1}^{4}$. In this
case, since $\rho_{\textsc{ca}}(\ell_{1},\ell_{1},\ell_{3})$ for general
$\ell_{3}$ is believed to be of the same order of magnitude as these two
values, we expect that $\rho_{\textsc{ca}}(\ell_{1},\ell_{1},\ell_{3})$ is a
slowly varying function of $\ell_{3}$ such that

$\displaystyle\rho_{\textsc{ca}}(\ell_{1},\ell_{1},\ell_{3})$
$\displaystyle=-\frac{1}{\ell_{1}^{4}}\,f(\frac{\ell_{1}}{\ell_{3}})\,,$ with
$\displaystyle\lim_{z\to 1}f(z)$ $\displaystyle=0.838\,,$
$\displaystyle\lim_{z\to 0}f(z)$ $\displaystyle=0.301\,.$ (84)

Thus, the dimensionless, coefficient function $f(z)$ is typically small for
$0\leq z\leq 1$. Then the Casimir effect contribution to the vacuum energy
$E_{\textsc{ca}}$ is given by

$E_{\textsc{ca}}=-\frac{\ell_{3}}{\ell_{1}^{2}}\,f(\frac{\ell_{1}}{\ell_{3}})\,,$
(85)

in turn, this yields the corresponding contribution to the pressure

$P_{\textsc{ca}}=-\frac{1}{\ell_{1}^{2}}\frac{\partial
E_{\textsc{ca}}}{\partial\ell_{3}}=\frac{1}{\ell_{1}^{4}}\Bigl{[}f(\frac{\ell_{1}}{\ell_{3}})-\frac{\ell_{1}}{\ell_{3}}\,f^{\prime}(\frac{\ell_{1}}{\ell_{3}})\Bigr{]}\simeq\frac{1}{\ell_{1}^{4}}\,f(\frac{\ell_{1}}{\ell_{3}})\,,$
(86)

where the prime here represents differentiation with respect to the argument
of the function. The slowly varying nature of $f(z)$ and the fact that
$\ell_{1}/\ell_{3}<1$ imply that the second term in (86) can be ignored. Next,
since the numerical values of $f(z)$ are already small, if we are interested
in physically reasonable cases $\ell_{3}\geq\ell_{1}$ such that $\ell_{1}$ is
not too small, then we can conclude the contribution of the Casimir energy to
the pressure is typically negligibly small, compared with the pressure caused
by particle creation, unless the latter is accidentally zero or falls under
the adiabatic limit. Therefore the Casimir energy can be ignored in the
calculation of the pressure. This will be further justified by Fig. 14 where
the vacuum energy due to particle creation is numerically much larger than
that from the Casimir effect.

For the term in the second line of (V)

$\displaystyle-\frac{a^{-\frac{4}{3}}}{2l^{3}}\sum_{\mathbf{k}}\frac{Q}{2\Omega_{\mathbf{k}}}=-\frac{a^{-\frac{4}{3}}}{4l^{3}}\sum_{n_{x}=-\infty}^{\infty}\sum_{n_{y}=-\infty}^{\infty}\sum_{n_{z}=-\infty}^{\infty}\frac{Q}{\sqrt{a^{\frac{2}{3}}\quantity[\frac{(2\pi
n_{x})^{2}}{l^{2}}+\frac{(2\pi n_{y})^{2}}{l^{2}}+\frac{(2\pi
n_{z})^{2}}{l^{2}}]}}\,,$ (87)

the zero mode in Eq. (87) causes infrared divergence, but we can exclude it
since it represents the mode with infinite wavelength. The dimensions of the
torus serve as an infrared cutoff of the spectrum, so the above expression can
be decomposed into the following three components: (a))

$\displaystyle-\frac{a^{-\frac{4}{3}}}{2l^{3}}\sum_{n_{x}=1}^{\infty}\sum_{n_{y}=1}^{\infty}\sum_{n_{z}=1}^{\infty}\frac{Q}{\sqrt{a^{\frac{2}{3}}\quantity[\frac{(2\pi
n_{x})^{2}}{l^{2}}+\frac{(2\pi n_{y})^{2}}{l^{2}}+\frac{(2\pi
n_{z})^{2}}{l^{2}}]}}\,,$ (88)

and b)

