# Polarization of the conformal vacuum in the Milne universe

T. A. Petrosyan

Department of Physics, Yerevan State University,
1 Alex Manoogian Street, 0025 Yerevan, Armenia

###### Abstract

The vacuum expectation values (VEVs) of the field squared and energy-momentum
tensor for a massless scalar field are investigated in the Milne universe with
general number of spatial dimensions. The vacuum state depends on the choice
of the mode functions in the canonical quantization procedure and we assume
that the field is prepared in the conformal vacuum. As the first step an
integral representation for the difference of the Wightman functions
corresponding to the conformal and Minkowski vacua is derived. The mean field
squared and energy-momentum tensor are obtained in the coincidence limit. It
is shown that the Minkowski vacuum state is interpreted as a thermal one with
respect to the conformal vacuum. The thermal factor is of the Bose-Einstein
type in odd dimensional space and of the Fermi-Dirac type in even number of
spatial dimensions.

Keywords: Vacuum polarization, Milne universe, conformal vacuum

##  1 Introduction

It is well-known that the vacuum state in quantum field theory, in general,
depends on the choice of the mode functions used in the canonical quantization
procedure. If the Bogoliubov $\beta$-coefficient relating two sets of modes is
different from zero then the corresponding vacuum states are inequivalent. An
example, widely considered in the literature (see [1, 2, 3] and references
therein), is provided by the Minkowski and Fulling-Rindler vacua in flat
spacetime. The Minkowski vacuum corresponds to the modes of inertial observer
and it preserves all the symmetries of the Minkowski spacetime (is maximally
symmetric). The Fulling-Rindler vacuum is realized in the quantization
procedure based on the modes of uniformly accelerated observers. For those
observers horizons are present corresponding to the light cones that divide
the spacetime into four sections. The right and left patches are referred to
as the right and left Rindler wedges, respectively. The upper and lower
patches are covered by the Milne coordinates and are referred to as the Milne
universe.

The line element of the flat spacetime in the Milne coordinates is of the
Friedmann-Robertson-Walker type with a scale factor being a linear function of
the corresponding time coordinate. The spacetime is foliated by negative
curvature spatial sections. The corresponding geometry serves as a simple
model for the investigation of quantum-field-theoretical effects in time-
dependent backgrounds. A relatively large number of problems are exactly
solvable and motivated by that various aspects of the dynamics of quantum
fields in the Milne universe have been discussed in [4]-[13]. In the present
paper we consider the local characteristics of the conformal vacuum state
(referred to as the C-vacuum) for a massless scalar field in the Milne
universe.

The paper is organized as follows. In the next section we present the normal
modes and the Wightman functions for the C- and Minkowski vacua in the case of
a massive scalar field with general curvature coupling parameter. The general
results are specified for a massless field in section 3. The expression for
the mean field squared and energy-momentum tensor are discussed in section 4.
The main results are summarized in section 6.

##  2 Wightman functions in conformal and Minkowski vacua

The line element for the $(D+1)$-dimensional Milne universe is expressed as

$ds^{2}=dt^{2}-t^{2}(dr^{2}+\sinh^{2}rd\Omega_{D-1}^{2}),$ (1)

where for the time and dimensionless radial coordinates one has $0\leq
t<\infty$, $0\leq r<\infty$, and $d\Omega_{D-1}^{2}$ is the line element on a
$(D-1)$-dimensional sphere. The spatial part corresponds to a constant
negative curvature space covered by the hyperspherical coordinates
$(r,\vartheta,\phi)$. For the set of angular coordinates we have
$\vartheta=(\theta_{1},\theta_{2},\ldots\theta_{n})$,
$0\leq\theta_{k}\leq\pi$, $k=1,2,\ldots,n$ and $0\leq\phi\leq 2\pi$, where
$n=D-2$. Note that the spacetime described by the line element (1) is flat.
That is explicitly seen introducing new coordinates $(T,\mathbf{R})$, with
$\mathbf{R}=(R,\vartheta,\phi)$, in accordance with

$T=t\cosh r,\;R=t\sinh r.$ (2)

The line element (1) takes the Minkowskian form
$ds^{2}=dT^{2}-dR^{2}-R^{2}d\Omega_{D-1}^{2}$ in hyperspherical spatial
coordinates. As seen from (2), $T>R$ and the Milne coordinates
$(t,r,\vartheta,\phi)$ cover the patch of the Minkowski spacetime inside the
future light cone. In the region corresponding to the past light cone we take
$T=t\cosh r$, $R=-t\sinh r$ with the time coordinate $-\infty<t\leq 0$. The
remaining regions of the Minkowski spacetime, $R>|T|$, correspond to the
Rindler patches.

Passing to new coordinates $(\eta,\bar{r})$, $-\infty<\eta<\infty$, in
accordance with

$t=ae^{\eta/a}\equiv ae^{\bar{\eta}},\;r=\bar{r}/a,$ (3)

where $a$ is a constant with dimension of length, the line element (1) is
presented in a conformally-static form

$ds^{2}=\left(t/a\right)^{2}\left[d\eta^{2}-d\bar{r}^{2}-a^{2}\sinh^{2}\left(\bar{r}/a\right)d\Omega_{D-1}^{2}\right].$
(4)

The line element in the square brackets describes a static spacetime with a
constant negative curvature space. The curvature radius of the latter is
determined by $a$ and in the limit $a\rightarrow\infty$ for fixed $\eta$ and
$\bar{r}$, from (4) the Minkowskian line element in spherical coordinates is
obtained. For two spacetime points $(T^{\prime},\mathbf{R}^{\prime})$ and
$(T,\mathbf{R})$ the spacetime interval between them is expressed as

$\left(\Delta
T\right)^{2}-\left|\Delta\mathbf{R}\right|^{2}=2tt^{\prime}\left(\cosh\Delta\bar{\eta}-\cosh\zeta\right),$
(5)

where $\Delta T=T^{\prime}-T$,
$\Delta\mathbf{R}=\mathbf{R}^{\prime}-\mathbf{R}$,
$\Delta\bar{\eta}=\bar{\eta}^{\prime}-\bar{\eta}$, and $\zeta$ is defined by
the relation

$\bar{u}=\cosh r\cosh r^{\prime}-\sinh r\sinh
r^{\prime}\cos\theta=\cosh\zeta.$ (6)

Here, $\theta$ is the angle between directions
$\left(\vartheta^{\prime},\phi^{\prime}\right)$ and
$\left(\vartheta,\phi\right)$. In the special case $\theta=0$, corresponding
to points on the same radial directions, we have
$\zeta=r^{\prime}-r\equiv\Delta r$.

Having described the background geometry we turn to the field content. We
consider a quantum scalar field $\varphi(x)$ with curvature coupling parameter
$\xi$. The field equation reads

$\left(\Box+m^{2}+\xi R\right)\varphi=0,$ (7)

where $\Box$ is the d’Alembert operator and for background under consideration
the Ricci scalar is zero, $R=0$. Let
$\\{\varphi_{\sigma}\left(x\right),\varphi_{\sigma}^{\ast}\left(x\right)\\}$
be the complete set of mode functions obeying the field equation and specified
by the quantum numbers $\sigma$. For the modes corresponding to the C-vacuum
$\sigma=\left(z,m_{p}\right)$ and

$\varphi_{\sigma}\left(x\right)=N_{\sigma}\frac{J_{-iz}(mt)}{t^{(D-1)/2}}\frac{P_{iz-1/2}^{1-D/2-l}(\cosh
r)}{\sinh^{D/2-1}r}Y(m_{p};\vartheta,\phi),$ (8)

where $J_{\nu}\left(x\right)$ is the Bessel function, $P_{\rho}^{\gamma}(x)$
is the associated Legendre function of the first kind and
$Y(m_{p};\vartheta,\phi)$ are the hyperspherical harmonics. In (8), $0\leq
z<\infty$ and $m_{p}=(l,m_{1},\ldots,m_{n})$ with $l=0,1,2,\ldots$. For the
integers $m_{1},m_{2},\ldots,m_{n}$ one has $-m_{n-1}\leqslant m_{n}\leqslant
m_{n-1}$ and $0\leqslant m_{n-1}\leqslant m_{n-2}\leqslant\cdots\leqslant
m_{1}\leqslant l$. The coefficient $N_{\sigma}$ is determined by the standard
normalization condition for the equation (7) and is determined from

$\left|N_{\sigma}\right|^{2}=\frac{z\left|\Gamma\left((D-1)/2+l+iz\right)\right|^{2}}{2N\left(m_{p}\right)}.$
(9)

The factor $N\left(m_{p}\right)$ comes from the normalization condition for
the hyperspherical harmonics and its explicit form will not be required in the
discussion below.

