# Anisotropic particle creation from $T-$vacuum in the radiation dominated
universe

Dhamar S. Astilla<EMAIL_ADDRESS>Facultad de Ciencias - CUICBAS,
Universidad de Colima, Colima, C.P. 28045, México  Sujoy K. Modak
<EMAIL_ADDRESS>Facultad de Ciencias - CUICBAS, Universidad de Colima, Colima,
C.P. 28045, México  Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de México, POB 70-543, 04510, CDMX, México  Enrique Salazar
<EMAIL_ADDRESS>Instituto de Física, Universidad Autónoma de San Luis
Potosí, C.P. 78295, San Luis Potosí, México

(August 28, 2024)

###### Abstract

We further investigate novel features of the $T-$vacuum state, originally
defined in the context of quantum field theory in a (1+1) dimensional
radiation dominated universe Modak:2018usa . Here we extend the previous work
to a realistic (3+1) dimensional set up and show that $T-$vacuum causes an
_anisotropic particle creation_ in the radiation dominated early universe.
Unlike the Hawking or Unruh effect, where the particle content is thermal and
asymptotically defined, here it is non-thermal and time dependent. This novel
example of particle creation is interesting because these particles are
detected in the frame of physical/cosmological observers, who envision the
$T-$vacuum as a particle excited state, and therefore may eventually be
compared with observations.

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###### Contents

  1. I Introduction
  2. II Novel geometric features of the radiation dominated early universe
    1. II.1 Conformal vs spherically symmetric forms of the radiation stage
    2. II.2 Identification of cosmological observers in the $(T,R)$ frame
    3. II.3 Cosmological foliation in $(T,R)$ frame
    4. II.4 Hubble parameter and horizon behaviour in $(T,R)$ frame
    5. II.5 The relationship of $(T,R)$ frame with the static de Sitter frame
  3. III Field modes in conformally flat FRW
    1. III.1 Field modes for the $s-$wave approximation
    2. III.2 Field modes beyond the $s-$wave approximation
  4. IV Field modes in spherically symmetric FRW
    1. IV.1 Sub-Hubble region
      1. IV.1.1 Field modes for the s-wave approximation
      2. IV.1.2 Field modes beyond the s-wave approximation
    2. IV.2 Super-Hubble region
      1. IV.2.1 Field modes for the s-wave approximation
      2. IV.2.2 Field modes beyond the s-wave approximation
  5. V Particle creation from the $T-$vacuum
    1. V.1 Particle number density for the s-wave approximation
    2. V.2 Particle number density beyond the s-wave approximation
  6. VI Discussions and Outlook
  7. A The normalization constant
  8. B Bogolyubov coefficient for $\ell>0$

##  I Introduction

Within the framework of Quantum Field Theory in Curved Spacetime (QFTCS),
under certain symmetries of the spacetime, there may exist non-unique,
unitarily in-equivalent decomposition of quantum fields, which are usually
expressed using distinct coordinates. It is not guaranteed that every curved
spacetime would offer such quantizations in Fock space, but when they do, a
direct notable consequence of this is the existence of non-unique vacuum
states corresponding to different observers who are, in general, non-inertial
with respect to each other. Therefore, the meaning of particle is not
universally accepted – vacuum states for a set of observers could be
manifested as particle excited quantum states for other set of observers. The
only observers that agree with the notion of particle are the inertial
observers in the Minkowski/flat spacetime, who are related by Lorentz
transformations. However, no such unique global notion of particles can be
attributed for observers in a curved spacetime including the cosmological
setting.

This novel difference between the standard description of quantum field theory
(QFT) in a flat and curved space has inspired a generation of physicists to
look into various aspects of QFT in curved space and obtaining interesting
results of particle excitation in various settings. Among them, most well
known are particle creation by black holes hawk1 ; hawk2 , in a cosmological
settings park1 -park3  as well as for the accelerated observers in flat
spacetime (i.e., the Unruh effect unruh ). The work of Unruh pointed out that
an accelerated observer in Minkowski spacetime (with uniform four-acceleration
$a^{\mu}$) detects particles which are created by the excitation of the
Minkowski vacuum. Further, the resulting radiation flux follow a thermal
distribution with a well-defined temperature $T=|a|/2\pi$ (in natural units).
It is understood that the energy required to put the observer in an
accelerated motion is also responsible for disturbing the Minkowski vacuum and
produce particles. Although important, the details of how to put the observer
and the detector in accelerated motion is not essential for understanding the
basic result of the Unruh effect. A nice review of Unruh effect and related
works can be found in rev2 .

Recently, one of us showed Modak:2018usa  that an analogous situation can be
contemplated in the radiation dominated early universe, where
physical/cosmological observers detect radiation flux as a result of an
excitation of a novel “$T-$vacuum state”. This is analogous to the case where
accelerated observers detect particles created by the Minkowski vacuum state.
Although, similar, there exists an important difference with the Unruh effect
– in Unruh effect the source of energy driving the accelerated observer is not
included, but in the radiation dominated universe there is a gravitational
force and various observers have their own proper frames that are not globally
inertial. In fact, this $T-$vacuum is a natural state for a new set of
observers first introduced in Modak:2018crw  who experience the radiation
dominated universe as a spherically symmetric metric, inhomogeneous and
anisotropic spacetime. Various geometric aspects of the said metric were also
highlighted in Modak:2018crw ; Modak:2018usa . In Salazar:2021pfm , in a
relatively different study, we provided a cosmic identity of the observers for
whom such a metric appear natural. We showed that the natural observers
inhabiting such a spherically symmetric form of the metric can be backtracked
and identified with the static de Sitter observers in the previous
(inflationary) epoch. In other words, $T-$vacuum state turns out to be also a
natural vacuum state for the static de Sitter observers when, the universe has
made a transition to the next (radiation) epoch and, the spacetime is no
longer inflationary but dominated by radiation. On the field theory side,
regarding particle creation, a point-wise comparison was also made in a (1+1)
dimensional case Modak:2018usa  to clearly state the similarity and
differences with the Unruh effect. These studies on the geometric and field
theoretic aspects of the radiation dominated universe is a part of a broader
motivation of understanding various subtleties associated with the QFT in a
realistic universe with multiple epochs Singh:2013bsa ; Modak:2019jbg  of
expansion. In Modak:2018usa  we also showed that $T-$vacuum state is Hadamard
(no divergences or other pathological behaviors present) in (1+1) dimensions.

