# Heat Capacity and Quantum Compressibility of
Dynamical Spacetimes with Thermal Particle Creation

Jen-Tsung Hsiang<EMAIL_ADDRESS>College of Electrical Engineering and
Computer Science, National Taiwan University of Science and Technology, Taipei
City, Taiwan 106, ROC  Yu-Cun Xie<EMAIL_ADDRESS>Department of Physics,
University of Maryland, College Park, Maryland 20742, USA  Bei-Lok Hu
<EMAIL_ADDRESS>Maryland Center for Fundamental Physics and Joint Quantum
Institute, University of Maryland, College Park, Maryland 20742, USA

(v1 Sept 3, 2023. v2 Jan 24, v3 March 29, v4 April 24, 2024)

###### Abstract

This work continues the investigation in two recent papers on the quantum
thermodynamics of spacetimes, 1) placing what was studied in [1] for thermal
quantum fields in the context of early universe cosmology, and 2) extending
the considerations of vacuum compressibility of dynamical spaces treated in
[2] to dynamical spacetimes with thermal quantum fields. We begin with a
warning that thermal equilibrium condition is not guaranteed to exist or
maintained in a dynamical setting and thus finite temperature quantum field
theory in cosmological spacetimes needs more careful considerations than what
is often described in textbooks. A full description requires nonequilibrium
quantum field theory in dynamical spacetimes using ‘in-in’ techniques. A more
manageable subclass of dynamics is where thermal equilibrium conditions are
established at both the beginning and the end of evolution, where the in-
thermal state and the out-thermal state are both well defined. Particle
creation in the full history can then be calculated in this in-out
asymptotically-stationary setup via the $S$-matrix transition amplitudes. Here
we shall assume an in-vacuum state. It has been shown that if the intervening
dynamics has an initial period of exponential expansion, such as in
inflationary cosmology, particles created from the parametric amplification of
the vacuum fluctuations in the initial vacuum will have a thermal spectrum
measured at the out-state. Under these conditions finite temperature field
theory can be applied to calculate the quantum thermodynamic quantities. Here
we consider a massive conformal scalar field in a closed four-dimensional
Friedmann-Lemaître-Robertson-Walker universe based on the simple analytically
solvable Bernard-Duncan model. We calculate the energy density of particles
created from an in-vacuum and derive the partition function. From the free
energy we then derive the heat capacity and the quantum compressibility of
these spacetimes with thermal particle creation. We end with some discussions
and suggestions for further work in this program of studies.

###### Contents

  1. I Introduction
    1. I.1 Dynamics and Thermal Field Theory
      1. I.1.1 Finite temperature quantum field theory in cosmology
      2. I.1.2 Casimir, trace anomaly and particle creation: from vacuum to radiation dominated
    2. I.2 Levels of approximation in treating a basically nonequilibrium problem
  2. II Thermal particle creation in inflationary universe
    1. II.1 General properties of cosmological scale function for thermal production
    2. II.2 Heat capacity and quantum compressibility
  3. III Quantum thermodynamics of an inflationary universe
    1. III.1 Asymptotically-stationary universe with a period of exponential expansion
    2. III.2 Thermal particle production originating from the in-vacuum
    3. III.3 Energy, pressure, entropy, heat capacity, quantum compressibility
  4. IV Brief summary and further developments

##  I Introduction

This is a sequel to two recent papers on the quantum thermodynamics of
spacetimes, 1) placing the work of [1] for the heat capacity and quantum
compressibility of static spaces containing quantum fields at finite
temperatures in the context of early universe cosmology, and 2) extending the
dynamic compressibility of spaces containing quantum fields in a vacuum state
considered in [2] to thermal quantum fields. We have considered in [2] three
types of early universe vacuum quantum processes: the Casimir effect, the
trace anomaly [3, 4, 5] and vacuum particle creation [6, 7, 8, 9, 10, 11] in a
number of dynamical spaces. Here, we shall focus only on thermal particle
creation in a Friedmann-Lemaître-Robertson-Walker (FLRW) universe, with
special attention paid to inflationary dynamics with a period of exponential
expansion.

Unlike the settings in the previous two papers, either thermal quantum fields
in a static spacetime or vacuum quantum field processes in a dynamical
spacetime, finite temperature is predicated upon the existence of a thermal
equilibrium condition, which cannot be assumed a priori for any dynamical
spacetime. Even when such a condition is fulfilled at one time, particle
creation at that moment of time will steer the system away from equilibrium in
a later moment. Thus before one discusses the thermodynamic properties of the
universe one needs to first address this basic conceptual issue of when in a
dynamical spacetime could thermal equilibrium be conditionally established,
and under what conditions can a thermal equilibrium state be maintained
throughout its evolution.

###  I.1 Dynamics and Thermal Field Theory

Thermal field theory deals with systems at finite temperatures. This is
predicated upon an underlying assumption that the system can be kept in
thermal equilibrium with a finite temperature bath. For time-dependent systems
involved in cosmological particle creation (CPC) and dynamical Casimir effect
(DCE), one key issue which need be addressed ab initio is: how can one ensure
cosmological particle creation or enforce dynamical Casimir effect such that
the equilibrium condition is maintained in the temporal evolution of the
system. This is possible if there is no energy input into, or entropy
generated from, the evolving system in the process. Otherwise thermal field
theory is inadequate and one needs to use concepts in nonequilibrium (NEq)
statistical mechanics and techniques in nonequilibrium quantum field theory.
In the cosmology setting, this issue has been addressed in a series of papers
[12, 13, 14, 15] by one of the present authors. We summarize the key points
from that body of work relevant to our investigation here:

####  I.1.1 Finite temperature quantum field theory in cosmology

In an evolving universe, allowing for particle creation from the vacuum, the
number of particles, the energy and entropy of quantum matter may all change.
If the expansion (or contraction) is isotropic such as in a FLRW universe, and
if the particles described by a free field obey conformally-invariant field
equations (e.g., photons), it can be shown that no entropy is generated and
the gas remains in thermal equilibrium throughout the evolution 111This is the
situation where there is no particle creation. Mathematically, the conformal
flatness of the FLRW metric and the conformal invariance of the matter field
allow one to perform a conformal transformation on the field and on the
spacetime, giving rise to a field equation in a flat space where there is no
particle creation..

In the radiation-dominated era one often encounters the statement that the
(Tolman) temperature $T\propto 1/a$, where $a(t)$ is the scale factor and $t$
the cosmic time. This assumes an isentropic expansion of the universe, where
the total number of photons (assuming that is the dominant particle species
present) measured by $\Theta\equiv T(t)a(t)$ remains a constant.

For interacting particles, thermal equilibrium can be maintained at a
temperature $T(t)$ provided that the characteristic rate of thermal
interaction $R$ is much greater than the expansion rate of the universe
$H=(da/dt)/a$, i.e., $R\gg H$. We shall introduce the criterion of
nonadiabaticity and the notion of quasi-equilibrium in this context in the
next subsection.

####  I.1.2 Casimir, trace anomaly and particle creation: from vacuum to
radiation dominated

As a simple illustration of the inter-knitted relation between vacuum effects
due to the Casimir energy, the trace anomaly and particle creation, such as
studied in [16, 2], and thermal effects, we mention the following relation
[14]. Consider a massless conformal scalar field at temperature $T$ in a
closed FLRW universe, the total energy density $\rho$ has vacuum contributions
$\rho_{0}$ comprising the Casimir (CA) energy density and the trace anomaly
(TA) plus contributions from thermal fields $\rho_{\textsc{t}}$

$\rho=\rho_{0}+\rho_{\textsc{t}}=(\rho_{\textsc{ca}}+\rho_{\textsc{ta}})+(\rho_{\textsc{r}}+\rho_{\textsc{c}})\,.$
(1)

where $\rho_{\textsc{r}}=\pi^{2}T^{4}/30$ is the Planck thermal energy
density, and $\rho_{\textsc{c}}$ is a correction term which takes on the forms
given below at high and low temperatures. The controlling parameter is the
photon number $\Theta=Ta$, which is a constant for isentropic expansion.

i) In the high-temperature $T$ (or small $a$) limit, large $\Theta$:
$\rho_{\textsc{c}}\to-\rho_{\textsc{ca}}$ (minus the Casimir energy), and thus
$\rho\to\rho_{\textsc{ta}}+\rho_{\textsc{r}}$.

ii) In the low-temperature $T$ (or large $a$) limit, small $\Theta$:
$\rho_{\textsc{c}}\to-\rho_{\textsc{r}}$ (minus the Planck energy) and thus
$\rho\to\rho_{\textsc{ta}}+\rho_{\textsc{ca}}$.

We see the former case corresponds to a universe containing hot radiating gas
while the latter corresponds to a cold quantum universe. Any finite value of
$\Theta$ corresponds to a universe with some radiation content. The finite-
temperature theory of quantum fields thus provides a natural framework for the
discussion of the interplay of quantum versus classical processes with vacuum
or matter contents, as well as the occurrence and transition from the vacuum-
dominated to the radiation-dominated regimes of the early universe.