The correlations of the vacuum fluctuations of quantum fields at different
spacetime points $x$ and $x^{\prime}$ are determined by the two-point
functions. Here we consider the Wightman function. The latter can be evaluated
by using the mode-sum formula

$W(x,x^{\prime})=\sum_{\sigma}\varphi_{\sigma}\left(x\right)\varphi_{\sigma}^{\ast}\left(x^{\prime}\right).$
(10)

The expression of the Wightman function for the conformal vacuum (denoted here
as $W_{\text{{C}}}(x,x^{\prime})$) can be obtained by using the corresponding
formula for the Wightman function from [13]:

$\displaystyle W_{\text{{C}}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{2nS_{D}}\sum_{l=0}^{\infty}\frac{\left(2l+n\right)C_{l}^{n/2}(\cos\theta)}{(\sinh
r\sinh
r^{\prime})^{D/2-1}}\int_{0}^{\infty}dy\,y|\Gamma(l+\left(D-1\right)/2+iy)|^{2}$
(11) $\displaystyle\times
J_{-iy}(mt)J_{iy}(mt^{\prime})P_{iy-1/2}^{-\mu}(u)P_{iy-1/2}^{-\mu}(u^{\prime}),$

where $S_{D}=2\pi^{D/2}/\Gamma\left(D/2\right)$, $C_{l}^{n/2}(\cos\theta)$ is
the Gegenbauer polynomial, and

$u=\cosh r,\,u^{\prime}=\cosh r^{\prime}.$ (12)

By using the addition theorem for the associated Legendre functions from [14]
(for the correction of the missprint see [19]) it can be shown that

$\displaystyle\overset{\infty}{\underset{l=0}{\sum}}\left(l+\frac{n}{2}\right)C_{l}^{n/2}\left(\cos\theta\right)\left|\Gamma\left(l+\left(D-1\right)/2+iy\right)\right|^{2}P_{iy-1/2}^{-l-n/2}\left(u\right)P_{iy-1/2}^{-l-n/2}\left(u^{\prime}\right)$
$\displaystyle\qquad=\frac{2^{-n/2}}{\Gamma\left(n/2\right)}\left[\left(u^{2}-1\right)\left(u^{\prime
2}-1\right)\right]^{n/4}\left|\Gamma\left(\left(D-1\right)/2+iy\right)\right|^{2}\frac{P_{iy-1/2}^{-n/2}\left(\bar{u}\right)}{\left(\bar{u}^{2}-1\right)^{n/4}},$
(13)

with $\bar{u}$ given by (6). By using this result, the expression (11) is
further simplified as

$W_{\text{{C}}}(x,x^{\prime})=\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{2\left(2\pi\right)^{D/2}}\int_{0}^{\infty}dy\,yJ_{-iy}(mt)J_{iy}(mt^{\prime})\left|\Gamma\left(iy+\left(D-1\right)/2\right)\right|^{2}\frac{P_{iy-1/2}^{-n/2}\left(\bar{u}\right)}{\left(\bar{u}^{2}-1\right)^{n/4}}.$
(14)

The VEVs of the field squared and energy-momentum tensor are obtained from
two-point functions in the coincidence limit. That limit is divergent and a
renormalization is required. In the present paper we are interested in the
difference of the local characteristics of the C- and Minkowski vacua. The
latter is obtained from the difference $\Delta
W(x,x^{\prime})=W_{\text{{C}}}(x,x^{\prime})-W_{\text{{M}}}(x,x^{\prime})$,
where

$W_{\text{{M}}}(x,x^{\prime})=\frac{m^{\left(D-1\right)/2}}{\left(2\pi\right)^{\left(D+1\right)/2}}\frac{K_{\left(D-1\right)/2}(m\sqrt{\left|\Delta\mathbf{R}\right|^{2}-\left(\Delta
T\right)^{2}})}{\left[\left|\Delta\mathbf{R}\right|^{2}-\left(\Delta
T\right)^{2}\right]^{\left(D-1\right)/4}},$ (15)

is the Wightman function for the Minkowski vacuum. Here, we assume that
$\left|\Delta\mathbf{R}\right|>|\Delta T|$. Note that, in accordance with (5),
this corresponds to $|\zeta|>|\Delta\bar{\eta}|$. The expressions in the other
regions of the Minkowski spacetime are obtained by the analytical
continuation. The local geometry for both the C-vacuum in the Milne patch and
for the Minkowski vacuum is the same and, hence, the difference $\Delta
W_{\text{{C}}}(x,x^{\prime})$ is finite and can be directly used for the
evaluation of the local VEVs.

##  3 Wightman function for a massless field

The expression (14) for the Wightman function is further simplified for a
massless field. In the limit $m\rightarrow 0$ for the product of the Bessel
functions one has

$J_{-iy}(mt)J_{iy}(mt^{\prime})\approx\frac{\sinh\left(\pi y\right)}{\pi
y}e^{iy\Delta\bar{\eta}},$ (16)

and the formula takes the form

$W_{\text{{C}}}(x,x^{\prime})=\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{\left(2\pi\right)^{D/2+1}}\int_{0}^{\infty}dy\,\sinh\left(\pi
y\right)e^{iy\Delta\bar{\eta}}\left|\Gamma\left(iy+\frac{D-1}{2}\right)\right|^{2}\frac{P_{iy-1/2}^{-n/2}\left(\bar{u}\right)}{\left(\bar{u}^{2}-1\right)^{n/4}}.$
(17)

By using the properties of the associated Legendre function this expression
can be rewritten as

$\displaystyle W_{\text{{C}}}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\left(-1\right)^{q}\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{\left(2\pi\right)^{D/2+1}}\partial_{\bar{u}}^{q}\int_{0}^{\infty}dy\,\sinh\left(\pi
y\right)e^{iy\Delta\bar{\eta}}$ (18)
$\displaystyle\times\left|\Gamma\left(\frac{D-1}{2}-q+iy\right)\right|^{2}\frac{P_{iy-1/2}^{1-D/2+q}\left(\bar{u}\right)}{\left(\bar{u}^{2}-1\right)^{(D-2q-2)/4}}.$

with $q$ being a non-negative integer. In the massless limit, by taking into
account the relation (5), for the Minkowski vacuum one gets

$W_{\text{{M}}}(x,x^{\prime})=\frac{\left(tt^{\prime}\right)^{\left(1-D\right)/2}}{2\left(2\pi\right)^{\left(D+1\right)/2}}\frac{\Gamma\left(\left(D-1\right)/2\right)}{\left(\cosh\zeta-\cosh\Delta\bar{\eta}\right)^{\left(D-1\right)/2}}.$
(19)

For the further transformation we will consider the odd and even values for
the spatial dimension $D$ separately.