While the geometric construction of the above studies in the context of the
radiation dominated universe was performed considering (3+1) dimensions, the
field theoretic formulation on particle creation and renormalization was
confined only in (1+1) dimensions. In this work we extend the particle
creation by $T-$vacuum in a (3+1) dimensional set-up and leave the
renormalization properties for a future work ams . First, considering a
massless scalar field we derive various field equations and quantize the field
separately in the cosmological and spherically symmetric metrics describing
the radiation dominated universe. Some relational aspects of the Hubble
horizon in two metrics are carefully mentioned. We then explicitly calculate
the Bogolyubov coefficients and particle number density within the $s-$wave
approximation and compare the result with the two dimensional case. Finally,
we consider waves with arbitrary $l>0$ and manage to calculate particle number
densities analytically for the deep sub-Hubble modes. Naturally, the particle
distribution is a function of the conformal time $\eta$ (or the scale factor
$a$). Our result thus confirm an anisotropic, time-dependent particle creation
by $T-vacuum$ in the early universe and in the radiation dominated universe.
In principle, this particle creation should have a signature on the Cosmic-
Microwave-Background (CMB) especially because there is no such process like
inflation anymore that would wipe out tiny anisotropies. It will indeed be
interesting to see what cosmological implications can be reached by including
this gravitational particle creation but before enduring such a proposal one
might need to study the Hadamard property of $T-$vacuum to conclusively prove
that there are no pathologies involved with this new state which will be
addressed separately elsewhere ams .

The paper is organized as follows: in section II we present a brief review of
various geometric features related to the spherically symmetric form of the
radiation dominated universe. It includes the coordinate transformation,
identification of cosmological observers’ trajectory, spacetime foliation,
comparison of Hubble radius and connection with static de Sitter metric. In
the following sections, III and IV, we quantize massless scalar field in the
two frames relevant for this study. Section V provides the main result of
particle creation separately for the $s-$wave approximation and beyond the
$s-$wave. Finally, we conclude in VI. There are two useful appendices A and B
where we calculate the normalization constants and Bogolyubov coefficients.

##  II Novel geometric features of the radiation dominated early universe

In this section we review the key results of some of the our recent works
which are essential for the main study of particle creation in the radiation
dominated universe to be presented in the subsequent sections of this article.

###  II.1 Conformal vs spherically symmetric forms of the radiation stage

Let us start from the spatially flat FRW metric in comoving coordinates,

$ds^{2}=dt^{2}-a^{2}(t)[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})],$
(1)

which can be expressed in a conformal flat form

$ds^{2}=a^{2}(\eta)[d\eta^{2}-dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})].$ (2)

which we call here the “cosmological frame” $(\eta,r,\theta,\phi)$ where
$\eta=\int\frac{dt}{a(t)}$ is the conformal/cosmological time. In our
convention the scale factor $a(t)$ has a dimension of length while $\eta$ is
dimensionless ($t$ has dimension of time). Thus except the conformal factor
everything is dimensionless in (2).

We are working in a radiation dominated stage of the universe which comes
after the inflationary universe. Considering scale factors for these two
stages one can determine a time where this transition takes place
Singh:2013bsa . The idea is to make the scale factor equal and differentiable
at some transition point. that is, if we consider $a_{inf}=e^{{\cal H}t}$ and
$a_{rd}=a_{0}t^{1/2}$ and make them continuous and differentiable at some
$t=t_{r}$, we can fix $t_{r}=1/2{\cal H}$ and $a_{0}=\sqrt{2\mathcal{H}e}$
(where $\mathcal{H}$ is the inflationary Hubble constant). Substituting both
we can obtain the value of scale factor $a_{inf}=a_{rd}=\sqrt{e}$, the length
scale where the early universe transits from the inflationary to the radiation
dominated stage. The metric (2) will later be used for the construction of a
quantum field theory.

Now we want to express the radiation dominated universe in a spherically
symmetric and conformally static form which was discovered in Modak:2018crw .
First rewrite (2) by defining the lightcone coordinates $u=\eta-r$ and
$v=\eta+r$, so that equation (2) becomes

$ds^{2}=a^{2}(u,v)\Big{[}dudv-\frac{(v-u)^{2}}{4}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\Big{]}.$ (3)

Then perform the following conformal transformation of the cosmological null
coordinates for $a(t)\propto t^{1/2}$, given by Modak:2018crw

$U\equiv T-R=\pm\frac{\mathcal{H}e}{2}u^{2},\ \ \ V\equiv
T+R=\frac{\mathcal{H}e}{2}v^{2},$ (4)

where $+$ and $-$ signs stand for $u\geq 0$ and $u\leq 0$, respectively. This
transformation gives the following metrics for the sub-Hubble ($I$) and super-
Hubble ($II$) regions, described by

$ds^{2}=F_{I}(T,R)(dT^{2}-dR^{2})-R^{2}d\Omega^{2}$ (5)

for $U\geq 0$ ($T\geq R$), and

$ds^{2}=F_{II}(T,R)(dT^{2}-dR^{2})-T^{2}d\Omega^{2},$ (6)

for $U\leq 0$ ($T\leq R$). With the functions $F_{I}$ and $F_{II}$ as

$F_{I}(T,R)=\frac{(\sqrt{T+R}+\sqrt{T-R})^{2}}{4\sqrt{T^{2}-R^{2}}}$ (7)
$F_{II}(T,R)=\frac{(\sqrt{R+T}-\sqrt{R-T})^{2}}{4\sqrt{R^{2}-T^{2}}}$ (8)

The relationship between $(T,R)$ and $(\eta,r)$ frames is given by

$\displaystyle T$
$\displaystyle=\frac{V+U}{2}=\frac{\mathcal{H}e}{2}(\eta^{2}+r^{2})$ (9)
$\displaystyle R$ $\displaystyle=\frac{V-U}{2}=\mathcal{H}e\eta r$

for region $I$, and

$\displaystyle T$ $\displaystyle=\frac{V+U}{2}=\mathcal{H}e\eta r$ (10)
$\displaystyle R$
$\displaystyle=\frac{V-U}{2}=\frac{\mathcal{H}e}{2}(\eta^{2}+r^{2})$

for region $II$.