$\displaystyle-\frac{a^{-\frac{4}{3}}}{2l^{3}}\sum_{n_{y}=1}^{\infty}\sum_{n_{z}=1}^{\infty}\frac{Q}{\sqrt{a^{\frac{2}{3}}\quantity[\frac{(2\pi
n_{y})^{2}}{l^{2}}+\frac{(2\pi
n_{z})^{2}}{l^{2}}]}}\;\Bigg{|}_{n_{x}=0}+\cdots\,,$ (89)

where $\cdots$ are the two other terms with $n_{x}=0$ replaced by $n_{y}=0$,
$n_{z}=0$, and (c)

$\displaystyle-\frac{a^{-\frac{4}{3}}}{2l^{3}}\Biggl{\\{}\sum_{n_{z}=1}^{\infty}\frac{Q}{a^{\frac{1}{3}}\,\frac{2\pi
n_{z}}{l}}\;\Bigg{|}_{n_{x}=n_{y}=0}+\sum_{n_{y}=1}^{\infty}\frac{Q}{a^{\frac{1}{3}}\,\frac{2\pi
n_{y}}{l}}\;\Bigg{|}_{n_{x}=n_{z}=0}+\sum_{n_{x}=1}^{\infty}\frac{Q}{a^{\frac{1}{3}}\,\frac{2\pi
n_{x}}{l}}\;\Bigg{|}_{n_{y}=n_{z}=0}\Biggr{\\}}\,.$ (90)

Eq. (88) can be renormalized with the help of the Abel-Plana formula Bordag
_et al._ (2009)

$\sum_{n=0}^{\infty}f(n)=\frac{1}{2}f(0)+\int_{0}^{\infty}f(x)\,dx+i\int_{0}^{\infty}\frac{f(it)-f(-it)}{e^{2\pi
t}-1}\,,$ (91)

iteratively applied to the triple sums in (88). Thus, Eq. (89), (90) will be
numerically renormalized by subtract the corresponding integrals

$\displaystyle-\frac{a^{-\frac{4}{3}}}{2}\int_{0}^{1}\\!\int_{1}^{\infty}\\!\int_{1}^{\infty}\\!\frac{d^{3}k}{(2\pi)^{3}}\;\frac{Q}{\sqrt{a^{\frac{2}{3}}\quantity[\frac{k_{x}^{2}}{a^{2}}+k_{y}^{2}+k_{z}^{2}]}}+\text{permutations
of $ijk$}\,,$ (92)

and

$\displaystyle-\frac{a^{-\frac{4}{3}}}{2}\int_{0}^{1}\\!\int_{0}^{1}\\!\int_{1}^{\infty}\\!\frac{d^{3}k}{(2\pi)^{3}}\;\frac{Q}{\sqrt{a^{\frac{2}{3}}\quantity[\frac{k_{x}^{2}}{a^{2}}+k_{y}^{2}+k_{z}^{2}]}}+\text{permutations
of $ijk$}\,.$ (93)

After renormalization, Eq. (88), (89) and (90) in a physically reasonable case
(length scale much higher than the Planck scale) gives very small values,
compared to the other contributions in the vacuum energy density, so we
neglect these terms.