For even values of spatial dimension, taking $q=D/2-1$, the Wightman function
is expressed as

$W_{\text{{C}}}(x,x^{\prime})=-\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{2\left(-2\pi\right)^{D/2}}\left(\frac{\partial_{\zeta}}{\sinh\zeta}\right)^{D/2-1}\int_{0}^{\infty}dy\,\tanh\left(\pi
y\right)e^{iy\Delta\bar{\eta}}P_{iy-1/2}\left(\cosh\zeta\right).$ (20)

The corresponding function for the Minkowski vacuum is presented in the form

$W_{\text{{M}}}(x,x^{\prime})=-\frac{\left(tt^{\prime}\right)^{\left(1-D\right)/2}}{2^{3/2}\left(-2\pi\right)^{D/2}}\left(\frac{\partial_{\zeta}}{\sinh\zeta}\right)^{D/2-1}\frac{1}{\sqrt{\cosh\zeta-\cosh\Delta\bar{\eta}}}.$
(21)

For the evaluation of the difference of the Wightman functions it is
convenient to present $W_{\text{{M}}}(x,x^{\prime})$ in an integral form

$W_{\text{{M}}}(x,x^{\prime})=-\frac{\left(tt^{\prime}\right)^{\left(1-D\right)/2}}{2\left(-2\pi\right)^{D/2}}\left(\frac{\partial_{\zeta}}{\sinh\zeta}\right)^{D/2-1}\int_{0}^{\infty}dy\,\cos\left(y\Delta\bar{\eta}\right)P_{iy-1/2}\left(\cosh\zeta\right).$
(22)

Here we have used the relation [16]

$\int_{0}^{\infty}dy\,\cos\left(yv\right)P_{iy-1/2}\left(w\right)=\frac{1}{\sqrt{2}\sqrt{w-\cosh
v}},$ (23)

valid for $w>\cosh v$. Thus, for the difference of the Wightman functions we
get

$\Delta
W(x,x^{\prime})=\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{2\left(-2\pi\right)^{D/2}}\int_{0}^{\infty}dy\left[\frac{2\cos\left(y\Delta\bar{\eta}\right)}{e^{2\pi
y}+1}-i\tanh\left(\pi
y\right)\sin\left(y\Delta\bar{\eta}\right)\right]\frac{P_{iy-1/2}^{D/2-1}\left(\bar{u}\right)}{(\bar{u}^{2}-1)^{(D-2)/4}},$
(24)

where the relation
$\partial_{\bar{u}}^{D/2-1}P_{iz-1/2}\left(\bar{u}\right)=(\bar{u}^{2}-1)^{(2-D)/4}P_{iz-1/2}^{D/2-1}\left(\bar{u}\right)$
was used.

Now we pass to the odd values of $D$. For this case in (18) we take
$q=(D-3)/2$ and the associated Legendre function in the integrand of (18) is
reduced to the function $P_{iy-1/2}^{-1/2}\left(\bar{u}\right)$. The latter is
expressed in terms of elementary functions and for
$\left(\Delta\bar{\eta}\right)^{2}\neq$ $\zeta^{2}$ one gets

$W_{\text{{C}}}(x,x^{\prime})=\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{2\left(-2\pi\right)^{\left(D+1\right)/2}}\left(\frac{\partial_{\zeta}}{\sinh\zeta}\right)^{\left(D-3\right)/2}\frac{2\zeta/\sinh\zeta}{\zeta^{2}-\left(\Delta\bar{\eta}\right)^{2}},$
(25)

where $\zeta$ is defined in accordance with (6). For a massless field and for
odd values of $D$ the Wightman function (19) for the Minkowski vacuum is
presented as

$W_{\text{{M}}}(x,x^{\prime})=\frac{\left(tt^{\prime}\right)^{\left(1-D\right)/2}}{2\left(-2\pi\right)^{\left(D+1\right)/2}}\left(\frac{\partial_{\zeta}}{\sinh\zeta}\right)^{\left(D-3\right)/2}\frac{1}{\cosh\zeta-\cosh\Delta\bar{\eta}}.$
(26)

Here we have used the relation (5). For the difference of the Wightman
functions, that determines the difference in the local VEVs, we obtain

$\Delta
W(x,x^{\prime})=\frac{\left(tt^{\prime}\right)^{(1-D)/2}}{2\left(-2\pi\right)^{\left(D+1\right)/2}}\left(\frac{\partial_{\zeta}}{\sinh\zeta}\right)^{\left(D-3\right)/2}\left[\frac{2\zeta/\sinh\zeta}{\zeta^{2}-\left(\Delta\bar{\eta}\right)^{2}}-\frac{1}{\cosh\zeta-\cosh\Delta\bar{\eta}}\right].$
(27)

This representation is well adapted for the evaluation of the differences
between local VEVs of the field squared and energy-momentum tensor.

In the discussion above we have considered the difference in the Wightman
functions for the C- and Minkowski vacua. Similar expressions are obtained for
the differences of other two-point functions. In particular the VEVs of the
field squared are obtained from the Hadamard function $G(x,x^{\prime})$. For
odd $D$ the corresponding expression is obtained from (27) with an additional
coefficient 2. In the case of even $D$ from (24) for the difference $\Delta
G(x,x^{\prime})=G_{\text{{C}}}(x,x^{\prime})-G_{\text{{M}}}(x,x^{\prime})$ we
get

$\Delta
G(x,x^{\prime})=\frac{2\left(tt^{\prime}\right)^{(1-D)/2}}{\left(-2\pi\right)^{D/2}}\int_{0}^{\infty}dy\frac{\cos\left(y\Delta\bar{\eta}\right)}{e^{2\pi
y}+1}\frac{P_{iy-1/2}^{D/2-1}\left(\bar{u}\right)}{(\bar{u}^{2}-1)^{(D-2)/4}}.$
(28)

##  4 VEV of the field squared

As a local characteristic of the C-vacuum first let us consider the VEV of the
field squared. It is obtained from the Hadamard function in the coincidence
limit as
$\Delta\left\langle\varphi^{2}\right\rangle=\lim_{x^{\prime}\rightarrow
x}\Delta G(x,x^{\prime})/2$. If we renormalize the corresponding VEV for the
Minkowski vacuum to zero,
$\left\langle\varphi^{2}\right\rangle_{\mathrm{M}}^{\mathrm{(ren)}}=0$, then
$\Delta\left\langle\varphi^{2}\right\rangle$ gives the renormalized VEV for
the C-vacuum,
$\Delta\left\langle\varphi^{2}\right\rangle=\left\langle\varphi^{2}\right\rangle_{\mathrm{C}}^{\mathrm{(ren)}}$.
For even values of $D$ we use the relation (see, for example, [17])

$\lim_{\bar{u}\rightarrow
1}\frac{P_{iy-1/2}^{D/2-1}\left(\bar{u}\right)}{\left(\bar{u}^{2}-1\right)^{(D-2)/4}}=\frac{2^{1-D/2}\Gamma\left(iy+\left(D-1\right)/2\right)}{\Gamma(D/2)\Gamma\left(iy-(D-3)/2\right)}.$
(29)

The expression for the VEV can be transformed to the form

$\Delta\left\langle\varphi^{2}\right\rangle=-\frac{2^{-D}t^{1-D}}{\pi^{D/2+1}\Gamma(D/2)}\int_{0}^{\infty}dy\,e^{-\pi
y}\left|\Gamma\left(iy+\left(D-1\right)/2\right)\right|^{2}.$ (30)

Alternatively, by using the properties of the gamma function we can see that

$\Delta\left\langle\varphi^{2}\right\rangle=-\frac{2(4\pi)^{-D/2}}{\Gamma(D/2)t^{D-1}}\int_{0}^{\infty}dy\,\frac{y^{D-2}A_{D}\left(y\right)}{e^{2\pi
y}+1},$ (31)

where for even $D$ we have defined

$A_{D}\left(y\right)=\prod\limits_{l=0}^{D/2-2}\left[\left(l+1/2\right)^{2}/y^{2}+1\right].$
(32)

As seen, the VEV is always negative.