We can also express the conformal factors $F_{I}(T,R)$ and $F_{II}(T,R)$ as
functions of the Hubble parameter for radiation stage
$H=(\frac{\dot{a}}{a})_{RD}$ Modak:2018crw . These are

$F_{I}(H,R)=\frac{1}{1-H^{2}R^{2}}$ (11)

and

$F_{II}(H,T)=\frac{1}{H^{2}T^{2}-1}.$ (12)

The lightcone boundary $T=R$ for these observers is translated as the comoving
Hubble radius at $R=1/H$.

The metrics introduced in (5) and (6) are static up to a dynamical conformal
factor multiplying the $R-T$ sector and they exhibit a spherical symmetry. We
shall refer to this coordinate system as the $(T,R)$ frame to meaningfully
express our thoughts. The spherical symmetry in $(T,R)$ frame have some
interesting geometric and field theoretic implications. We shall discuss them
in detail in the remaining part of this work.

###  II.2 Identification of cosmological observers in the $(T,R)$ frame

We want to identify the physical cosmological observers in the $(T,R)$ since
their frame is often used in cosmology. It was shown in Modak:2018usa  that
for the sub-Hubble region ($I$) a worldline of the form

$T=G(R)=\alpha_{1}R^{2}+\beta_{1}$ (13)

with $\beta_{1}=1/4\alpha_{1}=\mathcal{H}e\eta^{2}_{0}/2$ is identified as the
cosmological static frame (at rest at an arbitrary but sub-Hubble point
$r=r_{0}$). These observers have a radial velocity
$\frac{dR}{dT}=\frac{1}{2\alpha_{1}R}$ and a non-zero position dependent
deceleration $\frac{d^{2}R}{dT^{2}}=-\frac{1}{4\alpha^{2}_{1}R^{3}}$. For the
super-Hubble region ($II$) cosmological observers at constant $r=r_{0}$ are
described by the following worldline

$R=G(T)=\alpha_{1}T^{2}+\beta_{1}$ (14)

with $\beta_{1}=1/4\alpha_{1}=\mathcal{H}er^{2}_{0}/2$. The radial velocity is
time dependent, $\frac{dR}{dT}=2\alpha_{1}T$, while the acceleration is
constant $\frac{d^{2}R}{dT^{2}}=2\alpha_{1}$ for this case. Therefore, if we
see the journey of the cosmological observer, in $(T,R)$ frame, starting from
the super-Hubble region, we find the said observers are accelerating at a
constant rate which lasts up to the Hubble entry. After the Hubble entry they
start decelerating and come to a rest, even in $(T,R)$ frame, as they approach
$\mathcal{I}^{+}$. As expected, the cosmological observers do not encounter
any horizon (due to a coordinate singularity) although the metric (5) and (6)
is fragmented into the sub-Hubble and super-Hubble regions.

###  II.3 Cosmological foliation in $(T,R)$ frame

While cosmological frame in $(\eta,r,\theta,\phi)$ has a straightforward
foliation of time and space slices with $\eta=$const. and $r=$const., one has
a slightly nontrivial case in $(T,R)$ frame where new time and space slices
are identified with $T=$const. and $R=$const. For a well defined initial value
formulation it is necessary that any spacetime is foliated by Cauchy slices.
Since we already have Cauchy problem well defined in the cosmological frame we
can prove that the same is true in the $(T,R)$ frame if we can embed
cosmological time and space slices using the $(T,R)$ metric. This task was
also completed in Modak:2018usa .

The first step is to identify timeslices $\eta=\eta_{0}$, for the sub-Hubble
region,

$T=\frac{R^{2}}{2\mathcal{H}e\eta^{2}_{0}}+\frac{1}{2}\mathcal{H}e\eta^{2}_{0},$
(15)

taking into account only the portion below the Semi-Latus-Rectum (SLR) of the
above parabola. For the super-Hubble region the timeslices will be a portion
of the following parabola

$R=\frac{T^{2}}{2\mathcal{H}e\eta^{2}_{0}}+\frac{1}{2}\mathcal{H}e\eta^{2}_{0}$
(16)

starting from the point where the SLR of (15) meet the Hubble horizon. Joining
these two slices along the Hubble horizon provides an embedding of the full
cosmological time slice in the $(T,R)$ frame. The spaceslices (timelike
hypersurfaces) are constructed in a similar way, just by taking $r=r_{0}$. In
region-$I$ the spaceslice is given by

$T=\frac{R^{2}}{2\mathcal{H}er^{2}_{0}}+\frac{1}{2}\mathcal{H}er^{2}_{0}$ (17)

while for the region-$II$ the spaceslice is

$R=\frac{T^{2}}{2\mathcal{H}er^{2}_{0}}+\frac{1}{2}\mathcal{H}er^{2}_{0},$
(18)

again joining at the Hubble radius. The union of these two portions is made by
joining the upper portion (of the SLR) of the parabola (17) with the left
portion (of the SLR) of the parabola (18).

1.

Figure 1: Full spacetime foliation of the $(T,R)$ plane with respect to
cosmological time and spaceslices as shown in Modak:2018usa . The blue slices
are the geodesics for a cosmological observer with $r=r_{0}=\text{const.}$ and
the red ones are the spacelike trajectories ($\eta=\eta_{0}=\text{const.}$).
The dashed line is the comoving Hubble radius.

With these cosmological Cauchy slices we can now foliate the full spacetime in
the $(T,R)$ plane. There is no horizon for cosmological observers as expected.
Looking into the blue curves in the diagram 1, now we can confirm that
cosmological observers start radially accelerating in asymptotic past while
looked at from a $(T,R)$ frame which is valid in the super-Hubble region.
After entering the Hubble radius they start decelerating and come to a rest in
the asymptotic future at $\mathcal{I}^{+}$. Here we first get an impression
that a vacuum state in $(T,R)$ frame cannot be vacuum for cosmological
observers since these observers are non-inertially connected with each other.

###  II.4 Hubble parameter and horizon behaviour in $(T,R)$ frame

It is important to realize that the transformed metrics (5) and (6)
representing the radiation dominated universe are both inhomogeneous and
anisotropic since they dependent on both $T$ and $R$. A direct consequence of
that will be the value of the Hubble parameter depending not only on new time
$T$ but also on the position of the observer $R$. We expect the Hubble horizon
to be manifested in a nontrivial manner while using these new coordinates.