###  I.2 Levels of approximation in treating a basically nonequilibrium
problem

In evolutionary cosmology when the temperature of the universe at a certain
time in history is mentioned one has already made the assumption that matter
at that time has come to thermal equilibrium, which needs justification.
Cosmological particle creation is fundamentally a nonequilibrium quantum field
process.

For non-conformally invariant quantum fields, be it with mass, or massless but
minimally-coupled (e.g., the two polarizations of gravitons can thus be
described), or for spacetimes which are not conformally-static (e.g., the
Kasner or mixmaster universes), particle production can happen which generates
entropy and disrupts the thermal equilibrium condition. A more accurate and
thorough treatment would be by way of nonequilibrium quantum field theory
[17], via the ‘in-in’ formalism as an initial value problem, as mentioned
previously.

Having enunciated the conceptual and technical challenges for treating this
intrinsically nonequilibrium problem, we now ask, are there situations which
allows for a $S$-matrix or ‘in-out’ treatment via finite temperature quantum
field theory? A subclass of dynamics, albeit somewhat artificial, is known as
the asymptotically-stationary or statically-bounded setup. It has the distinct
advantage that the quantum field theory in both asymptotic regions, thanks to
their endowed global timelike Killing vector, are well defined. Here, one
assumes that the in-state and the out-state of the system are in thermal
equilibrium at temperatures $T_{i}$, $T_{f}$. In the intervening time there is
particle production, which can be calculated from the transition amplitudes of
a $S$-matrix. For thermal scalar fields, the in-state is an $n$-particle state
obeying the Bose-Einstein distribution. Particle creation from the vacuum
state is known as spontaneous production while from an $n$-particle state is
known as stimulated production. (See, e.g., [18]). The assumption that the
out-state is also a thermal state implies that the particles created either
spontaneously or by stimulation have had enough time and channels to come to
equilibrium. The temperature of the out-state should be calculable as it is
related to the number of particles created, assuming that is the only entropy
generating process. A simple-minded way to see this is by way of the Stefan-
Boltzmann law, where the energy density is proportional to $T^{4}$ for a
photon gas. In fact, we can track the increase in energy associated with the
number of particles created in real time, from the in- to the out-state, even
though temperature is not a well-defined notion in a time-dependent setting.

##  II Thermal particle creation in inflationary universe

In the asymptotically-stationary set up, there is a special class of
evolutionary cosmology where the particles created from the in-vacuum has a
thermal spectrum measured in the out state, namely, when there is a period of
exponential expansion. Since this includes the widely popular inflationary
universe we shall in this paper focus on this class of cosmology for studying
the thermodynamic properties of spacetimes due to quantum field processes.
Thermal particle creation was shown in the 70s in several statically-bounded
models which are completely solvable so one can closely examine all the
details (see, e.g., [19]): One model worked out by Bernard and Duncan [20] is
for a massive conformal scalar field in a two-dimensional spatially-flat FLRW
universe with a scale factor $a(\eta)$ (conformal time $dt=a\,d\eta$)
following a hyperbolic-tangent (conformal) time dependence,

$a^{2}(\eta)=A+B\tanh\rho\eta\,,$ (2)

which tends to constant values $a^{2}_{\pm}\equiv A\pm B$ at asymptotic times
$\eta\rightarrow\pm\infty$. We see that from an in-static spacetime with scale
factor $a_{-}$ there is an initial exponential rise measured by the rise
parameter $\rho$, followed by a nearly linear mid-segment and a smooth exit to
an out-static spacetime with scale factor $a_{+}$. These authors found that if
$a_{+}\gg a_{-}$ to a good approximation the ratio of the modulus of the
Bogoliubov coefficients

$|\beta_{k}/\alpha_{k}|^{2}=\exp(-2\pi\omega_{\textsc{in}}/\rho)\,.~{}~{}$ (3)

where $\omega_{\textsc{in}}$ is the mode frequency of the field of the in-
state at $\eta\to-\infty$. For high momentum modes, one can recognize the
Planckian distribution with temperature given by

$k_{B}T_{\eta}=\frac{\rho}{2\pi a_{+}}\,.$ (4)

The Bogoliubov coefficients $\alpha_{k}$, $\beta_{k}$ are defined as follows:
Call $\Phi_{k}^{\textsc{in},\,\textsc{out}}(t,x)$ the positive frequency mode
function of the $k^{\text{th}}$ mode of the quantum field in the in- and out-
regions at $t=-\infty$ and $t=+\infty$, respectively. They are related by the
Bogoliubov coefficients $\alpha_{k}$, $\beta_{k}$ defined by

$\Phi_{k}^{\textsc{in}}(t,x)=\alpha_{k}\Phi_{k}^{\textsc{out}}(t,x)+\beta_{k}\Phi_{-k}^{\textsc{out}*}(t,x)\,.$
(5)

The probability $P_{n}(\vec{k})$ of observing $n$ particles in mode $\vec{k}$
at late times is [21]:

$P_{n}(\vec{k})=|\beta_{k}/\alpha_{k}|^{2n}|\alpha_{k}|^{-2}.$ (6)

From this one can find the average number of particles $\langle
N_{\vec{k}}\rangle$ created in mode $\vec{k}$ (in a comoving volume) at late
times to be

$N_{\vec{k}}\rangle=\sum_{n=0}^{\infty}n\,P_{n}(\vec{k})=|\beta_{k}|^{2}\,.$
(7)

The modulus of their ratio is particularly useful for identifying the thermal
character of particle creation.

In another analytically solvable model, Parker [22] treated a massless
minimally coupled scalar field in a FLRW universe using a more complicated
functional form for the scale factor’s $\tau$ time defined by $d\tau=a^{3}dt$
dependence. He showed that significant particle creation occurs during the
early period when the universe expands exponentially fast. In fact the ratio
of the modulus of the Bogoliubov coefficients obtained in a simpler model
where the scale factor takes on an initial exponential rise has the same form
as in this more complicated model (see also Berger [23]). Both cases studied
give rise to a thermal spectrum with temperatures proportional to the rise
function, e.g., $\rho$ in conformal time $\eta$ in the Bernard-Duncan model,
and $\sigma$ in $\tau$ time in the Parker-Berger model.

###  II.1 General properties of cosmological scale function for thermal
production

In more general terms, Parker speculated that the exponential form in
$|\beta/\alpha|$ should hold for a general class of scale functions which
possess the properties that: 1) they smoothly approach a constant at early
time, 2) their values at late times are much larger than at initial times, and
3) they and their derivatives are continuous functions. The exponential factor
contained in the scale functions at early times is responsible for the thermal
property of particle creation, with the temperature proportional to the rise
factor in the exponential function. More noteworthy is that this property is
quite insensitive to the late time asymptotically static behavior of
$a(\tau)$, namely, it could be the asymptotically flat behavior of a $\tanh$
function, or the continuing rising behavior of an $\exp$ function.

###  II.2 Heat capacity and quantum compressibility

We have specified the setting and identified the focus of our problem, namely,
calculate the energy density of quantum field processes ranging from the
Casimir effect to the trace anomaly and particle creation. We have chosen to
work with the class of dynamics where there is a period of exponential
expansion which gives rise to a thermal spectrum of particle production. This
thermality condition facilitates a simpler calculation of the heat capacity
and the quantum compressibility of this class of cosmology. Focusing only on
particle production we can break this down into several steps:

1) For the in-out equilibrium state set up, assuming we begin with the vacuum,
i.e., temperature $T_{\textsc{in}}=0$, calculate the energy density of
particles created from this in-vacuum. For the class of dynamics with a period
of exponential expansion, call it inflationary cosmology, the particles
produced has a thermal spectrum, the temperature of which is determined by the
expansion rate. Even for a dynamics which does not become asymptotically-
stationary at late times, per what was said above, an initial exponential
expansion period would produce particles with a thermal spectrum the
temperature of which we can use for our quantum thermodynamics investigations.

2 ) Heat capacity is defined under two separate conditions, constant volume
and constant pressure. For static spacetimes these are well defined, see,
e.g., [1]. For spaces whose volumes change with time, we have to make sure
that all cases in the exponential expansion family (we may call this
inflationary universe) with different rise parameters expand to the same
volume in the out-state. Integrating the energy density over the same volume
gives the energies in different inflationary universes with varying expansion
rates. The faster the expansion the larger amount of particles produced, the
bigger the energy and the higher the temperature. The heat capacity of an
inflationary universe is obtained by calculating the change of the energy from
one member of the family to another with respect to the temperature change
across those two relevant members of the family.