For odd values of $D$ it is convenient firstly to put $\Delta\bar{\eta}=0$,
$\theta=0$ in the expression (27). With this choice we have $\zeta=\Delta r$.
The VEV of the field squared is presented as

$\Delta\left\langle\varphi^{2}\right\rangle=-\frac{b_{D}t^{1-D}}{12(2\pi)^{\left(D+1\right)/2}}\,,$
(33)

where the coefficient $b_{D}$ is defined by the relation

$b_{D}=6(-1)^{\frac{D-1}{2}}\lim_{\bar{u}\rightarrow
1}\partial_{\bar{u}}^{\left(D-3\right)/2}\left[\frac{2\left(\bar{u}^{2}-1\right)^{-1/2}}{\text{{arccosh}}\left(\bar{u}\right)}-\frac{1}{\bar{u}-1}\right].$
(34)

In particular,

$b_{3}=1,\;b_{5}=\frac{11}{30},\;b_{7}=\frac{191}{630},\;b_{9}=\frac{2497}{6300}.$
(35)

It is of interest to note that the expression (33) can also be written in the
form

$\Delta\left\langle\varphi^{2}\right\rangle=-\frac{2(4\pi)^{-D/2}}{\Gamma(D/2)t^{D-1}}\int_{0}^{\infty}dy\,\frac{y^{D-2}A_{D}\left(y\right)}{e^{2\pi
y}-1},$ (36)

where for odd $D$

$A_{D}\left(y\right)=\prod\limits_{l=0}^{\left(D-3\right)/2}\left(l^{2}/y^{2}+1\right).$
(37)

As before, the VEV is negative. We can combine the expressions for even and
odd values of the spatial dimension in a single formula

$\Delta\left\langle\varphi^{2}\right\rangle=-\frac{B_{D}}{t^{D-1}},$ (38)

where

$B_{D}=\frac{2(4\pi)^{-D/2}}{\Gamma(D/2)}\int_{0}^{\infty}dy\,\frac{y^{D-2}A_{D}\left(y\right)}{e^{2\pi
y}+\left(-1\right)^{D}}.$ (39)

##  5 VEV of the energy-momentum tensor

In this section we will consider the VEV of the energy-momentum tensor.
Through the Einstein semiclassical equations it determines the backreaction of
quantum effects on the background geometry. By taking into account that for
the background under consideration the Ricci tensor is zero, the difference in
the VEVs for the C- and Minkowski vacua, $\Delta\left\langle
T_{ik}\right\rangle=\left\langle T_{ik}\right\rangle_{\text{{C}}}-\left\langle
T_{ik}\right\rangle_{\mathrm{M}}$, is evaluated on the basis of the formula

$\Delta\left\langle
T_{ik}\right\rangle=\frac{1}{2}\underset{x^{\prime}\rightarrow
x}{\lim}\partial_{i^{\prime}}\partial_{k}\Delta
G\left(x,x^{\prime}\right)+\left[\left(\xi-\frac{1}{4}\right)g_{ik}\Box-\xi\nabla_{i}\nabla_{k}\right]\Delta\left\langle\varphi^{2}\right\rangle,$
(40)

where $\nabla_{i}$ is the covariant derivative operator and $\xi$ is the
curvature coupling parameter. If we renormalize the VEV for the Minkowski
vacuum to zero, then $\Delta\left\langle T_{ik}\right\rangle$ gives the mean
energy-momentum tensor for the C-vacuum. Relations between the separate
components follow from the symmetry of the problem and from general relations.
First of all, the spatial geometry is isotropic and the stresses are equal,
$\Delta\left\langle T_{1}^{1}\right\rangle=\Delta\left\langle
T_{2}^{2}\right\rangle=\cdots=\Delta\left\langle T_{D}^{D}\right\rangle$.
Next, by taking into account that all the components depend on the time
coordinate only, from the covariant continuity equation
$\nabla_{k}\left\langle T_{i}^{k}\right\rangle=0$ we get

$\Delta\left\langle
T_{1}^{1}\right\rangle=\frac{1}{Dt^{D-1}}\partial_{t}\left(t^{D}\Delta\left\langle
T_{0}^{0}\right\rangle\right).$ (41)

Additionally, one has the trace relation

$\Delta\left\langle
T_{i}^{i}\right\rangle=D(\xi-\xi_{D})\Box\Delta\left\langle\varphi^{2}\right\rangle,$
(42)

where $\xi_{D}=(D-1)/(4D)$ is the curvature coupling parameter for a
conformally coupled field. By taking into account that
$\Delta\left\langle\varphi^{2}\right\rangle\propto 1/t^{D-1}$ it is easy to
see that $\Box\Delta\left\langle\varphi^{2}\right\rangle=0$ and, hence, the
VEV of the energy-momentum tensor is traceless. This leads to the relation
$\Delta\left\langle T_{0}^{0}\right\rangle=-D\Delta\left\langle
T_{1}^{1}\right\rangle$ between the energy density and the stresses. From this
relation and from (41) we get

$\Delta\left\langle
T_{i}^{k}\right\rangle=\frac{C_{D}}{t^{D+1}}\text{{diag}}\left(1,-1/D,\ldots,-1/D\right).$
(43)

The problem is reduced to the evaluation of the constant $C_{D}$.

We will evaluate the component $\Delta\left\langle T_{11}\right\rangle$. For
the derivative in the last term of (40) one obtains

$\nabla_{1}\nabla_{1}\Delta\left\langle\varphi^{2}\right\rangle=\left(D-1\right)\Delta\left\langle\varphi^{2}\right\rangle.$
(44)

For the evaluation of the first term we consider the cases of even and odd $D$
separately. For even $D$ the difference of the Hadamard functions is given by
(28). Taking $\theta=0$, $\Delta\eta=0$, in the coincidence limit we get

$\frac{1}{2}\lim_{x^{\prime}\rightarrow
x}\partial_{1^{\prime}}\partial_{1}\Delta
G\left(x,x^{\prime}\right)=-\frac{t^{1-D}}{\left(-2\pi\right)^{D/2}}\int_{0}^{\infty}dy\,\frac{1}{e^{2\pi
y}+1}\lim_{\bar{u}\rightarrow 1}\partial_{\Delta
r}^{2}\frac{P_{iy-1/2}^{D/2-1}\left(\cosh\Delta r\right)}{\sinh^{D/2-1}\Delta
r}.$ (45)

From the recurrence relations for the associated Legendre function the
following relation can be shown:

$\partial_{\Delta r}^{2}\frac{P_{iy-1/2}^{D/2-1}\left(\cosh\Delta
r\right)}{\sinh^{D/2-1}\Delta r}=\frac{P_{iy-1/2}^{D/2+1}\left(\cosh\Delta
r\right)}{\sinh^{D/2-1}\Delta r}+\cosh\Delta
r\frac{P_{iy-1/2}^{D/2}\left(\cosh\Delta r\right)}{\sinh^{D/2}\Delta r}.$ (46)

The contribution of the first term in the right-hand side tends to zero in the
limit $\Delta r\rightarrow 0$ and we get

$\frac{1}{2}\lim_{x^{\prime}\rightarrow
x}\partial_{1^{\prime}}\partial_{1}\Delta
G\left(x,x^{\prime}\right)=-\frac{2^{-D}\pi^{-D/2-1}}{D\Gamma(D/2)t^{D-1}}\int_{0}^{\infty}dy\,e^{-\pi
y}\left|\Gamma\left(iy+\left(D+1\right)/2\right)\right|^{2}.$ (47)

Substituting (44) and (47) into (40) with $i=k=1$ and comparing with (43) for
even values of $D$ one finds

$\displaystyle C_{D}$ $\displaystyle=$
$\displaystyle-\frac{\pi^{-D/2-1}}{2^{D}\Gamma(D/2)}\int_{0}^{\infty}dy\,e^{-\pi
y}\left|\Gamma\left(iy+\left(D-1\right)/2\right)\right|^{2}\left[y^{2}+D\left(D-1\right)\left(\xi_{D}-\xi\right)\right]$
(48) $\displaystyle=$
$\displaystyle-\frac{2^{1-D}\pi^{-D/2}}{\Gamma(D/2)}\int_{0}^{\infty}dy\,\frac{y^{D-2}A_{D}\left(y\right)}{e^{2\pi
y}+1}\left[y^{2}+D\left(D-1\right)\left(\xi_{D}-\xi\right)\right],$

where $A_{D}(y)$ is given by (32).