Hubble parameter is calculated from the knowledge of the scalar factor $a(t)$
and its derivative ($H=\frac{\dot{a}}{a}$), meaning that an observer in any
part of the universe can measure the same value if they are in the same
timeslice. In the case of the radiation dominated universe we have
$H(t)=\frac{1}{2t}$, but since we are working on the cosmological frame
($\eta=\frac{2}{a_{0}}t^{1/2}$, where we are using
$a_{0}=\sqrt{2\mathcal{H}e}$) then the value of this parameter becomes

$H=\frac{2}{a^{2}_{0}\eta^{2}}=\frac{1}{\mathcal{H}e\eta^{2}}.$ (19)

Since $\eta$ has different expressions in sub and super Hubble regions due to
(9) and (10), in the $(T,R)$ frame we will have different expressions for the
Hubble parameter in those regions. In region $I$, using the coordinate
transformations in (9), we can express $\eta$ as

$\eta=\frac{1}{\sqrt{2\mathcal{H}e}}\Big{(}\sqrt{T+R}+\sqrt{T-R}\Big{)}.$ (20)

Using (19) and simplifying we obtain

$H_{I}(T,R)=\frac{1}{T+\sqrt{T^{2}-R^{2}}}$ (21)

for the region $I$, i.e., for $R<T$. For the region $II$, we use coordinate
transformation given in (10), and obtain a similar expression for $\eta$, and
by doing a similar process, we can obtain the value of the Hubble parameter
for this region, as

$H_{II}(T,R)=\frac{1}{R+\sqrt{R^{2}-T^{2}}}.$ (22)

This expression is valid for $R>T$.

We can easily observe that these expressions are continuous on $T=R$,
coinciding in the value $H=\frac{1}{T}=\frac{1}{R}$. Also we can observe, as
we expected, that these expressions are function of $T$ and $R$ coordinates,
so in general any two observers in different points on space or time measure
different values of Hubble parameter, this is due to the nature of the metric
which defines the $(T,R)$ spacetime which breaks homogeneity and isotropy.

As we have the expression for the Hubble parameter in terms of new $T$ and $R$
coordinates, we can express the radii for these two regions in both the sub-
Hubble and super-Hubble spacetime sections which are simply

$L^{I}_{H}=L^{I}_{P}=T_{0}+\sqrt{T^{2}_{0}-R^{2}_{0}}$ (23)

for region-I ($R<T$) and

$L^{II}_{H}=L^{II}_{P}=R_{0}+\sqrt{R^{2}_{0}-T^{2}_{0}}$ (24)

for the region-II ($R>T$). We can observe that now we need to measure both the
current time and radial position to get radii values instead of just measuring
the current time in cosmological frame. In the new frame the Hubble radius and
particle horizon depend on the position where and when they are measured.

###  II.5 The relationship of $(T,R)$ frame with the static de Sitter frame

In a previous work involving one of us Salazar:2021pfm  we showed an
interesting property of the metric (5) representing the $(T,R)$ frame. The
point is (5) can be backtracked and matched with the static de Sitter frame in
the preceding inflationary epoch with a time redefinition. Therefore, static
de Sitter observers in the inflationary epoch will find themselves closely
related with the $(T,R)$ frame when a transition to the radiation epoch has
already taken place. This identification will be helpful in a way that will be
made more clearer later on when we discuss particle creation. Most
importantly, this observation provides a valuable “cosmic identity” of these
new observers in the radiation dominated universe. To demonstrate how this
happens mathematically it was shown in Salazar:2021pfm  that de Sitter
universe in the static frame, given by

$ds^{2}=(1-{\cal H}^{2}R^{2}){d{\tau}^{2}}-\frac{dR^{2}}{1-{\cal
H}^{2}R^{2}}-R^{2}d\Omega^{2},$ (25)

smoothly transits into the following metric in the radiation epoch,

$ds^{2}=\frac{4e^{2}e^{4{\cal
H}\tau}d\tau^{2}-dR^{2}}{1-H^{2}R^{2}}-R^{2}d\Omega^{2}.$ (26)

a transition which takes place at the comoving time $t_{r}=1/2{\cal H}$ which
in $\tau,R$ coordinates is translated to $R=a(t_{r})r$ and
$\tau=\frac{1}{2{\cal H}}\log|\frac{1}{2}({\cal H}^{2}r^{2}-1/e)|$. We may
call (26) as the de Sitter “post-static” frame as described in the radiation
stage of the universe. We suggest an interested reader to read section 5 of
Salazar:2021pfm  for technical details. The static de sitter metric, as
represented by (26), has a striking similarity with the metric (5). If we
redefine the de Sitter time

$\tau=\frac{1}{2{\cal H}}\log({{\cal H}T}/{e})$ (27)

then we see that (26) matches exactly with (5). Therefore, one may interpret
that (5) is actually the de Sitter static frame living in the radiation epoch
with a logarithmic scaling of time. Note that (27) is purely temporal and does
not mix the time with space coordinates which will have a crucial role to
interpret the particle creation in even in this de Sitter static post-static
frame, to be discussed in the subsequent sections of the paper.

##  III Field modes in conformally flat FRW

Now that we have a geometric understanding of the radiation dominated universe
in both frames we turn our attention on the construction of quantum field
theory on them. In this section we consider the cosmological frame (2) where
the metric is conformally flat.

Consider a massless scalar field with arbitrary coupling in the radiation
stage and solve the Klein-Gordon equation $\Box\Phi=0$ (simply because $R=0$
and all fields become minimally coupled). We can express the scalar field
ansatz using the separation of variables as

$\Phi(\eta,r,\theta,\phi)=\sum_{\ell,m}\frac{f_{\ell}(r)}{r}g(\eta)Y_{\ell
m}(\theta,\phi)$ (28)

where the angular part $Y_{\ell m}$ are the spherical harmonics and the
$(\eta,r)$ dependent parts satisfy the following equations

$\eta^{2}\frac{d^{2}g}{d\eta^{2}}+2\eta\frac{dg}{d\eta}+\omega^{2}\eta^{2}g=0$
(29)

$\frac{d^{2}f_{\ell}}{dr^{2}}+\Bigg{(}\omega^{2}-\frac{\ell(\ell+1)}{r^{2}}\Bigg{)}f_{\ell}=0.$
(30)

At this point we shall separate our study for the $\ell=0$ or the $s-$waves
and for higher values of $\ell>0$ to make the comparison clear with the
spherically symmetric case.