3) To calculate the thermal compressibility, one needs the Helmholtz free
energy. From the particle number one can calculate the entropy222In a textbook
description it is often stated that the entropy of a photon gas is given by
the number of photons. Note this assumes many layers of coarse graining from
its point of origin, namely, the vacuum. Entropy of particles created from the
vacuum is a conceptually subtle issue. Since the vacuum is a pure state
particle pairs created from it remains so – there should be no entropy
generation in vacuum particle production. Only after some coarse-graining
measure, such as paying attention only to the particle number but ignoring the
phase coherence relation or correlations between the particles [24, 25, 26] or
if only one particle in the pair is observable, would such an observer report
on entropy being generated [27, 28]. For a summary description of this
subject, see [29]. Combining the two one obtains the Helmholtz free energy and
from there one can calculate all the thermodynamic quantities.

In the next section we proceed to analyze thermal particle creation of a
massive conformal scalar field in a four-dimensional closed Friedmann-
Lemaître-Robertson-Walker universe based on the simple analytically solvable
Bernard-Duncan model which is statically-bounded with an initial period of
exponential expansion. We calculate the energy density of particles created
from the in-vacuum and derive the partition function. From the Helmholtz free
energy we then derive the heat capacity and the quantum compressibility of
these spacetimes with thermal particle creation. In the last section we
conclude with some discussions and suggestions for further work in this
program of studies.

##  III Quantum thermodynamics of an inflationary universe

Consider a massive ($m$) scalar field $\phi$ in an expanding four-dimensional
closed Friedmann-Lemaître-Robertson-Walker (FLRW) universe with line element

$ds^{2}=-dt^{2}+a^{2}(t)\,d\Omega^{2}_{S^{3}}=a^{2}(\eta)\,(-d\eta^{2}+d\Omega^{2}_{S^{3}})\,,$
(8)

where $d\Omega^{2}_{S^{3}}$ is the line element for a 3-sphere $S^{3}$, $t$ is
the cosmic time and $\eta\equiv\displaystyle\int\\!dt/a(t)$ is the conformal
time, This field obeys the wave equation:

$\displaystyle\frac{1}{\sqrt{-g}}\,\partial_{\mu}\biggl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}\phi\biggr{]}-m^{2}\phi-\xi
R\,\phi=0\,,$ (9)

where $\sqrt{-g}=a^{4}$, $g^{\mu\nu}=a^{-2}\mathsf{g}^{\mu\nu}$ and
$\mathsf{g}^{\mu\nu}$ is the metric corresponding to the line element
$-d\eta^{2}+d\Omega^{2}_{S^{3}}$. The Ricci scalar $R$ is given by

$R=\frac{6}{a^{2}}\biggl{(}1+\frac{a^{\prime\prime}}{a}\biggr{)}\,,$ (10)

where a prime denotes taking the derivative with respect to the conformal time
$\eta$. We shall consider conformally-coupled fields where $\xi=1/6$ in four
dimension.

Spelling out (9) explicitly gives

$\displaystyle-\frac{1}{a^{4}}\,\partial_{\eta}\Bigl{(}a^{2}\,\partial_{\eta}\phi\Bigr{)}+\frac{1}{a^{2}}\bm{\nabla}_{S^{3}}^{2}\phi-m^{2}\phi-\frac{1}{a^{2}}\Bigl{(}1+\frac{a^{\prime\prime}}{a}\Bigr{)}\,\phi$
$\displaystyle=0\,,$ (11)

where

$\bm{\nabla}_{S^{3}}^{2}=\partial_{i}\bigl{(}\mathsf{g}^{ij}\partial_{j}\bigr{)}\,.$
(12)

Now let $\phi=\chi/a$, and Eq. (9) becomes

$\displaystyle\chi^{\prime\prime}-\bm{\nabla}_{S^{3}}^{2}\chi+\bigl{(}m^{2}a^{2}+1\bigr{)}\chi$
$\displaystyle=0\,.$ (13)

Since the eigenvalues $\lambda_{l}$ corresponding to $\bm{\nabla}_{S^{D}}^{2}$
are $\lambda_{l}=-l(l+n-1)$ for $l=0$, 1, 2, $\cdots$. In the case $D=3$, Eq.
(13) reduces to

$\displaystyle\chi^{\prime\prime}(\eta)+\omega_{l}^{2}\,\chi(\eta)$
$\displaystyle=0\,,$ $\displaystyle\omega_{l}^{2}=(l+1)^{2}+m^{2}a^{2}\,.$
(14)

The corresponding physical frequency is
$\varpi_{l}=\dfrac{\omega_{l}}{a}=\sqrt{\dfrac{(l+1)^{2}}{a^{2}}+m^{2}}$.

###  III.1 Asymptotically-stationary universe with a period of exponential
expansion

As an example of an asymptotically-stationary universe, we let the scale
factor $a(\eta)$ evolve in conformal time as

$a^{2}(\eta)=\frac{a_{f}^{2}+a_{i}^{2}}{2}+\frac{a_{f}^{2}-a_{i}^{2}}{2}\,\tanh\frac{\eta}{\Delta}\,,$
(15)

where the scale factor transits from one constant value $a_{i}$ in the
asymptotic past to another constant $a_{f}$ in the asymptotic future. In the
transition time of duration $\Delta$ (inverse of the rise function), the
universe expands exponentially fast, enabling us to explore the thermodynamic
properties. Note, however, the exponential expansion considered here is in
conformal time, not in cosmic time. (For the standard inflationary cosmology
$a(t)=e^{Ht}=-1/(H\eta)=a(\eta)$ so the exponential form is not reserved.)

Since the equation of motion for $\chi$ is exactly the same as in the Bernard-
Duncan model [20] in a two-dimensional, spatially-flat FLRW spacetime. With
the continuous $k$ there replaced by the discrete values $k_{l}=l+1$ here,
their results apply here. Let

$\displaystyle\omega_{l}^{\textsc{in}}$
$\displaystyle=\sqrt{k_{l}^{2}+m^{2}a_{i}^{2}}\,,$
$\displaystyle\omega_{l}^{\textsc{out}}$
$\displaystyle=\sqrt{k_{l}^{2}+m^{2}a_{f}^{2}}\,,$
$\displaystyle\omega_{l}^{\pm}$
$\displaystyle=\frac{\omega_{l}^{\textsc{out}}\pm\omega_{l}^{\textsc{in}}}{2}\,,$
(16)

the Bogoliubov coefficients $\alpha_{k_{l}}$, $\beta_{k_{l}}$ in the
asymptotic future are given by

$\displaystyle\lvert\alpha_{k_{l}}\rvert^{2}$
$\displaystyle=\frac{\sinh^{2}(\pi\omega_{l}^{+}\Delta)}{\sinh(\pi\omega_{l}^{\textsc{in}}\Delta)\,\sinh(\pi\omega_{l}^{\textsc{out}}\Delta)}\,,$
$\displaystyle\lvert\beta_{k_{l}}\rvert^{2}$
$\displaystyle=\frac{\sinh^{2}(\pi\omega_{l}^{-}\Delta)}{\sinh(\pi\omega_{l}^{\textsc{in}}\Delta)\,\sinh(\pi\omega_{l}^{\textsc{out}}\Delta)}\,,$
(17)

with the Wronskian condition,
$\lvert\alpha_{k_{l}}\rvert^{2}-\lvert\beta_{k_{l}}\rvert^{2}=1$. This follows
from the Bogoliubov transformation

$u_{k_{l}}^{\textsc{in}}(\eta,\bm{x})=\alpha_{k_{l}}\,u_{k_{l}}^{\textsc{out}}(\eta,\bm{x})+\beta_{k_{l}}\,u_{-k_{l}}^{\textsc{out}*}(\eta,\bm{x})\,,$
(18)

where the positive frequency mode $u_{k}(\eta,\bm{x})\propto
e^{-i\omega_{l}\eta}$ in both asymptotic regions for the suitable
$\omega_{l}$. The particle number generated during the transition is given by
$\lvert\beta_{k_{l}}\rvert^{2}$ for mode $k_{l}$.

The energy associated with particle creation is obtained by summing the
contributions from all modes

$E=\sum_{l=0}^{\infty}d_{l}^{(4)}\Bigl{(}\lvert\beta_{k_{l}}\rvert^{2}+\frac{1}{2}\Bigr{)}\,\varpi_{l}$
(19)

where the degeneracy $d_{l}^{(D+1)}$ in $R\times S^{D}$ is

$d_{l}^{(D+1)}=(2l+D-1)\,\frac{(l+D-2)!}{l!(D-1)!}=(l+1)^{2}\,,$ (20)

for $D=3$ and $l=0$, 1, 2, $\cdots$. This quantity associated with vacuum
particle creation is the source of our present investigation into the quantum
thermodynamics of dynamical spacetimes. Note, however, this expression does
not include other contributions to the vacuum energy such as the trace
anomaly. In fact, there is no contribution from the trace anomaly in the
asymptotically stationary out-state because it only depends on different
orders of time derivatives of the scale factor.