In the case of odd $D$, as in the case of the mean field squared, we can take
$\Delta\bar{\eta}=0$ and $\theta=0$. For the coincidence limit of the
derivative of the Hadamard function we get

$\lim_{x^{\prime}\rightarrow x}\partial_{1^{\prime}}\partial_{1}\Delta
G\left(x,x^{\prime}\right)=\frac{t^{1-D}}{\left(-2\pi\right)^{\left(D+1\right)/2}}\lim_{r^{\prime}\rightarrow
r}\partial_{\Delta
r}^{2}\partial_{\bar{u}}^{\left(D-3\right)/2}\left[\frac{1}{\bar{u}-1}-\frac{2\left(\bar{u}^{2}-1\right)^{-1/2}}{\text{{arccosh}}\left(\bar{u}\right)}\right].$
(49)

By taking into account that $\bar{u}=\cosh\Delta r$, we see that
$\partial_{\Delta
r}^{2}=\bar{u}\partial_{\bar{u}}+(\bar{u}^{2}-1)\partial_{\bar{u}}^{2}$. The
contribution of the last term vanishes in the limit $r^{\prime}\rightarrow r$
($\bar{u}\rightarrow 1$) and by using the definition (34) one obtains

$\lim_{x^{\prime}\rightarrow x}\partial_{1^{\prime}}\partial_{1}\Delta
G\left(x,x^{\prime}\right)=-\frac{b_{D+2}t^{1-D}}{6\left(2\pi\right)^{\left(D+1\right)/2}}.$
(50)

This result with the combination of (33) and (44) leads to the following
expression for the coefficient in (43):

$C_{D}=D\frac{\xi\left(D-1\right)b_{D}-b_{D+2}}{12(2\pi)^{\left(D+1\right)/2}}.$
(51)

It can be checked that the latter expression, valid for odd values of $D$, may
also be written in the integral form

$C_{D}=-\frac{2^{1-D}}{\pi^{D/2}\Gamma(D/2)}\int_{0}^{\infty}dy\,\frac{y^{D-2}A_{D}\left(y\right)}{e^{2\pi
y}-1}\left[y^{2}+D\left(D-1\right)\left(\xi_{D}-\xi\right)\right],$ (52)

with $A_{D}(y)$ defined by (32).

Let us compare the differences $\Delta\left\langle\varphi^{2}\right\rangle$
and $\Delta\left\langle T_{ik}\right\rangle$ with the differences in the
corresponding VEVs between the Fulling-Rindler and Minkowski vacua. The right
Rindler wedge is covered by the coordinates
$(\tau_{\mathrm{R}},\chi,\mathbf{x}_{\mathrm{R}})$, with
$-\infty<\tau_{\mathrm{R}}<+\infty$, $0\leq\chi<\infty$,
$\mathbf{x}_{\mathrm{R}}=(x_{\mathrm{R}}^{2},\ldots,x_{\mathrm{R}}^{D})$, and
the corresponding line element has the form

$ds_{\mathrm{M}}^{2}=\chi^{2}d\tau_{\mathrm{R}}^{2}-d\chi^{2}-d\mathbf{x}_{\mathrm{R}}^{2}.$
(53)

For a massless field, the difference of the mean field squared in the Fulling-
Rindler and Minkowski vacua is given by the expression [18]

$\left\langle\varphi^{2}\right\rangle_{\mathrm{FR}}-\left\langle\varphi^{2}\right\rangle_{\mathrm{M}}=-\frac{B_{D}}{\chi^{D-1}},$
(54)

where the coefficient $B_{D}$ is the same as in (39). The difference in VEV of
the energy-momentum tensor is expressed as (no summation over $i$)

$\left\langle T_{i}^{k}\right\rangle_{\mathrm{FR}}-\left\langle
T_{i}^{k}\right\rangle_{\mathrm{M}}=-\frac{2\delta_{i}^{k}(4\pi)^{-D/2}}{\Gamma(D/2)\chi^{D+1}}\int_{0}^{\infty}dy\,\frac{y^{D-2}A_{D}\left(y\right)}{e^{2\pi
y}+(-1)^{D}}f_{0}^{(i)}(y),$ (55)

where

$\displaystyle f_{0}^{(0)}(y)$ $\displaystyle=$ $\displaystyle-
Df_{0}^{(1)}(y)=y^{2}+D(D-1)\left(\xi_{D}-\xi\right),$ $\displaystyle
f_{0}^{(i)}(y)$ $\displaystyle=$
$\displaystyle-y^{2}/D+(D-1)^{2}\left(\xi_{D}-\xi\right),\;i=2,3,\ldots,D.$
(56)

For a conformally coupled massless field one gets

$\left\langle T_{i}^{k}\right\rangle_{\mathrm{FR}}-\left\langle
T_{i}^{k}\right\rangle_{\mathrm{M}}=C_{D}\chi^{-D-1}{\mathrm{diag}}\left(1,-1/D,\ldots,-1/D\right),$
(57)

with the same coefficient as in (43). For non-conformally coupled fields the
stresses for the Fulling-Rindler vacuum are anisotropic.

##  6 Conclusion

The Milne universe is well suited for studying various aspects of the
influence of background geometry on the properties of quantum fields. In
particular, the study of properties of vacuum state is of special interest. We
have investigated the local properties of the C-vacuum for a massless scalar
field. Among the most important local characteristics are the expectation
values of the field squared and energy-momentum tensor. They are obtained from
the two-point functions in the coincidence limit of the arguments. For the
renormalization of the VEVs the subtraction of the corresponding VEVs for the
Minkowski vacuum is sufficient. This is related to the fact that the Milne
universe is flat and the divergences in the VEVs for C- and Minkowski vacua
are the same.

For a massless scalar field we have derived relatively simple representations
for the difference in the Wightman and Hadamard functions for C- and Minkowski
vacua. The renormalized mean field squared for C-vacuum is directly obtained
from the difference taking the coincidence limit. For the evaluation of the
renormalized mean energy-momentum tensor we have used the formula (40). The
mean field squared is given by (38) with the coefficient (39). From the
symmetry of the problem it follows that VEV of the energy-momentum tensor
should have the form (43). In order to obtain the expression for the
coefficient $C_{D}$ we have evaluated the 11-component. The coefficient is
given by (48) for even $D$ and by (52) in odd spatial dimensions. The formulas
for the mean field squared and energy-momentum tensor show that, from the
point of view of the quantization procedure in terms of the mode functions
based on the line element (1), the Minkowski vacuum appears as a thermal
state. It is of interest to note that thermal factor is the Bose-Einstein one
for odd number of spatial dimensions and Fermi-Dirac type in even number of
spatial dimensions. We have emphasized that this feature is also present in
the relations between the VEVs in Fulling-Rindler and Minkowski vacua. Similar
features between the hyperbolic and Bunch-Davies vacua in de Sitter spacetime
have been discussed in [15, 19].

## Acknowledgments

I am indebted to Prof. Aram A. Saharian for constant support and valuable
advice. I am also grateful to Prof. Levon Sh. Grigoryan and Prof. Roland M.
Avagyan for encouraging discussions and productive colaboration.

## Funding

The work was supported by the Committee of Science of the Ministry of
Education, Science, Culture and Sport RA in the frames of the research
projects No. 20AA-1C005 and No. 21AG-1C047. The work was partly supported by
the ”Faculty Research Funding Program 2020” (PMI Science and Enterprise
Incubator Foundation).

## References

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  * [10] K. Yamamoto, T. Tanaka, M. Sasaki, Particle spectrum created through bubble nucleation and quantum field theory in the Milne universe. Phys. Rev. D 51, 2968 (1995).
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  * [12] A. Higuchi, S. Iso, K. Ueda, K. Yamamoto, Entanglement of the vacuum between left, right, future, and past: The origin of entanglement-induced quantum radiation. Phys. Rev. D 96, 083531 (2017).
  * [13] A.A. Saharian, T.A. Petrosyan, The Casimir densities for a sphere in the Milne universe. Symmetry 12, 619 (2020).
  * [14] P. Henrici, Addition theorems for teneral Legendre and Gegenbauer tunctions, Journal of Rational Mechanics and Analysis 4, 983 (1955).
  * [15] A. A. Saharian, T. A. Petrosyan and V. S. Torosyan, Mean field squared and energy-momentum tensor for the hyperbolic vacuum in dS spacetime (2021) arXiv:2110.06662 [hep-th].
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  * [18] A. A. Saharian, Polarization of the Fulling-Rindler vacuum by a uniformly accelerated mirror. Class. Quantum Grav. 19, 5039 (2002).
  * [19] A. A. Saharian and T. A. Petrosyan, Casimir densities induced by a sphere in the hyperbolic vacuum of de Sitter spacetime. Phys. Rev. D 104, 065017 (2021).