###  III.1 Field modes for the $s-$wave approximation

Considering the s-wave approximation ($\ell=0$) in (30) and solving the
differential equations we can express the field $\Phi$ as

$\displaystyle\Phi$
$\displaystyle=\sum_{\omega}\mathcal{N}\frac{f_{0}(r)}{r}g_{\omega}(\eta)Y_{00}(\theta,\phi)$
(31)
$\displaystyle=\sum_{\omega}\mathcal{N}\Bigg{(}\frac{e^{-i\omega(\eta-r)}}{2\sqrt{\pi}\omega\eta
r}a^{in}_{\omega}+\frac{e^{-i\omega(\eta+r)}}{2\sqrt{\pi}\omega\eta
r}a^{out}_{\omega}\Bigg{)}+\text{h.c.}$

where $\mathcal{N}$ is a normalization constant determined by the
orthogonality condition between the field modes which is simply
$\mathcal{N}=\frac{\sqrt{\omega}}{\sqrt{4\pi}\mathcal{H}e}$. The superscripts
in and out implies the usual ingoing and outgoing modes. The field expansion
for $\Phi$ in the integral form can be written as

$\Phi=\int
d^{3}\omega(u_{\omega}a^{in}_{\omega}+v_{\omega}a^{out}_{\omega})+\text{h.c.}$
(32)

with the field modes $u_{\omega}$ and $v_{\omega}$ being

$u_{\omega}=\frac{e^{-i\omega(\eta-r)}}{4\pi\mathcal{H}e\sqrt{\omega}\eta r}$
(33)

and

$v_{\omega}=\frac{e^{-i\omega(\eta+r)}}{4\pi\mathcal{H}e\sqrt{\omega}\eta r}.$
(34)

###  III.2 Field modes beyond the $s-$wave approximation

The general solution for equation (30) when $\ell\neq 0$ has the form of

$f(r)=\sqrt{r}(C_{1}J_{\ell+1/2}(\omega r)+C_{2}Y_{\ell+1/2}(\omega r))$ (35)

where $J_{\ell+1/2}$ and $Y_{\ell+1/2}$ are Bessel functions of first and
second kind, respectively. The Bessel function of second kind
$Y_{\ell+1/2}(r)$ blows up at $r\to 0$, so we must choose $C_{2}=0$ and let us
rename $C_{1}={\mathcal{N}}$ so that that

$f_{\ell}(r)=\mathcal{N}\sqrt{r}J_{\ell+1/2}(\omega r),$ (36)

where $\mathcal{N}$ is the normalization constant determined by the
orthogonality condition between the field modes. The value of $\mathcal{N}$ is
calculated in the appendix A, giving,
$\mathcal{N}={\omega}/{\sqrt{2}\mathcal{H}e}$. For further simplification, the
Bessel function can be separated in terms in spherical Hankel functions such
that

$\displaystyle J_{\ell+1/2}(\omega r)$ $\displaystyle=\sqrt{\frac{2\omega
r}{\pi}}j_{\ell}(\omega r)$ $\displaystyle=\sqrt{\frac{2\omega
r}{\pi}}\frac{1}{2}\left(h^{(1)}_{\ell}(\omega r)+h^{(2)}_{\ell}(\omega
r)\right)$ $\displaystyle=\sqrt{\frac{\omega r}{2\pi}}\left(\frac{e^{i\omega
r}}{\omega r}Q_{\ell}(\omega r)+\frac{e^{-i\omega r}}{\omega
r}Q_{\ell}^{*}(\omega r)\right)$

Where the spherical Hankel functions were expanded as

$\displaystyle h^{(1)}_{\ell}(x)$
$\displaystyle=(-i)^{\ell+1}\frac{e^{ix}}{x}\sum_{s=0}^{\ell}\frac{(i/2)^{s}}{s!}\frac{(\ell+s)!}{(\ell-s)!}x^{-s}\equiv\frac{e^{ix}}{x}Q_{\ell}(x),$
(37) $\displaystyle h^{(2)}_{\ell}(x)$
$\displaystyle=(i)^{\ell+1}\frac{e^{-ix}}{x}\sum_{s=0}^{\ell}\frac{(-i/2)^{s}}{s!}\frac{(\ell+s)!}{(\ell-s)!}x^{-s}=\frac{e^{-ix}}{x}Q_{\ell}^{*}(x)$
(38)

Then, the complete field expansion is given by

$\Phi=\frac{1}{\mathcal{H}e\eta}\sum_{\ell,m}\int
d^{3}\omega\,\sqrt{\frac{\omega}{4\pi}}\left(a^{in}_{\omega
lm}\frac{e^{-i\omega(\eta-r)}}{\omega r}Q_{\ell}(\omega r)+a^{out}_{\omega
lm}\frac{e^{-i\omega(\eta+r)}}{\omega r}Q_{\ell}^{*}(\omega r)\right)Y_{\ell
m}(\theta,\phi)+h.c.$ (39)

From the above expression we can read-off the mode functions

$\displaystyle u_{\omega lm}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\omega}{4\pi}}\frac{e^{-i\omega
u}}{\mathcal{H}e\omega\eta r}Q_{\ell}(\omega r)Y_{\ell m}(\theta,\phi),$ (40)
$\displaystyle v_{\omega lm}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\omega}{4\pi}}\frac{e^{-i\omega
v}}{\mathcal{H}e\omega\eta r}Q^{*}_{\ell}(\omega r)Y_{\ell m}(\theta,\phi).$
(41)

The above two equations show that unlike the $s-$wave sector, here the
complete set of modes cannot be separated into a incoming and outgoing
sectors. The functions $Q_{\ell}(\omega r)$ and $Q_{\ell}^{*}(\omega r)$ are
multiplying the incoming and outgoing modes which is expected for a situation
where there is no spherical symmetry. This process of back-scattering is
nicely captured by $Q_{\ell}/Q_{\ell}^{*}$ multiplication to the incoming and
outgoing sectors.

##  IV Field modes in spherically symmetric FRW

In this section we quantize massless scalar fields in the background
geometries, given by (5) and (6) representing spherically symmetric forms of
the radiation dominated universe in the sub and super Hubble scales. We
present this calculation separately using the above metrics as backgrounds,
also separating the $s-$waves and $\ell>0$ cases.