###  III.2 Thermal particle production originating from the in-vacuum

Let us try to understand the physics of particle creation associated with a
universe undergoing exponential expansion by examining
$\lvert\beta_{k_{l}}\rvert^{2}$ given by Eq. (17). In the case of a large
$\Delta$, corresponding to a very long duration of slow transition or gradual
rise:

$\displaystyle\pi\omega_{l}^{\textsc{out}}\Delta$ $\displaystyle\gg 1\,,$ (21)

we can approximate $\lvert\beta_{k_{l}}\rvert^{2}$ by

$\displaystyle
N_{l}=\lvert\beta_{k_{l}}\rvert^{2}\sim\frac{1}{2}\,\coth(\pi\omega_{l}^{\textsc{in}}\Delta)-\frac{1}{2}\,.$
(22)

We recognize this as the distribution of the mean particle number of bosons at
finite temperatures. Thus at late times, in a sufficiently high frequencies in
the out-state, we may identify an effective temperature from the statistical
distribution of the created particles in the out state

$\frac{\beta_{\textrm{eff}}\,\varpi_{l}}{2}=\pi\omega_{l}^{\textsc{in}}\Delta\,,$
(23)

given by

$T_{\textrm{eff}}=\frac{1}{2\pi
a_{f}\,\Delta}\frac{\omega_{l}^{\textsc{out}}}{\omega_{l}^{\textsc{in}}}\,.$
(24)

In particular, for lower $l$ modes, $l\ll m\,a_{i}$, the effective temperature
$T_{\textrm{eff}}\sim\dfrac{1}{2\pi a_{i}\,\Delta}$ becomes independent of
field parameters.

As a check, that there is no particle creation for a massless field, notice
Eq. (16) gives $\omega_{l}^{\textsc{in}}=\omega_{l}^{\textsc{out}}$ for all
values of $l$, so we have $\omega_{l}^{-}=0$ and
$\lvert\beta_{l}\rvert^{2}=0$. This is a well-known fact, that there is no
particle creation for a conformally-coupled massless field in a conformally-
static spacetime.

In the other extreme, for a very small $\Delta$, that is, a remarkably rapid
transition or a very steep rise, with

$\displaystyle\pi\omega_{l}^{\textsc{out}}\Delta$ $\displaystyle\ll 1\,,$ (25)

we find

$N_{l}=\lvert\beta_{k_{l}}\rvert^{2}\sim\frac{(\omega_{l}^{\textsc{out}}-\omega_{l}^{\textsc{in}})^{2}}{4\omega_{l}^{\textsc{out}}\omega_{l}^{\textsc{in}}}\,.$
(26)

This is a rather unusual situation because it yields a constant independent of
$\Delta$, hence independent of the effective temperature. However, with this
fairly strong constraint, it automatically implies
$\pi\omega_{l}^{\textsc{in}}\Delta\ll 1$. If we write Eq. (25) into

$\displaystyle\frac{1}{\pi\omega_{l}^{\textsc{out}}\Delta}=\biggl{(}\frac{1}{2\pi
a_{f}\,\Delta}\frac{\omega_{l}^{\textsc{out}}}{2\omega_{l}^{\textsc{in}}}\biggr{)}\frac{\omega_{l}^{\textsc{in}}a_{f}}{\omega_{l}^{\textsc{out}}{}^{2}}\gg
1\,,$ (27)

and note that the expression inside the parentheses may be identified as the
effective temperature $T_{\textrm{eff}}$ of the field at late time after the
transition stops, then Eq. (25) is equivalent to requiring

$T_{\textrm{eff}}\,\frac{\omega_{l}^{\textsc{in}}a_{f}}{\omega_{l}^{\textsc{out}}{}^{2}}\gg
1.$ (28)

The second factor is typically very tiny on considering the extreme expansion
of the spacetime. This then infers that the effective temperature of the
environment is exceptionally high, which may not be a sensible scenario for
the theory under consideration. Alternatively we can imagine that the
spacetime in this limit needs to expand with a very large scale change over an
exceeding short time in a way the scale factor behaves like a step function in
time. Thus, it is unlikely that the field can still be assumed to be globally
spatially homogeneous after transition, and expectantly large field
fluctuations may be induced. They can persist for a long time if there is no
interacting environment to relax them.

Figure 1: Dependence of $\lvert\beta_{k_{l}}\rvert^{2}$ on $\Delta$ is shown
in the blue curve. For comparison, the corresponding result according to the
Bose-Einstein distribution is shown in the orange dashed curve. They deviate
at very small $\Delta$, given by Eq. (26). Here we choose
$\omega_{l}^{\textsc{out}}=10$ and $\omega_{l}^{\textsc{in}}=1$.

Instead we may ease up the limit a bit by demanding
$\pi\omega_{l}^{\textsc{in}}\Delta\ll 1$ only, and then we obtain

$N_{l}=\lvert\beta_{k_{l}}\rvert^{2}\sim\tanh\frac{\pi\omega_{l}^{\textsc{out}}\Delta}{2}\,\frac{1}{2\pi\omega_{l}^{\textsc{in}}\Delta}\,.$
(29)

The second factor is nothing but

$\frac{1}{2\pi\omega_{l}^{\textsc{in}}\Delta}=\frac{\omega_{l}^{\textsc{out}}}{2\pi
a_{f}\Delta\omega_{l}^{\textsc{in}}}\frac{a_{f}}{\omega_{l}^{\textsc{out}}}=\frac{T_{\textrm{eff}}}{\varpi_{l}}\,,$
(30)

Here $\varpi_{l}=\omega_{l}^{\textsc{out}}/a_{f}$ is the physical frequency of
mode $l$ at late times after the transition ends. This is the expression we
typically see for the particle number in the high temperature limit. Thus, if
Eq. (29) is to describe the well known high temperature limit, we should
require

$\tanh\frac{\pi\omega_{l}^{\textsc{out}}\Delta}{2}\simeq 1\,.$ (31)

In other words, by the high temperature limit in the conventional sense, we
mean $\pi\omega_{l}^{\textsc{in}}\Delta\ll 1$ as well as
$\pi\omega_{l}^{\textsc{out}}\Delta\gg 1$. Thus, in summary, under the
assumption $\pi\omega_{l}^{\textsc{out}}\Delta\gg 1$, the various temperature
limits are determined by $\pi\omega_{l}^{\textsc{in}}\Delta$: The low
temperature regime corresponds to $\pi\omega_{l}^{\textsc{in}}\Delta\gg 1$
while the high temperature regime occurs when
$\pi\omega_{l}^{\textsc{in}}\Delta\ll 1$.

###  III.3 Energy, pressure, entropy, heat capacity, quantum compressibility

Having established the linkage in Eq. (23) with thermal field theory, we may
now explore the quantum thermodynamics of exponentially expanding spacetimes
with thermal particle creation. Since we are working with the out-state where
spacetime is stationary the partition function would be similar to that in
Minkowski space thermal field theory:

$\mathcal{Z}=\prod_{l=0}^{\infty}\biggl{[}\sum_{N_{l}=0}^{\infty}e^{-(N_{l}+1/2)\beta_{\textrm{eff}}\varpi_{l}}\biggr{]}^{d_{l}^{(4)}}=\prod_{l=0}^{\infty}\biggl{[}\frac{e^{\frac{\beta_{\textrm{eff}}\varpi_{l}}{2}}}{e^{\beta_{\textrm{eff}}\varpi_{l}}-1}\biggr{]}^{d_{l}^{(4)}}\,.$
(32)

The summation is over all modes even though the high frequency modes are less
easily excited while their contributions are suppressed by a Boltzmann-like
factor. The Helmholtz free energy $\mathcal{F}$ associated with thermal
particle production then follows

$\displaystyle\mathcal{F}$
$\displaystyle=-\frac{1}{\beta_{\textrm{eff}}}\,\ln\mathcal{Z}=\sum_{l=0}^{\infty}d_{l}^{(4)}\biggl{[}\frac{\varpi_{l}}{2}+\frac{1}{\beta_{\textrm{eff}}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\biggr{]}\,.$
(33)

The first term in the square brackets gives the vacuum contribution from the
zero-point energy of each mode at the out-state. The finite-temperature
contribution is from the second term. Notice that the free energy is always
negative, becoming more negative with increasing temperature, due to the
increasingly significant role of entropy.

(a) thermal energy $\mathcal{E}_{\beta}$ (b) von Neumann entropy $\mathcal{S}$
(c) heat capacity $\mathcal{C}_{\textsc{v}}$

Figure 2: Mode dependence of thermal energy, entropy and heat capacity at
constant volume. In each plot, we show the curves corresponding to three
choices of $\Delta$, whose inverse is proportional to the effective
temperature $T_{\textrm{eff}}$ of the field after the transition is completed.
These plots are consistent with the low-temperature approximations in Eqs.
(37)–(39). The values of the parameters used here are $m=1$, $a_{1}=1$,
$a_{f}=20$. For these choices, most of the modes fall within the low
temperature regime $\beta_{\textrm{eff}}\varpi_{l}\gg 1$.