(almost) all $t\in(0,T]$. Thanks to Theorem 1.2, only the following two cases
may occur.

Case 1: $g(t,0+)=+\infty$. In this case, there exists $r_{*}>0$ such that
$\xi^{\prime}(g(t,r))=0$ for all $r\in(0,r_{*})$. Hence, we trivially have
$\lim_{\rho\to 0}A(t,\rho)=0$.

Case 2: $g(t,\cdot)\in L^{\infty}$. In this case, by Theorem 1.2, there exists
a neighbourhood $J$ of $t$ such that $f_{|J\times\mathbb{R}^{d}}$ is smooth.
In particular, $\partial_{r}g(t,\cdot)\in L^{\infty}(0,1)$. If $d>1$, the
conclusion $\lim_{\rho\to 0}A(t,\rho)=0$ then directly follows from the
definition of $A(t,\rho)$, while for $d=1$ we resort to the fact that
$\sup_{r\in(0,\rho)}|\partial_{r}g(t,r)|\to 0$ as $\rho\to 0$. ∎

###  4.3. Energy dissipation identity

An argument similar to that in the proof of Theorem 1.5 shows that isotropic
solutions satisfy the energy dissipation balance. In the anisotropic case, we
obtain an inequality.

###### Proof of Proposition 1.6.

Combining Lemmas 4.1 and 4.2 with the convergence properties of
$f_{\varepsilon}$ to $f$ in Proposition 1.1 (iii), we readily infer for all
$t>0$ the inequality

$\displaystyle\mathcal{H}(f(t))+\int_{0}^{t}\mathcal{D}(f(\tau))\,\mathrm{d}\tau\leq\mathcal{H}(f_{\mathrm{in}}),$
(4.24)

where

$\mathcal{D}(f):=\int_{\mathbb{R}^{d}}\tfrac{1}{h(f)}|\nabla
f+h(f)v|^{2}\,\mathrm{d}v.$

It remains to prove that in the isotropic case the above inequality holds with
an equality. Then, the asserted identity (1.14) follows by subtracting on both
sides the quantity $\mathcal{H}(f(s))$, which is then known to equal
$\mathcal{H}(f_{\mathrm{in}})-\int_{0}^{s}\mathcal{D}(f(\tau))\,\mathrm{d}\tau$.
Thus, in the remainder, we assume that $f_{\mathrm{in}}$ is isotropic.
Moreover, without loss of generality, we may assume hypotheses (H2’).
Otherwise we replace $f_{\mathrm{in}}$ by $f(t_{0})$ for small $t_{0}>0$. From
the arguments below we will then obtain the identity
$\mathcal{H}(f(t))+\int_{t_{0}}^{t}\mathcal{D}(f(\tau))\,\mathrm{d}\tau=\mathcal{H}(f(t_{0})),$
and taking the limit $t_{0}\downarrow 0$, using monotone convergence for the
term $\int_{t_{0}}^{t}\mathcal{D}(f(\tau))\,\mathrm{d}\tau$ and dominated
convergence for $\mathcal{H}(f(t_{0}))$ we will arrive at the assertion.

As in the proof of Theorem 1.5 (cf. Section 4.2), we pick a non-decreasing
function $\varphi\in C^{\infty}([0,\infty);[0,1])$ satisfying $\varphi(r)=0$
for $r\in[0,1]$ and $\varphi(r)=1$ for $r\geq 2$, and abbreviate
$\varphi_{\rho}(r)=\varphi(r/\rho)$ for $\rho\in(0,1]$. Then, defining

$\displaystyle\mathcal{H}^{(\rho)}(f):=\int_{{\mathbb{R}^{d}}}[\,\tfrac{1}{2}|v|^{2}f+\Phi(f)\,]\varphi_{\rho}(|v|)\,\mathrm{d}v,$
(4.25)

one has

$\displaystyle\mathcal{H}^{(\rho)}(f(t))-\mathcal{H}^{(\rho)}(f_{\mathrm{in}})$
$\displaystyle=\int_{0}^{t}\\!\int_{{\mathbb{R}^{d}}}[\,\tfrac{1}{2}|v|^{2}+\Phi^{\prime}(f)\,]\,\mathrm{div}(\nabla
f+h(f)v)\varphi_{\rho}(|v|)\,\mathrm{d}v\mathrm{d}\tau$ (4.26)
$\displaystyle=-\int_{0}^{t}\\!\int_{{\mathbb{R}^{d}}}\frac{1}{h(f)}|\nabla
f+h(f)v|^{2}\varphi_{\rho}(|v|)\,\mathrm{d}v\mathrm{d}\tau$ (4.27)
$\displaystyle\qquad-\int_{0}^{t}\\!\int_{{\mathbb{R}^{d}}}[\,\tfrac{1}{2}|v|^{2}+\Phi^{\prime}(f)\,]\,(\nabla
f+h(f)v)\cdot\tfrac{v}{|v|}\varphi_{\rho}^{\prime}(|v|)\,\mathrm{d}v\mathrm{d}\tau.$
(4.28)

We note that

$\displaystyle\lim_{\rho\to
0}\mathcal{H}^{(\rho)}(f(t))=\mathcal{H}(f(t))\text{ for all }t\geq 0.$ (4.29)

Furthermore, monotone convergence gives

$\displaystyle\lim_{\rho\to
0}\int_{0}^{t}\\!\int_{{\mathbb{R}^{d}}}\frac{1}{h(f)}|\nabla
f+h(f)v|^{2}\varphi_{\rho}(|v|)\,\mathrm{d}v\mathrm{d}\tau=\int_{0}^{t}\\!\int_{{\mathbb{R}^{d}}}\frac{1}{h(f)}|\nabla
f+h(f)v|^{2}\,\mathrm{d}v\mathrm{d}\tau.$ (4.30)

Hence, it remains to prove that the quantity

$\displaystyle
B(\tau,\rho):=\int_{{\mathbb{R}^{d}}}[\,\tfrac{1}{2}|v|^{2}+\Phi^{\prime}(f)\,]\,(\nabla
f+h(f)v)\cdot\tfrac{v}{|v|}\varphi_{\rho}^{\prime}(|v|)\,\mathrm{d}v$ (4.31)

satisfies

$\displaystyle\lim_{\rho\to 0}\int_{0}^{t}B(\tau,\rho)\,\mathrm{d}\tau=0.$
(4.32)

Using the isotropy of $f(\tau,\cdot)$ we write

$\displaystyle
B(\tau,\rho)=c_{d}\int_{0}^{2\rho}[\,\tfrac{1}{2}r^{2}+\Phi^{\prime}(g)\,]\,(\partial_{r}g+h(g)r)\varphi_{\rho}^{\prime}(r)\,r^{d-1}\mathrm{d}r,$
(4.33)

where $c_{d}$ denotes the area of the unit sphere. We can now argue similarly
as in the proof of Theorem 1.5 (see page 4.2). The function $|B(\tau,\rho)|$
is uniformly bounded for $(\tau,\rho)\in[0,t]\times(0,1]$ thanks to the
estimate $|(\partial_{r}g+rh(g))r^{d-1}|\leq K_{*}$ and the fact that, by
(H2’) and Proposition 1.1 (ii), $\inf_{[0,t]\times(0,1]}g=:\iota>0$. Indeed,
note that for $(\tau,\rho)\in[0,t]\times(0,1]$

$\displaystyle|B(\tau,\rho)|\leq
C(\Phi^{\prime}(\iota))K_{*}\int_{0}^{2\rho}|\varphi^{\prime}(\rho^{-1}r)|\rho^{-1}\mathrm{d}r=C(\Phi^{\prime}(\iota))K_{*}\int_{0}^{2}|\varphi^{\prime}(\hat{r})|\mathrm{d}\hat{r}.$
(4.34)

Identity (4.32) therefore follows from the dominated convergence theorem
provided we can prove the pointwise convergence $\lim_{\rho\to
0}B(\tau,\rho)=0$ for a.e. $\tau>0$.