###  IV.1 Sub-Hubble region

Since (5) is spherically symmetric, we can write the scalar field ansatz as

$\Phi_{\Omega}=\sum_{\ell,m}\frac{\Phi^{\ell m}_{\Omega}(T,R)}{R}Y_{\ell
m}(\theta,\phi)$ (42)

Then, the Klein-Gordon equation is

$\Bigg{(}\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
T^{2}}-\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
R^{2}}\Bigg{)}+\frac{\ell(\ell+1)}{R^{2}(1-H^{2}R^{2})}\Phi^{\ell
m}_{\Omega}=0,$ (43)

where we find an effective potential

$V_{\ell}(R,H)=\frac{\ell(\ell+1)}{R^{2}(1-H^{2}R^{2})}.$ (44)

For any $\ell\neq 0$ modes, this potential has an interesting behavior – it
diverges at $R=0$ and $R=1/H$, i.e., at the origin of the coordinate system
and also at the Hubble radius. While the divergence at the origin is a
characteristic of the spherical waves because of the choice of coordinate
system, the other divergence at $R=1/H$ is nontrivial. We need to understand
this in a finer detail. We know that whenever the potential tends to infinity
at some value of the coordinate, it means that the mode functions are confined
within this limit. That is, no $\ell\neq 0$ modes can pass to the super-Hubble
region, they will always be trapped inside the Hubble radius, i.e., in the
sub-Hubble region. Note that although our background spacetime is defined in
the sub-Hubble region, mode functions are global and there is, in principle,
no restriction to extend them beyond the Hubble scale. Peculiarity of the
potential (44) is interesting in this regard. It must also be added that
although in this new coordinates, it may appear that the Hubble horizon is
behaving just like a black hole event horizon where modes cannot get outside
from the inside of the event horizon, there is a crucial difference. The
potential barrier is mode specific – i.e., (44) tells us $V_{\ell}(R,H)=0$ for
$\ell=0$ modes and hence the $s-$wave modes see no potential whatsoever but
other modes do feel the potential barrier which is not a characteristic of an
event horizon.

Given the peculiarity of the potential (44) we now part ways in discussing the
$\ell=0$ and $\ell>0$ modes in the following discussion.

####  IV.1.1 Field modes for the s-wave approximation

If we consider the $\ell=0$ modes only we have from (43),

$\Bigg{(}\frac{\partial^{2}\Phi^{00}_{\Omega}}{\partial
T^{2}}-\frac{\partial^{2}\Phi^{00}_{\Omega}}{\partial R^{2}}\Bigg{)}=0.$ (45)

The field expansion is of the following form

$\Phi=\int
d^{3}\Omega(U^{\text{sub}}_{\Omega}b^{in}_{\Omega>\Omega_{H}}+V^{\text{sub}}_{\Omega}b^{out}_{\Omega>\Omega_{H}})+\text{h.c.}$
(46)

where the mode functions are

$U^{\text{sub}}_{\Omega}=\frac{1}{4\pi\sqrt{\Omega}R}e^{-i\Omega(T-R)}$ (47)

and

$V^{\text{sub}}_{\Omega}=\frac{1}{4\pi\sqrt{\Omega}R}e^{-i\Omega(T+R)}$ (48)

and are valid for $R\leq 1/H$. The operator $b^{in}_{\Omega>\Omega_{H}}$ is
defined for the incoming sub-Hubble modes and the state which is annihilated
by this operator is called the $T-$vacuum (for this sector) which satisfy
$b^{in}_{\Omega>\Omega_{H}}|0\rangle_{T}^{in}=0$. Similarly, one can define a
vacuum $b^{out}_{\Omega>\Omega_{H}}|0\rangle_{T}^{out}=0$ for the outgoing
sector.

Our focus in this paper will be to calculate particle creation from the
$T-$vacuum state when looked into from the cosmological frame described in the
last section.

####  IV.1.2 Field modes beyond the s-wave approximation

If we want to go beyond the $s-$waves, we need to solve (43) for arbitrary
$\ell>0$ which is clearly a difficult task because the Hubble parameter
$H\equiv H(T,R)$. It makes the differential equation quite non-trivial to
solve analytically, in general. Even if one uses a numerical method to solve
this equation, it is most likely to provide a solution that is not separable
in the positive and negative frequency modes and therefore, the solution does
not lead us to a Fock representation of particles. Therefore, intuitively
speaking there is not much motivation of doing that if we are interested in
the process of particle creation.

However, we can consider a curious case by restricting our interest in the
deep sub-Hubble region $R<<1/H$. In this case (43) becomes

$\Bigg{(}\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
T^{2}}-\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
R^{2}}\Bigg{)}+\frac{\ell(\ell+1)}{R^{2}}\Phi^{lm}_{\Omega}=0,$ (49)

By decomposing $\Phi^{\ell m}_{\Omega}(T,R)=\psi(T)\chi(R)$ and solve (49) it
is found that

$\displaystyle\frac{d^{2}\psi}{dT^{2}}$ $\displaystyle+\Omega^{2}\psi=0$ (50)
$\displaystyle\frac{d^{2}\chi}{dR^{2}}$
$\displaystyle+(\Omega^{2}-\frac{\ell(\ell+1)}{R^{2}})\chi=0$ (51)

it is clear that the solutions for $\psi(T)\propto e^{\pm i\Omega T}$, while
the differential equation for $\chi(T)$ it is the exact same form of (30), and
its solution is the Bessel function of first kind $\sqrt{R}J_{\ell+1/2}(\Omega
R)$. The normalization constant can be solved in a similar way as we did in
the previous case. This calculation is also shown in appendix A . The field
decomposition is therefore given by

$\Phi=\sum_{\ell,m}\int
d^{3}\Omega\,\sqrt{\frac{\Omega}{4\pi}}\left(b^{in}_{\Omega\ell
m}\frac{e^{-i\Omega(T-R)}}{\Omega R}Q_{\ell}(\Omega R)+b^{out}_{\Omega\ell
m}\frac{e^{-i\Omega(T+R)}}{\Omega R}Q_{\ell}^{*}(\Omega R)\right)Y_{\ell
m}(\theta,\phi)+\text{h.c.}$ (52)

We can also extract the mode functions for the above expansion which are

$\displaystyle U_{\Omega\ell m}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Omega}{4\pi}}\frac{e^{-i\Omega U}}{\Omega
R}Q_{\ell}(\Omega R)Y_{\ell m}(\theta,\phi)$ (53) $\displaystyle V_{\Omega\ell
m}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{\Omega}{4\pi}}\frac{e^{-i\Omega
V}}{\Omega R}Q^{*}_{\ell}(\Omega R)Y_{\ell m}(\theta,\phi).$ (54)

The functions $Q_{\ell}$ and $Q_{\ell}^{*}$ are already defined in (37). Once
again we see these coefficients represent back-scattering between the incoming
and outgoing field modes in (52).