Since at late times we essentially have an effective thermal field in a
constant radius $S^{3}$ with fixed volume $\mathcal{V}$, we can use this free
energy to find the internal energy $\mathcal{E}$, entropy $\mathcal{S}$, and
the other thermodynamic quantities

$\displaystyle\mathcal{E}$
$\displaystyle=-\frac{\partial}{\partial\beta_{\textrm{eff}}}\ln\mathcal{Z}=\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{\varpi_{l}}{2}\,\coth\frac{\beta_{\textrm{eff}}\varpi_{l}}{2}\,,$
(34) $\displaystyle\mathcal{S}$
$\displaystyle=\beta_{\textrm{eff}}^{2}\biggl{(}\frac{\partial\mathcal{F}\hphantom{|_{\textsc{ef}}}}{\partial\beta_{\textrm{eff}}}\biggr{)}_{\mathcal{V}}=\sum_{l=0}^{\infty}d_{l}^{(4)}\Bigl{[}\frac{\beta_{\textrm{eff}}\,\varpi_{l}}{e^{\beta_{\textrm{eff}}\varpi_{l}}-1}-\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\Bigr{]}\,,$
(35) $\displaystyle\mathcal{C}_{\textrm{V}}$
$\displaystyle=-\beta_{\textrm{eff}}^{2}\biggl{(}\frac{\partial\mathcal{E}}{\partial\beta_{\textrm{eff}}}\biggr{)}_{\mathcal{V}}=-\sum_{l=0}^{\infty}d_{l}^{(4)}\frac{\beta_{\textrm{eff}}^{2}\varpi_{l}^{2}}{4}\,\operatorname{csch}^{2}\frac{\beta_{\textrm{eff}}\varpi_{l}}{2}\,,$
(36)

where Eq. (23) can be used for the expression $\beta_{\textrm{eff}}\varpi_{l}$

The contributions of each mode in the thermodynamic quantities, Eqs. (34)-(36)
take on the familiar forms in the low- and the high-temperature limits, as
follows

  * •

low-temperature, $\beta_{\textrm{eff}}\varpi_{l}\gg 1$:

$\displaystyle\mathcal{E}_{l}$
$\displaystyle\simeq\frac{\varpi_{l}}{2}+\omega_{l}\,e^{-\beta_{\textrm{eff}}\varpi_{l}}+\cdots\,,$
(37) $\displaystyle\mathcal{S}_{l}$
$\displaystyle\simeq\bigl{(}1+\beta_{\textrm{eff}}\varpi_{l}+\cdots\bigr{)}\,e^{-\beta_{\textrm{eff}}\varpi_{l}}\,,$
(38) $\displaystyle\mathcal{C}_{\textrm{V}}$
$\displaystyle\simeq\bigl{(}\beta_{\textrm{eff}}^{2}\varpi_{l}^{2}+\cdots\bigr{)}\,e^{-\beta_{\textrm{eff}}\varpi_{l}}\,,$
(39)

  * •

high-temperature, $\beta_{\textrm{eff}}\varpi_{l}\ll 1$:

$\displaystyle\mathcal{E}_{l}$
$\displaystyle\simeq\frac{1}{\beta_{\textrm{eff}}}+\frac{\beta_{\textrm{eff}}\varpi_{l}^{2}}{12}+\cdots\,,$
(40) $\displaystyle\mathcal{S}_{l}$
$\displaystyle\simeq\ln\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}+1+\frac{\beta_{\textrm{eff}}^{2}\varpi_{l}^{2}}{24}+\cdots\,,$
(41) $\displaystyle\mathcal{C}_{\textrm{V}}$ $\displaystyle\simeq
1-\frac{\beta_{\textrm{eff}}^{2}\varpi_{l}^{2}}{12}+\cdots\,,$ (42)

These are consistent with results shown in Fig. 2 and Fig. 3, where we have
chosen the parameters such that $\pi\omega_{l}^{\textsc{out}}\Delta\gg 1$ as
discussed earlier, and generated the result for each mode according to Eqs.
(34) to (36) with $\beta_{\textrm{eff}}\varpi_{l}$ replaced by
$2\pi\omega_{l}^{\textsc{in}}\Delta$.

(a) thermal energy $\mathcal{E}_{\beta}$ (b) von Neumann entropy $\mathcal{S}$
(c) heat capacity $\mathcal{C}_{\textsc{v}}$

Figure 3: Mode dependence of thermal energy, entropy and heat capacity at
constant volume, similar to Fig. 2, except for a different $a_{f}=200$. For
these choices, the contributions of the modes within the high temperature
regime $\beta_{\textrm{eff}}\varpi_{l}\ll 1$ can be better seen. They are
consistent with the approximations in Eqs. (40)–(42).

To find the thermodynamic quantities of the field, for simplicity we will
assume $\pi ma_{f}\Delta\gg 1$, which can be translated into

$\frac{a_{f}}{a_{i}}\frac{m}{T_{\textrm{eff}}}\gg 1\,.$ (43)

This seems plausible for a quantum field of finite mass, in consideration of
the extreme change of the scale factor under an exponential expansion. It
allows the lower bound of the summation to start from $l=0$. When high
frequency modes are included in the sum over all modes, the thermodynamic
quantities of the field may diverge and regularization measures need be
introduced, as is usually done with the calculation of the field energy333For
example, the vacuum contribution to the renormalized internal energy in the
out-state is given by
$\mathcal{E}_{\textsc{vac}}=\int_{ma_{f}}^{\infty}\\!d\xi\;\frac{\xi^{2}}{e^{2\pi\xi}-1}\,\sqrt{\frac{\xi^{2}}{a_{f}^{2}}-m^{2}}\,.$
(44) . On a closer inspection, except for the internal energy, we note that
for a fixed effective temperature, the high-frequency contributions to the
entropy and the heat capacity are all exponentially suppressed, so when we add
up the contributions of all modes, the entropy and the heat capacity of the
field are well defined.

As for the internal energy, if we leave out the contributions of vacuum
fluctuations in the out-state, and consider only the thermal energy
$\mathcal{E}_{\beta}$ of created particles from the parametric amplification
of the vacuum fluctuations in the in state, then

$\mathcal{E}_{\beta}=\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{\varpi_{l}}{e^{\beta_{\textrm{eff}}\varpi_{l}}-1}\,,$
(45)

the result is also well defined. Since the contributions to the thermodynamic
quantities from sufficiently high frequency modes are exponentially
suppressed, we may truncate the infinite series to finite sums in performing
numerical computations. For very small $\Delta$, many more terms must be
included in the summation in order to obtain stably convergent results.

(a) thermal energy $\mathcal{E}_{\beta}$ (b) von Neumann entropy $\mathcal{S}$
(c) heat capacity $\mathcal{C}_{\textsc{v}}$

Figure 4: Here we show $\mathcal{E}_{\beta}$, $\mathcal{S}$, and
$\mathcal{C}_{\textsc{v}}$ of the whole field after we add up the
contributions of all modes. We see the familiar results of
$\mathcal{E}_{\beta}$ growing like $T_{\textrm{eff}}^{4}$ (Stefan-Boltzmann)
while $\mathcal{S}$, and $\mathcal{C}_{\textsc{v}}$ grow like
$T_{\textrm{eff}}^{3}$ for sufficiently large $T_{\textrm{eff}}$. Here we
choose $a_{i}=1$, $a_{f}=200$, $m=1$,

The results are shown in Fig. 4. At late times after the transition ends, in
the regime of high effective temperature, the energy scales like
$T_{\textrm{eff}}^{4}$ and the constant volume heat capacity is proportional
to $T_{\textrm{eff}}^{3}$. Meanwhile the entropy grows with
$T_{\textrm{eff}}^{3}$. They behave very similar to those of bosonic particles
in the relativistic limit in thermal equilibrium [30] at temperature
$T_{\textrm{eff}}$.

To calculate thermodynamic quantities such as the pressure, and from it the
compressibility, we need variable volumes. For this purpose we consider a
different scenario, namely, We fix the initial scale factor $a_{i}$ and the
duration of transition $\Delta$, and then examine the dependence of the
Helmholtz free energy $\mathcal{F}$ on the final scale factor $a_{f}$. Then
the pressure will be given by

$\mathcal{P}=-\biggl{(}\frac{\partial\mathcal{F}}{\partial\mathcal{V}}\biggr{)}_{T}\,,$
(46)

and the compressibility $\kappa_{\textsc{t}}$

$\kappa_{\textsc{t}}=-\frac{1}{\mathcal{V}}\biggl{(}\frac{\partial\mathcal{V}}{\partial\mathcal{P}}\biggr{)}_{T}\,.$
(47)

The volume $\mathcal{V}$ of $S^{3}$ is $2\pi^{2}a^{3}$, so at late times after
the end of transition, we formally obtain

$\displaystyle\mathcal{P}$
$\displaystyle=-\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{\left(\dfrac{\partial\varpi_{l}}{\partial
a_{f}}\right)_{T}}{\left(\dfrac{\partial\mathcal{V}}{\partial
a_{f}}\right)_{T}}\,\biggl{[}\frac{1}{2}+\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\biggr{]}$
$\displaystyle=\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{(l+1)^{2}}{6\pi^{2}a_{f}^{5}\varpi_{l}}\,\biggl{[}\frac{1}{2}+\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\biggr{]}\,,$
(48)

where the factor
$\beta_{\textrm{eff}}\varpi_{l}=2\pi\omega_{l}^{\textsc{in}}\Delta$ is
independent of $a_{f}$ and $\varpi=\sqrt{(l+1)^{2}/a_{f}^{2}+m^{2}}$. In fact,
Eq. (III.3) gives only the partial pressure due to particle creation. Here we
can see for the very high modes $l\gg ma_{f}$, we may approximate the
fractional before the square brackets by

$\frac{(l+1)^{2}}{6\pi^{2}a_{f}^{5}\varpi_{l}}\sim\frac{1}{3}\frac{\varpi_{l}}{2\pi^{2}a_{f}^{3}}\,,$
(49)

and formally obtain

$\mathcal{P}\sim\frac{\rho_{\mathcal{F}}}{3}\,,$ (50)

where $\rho_{\mathcal{F}}$ is the free-energy density
$\mathcal{F}/\mathcal{V}$.