Now, the remaining part of energy density can be written as

$\displaystyle\expectationvalue{\hat{T}_{\mu\nu\;\text{reg}}}{0_{A}}U^{\mu}U^{\nu}={\frac{a^{-\frac{4}{3}}}{l^{3}}}\sum_{\mathbf{k}}\bqty{s_{\mathbf{k}}\Omega_{\mathbf{k}}-\frac{Q}{4\Omega_{\mathbf{k}}}(2s_{\mathbf{k}}+p_{\mathbf{k}})}$
(94)
$\displaystyle\qquad-a^{-\frac{4}{3}}\int\\!\frac{d^{3}k}{(2\pi)^{3}}\;\quantity[\frac{\Omega^{\prime
2}_{\mathbf{k}}}{16\Omega_{\mathbf{k}}^{3}}-\frac{Q}{4\Omega^{\vphantom{2}}_{\mathbf{k}}}\quantity(\frac{\Omega^{\prime
2}_{\mathbf{k}}}{8\Omega_{\mathbf{k}}^{4}}+\frac{\Omega^{\prime\prime}_{\mathbf{k}}}{4\Omega^{3}_{\mathbf{k}}}-\frac{\Omega^{\prime}_{\mathbf{k}}}{2\Omega_{\mathbf{k}}^{4}}-\frac{Q}{2\Omega_{\mathbf{k}}^{2}})+\Omega_{\mathbf{k}}^{\vphantom{2}}s_{\mathbf{k}}^{(4)}]\,,$

where 555The notation $s_{\mathbf{k}}^{(4)}$ gives the fourth-adiabatic order
contribution of $s_{\mathbf{k}}$, but it is not relevant to the current
discussion.

$\displaystyle s_{\mathbf{k}}^{(4)}$
$\displaystyle=\frac{1}{16}\Biggl{\\{}\Biggr{.}\frac{Q^{2}}{\Omega_{\mathbf{k}}^{4}}+\frac{Q^{\prime}\Omega^{\prime}_{\mathbf{k}}}{\Omega^{5}_{\mathbf{k}}}-\frac{Q}{\Omega^{3}_{\mathbf{k}}}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}})^{\prime}-\frac{3Q}{\Omega^{2}_{\mathbf{k}}}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}})^{2}-\frac{1}{2\Omega^{2}_{\mathbf{k}}}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}})^{\prime\prime}\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{4\Omega^{2}_{\mathbf{k}}}\quantity[\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}})^{\prime}]^{2}+\frac{1}{2\Omega^{\vphantom{2}}_{\mathbf{k}}}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}})^{\prime}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}})^{2}+\frac{3}{16}\quantity(\frac{\Omega^{\prime}_{\mathbf{k}}}{\Omega^{2}_{\mathbf{k}}})^{4}\Biggl{.}\Biggr{\\}}\,.$
(95)

The integral in the second line of Eq. (94) associate with trace anomaly is
divergent, and it must cancel the UV divergence in $s_{\mathbf{k}}$ and
$p_{\mathbf{k}}$ of the first line. Although it is hard to further decompose
$s_{\mathbf{k}}$ and $p_{\mathbf{k}}$ to find the exact divergent
contribution, we could make the following observations: At large momentum, the
integrand in the second line of Eq. (94) will be suppressed by
$\Omega_{\mathbf{k}}$ in the denominators, and it goes like $1/k$ as
$k\to\infty$, then it will cancel the divergent part in $s_{\mathbf{k}}$ and
$p_{\mathbf{k}}$ for large momentum, and give a finite part. Therefore, if we
can estimate the value of the integral in the small momentum range, and show
it is small compared to the summation in the small momentum range in the first
line of Eq. (94), then we can neglect the whole thing in the second line of
Eq. (94). For the scale factor $a(t)$ of our interest, the integral indeed is
a few orders of magnitude smaller than the rest of energy density expressions
for $k_{i}<20$. Therefore we can just use the first line of Eq. (94) and sum
up to $k_{i}<20$ to give an pretty accurate approximation of energy. The
corresponding plot is shown in Fig. 14.

Figure 14: The volume, pressure, energy, and vacuum compressibility of a
massless conformal field in a rectangular $T^{3}$ box when one side of the box
is allowed to expand and contract in cosmic time. The scalar factor is
described by the Gaussian function, Eq. (20) with $a_{0}=1$, $\Delta=2$,
$t_{0}=1$ and $1/\tau^{2}=5$.

In the current case, the major component of the vacuum energy come from the
contribution caused by spontaneous particle production induced by a mirror
which moves in a specific direction and has a trajectory described by a
Gaussian, or equivalently induced by anisotropic variation of the scale factor
in $\mathbf{R}\times T^{3}$ spacetime. The result in Fig. 14 shows the energy
does not monotonically grows with time even when the volume is expanding. A
local maximum does occurs at the moment when the expansion rate is largest.