###  IV.2 Super-Hubble region

We follow the same procedure as in the previous case, now with the metric (6)
to obtain the following field equation for the super-Hubble region

$\Phi_{\Omega}=\sum_{\ell,m}\frac{\Phi^{\ell m}_{\Omega}(T,R)}{T}Y_{\ell
m}(\theta,\phi)$ (55)

since the radius of the two sphere is now given by $T$ and not $R$. With the
above expansion, the $R,T$ dependent part of the field equation is given by

$\Bigg{(}\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
T^{2}}-\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
R^{2}}\Bigg{)}+\frac{\ell(\ell+1)}{T^{2}(H^{2}T^{2}-1)}\Phi^{\ell
m}_{\Omega}=0.$ (56)

Notice that the potential has a different form, as is expected because of the
change in the metric coefficients in the super-Hubble region, given by

$V_{\ell}(T,H)=\frac{\ell(\ell+1)}{T^{2}(H^{2}T^{2}-1)}.$ (57)

Once again, the potential diverges at the origin of the radial coordinate
(which is simply $T$ for this region) and also at $T=1/H$. If we solve
$T=1/H(T,R)$ we shall end up getting $R=1/H$ which is again telling us that
there is a divergence at the Hubble radius. Consequently, once again, we see a
peculiar behaviour just like the sub-Hubble case – the $s=0$ modes do not see
this potential but for $\ell>0$ all modes are confined outside the Hubble
radius and cannot enter the sub-Hubble region.

####  IV.2.1 Field modes for the s-wave approximation

For $\ell=0$, the solution of (56) is straightforward and the mode expansion
is

$\Phi=\int
d^{3}\Omega(U^{\text{sup}}_{\Omega}b^{in}_{\Omega<\Omega_{H}}+V^{\text{sup}}_{\Omega}b^{out}_{\Omega<\Omega_{H}})+\text{h.c.}$
(58)

which is natural in the super-Hubble region. Particular expressions for the
incoming and outgoing modes are

$U^{\text{sup}}_{\Omega}=\frac{1}{4\pi\sqrt{\Omega}T}e^{-i\Omega(T-R)}$ (59)

and

$V^{\text{sup}}_{\Omega}=\frac{1}{4\pi\sqrt{\Omega}T}e^{-i\Omega(T+R)}.$ (60)

Once again the quantum state annihilated by $b^{in}_{\Omega<\Omega_{H}}$ and
$b^{in}_{\Omega<\Omega_{H}}$ is the definition of $T-$vacuum in super-Hubble
scale for respective in and out sectors.

####  IV.2.2 Field modes beyond the s-wave approximation

Just like the sub-Hubble case the form of the potential (57) makes the
differential equation hard to solve in general. But it is simpler if we are
interested in deep super-Hubble modes for which $T<<1/H$, and (56) takes the
following form,

$\Bigg{(}\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
T^{2}}-\frac{\partial^{2}\Phi^{\ell m}_{\Omega}}{\partial
R^{2}}\Bigg{)}-\frac{\ell(\ell+1)}{T^{2}}\Phi^{\ell m}_{\Omega}=0.$ (61)

This is identical to (49) with an exchange $R\leftrightarrow T$, and therefore
has the same solution (52) – we just need to replace $R$ with $T$. The final
expression is

$\Phi=\sum_{\ell,m}\int
d^{3}\Omega\,\sqrt{\frac{\Omega}{4\pi}}\left(b^{in}_{\Omega\ell
m}\frac{e^{-i\Omega(T-R)}}{\Omega T}Q_{\ell}^{*}(\Omega T)+b^{out}_{\Omega\ell
m}\frac{e^{-i\Omega(T+R)}}{\Omega T}Q_{\ell}(\Omega T)\right)Y_{\ell
m}(\theta,\phi)+\text{h.c.}$ (62)

We can also extract the mode functions for the above expansion which are

$\displaystyle U_{\Omega\ell m}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Omega}{4\pi}}\frac{e^{-i\Omega U}}{\Omega
T}Q_{\ell}^{*}(\Omega T)Y_{\ell m}(\theta,\phi)$ (63) $\displaystyle
V_{\Omega\ell m}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Omega}{4\pi}}\frac{e^{-i\Omega V}}{\Omega
T}Q_{\ell}(\Omega T)Y_{\ell m}(\theta,\phi).$ (64)

To conclude, we now have complete set of mode functions for the spherically
symmetric form of the radiation dominated universe for the following cases –
(i) exact expressions (47), (48) and (59), (60) for the $s-$waves both in the
sub and super-Hubble regions, (ii) approximate analytical solutions (53), (54)
and (63), (64) for the deep sub-Hubble and deep super-Hubble modes for
$\ell>0$. We now move to the next section to use these results for calculating
particle creation from $T-$vacuum.

##  V Particle creation from the $T-$vacuum

We are interested to calculate the number density of particles, created by
$T-$vacuum, when viewed from a cosmological frame of reference. The first step
is to calculate the Bogolyubov coefficient $\beta_{\omega\Omega}$ and then use
it to get the particle number density.

By definition, the Bogolyubov coefficient has the following forms nsalas

$\beta_{\omega\Omega}^{in}=-(u_{\omega},U^{*}_{\Omega})=i\int_{\Sigma}d\Sigma^{\mu}\>(u_{\omega}\partial_{\mu}U_{\Omega}-U_{\Omega}\partial_{\mu}u_{\omega}).$
(65)

and

$\beta_{\omega\Omega}^{out}=-(v_{\omega},V^{*}_{\Omega})=i\int_{\Sigma}d\Sigma^{\mu}\>(v_{\omega}\partial_{\mu}V_{\Omega}-v_{\Omega}\partial_{\mu}v_{\omega}).$
(66)

for the incoming and outgoing sectors. In the above expressions $\Sigma$ is a
Cauchy hypersurface and $d\Sigma^{\mu}=n^{\mu}d\Sigma$; $d\Sigma$ is the
volume element associated with this Cauchy hypersurface and $n^{\mu}$ is a
future directed unit normal vector to $\Sigma$. In this case, we are taking
into account scalar field $\Phi$, which can be expanded into field modes as
expressed in cosmological and $(T,R)$ frames in section IV.