(a) fixed $a_{f}$ (b) fixed $\Delta$

Figure 5: Shown here are the behaviors of $a^{2}(t)$ for several scenarios we
have considered: In (a), the final scale factor $a_{f}$ is fixed while varying
$\Delta$, so that we can compare the results for different effective
temperatures. In (b) the rise time $\Delta$ is fixed, while $a_{f}$ takes on
different values. This corresponds to keeping the temperature constant while
varying the volume.

Formally, the redshift in the physical frequency due to the expansion of the
spacetime, for a fixed $\Delta$, will decrease the energy of each mode $l$, as
shown in Eq. (34). There, since the factor $\beta_{\textrm{eff}}\varpi_{l}$ is
independent of $a_{f}$, the mean particle number
$\dfrac{1}{e^{\beta_{\textrm{eff}}\varpi_{l}}-1}$ of each mode and the entropy
are independent of $a_{f}$. However, to see the full behavior of the pressure,
we need to include the zero-point contributions of all modes of the whole
field, and deal with the divergences of both the energy and the pressure. (See
[2]).

If we are interested in only the thermal contribution $\mathcal{F}_{\beta}$ of
the free energy, and $\mathcal{P}_{\beta}$ of the pressure, then by
subtracting off the zero-point contributions in (III.3), we find
$\mathcal{P}_{\beta}$ is always negative due to the effect of redshift. This
is the contribution to the isothermal compressibility from particles created
via the parametric amplification of the vacuum fluctuations in the in-vacuum
by the expansion of the spacetime. The result is rather counter-intuitive
against our understanding of photon gas in thermal equilibrium, and against
the similarities between the earlier results and the counterparts of the
bosonic particles. On a closer comparison, we note that the finite-temperature
part of the free energy of the photon gas is proportional to the volume with a
negative proportionality factor, so its
$\partial\mathcal{F}_{\beta}/\partial\mathcal{V}<0$. In contrast, here due to
the redshift in the frequency, we have
$\partial\mathcal{F}_{\beta}/\partial\mathcal{V}>0$ instead. That makes the
pressure $\mathcal{P}_{\beta}$ of the photon gas positive but renders
$\mathcal{P}_{\beta}$ of the massive field negative with increasing volume.

(a) fixed $a_{f}$ (b) fixed $l$

Figure 6: (a) We show the finite-temperature contribution of the partial
pressure of each mode. We choose three different effective temperatures but
fix the final scale factor $a_{f}$. Only the modes with physical frequencies
closely matching the effective temperature would be effectively excited. In
(b) the partial pressure of the mode drops quickly with the spatial volume
because the mean particle number is fixed by $\Delta$.

With this understanding, now we show in Fig. 6, the mode dependence of the
pressure for a fixed transition duration $\Delta$ and final scale factor
$a_{f}$ in (a) and the dependence of the pressure of a chosen mode on the
final scale factor for a fixed transition interval in (b). A smaller value of
$\Delta$ represents a higher value of the effective temperature, so in (a) the
curve showing the dependence of the pressure on the mode has a more prominent
feature for smaller $\Delta$, but drops rapidly with a greater value of $l$
because it is more difficult to excite the higher modes. In (b) we choose the
mode $l=10$, and see that the pressure, although negative, falls off to zero
very fast with $a_{f}$, Moreover, for larger values of $\Delta$ the pressure
becomes extremely small. The former can be understood from the key observation
that the mean number of particles produced
$\dfrac{1}{e^{\beta_{\textrm{eff}}\varpi_{l}}-1}$ due to expansion depends
only on $\Delta$ for a fixed $a_{i}$, and is independent of $a_{f}$, as seen
in Eq. (23). Thus as the volume expands, the particle density is quickly
diluted, leading to a decreasing pressure for each mode.

To compute the compressibility $\kappa_{\textsc{t}}$, we follow the same path
to first calculate $(\partial\mathcal{P}/\partial a_{f})_{T}$ at late times

$\displaystyle\biggl{(}\frac{\partial\mathcal{P}}{\partial
a_{f}}\biggr{)}_{T}$
$\displaystyle=\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{(l+1)^{2}}{6\pi^{2}}\biggl{(}\frac{\partial}{\partial
a_{f}}\frac{1}{a_{f}^{5}\varpi_{l}}\biggr{)}_{T}\,\biggl{[}\frac{1}{2}+\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\biggr{]}$
$\displaystyle=-\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{(l+1)^{2}(4\varpi_{l}^{2}+m^{2})}{6\pi^{2}a_{f}^{6}\varpi_{l}^{3}}\,\biggl{[}\frac{1}{2}+\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\biggr{]}\,.$
(51)

Then we calculate the factor

$\frac{1}{\mathcal{V}}\biggl{(}\frac{\partial\mathcal{V}}{\partial
a_{f}}\biggr{)}_{T}=\frac{3}{a_{f}}\,,$ (52)

finally arriving at the inverse of the isothermal compressibility, that is,
isothermal bulk modulus

$\displaystyle\kappa_{\textsc{t}}^{-1}=\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{(l+1)^{2}(4\varpi_{l}^{2}+m^{2})}{18\pi^{2}a_{f}^{5}\varpi_{l}^{3}}\,\biggl{[}\frac{1}{2}+\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\biggr{]}\,.$
(53)

Let us first examine the expressions inside the square brackets in the low and
the high effective temperature limits. The first term represents the zero-
point contribution, while the second term accounts for particle creation
resulting from the parametric amplification of the in-vacuum by the expansion
of the spacetime. We find

$\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})=\begin{cases}\dfrac{\ln\beta_{\textrm{eff}}\varpi_{l}}{\beta_{\textrm{eff}}\varpi_{l}}\,,&\beta_{\textrm{eff}}\varpi_{l}\ll
1\,,\;\text{high temperature regime}\,,\vspace{8pt}\\\
\dfrac{e^{-\beta_{\textrm{eff}}\varpi_{l}}}{\beta_{\textrm{eff}}\varpi_{l}}\,,&\beta_{\textrm{eff}}\varpi_{l}\gg
1\,,\;\text{low temperature regime}\,.\end{cases}$ (54)

Since the factor $\ln z/z$ gives a very large negative number as $z\to 0$, in
the high temperature regime, the vacuum contribution in
$\kappa_{\textsc{t}}^{-1}$ can be safely ignored for modes whose values of $l$
are not too high. However, with increasing $l$, the modes behave more like
that in the low temperature regime since
$\beta_{\textrm{eff}}\,\varpi_{l}=2\pi\omega_{l}^{\textsc{in}}\Delta$. The
summation in (53) eventually leads to divergences for a fixed $a_{f}$, and
renormalization procedures need be introduced. From (III.3), the renormalized
zero-point contribution of the pressure is given by [31]

$\mathcal{P}_{\textsc{vac}}=\frac{1}{6\pi^{2}a_{f}^{5}}\int_{ma_{f}}^{\infty}\\!dx\;\frac{x^{4}}{\bigl{(}e^{2\pi
x}-1\bigr{)}\sqrt{x^{2}/a_{f}^{2}-m^{2}}}$ (55)

by the Abel–Plana formula. If we suppose $ma_{f}\gg 1$, Eq. (55) is
approximately given by [32]

$\mathcal{P}_{\textsc{vac}}\simeq\frac{(ma_{f})^{\frac{7}{2}}}{12\pi^{2}a_{f}^{4}}\,e^{-2\pi
ma_{f}}\,.$ (56)

This is a very tiny positive quantity. The corresponding term in (III.3) is

$\biggl{(}\frac{\partial\mathcal{P}_{\textsc{vac}}}{\partial
a_{f}}\biggr{)}_{T}=-\frac{1}{2}\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{(l+1)^{2}(4\varpi_{l}^{2}+m^{2})}{6\pi^{2}a_{f}^{6}\varpi_{l}^{3}}\simeq-\frac{m^{3}\sqrt{ma_{f}}(4\pi
ma_{f}+1)}{24\pi^{2}a_{f}^{2}}\,e^{-2\pi ma_{f}}\,.$ (57)

Again this is exponentially small in the limit, $ma_{f}\gg 1$. Thus we only
need to consider the finite-temperature contribution in (53), which is given
by

$\displaystyle\kappa_{\textsc{t}}^{-1}\simeq\sum_{l=0}^{\infty}d_{l}^{(4)}\,\frac{(l+1)^{2}(4\varpi_{l}^{2}+m^{2})}{18\pi^{2}a_{f}^{5}\varpi_{l}^{3}}\times\frac{1}{\beta_{\textrm{eff}}\varpi_{l}}\ln\bigl(1-e^{-\beta_{\textrm{eff}}\varpi_{l}}\bigr{missing})\,.$
(58)

(a) temperature dependence (b) volume dependence

Figure 7: Shown in (a) is the temperature dependence of the isothermal
compressibility $\kappa_{\textsc{t}}$, computed by Eq. (58), leaving out the
renormalized zero-point contribution. For fixed $a_{f}$, the space behaves
‘stiffer’ with increasing temperature. Shown in (b) is the volume dependence
of $\kappa_{\textsc{t}}$. In contrast, the space with larger volume becomes
more compressible. In both plots we choose $m=1$, $a_{i}=1$.