The accompanying time variation of the pressure in Fig. 14 contains more
structure due to the expansion and contraction of the volume. The results show
that the pressure can takes on either positive or negative values. The
negative value occurs when the energy changes in the same trend as the volume
does, but if the energy changes in a way opposite to how the volume changes,
then the pressure can be positive. These can be seen by comparing the plots of
energy and pressure, as the consequence of the definition of the pressure;
however, it is not clear why a small hump shows up at early time in the figure
for the time evolution of the pressure. The zero pressure occurs when the
energy take on a local maximum. From the previous discussions, we expect the
compressibility can exhibit unusual behavior in the regions where $\Delta
V\neq 0$ but $\Delta P\approx 0$. This is rather baffling indeed and needs
more investigations.

##  VI Summary

We summarize our findings by first describing the quantum field theoretical
factors and then the quantum thermodynamic features.

###  VI.1 Quantum Processes

The quantum vacuum processes of interest to us include the familiar Casimir
effect, the dynamical Casimir effect which includes particle creation, and the
less familiar trace anomaly. The first depends on the way boundary conditions
are imposed on the quantum field and the topology of space it lives in. The
second entails particle creation due to the parametric amplification of vacuum
fluctuations of the quantum field. The third is specific to, and a
manifestation of, the quantum nature of a conformal field – ‘anomaly’ refers
to the fact that the trace of the stress energy tensor of a classical
conformal field is zero, but not so for a quantum conformal field. These three
types of quantum field processes/effects have been studied by researchers in
quantum field theory in curved spacetime since the 1970s. We have adopted the
methodologies developed there, such as using the adiabatic regularization for
the treatment of UV divergences. Here we have more extensive mathematical
usages and accompanying physical discussions. For particle creation in
$\mathbf{R}\times S^{2}$, $\mathbf{R}\times S^{3}$, and $\mathbf{R}\times
T^{3}$ treated here we solve the differential equations governing the
Bogoliubov coefficients without making any approximation. One new improvement
over earlier treatments is to obtain the Bogoliubov coefficients (two complex
or four real quantities minus one Wronskian condition) directly by solving the
three first-order ODE numerically with a specified time-dependence of the
scale factor.

Going into the specifics: We have treated the effect of trace anomaly in
$\mathbf{R}\times S^{1}$, $\mathbf{R}\times S^{3}$ spacetimes and particle
creation in $\mathbf{R}\times S^{2}$, $\mathbf{R}\times S^{3}$ and
$\mathbf{R}\times T^{3}$ spacetimes. In the first case the time dependence of
the scale factor is determined by self-consistently solving for the dynamics
of the quantum field and the spacetime. This was done in a recent paper Xie
_et al._ (2023). Results from there are imported into the present
calculations. We use this as an example of vacuum processes where the
backreaction effects are included in the consideration. In the other cases we
have considered two types of evolution of the scale factor. The marked
differences between the effects of the trace anomaly and particle creation
have been known much earlier (compare e.g., Fischetti  _et al._ (1979); Hartle
and Hu (1979)), and amply illustrated in Xie  _et al._ (2023): the
backreaction on $S^{1}$ due to the trace anomaly accelerates the collapse of
the ring while the backreaction of particle creation in $T^{3}$ retards the
expansion of the box. For the other cases studied without backreaction, one
distinction between the effects of particle creation and the trace anomaly is
that the energy due to (bosonic) particle creation overall monotonically
increases, while that due to trace anomaly can increase at one time and
decrease at another time. This is because the expression for the trace anomaly
contains terms of higher derivative orders with positive and negative signs.
As remarked in the Introduction we need to include the energy and the pressure
for all possible vacuum processes allowed before calculating the vacuum
dynamical compressibility of each spacetime. Detailed descriptions of how the
vacuum energy, the vacuum pressure and the vacuum dynamical compressibility
varies with time can be found in the plots given for each of the spacetimes
considered. Their overall features have a branding impact on the quantum
thermodynamic functions of spacetimes.