Case 1: $g(\tau,0+)=+\infty$. In this case, we estimate

$\displaystyle|B(\tau,\rho)|\leq
CK_{*}\int_{0}^{2}|\varphi^{\prime}(\hat{r})|\mathrm{d}\hat{r}\cdot\sup_{r\in(0,\rho)}|\tfrac{1}{2}r^{2}+\Phi^{\prime}(g(\tau,r))|$
(4.35)

and note that the sublinearity of $\Phi$ at infinity implies

$\displaystyle\sup_{r\in(0,\rho)}|\tfrac{1}{2}r^{2}+\Phi^{\prime}(g(\tau,r))|\to
0\quad\text{ as }\rho\to 0.$ (4.36)

Hence, $\lim_{\rho\to 0}|B(\tau,\rho)|=0$.

Case 2: $g(\tau,\cdot)\in L^{\infty}$. Here, the assertion $\lim_{\rho\to
0}|B(\tau,\rho)|=0$ is obtained similarly as in Case 2 of the proof of Theorem
1.5 using the regularity of $g(\tau,\cdot)$ shown in Theorem 1.2. ∎

##  5\. Long-time behaviour

###  5.1. Relaxation to equilibrium

###### Proof of Theorem 1.7.

Let

$\mathcal{D}(f):=\int_{\mathbb{R}^{d}}\frac{1}{h(f)}|\nabla
f+h(f)v|^{2}\,\mathrm{d}v=\int_{\mathbb{R}^{d}}h(f)|\nabla\delta\mathcal{H}(f)|^{2}\,\mathrm{d}v.$

Proposition 1.6 implies that
$\int_{0}^{\infty}\mathcal{D}(f(t))\,\mathrm{d}t\leq\mathcal{H}(f_{\mathrm{in}})-\inf_{\mathcal{M}_{+}}\mathcal{H}<\infty$,
and hence there exists an increasing sequence $t_{k}\to\infty$ such that

$\displaystyle\lim_{k\to\infty}\mathcal{D}(f(t_{k}))=0.$ (5.1)

The sequence $\\{\mu_{t_{k}}\\}_{k}\subset\mathcal{M}_{+}(\mathbb{R}^{d})$ of
measures of mass $m$ is tight since
$\sup_{k}\int|v|^{3}\mathrm{d}\mu_{t_{k}}\leq
C\|f_{\mathrm{in}}\|_{L^{1}_{3}}<\infty$. Prokhorov’s theorem thus ensures the
existence of a measure $\mu_{\infty}\in\mathcal{M}_{+}(\mathbb{R}^{d})$ with
$\int_{\mathbb{R}^{d}}\mathrm{d}\mu_{\infty}=m$ such that, along a subsequence
(not relabelled),

$\displaystyle\mu_{t_{k}}$
$\displaystyle\overset{\ast}{\rightharpoonup}\mu_{\infty}\quad\text{ in
}\mathcal{M}_{+}(\mathbb{R}^{d}),$ (5.2) $\displaystyle\text{ and moreover,
}\qquad\int|v|^{2}\mathrm{d}\mu_{t_{k}}$
$\displaystyle\to\int|v|^{2}\mathrm{d}\mu_{\infty}.$ (5.3)

At the same time, by Proposition 1.1 (ii),
$\mu_{t_{k}}=a(t_{k})\delta_{0}+f(t_{k})\mathcal{L}^{d}$ where $f$ satisfies a
time-uniform bound of the form $|f(t,r)|\leq C(\rho)$ for all $r\geq\rho>0$
(cf. L. 2.8 resp. Cor. 2.9). Thus, the sequence $f_{k}(t):=f(t_{k}+t)$ obeys
an estimate analogous to (2.50) for
$G\subset\subset(-1,\infty)\times(\mathbb{R}^{d}\setminus\\{0\\})$, where we
assume without loss of generality that $t_{1}\geq 1$. We may therefore argue
similarly as in the proof of Proposition 1.1 and invoke the Arzelà–Ascoli
theorem to infer the existence of $f_{\infty}\in
C^{2}(\mathbb{R}^{d}\setminus\\{0\\})\cap L^{1}_{3}(\mathbb{R}^{d})$ such
that, after possibly passing to another subsequence,

$\displaystyle f(t_{k})\to f_{\infty}\quad\text{ in
}C^{2}_{\mathrm{loc}}(\mathbb{R}^{d}\setminus\\{0\\}).$ (5.4)

Notice that $\mu_{\infty}^{\mathrm{reg}}=f_{\infty}\mathcal{L}^{d}$ and
$\mathop{\mathrm{supp}}\mu_{\infty}^{\mathrm{sing}}\subseteq\\{0\\}$, as a
consequence of (5.2) and (5.4).

We will now show that $\mu_{\infty}$ agrees with the minimiser of
$\mathcal{H}$ of mass $m$. To this end, let
$\lambda(s)=\exp(\Phi^{\prime}(s))=\frac{s}{(1+s^{\gamma})^{1/\gamma}}$ and
note that
$\frac{h(s)}{\lambda^{2}(s)}=\frac{s(1+s^{\gamma})^{1+2/\gamma}}{s^{2}}\geq(1+s^{\gamma})^{1+\frac{1}{\gamma}}\geq
1$ for all $s\in(0,\infty)$. Hence,

$\displaystyle\mathcal{D}(f)$
$\displaystyle=\int_{\mathbb{R}^{d}}h(f)|\nabla\Phi^{\prime}(f)+v|^{2}\,\mathrm{d}v$
(5.5)
$\displaystyle=\int_{\mathbb{R}^{d}}\tfrac{h(f)}{\lambda(f)^{2}}|\lambda(f)\nabla\Phi^{\prime}(f)+\lambda(f)v|^{2}\,\mathrm{d}v$
(5.6)
$\displaystyle\geq\int_{\mathbb{R}^{d}}|\nabla\lambda(f)+v\lambda(f)|^{2}\,\mathrm{d}v.$
(5.7)

Thanks to the convergence (5.1), the last estimate applied to $f:=f(t_{k})$
implies that

$\displaystyle\nabla\lambda(f(t_{k}))+v\lambda(f(t_{k}))\to 0\quad\text{ in
}L^{2}(\mathbb{R}^{d}).$ (5.8)

At the same time, using (5.4), we may pass to the limit in the sense of
distributions

$\displaystyle\nabla\lambda(f(t_{k}))+v\lambda(f(t_{k}))\to\nabla\lambda(f_{\infty})+v\lambda(f_{\infty})\quad\text{
in }\mathscr{D}^{\prime}(\mathbb{R}^{d}{\setminus}\\{0\\}).$ (5.9)

Thus, $\nabla\lambda(f_{\infty})+v\lambda(f_{\infty})=0$ in
$\mathscr{D}^{\prime}(\mathbb{R}^{d}{\setminus}\\{0\\})$, and hence
$\nabla\big{(}\mathrm{e}^{\frac{1}{2}|v|^{2}}\lambda(f_{\infty})\big{)}=0$ in
$\mathbb{R}^{d}{\setminus}\\{0\\}$. This implies that
$\Phi^{\prime}(f_{\infty})+\frac{1}{2}|v|^{2}\equiv-\theta$ for a constant
$\theta\in\mathbb{R}_{\geq 0}$, where the sign of $\theta$ follows from the
fact that $\Phi^{\prime}\leq 0$. Hence,
$f_{\infty}(v)=(\Phi^{\prime})^{-1}(-\frac{1}{2}|v|^{2}-\theta)=f_{\infty,\theta}(v)$.
To determine $\theta$, recall that $\mu_{\infty}(\mathbb{R}^{d})=m$. Thus, if
$\|f_{\infty,\theta}\|_{L^{1}(\mathbb{R}^{d})}<m$, then
$\mu_{\infty}(\\{0\\})>0$. In this case, the convergence (5.2) combined with
the time-uniform upper bound $\sup_{t>0}f(t,v)\leq C|v|^{-\frac{2}{\gamma}}\in
L^{1}(B_{r_{*}})$, which follows from Theorem 1.2 (resp. Cor. 3.4), implies
the existence of $\underline{k}\in\mathbb{N}$ such that
$\mu_{t_{k}}(\\{0\\})>0$ for all $k\geq\underline{k}$. Invoking once more
Theorem 1.2 (now using the radial symmetry assumption), we find that (1.10)
holds true for all such $t_{k}$ and, owing to the convergence (5.4), we
conclude that $\theta=0$. If on the other hand
$\|f_{\infty,\theta}\|_{L^{1}(\mathbb{R}^{d})}=m$, there is no excess mass and
we must have $\mu_{\infty}=f_{\infty,\theta}\mathcal{L}^{d}$. In conclusion,
we have shown that the measure $\mu_{\infty}$ coincides with the unique
minimiser $\mu_{\min}=\mu_{\min}^{(m)}$ of mass $m$.