In Fig. 7-(a), we see that the isothermal compressibility
$\kappa_{\textsc{t}}$ takes on smaller negative values if the effective
temperature is higher, associated with more rapid expansion for the same
$a_{f}$. The physical meaning of a negative $\kappa_{\textsc{t}}$ poses some
challenge444It is interesting to note that the expressions inside the square
brackets in (53) can be zero when
$e^{-\frac{\beta_{\textrm{eff}}\,\varpi_{l}}{2}}=\dfrac{\sqrt{5}-1}{2}<1$.
Since
$\dfrac{\beta_{\textrm{eff}}\,\varpi_{l}}{2}=\pi\omega_{l}^{\textsc{in}}\Delta$
from (23), we understand that this occurs only for a specific mode for a fixed
$\Delta$ and thus it will not render $\kappa_{\textsc{t}}^{-1}$ zero. In other
words, the isothermal compressibility $\kappa_{\textsc{t}}$ is well defined..
It would be totally counter-intuitive if one invokes examples of ordinary
matter such as the gas dynamics of a closed system, where the volume is
inversely related to the pressure such that the compressibility is positive.
In the current case, the system is not closed. Expansion of the universe (or
in analog gravity, an external agent which drives the system) imparts energy
into spacetime resulting in particle creation via the parametric amplification
of the vacuum fluctuations. (For a description of this process in terms of the
fluctuation-dissipation relation, see [33, 34].) Having this in mind, it may
not be too surprising to see the thermodynamic behavior of the spacetime under
study: the volume of the universe does not change as fast as the increase in
pressure caused by particle creation due to the rapid expansion of the
spacetime or, in analog gravity, driven by an external potential. In addition,
according to (54), a spacetime filled with massive quanta of a conformally-
coupled field would appear to be ‘stiffer’ at higher temperatures. In
contrast, as shown in Fig. 7-(b), at the same effective temperature, that is,
with a fixed $\Delta$ the space is more compressible if it reaches a larger
final volume. This characteristic is attributed to the simple fact that the
particle density is more diluted in a larger spatial volume. Note that it is
consistent with Fig. 6-(b), where the finite-temperature contribution of the
partial pressure of a specific mode drops off quickly with the final scale
factor $a_{f}$.

##  IV Brief summary and further developments

In this work we have treated a conformally-coupled massive scalar field in a
spacetime statically bounded in the asymptotic past and future. We assume the
scale factor rises exponentially fast in conformal time from one constant and
smoothly transits to anther one, as exemplified by the Bernard-Duncan model,
In the limit when the product of the mode frequency of the field in the out-
state and the transition time is much greater than unity, cf. Eq. (21), once
evolved from the in-vacuum in the asymptotic past, the quantum field in the
out-state after such an expansion behaves nicely like a massive thermal scalar
field in a Minkowski spacetime. The created particle obey the Bose-Einstein
distribution at a effective temperature which is inversely proportional to the
transition time. Thus, thermality results from the exponential expansion (or
contraction) of spacetime, and under the thermal equilibrium condition we can
work out the thermodynamics of the quantum field in the out-state. We
explicitly verify this point by examining the behavior of the Helmholtz free
energy, internal energy, entropy and the heat capacity for each mode of the
field and then for the field as a whole. They are consistent with the well-
known results in the thermal field theory. We further examine the pressure and
the isothermal compressibility. The latter is a less familiar notion in this
context. We find that thermal particle creation under the rapid expansion of
spacetime results in a negative value in the finite-temperature contribution
to the pressure. We see this as a consequence of the redshift in the physical
frequency of the modes of the field. Thus it gives a counteracting component
to the total pressure. The compressibility typically takes on small negative
values. This is understood as the fact that under the current scenario, the
volume does not increase as fast as the pressure does when the spacetime is
expanded to various sizes over a fixed transition time. Moreover spacetime
filled with massive quanta of a conformally-coupled field exhibits increased
stiffness at higher temperatures.

What we have presented here is only a model calculation, taking advantage of
the simple analytic form of the scale function
$a^{2}(\eta)\sim\tanh(\eta/\Delta)$ working in conformal time $\eta$. Two
extensions of more realistic significance can readily be made: One is to
consider a massive conformal field in spatially-flat FLRW universe with an
exponential expansion stage in cosmic time $t$. The other extension is to
treat a massless minimally-coupled scalar field, which depicts one
polarization of the graviton field, in the Poincare patch of de Sitter space,
also in cosmic time 555Note the thermal particle creation in an inflationary
universe with a period of exponential expansion where the scale factor
$a(t)\sim e^{Ht}$ in the flat FLRW coordinatization of de Sitter space is
different from the thermal radiance felt by an observer in a static
coordinatization of the de Sitter space, the former is due to parametric
amplification of vacuum fluctuations whereas the latter is due to the Gibbons-
Hawking effect akin to the Hawking effect in black holes or the Unruh effect
for uniformly accelerated detectors. For a broader viewpoint of de Sitter
thermality and calculations of thermodynamic functions, see, e.g., [35]. The
calculations will be a bit more involved, but the track laid down here toward
deriving the thermodynamic functions can be used just as well. The crucial
observation is that one does not need to impose the somewhat artificial static
out-state, because most of the particle production happens in the initial
exponential expansion stage. As long as the particle production has a thermal
spectrum the procedures illustrated here can be adopted. Alternatively, an
even better and conceptually clearer approach is to adopt the in-in or closed-
time-path or Schwinger-Keldysh formalism, as we alluded to in the beginning.
Only a well defined initial state, or unambiguous Fock state is needed and the
nonequilibrium dynamics will guide the ensuing evolution of the physical
observables with respect to the initial state. At least for Gaussian systems,
following the strategy outlined in [36], the final state is always in the form
of a squeezed state, from which one can extract the effective temperature and
the coherent, squeezed parameters up to a constant. The in-in formalism does
not require the spacetime to be asymptotically-stationary as it does in an in-
out $S$-matrix formulation.

Zooming out to the bigger picture, as we explained in our earlier paper [2]
(see references cited therein) spacetime thermodynamics offers a refreshing
and perhaps even correct way of viewing the nature of spacetime, namely, as an
effective theory which describes the low energy, long wavelength limit of
quantum gravity – theories about spacetime’s microscopic constituents. The
novel notions of quantum capacity and dynamical compressibility introduced
there and the theoretical results obtained herewith associated with thermal
particle production have bearings on two ends of a broad spectrum, namely, the
conceptualization of emergent gravity and the observational possibilities in
analog gravity, specifically, dynamical Casimir effect experiments in the
laboratory. Their physical meanings need be savored more and expounded further
in these contexts.

In the study of spacetime quantum thermodynamics of dynamical spacetimes with
particle creation, two processes we mentioned in the Introduction are worthy
of further investigation: For one, the inclusion of stimulated particle
creation. So far we have only considered spontaneous particle creation from
the vacuum, i.e., parametric amplification of the zero-point energy due to the
expansion (or contraction) of the universe. Even the thermal field in this
paper is of vacuum origin, albeit the spectrum of particles created obey a
Planck distribution, thanks to a period of exponential expansion. To be
complete, stimulated production, i.e., parametric amplification of particles
already present needs be included at every moment. This is because at any
moment whatever amount of particles created spontaneously from the vacuum at
an earlier moment will also be amplified. This will greatly enhance (for
bosons) the amount of energy produced. In fact, we think the great enhancement
factor reported in [37] whose authors regard as owing to finite temperature
effect, is fundamentally due to stimulated particle production. Here, one sees
the theoretical difference between thermal particle production and stimulated
particle production: the stimulated production mechanism is more basic and
general, even though it applies to thermal particle creation, it is not
limited to thermal fields. Thermal field refers to the state of the quantum
field while stimulated production refers to the dynamical process. It is this
stimulated amplification process, like stimulated emission in lasers, which
results in a much stronger effect than vacuum creation, like spontaneous
emission.