###  VI.2 Quantum Thermodynamics

The ending target from these QFTCST calculations is the energy density of the
quantum field in various quantum vacuum processes. It is also the starting
point for the derivation of the quantum thermodynamic functions of the system.
This can be seen from the four frames in the plots based on the results of
calculations in all the cases considered: the volume, energy, pressure and
compressibility. The first two from QFT calculations provide the raw materials
for ensuing thermodynamic considerations. If the challenge in the QFT part is
mainly technical, the QTD part is mainly conceptual. We refer the reader to
the descriptions of the results following each figure for details, but
highlight two overall qualitative features below which are radically different
from our traditional understanding of energy, pressure and compressibility
based on classical thermodynamics:

1) Energy could be negative: the energy due to the trace anomaly in $S^{1}$
and due to particle creation in $T^{3}$ are negative, whereas the energy due
to particle creation in $S^{2}$ and $S^{3}$ spaces are positive, as is in
$S^{3}$ due to the trace anomaly. Negative energy is a special hallmark of
vacuum quantum field processes, probably known first in Casimir effect: the
energy density in the space between two parallel plates is negative, resulting
in an attractive Casimir force whereas for a sphere it is positive, giving
rise to an expansion of the sphere.

2) Pressure could be negative. Recall the definition: $P=-\frac{\partial
E}{\partial V}$. If energy is negative as, e.g., in the $S^{1}$ case (see Fig.
2 & 3), then a) an increase in the volume with an increase in the (negative)
energy will give a negative pressure, while b) a decrease in the volume with
an increase of (negative) energy will give a positive pressure. Note that the
pressure calculated from the vacuum energy of a particular source, be it
Casimir, trace anomaly or particle creation, is only a partial pressure. That
is the reason for our earlier caution, that all partial pressures need be
added up to a total vacuum pressure before one proceeds to calculate the
compressibility.

3) Compressibility could be negative, even approach infinity at certain times.
Recall the definition $\kappa=-\frac{1}{V}\frac{\partial V}{\partial P}$.
First off, note as stated before, that the pressure in ‘taking the derivative
of volume with respect to the pressure’ refers to the total pressure of all
the partial pressures from different vacuum sources, like the three named
above, whichever is present in the different cases studied. i) In the $T^{3}$
example mentioned above, because of the cross-dependence both cases a) and b)
end up giving the same compressibility functional dependence on time, becoming
increasingly negative. ii) A more ‘alarming’ fact is that the compressibility
can become infinite (see Fig. 6 & 7) at certain points in time when the vacuum
pressure is a maximum or minimum while the volume is changing.

What we have encountered here is something very odd from the traditional
perspective of classical mechanics and thermodynamics. This is because the
sources of vacuum energy in the cases we have studied are all of quantum field
origins. Our calculations are meant to bring forth something novel in the
quantum thermodynamics of spacetimes, fundamentally different from spacetime
dynamics described by classical thermodynamics. It behooves us to try to
understand these processes from a different perspective, to expound / extend
the meanings of these physical quantities or, even go so far as coming up with
more cogent definitions for these quantum thermodynamic quantities to
encompass these new phenomena.

As extensions of the present work we shall study finite temperature quantum
fields in our next paper Xie  _et al._ . We will be able to include the
effects of stimulated particle creation and the considerations of heat
capacity of dynamical spacetimes due to various quantum field processes.

Acknowledgment YCX thanks Bo-Hao Chng and Yushan Su for help with the Riemann
sum in $T^{3}$, JTH and BLH thank Prof. H.-T. Cho for helpful discussions.
J.-T. Hsiang is supported by the National Science and Technology Council of
Taiwan, R.O.C. under Grant No. NSTC 112-2112-M-011-001-MY3. B.-L. Hu enjoyed
the warm hospitality of Prof. Hsiang-nan Li and colleagues at the Institute of
Physics, Academia Sinica, Taiwan, R.O.C. when this work began.

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