From the convergence properties established so far we infer
$\lim_{k\to\infty}\mathcal{H}(\mu_{t_{k}})=\mathcal{H}(\mu_{\min})$. Since
$t\mapsto\mathcal{H}(\mu_{t})$ is non-increasing, this immediately yields

$\displaystyle\lim_{t\to\infty}\mathcal{H}(\mu_{t})=\mathcal{H}(\mu_{\min}).$
(5.10)

Combining this result with the above compactness properties, mass
conservation, and the uniqueness of the minimiser $\mu_{\min}^{(m)}$, one can
easily deduce the remaining convergence properties along any sequence
$t\to\infty$ as asserted in Theorem 1.7. Here, also recall the bounds
$\sup_{t>0}f(t,v)\lesssim|v|^{-\frac{2}{\gamma}}$ for $|v|\leq r_{*}$,
$f(t,v)\lesssim 1$ for $|v|\geq r_{*}$ and
$\sup_{t>0}\|f(t)\|_{L^{1}_{3}}\lesssim\|f_{\mathrm{in}}\|_{L^{1}_{3}}$, which
guarantee that $\lim_{t\to\infty}\mu_{t}(\\{0\\})=\mu_{\infty}(\\{0\\})$ and
$\lim_{t\to\infty}\|f(t)-f_{\infty,\theta}\|_{L^{p}(\mathbb{R}^{d})}=0$ for
$p\in[1,\frac{\gamma d}{2})$. ∎

###  5.2. Long-time and transient properties

Let us briefly point out some implications of the above analysis on further
qualitative dynamical properties, restricting for consistency to the isotropic
case. If $m<m_{c}$, Theorem 1.7 along with Theorem 1.2 imply the eventual
regularity of $\mu_{t}$ after some sufficiently large time $T\gg 1$. However,
using a contradiction argument, finite-time blow-up and the formation of a
condensate (that is $\mu_{t}(\\{0\\})>0$ for some $t>0$) can be shown to occur
for any size of the mass $m>0$ by choosing the smooth initial data
sufficiently concentrated near the origin (cf. [CHR20, Tos12]). Hence, there
exist flows exhibiting transient condensates with singular parts compactly
supported in time. On the other hand, whenever $m>m_{c}$, the above theory
implies the eventual formation of a condensate: $\exists T\gg 1$ such that
$\mu_{t}(\\{0\\})>0$ for all $t\geq T$. This is a consequence of the
convergence $\lim_{t\to\infty}\mu_{t}(\\{0\\})=\mu_{\min}(\\{0\\})$. It is
also possible to infer information on the spatiotemporal features of
singularity formation and regularisation using rescaling methods. We refer to
[Hop19, Chapter 5.2], where such dynamics have been shown to be of ‘type II’
for the 1D case.

Finally, we note that finite-time condensation in the mass-supercritical case
and convergence to the entropy minimiser can also be deduced in the
anisotropic setting if $f_{\mathrm{in}}$ admits a mass-supercritical isotropic
lower barrier, i.e. $f_{\mathrm{in}}\geq f_{\mathrm{in}}^{\\#}$ for some non-
negative radially symmetric function $f_{\mathrm{in}}^{\\#}$ with $\int
f_{\mathrm{in}}^{\\#}>m_{c}$. In this case, the density $f(t,\cdot)$ is
squeezed between two isotropic barriers which, in virtue of Theorem 1.7, both
converge to $f_{c}$ as $t\to\infty$.

###  5.3. Concluding remark

The comparison principle structure provides us with a priori bounds that allow
for a detailed characterisation of the singularities which isotropic flows
starting from regular data may exhibit (and even gives uniqueness in the 1D
case [CHR20, Hop19] resp. convergence of the scheme to a unique limit in
higher dimensions). However, one may not expect such a structure to persist in
more complex situations. Particularly with regard to the study of uniqueness
and stability properties in the presence of singularities, it would be
interesting to see whether variational problems like (fpγ) allow for more
robust approaches.

## Appendix A

Recall that $\mathcal{F}(t,v,w)=\mathrm{e}^{dt}G_{\nu(t)}(\mathrm{e}^{t}v-w)$,
$\nu(t)=\mathrm{e}^{2t}-1$,
$G_{\lambda}(\xi)=(2\pi\lambda)^{-\frac{d}{2}}\mathrm{e}^{-\frac{|\xi|^{2}}{2\lambda}}$.

######  Lemma A.1.

Let $T\leq 1$ and let $\tilde{q}\in[1,\infty]$. There exists
$C=C(\tilde{q},d)<\infty$ such that for all $t\in(0,T]$

$\displaystyle\begin{aligned}
\bigg{\|}\int_{\mathbb{R}^{d}}|\nabla_{v}\mathcal{F}(t,v,w)||\mathrm{e}^{-t}w-v||f(w)|\,\mathrm{d}w\bigg{\|}_{L^{\tilde{q}}(\mathbb{R}^{d})}&\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{d})}\\\ &\leq
C\nu(t)^{-\frac{1}{2}}\|f\|_{L^{\tilde{q}}(\mathbb{R}^{d})}.\end{aligned}$
(A.1)

###### Proof.

The second bound in (A.1) is trivial.

To verify the first inequality, we compute for $t\in(0,T]$

$\displaystyle\int_{\mathbb{R}^{d}}|\nabla_{v}$
$\displaystyle\mathcal{F}(t,v,w)||\mathrm{e}^{-t}w-v||f(w)|\,\mathrm{d}w$
(A.2)
$\displaystyle=(2\pi\nu(t))^{-\frac{d}{2}}\mathrm{e}^{dt}\mathrm{e}^{t}(2\nu(t))^{-\frac{1}{2}}\int_{\mathbb{R}^{d}}2\tfrac{|\mathrm{e}^{t}v-w|}{\sqrt{2\nu(t)}}\mathrm{e}^{-\frac{|\mathrm{e}^{t}v-w|^{2}}{2\nu(t)}}|\mathrm{e}^{-t}w-v||f(w)|\,\mathrm{d}w$
(A.3)
$\displaystyle=(2\pi\nu(t))^{-\frac{d}{2}}\mathrm{e}^{2dt}\mathrm{e}^{t}(2\nu(t))^{-\frac{1}{2}}\int_{\mathbb{R}^{d}}2\frac{|\mathrm{e}^{t}(v-\tilde{w})|}{\sqrt{2\nu(t)}}\mathrm{e}^{-\frac{|\mathrm{e}^{t}(v-\tilde{w})|^{2}}{2\nu(t)}}|\tilde{w}-v||f(\mathrm{e}^{t}\tilde{w})|\,\mathrm{d}\tilde{w}$
(A.4)
$\displaystyle=(2\pi\nu(t))^{-\frac{d}{2}}\mathrm{e}^{2dt}\int_{\mathbb{R}^{d}}2\frac{|\mathrm{e}^{t}(v-\tilde{w})|^{2}}{2\nu(t)}\mathrm{e}^{-\frac{|\mathrm{e}^{t}(v-\tilde{w})|^{2}}{2\nu(t)}}|f(\mathrm{e}^{t}\tilde{w})|\,\mathrm{d}\tilde{w}.$
(A.5)

Now, the asserted inequality follows upon an application of Young’s
convolution inequality, $\|a\ast
b\|_{L^{\tilde{q}}}\leq\|a\|_{L^{1}}\|b\|_{L^{\tilde{q}}}$. ∎

## Acknowledgements

The author would like to thank Prof. John King for valuable discussions on
this topic.

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