The other task is the application of nonequilibrium quantum field theory
methods (e.g., [17]) for a fuller treatment of quantum field processes in the
early universe. Nonequilibrium treatments do not rely on thermal notions nor
require equilibrium states as pre-conditions. In fact there is no place for
the notion of temperature – temperature is a derived concept, taking shape
only after thermalization can be actuated. Quantum field processes like
particle creation in evolutionary cosmology or in dynamical Casimir effects
are intrinsically time-dependent processes, whereas finite temperature
theories are predicated upon the existence of a thermal equilibrium state
under stationarity conditions. It is now possible to treat the thermodynamics
of quantum field processes in dynamical spacetime from a nonequilibrium
quantum field theory approach because a nonequilibrium free energy density
functional related to the influence action has recently been established for
dynamical quantum systems interacting with a quantum field environment [36].
The approach and results reported there are useful for the investigations in
the present research program. We hope to report on progresses along these two
fronts in our future works [38, 39].

Acknowledgment 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. Chong-
Sun Chu of the National Center for Theoretical Sciences at National Tsing Hua
University, and Prof. Kin-Wang Ng of the Institute of Physics, Academia
Sinica, Taiwan, R.O.C. where part of this work was done.

## References

  * Cho  _et al._ [2022] H.-T. Cho, J.-T. Hsiang, and B. L. Hu, Quantum capacity and vacuum compressibility of spacetime: thermal fields, Universe 8, 291 (2022), arXiv:2204.08634 [gr-qc] .
  * Xie  _et al._ [2024] Y.-C. Xie, J.-T. Hsiang, and B.-L. Hu, Dynamical vacuum compressibility of space, Phys. Rev. D 109, 065027 (2024), arXiv:2312.09047 [gr-qc] .
  * Fischetti  _et al._ [1979] M. Fischetti, J. B. Hartle, and B. L. Hu, Quantum effects in the early universe. I. Influence of trace anomalies on homogeneous, isotropic, classical geometries, Phys. Rev. D 20, 1757 (1979).
  * Anderson [1983] P. Anderson, Effects of quantum fields on singularities and particle horizons in the early universe, Phys. Rev. D 28, 271 (1983).
  * Anderson [1984] P. R. Anderson, Effects of quantum fields on singularities and particle horizons in the early universe. II, Phys. Rev. D 29, 615 (1984).
  * Zel’Dovich and Starobinskiǐ [1972] Y. B. Zel’Dovich and A. Starobinskiǐ, Particle production and vacuum polarization in an anisotropic gravitational field, Sov. Phys. JETP 34, 1159 (1972).
  * Hu and Parker [1977] B. L. Hu and L. Parker, Effect of graviton creation in isotropically expanding universes, Phys. Lett. A 63 (1977).
  * Hu and Parker [1978] B. L. Hu and L. Parker, Anisotropy damping through quantum effects in the early universe, Phys. Rev. D 17, 933 (1978).
  * Hartle and Hu [1979] J. B. Hartle and B. L. Hu, Quantum effects in the early universe. II. Effective action for scalar fields in homogeneous cosmologies with small anisotropy, Phys. Rev. D 20, 1772 (1979).
  * Hartle and Hu [1980] J. B. Hartle and B. L. Hu, Quantum effects in the early universe. III. Dissipation of anisotropy by scalar particle production, Phys. Rev. D 21, 2756 (1980).
  * Anderson [1985] P. R. Anderson, Effects of quantum fields on singularities and particle horizons in the early universe. III. The conformally coupled massive scalar field, Phys. Rev. D 32, 1302 (1985).
  * Hu [1981] B. L. Hu, Effect of finite temperature quantum fields on the early Universe, Phys. Lett. B 103, 331 (1981).
  * Hu [1982a] B. Hu, Vacuum viscosity description of quantum processes in the early universe, Phys. Lett. A 90, 375 (1982a).
  * Hu [1982b] B. L. Hu, Finite temperature quantum fields in expanding universes, Phys. Lett. B 108, 19 (1982b).
  * Hu [1983] B. L. Hu, Quantum dissipative processes and gravitational entropy of the universe, Phys. Lett. A 97, 368 (1983).
  * Xie  _et al._ [2023] Y.-C. Xie, S. Butera, and B. L. Hu, Optomechanical backreaction of quantum field processes in dynamical Casimir effect, arXiv:2308.03129 [quant-ph] (2023).
  * Calzetta and Hu [2008] E. A. Calzetta and B.-L. Hu,  _Nonequilibrium Quantum Field Theory_ (Cambridge University Press, 2008).
  * Hu  _et al._ [1994] B. L. Hu, G. Kang, and A. Matacz, Squeezed vacua and the quantum statistics of cosmological particle creation, Int. J Mod. Phys. A 9, 991 (1994).
  * Koks  _et al._ [1997] D. Koks, B. L. Hu, A. Matacz, and A. Raval, Thermal particle creation in cosmological spacetimes: A stochastic approach, Phys. Rev. D 56, 4905 (1997).
  * Bernard and Duncan [1977] C. Bernard and A. Duncan, Regularization and renormalization of quantum field theory in curved space-time, Ann. Phys. 107, 201 (1977).
  * Parker [1976] L. Parker, Thermal radiation produced by the expansion of the universe, Nature 261, 20 (1976).
  * Parker [1977] L. Parker, The production of elementary particles by strong gravitational fields, in  _Asymptotic Structure of Space-Time_, edited by F. P. Esposito and L. Witten (Springer US, Boston, MA, 1977) pp. 107–226.
  * Berger [1975] B. K. Berger, Scalar particle creation in an anisotropic universe, Phys. Rev. D 12, 368 (1975).
  * Hu and Pavon [1986] B. L. Hu and D. Pavon, Intrinsic measures of field entropy in cosmological particle creation, Phys. Lett. B 180, 329 (1986).
  * Hu and Kandrup [1987] B. L. Hu and H. E. Kandrup, Entropy generation in cosmological particle creation and interactions: A statistical subdynamics analysis, Phys. Rev. D 35, 1776 (1987).
  * Kluger  _et al._ [1998] Y. Kluger, E. Mottola, and J. M. Eisenberg, Quantum Vlasov equation and its Markov limit, Phys. Rev. D 58, 125015 (1998).
  * Campo and Parentani [2003] D. Campo and R. Parentani, Space-time correlations within pairs produced during inflation: A wave-packet analysis, Phys. Rev. D 67, 103522 (2003).
  * Lin  _et al._ [2010] S.-Y. Lin, C.-H. Chou, and B. L. Hu, Quantum entanglement and entropy in particle creation, Phys. Rev. D 81, 084018 (2010).
  * Hsiang and Hu [2021a] J.-T. Hsiang and B.-L. Hu, Intrinsic entropy of squeezed quantum fields and nonequilibrium quantum dynamics of cosmological perturbations, Entropy 23, 1544 (2021a).
  * Kolb and Turner [1994] E. Kolb and M. Turner,  _The Early Universe_ (CRC Press, 1994).
  * Bordag  _et al._ [2009] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko,  _Advances in the Casimir effect_ , Vol. 145 (Oxford University Press, Oxford, 2009).
  * Mamayev  _et al._ [1976] S. G. Mamayev, V. M. Mostepanenko, and A. A. Starobinsky, Particle creation from vacuum near the homogeneous isotropic singularity, Sov. Phys. JETP 43, 823 (1976).
  * Calzetta and Hu [1987] E. Calzetta and B.-L. Hu, Closed-time-path functional formalism in curved spacetime: Application to cosmological back-reaction problems, Phys. Rev. D 35, 495 (1987).
  * Hu and Sinha [1995] B. L. Hu and S. Sinha, Fluctuation-dissipation relation for semiclassical cosmology, Phys. Rev. D 51, 1587 (1995).
  * Volovik [2023] G. E. Volovik, de Sitter state as thermal bath, arXiv:2312.02292 [gr-qc] (2023).
  * Hsiang and Hu [2021b] J.-T. Hsiang and B.-L. Hu, Nonequilibrium quantum free energy and effective temperature, generating functional, and influence action, Phys. Rev. D 103, 065001 (2021b).
  * Plunien  _et al._ [2000] G. Plunien, R. Schützhold, and G. Soff, Dynamical Casimir effect at finite temperature, Phys. Rev. Lett. 84, 1882 (2000).
  * Butera  _et al._ [2024] S. Butera, J.-T. Hsiang, and B.-L. Hu, Foundational issues in dynamical Casimir effect and analog features in cosmological particle creation (2024), in preparation.
  * Hsiang  _et al._ [2024] J.-T. Hsiang, H.-T. Cho, and B.-L. Hu, Quantum capacity and dynamic compressibility of the Universe: Nonequilibrium fields (2024), in preparation.