###  V.1 Particle number density for the s-wave approximation

Let us first consider the $s-$wave modes which are given by (33) and (34) for
the cosmological frame, whereas, for the $(T,R)$ frame we have relevant
expressions for the sub and super Hubble regions (appearing in (47), (48) and
(59), (60)). In both coordinates modes are separated into incoming and
outgoing sectors.

We first select the incoming modes which are functions of retarded light-cone
coordinates $u$ or $U$. Further, we consider the sub-Hubble modes for which we
should choose (33) and (47) for the calculation of Bogolyubov coefficient. It
is useful to consider the future null infinity $\mathcal{I}^{+}$ as the Cauchy
hypersurface. This choice leads us to
$d\Sigma=a^{2}(\eta)r^{2}dud\bar{\Omega}$ where $d\bar{\Omega}$ is the solid
angle element. Therefore,

$\beta_{\omega\Omega}^{in}=-(u_{\omega},U^{*}_{\Omega})=i\int_{\mathcal{I}^{+}}a^{2}(\eta)r^{2}dud\Omega\>(u_{\omega}\partial_{u}U_{\Omega}-U_{\Omega}\partial_{u}u_{\omega}).$
(67)

Doing an integration by parts and taking the limit corresponding the choice of
the hypersurface ($\eta\rightarrow+\infty$ and $r\rightarrow+\infty$), we can
write (67) as

$\beta^{in}_{\omega\Omega}=2i\int_{\mathcal{I}^{+}}a^{2}(\eta)r^{2}\>dud\bar{\Omega}\>u_{\omega}\partial_{u}(U_{\Omega}).$
(68)

with a scalar factor given by $a(\eta)=\mathcal{H}e\eta$. Simplifying (68) and
taking into account that $U=\pm\frac{\mathcal{H}e}{2}u^{2}$ for $u\geq 0$ and
$u\leq 0$ respectively, we can express $\beta_{\omega\Omega}$ as

$\beta^{in}_{\omega\Omega}=I_{1}+I_{2}$ (69)

where

$I_{1}=\frac{\mathcal{H}e}{2\pi}\sqrt{\frac{\Omega}{\omega}}\int^{\infty}_{0}du\>ue^{-i(\omega
u+\frac{\Omega\mathcal{H}e}{2}u^{2})}.$ (70)

and

$\displaystyle I_{2}$
$\displaystyle=-\frac{\mathcal{H}e}{2\pi}\sqrt{\frac{\Omega}{\omega}}\int^{0}_{-\infty}du\>ue^{-i(\omega
u-\frac{\Omega\mathcal{H}e}{2}u^{2})}$ (71) $\displaystyle=I^{*}_{1}.$ (72)

Now using the identity

$\int^{\infty}_{x_{0}}dx\ x^{s-1}e^{-bx}=e^{-s\log b}\Gamma[s,x_{0}];\ \
\text{Re}(b),\ \text{Re}(s)>0$ (73)

where $b=-i+\epsilon$ and $\log b=\log(1)-i\pi/2=-i\pi/2$, we can integrate
$I_{1}$ to to obtain

$\displaystyle\beta^{in}_{\omega\Omega}$
$\displaystyle=\frac{1}{\pi\sqrt{\Omega\omega}}\sin\Big(\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{missing})\Gamma\Big{[}1;\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{]}$
(74)
$\displaystyle-\frac{1}{2\pi\Omega}\sqrt{\frac{\omega}{\mathcal{H}e}}\Big{[}1+\sin\Big(\frac{\omega^{2}}{\Omega\mathcal{H}e}\Big{missing})\Big{]}^{1/2}\Gamma\Big{[}\frac{1}{2};\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{]}.$

The squared value is now given by

$\displaystyle|\beta_{\omega\Omega}^{in}|^{2}$
$\displaystyle=\frac{1}{\pi^{2}\Omega\omega}\sin^{2}\Big{(}\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{)}\Gamma^{2}\Big{[}1;\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{]}$
(75)
$\displaystyle+\frac{\omega}{4\pi^{2}\Omega^{2}\mathcal{H}e}\Big{[}1+\sin\Big(\frac{\omega^{2}}{\Omega\mathcal{H}e}\Big{missing})\Big{]}\Gamma^{2}\Big{[}\frac{1}{2};\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{]}$
$\displaystyle-\frac{1}{\pi^{2}\Omega^{3/2}\sqrt{\mathcal{H}e}}\sin\Big(\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{missing})\Big{[}1+\sin\Big(\frac{\omega^{2}}{\Omega\mathcal{H}e}\Big{missing})\Big{]}^{1/2}\Gamma\Big{[}1;\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{]}\Gamma\Big{[}\frac{1}{2};\frac{\omega^{2}}{2\Omega\mathcal{H}e}\Big{]}.$

We can notice that the above result for $|\beta_{\omega\Omega}^{in}|^{2}$ is
close (but not identical) to the (1+1) dimensional problem which was
calculated earlier in Modak:2018usa , which is expected since radial modes in
(3+1) dimensions is most close to a (1+1) dimensional problem. The cross terms
in (75) is a characteristic of the (3+1) dimensional case.

Figure 2: Plot of the average particle number density for modes fields in 3+1
dimensions versus the frequency $\omega$, from 0 to 0.1. This numerical
solution is made using an infrared cut-off of $\Omega=0.0001$ and taking
$\mathcal{H}=1$.

The final step to get the particle number density is to integrate the last
equation

$\langle 0|n_{\omega}|0\rangle_{T}=\int|\beta_{\omega\Omega}^{in}|^{2}d\Omega$
(76)

which unfortunately does not provide us a complete analytical solution. But it
is easy to plot the average particle number density by performing a numerical
integration. It is evident from (75) that the particle number density has a
infrared (IR) divergence which also appear in other cases of particle creation
such as in moving mirrors mir1 ; mir2 . We need to neglect modes with infinite
wavelengths, and an IR cut-off is used for performing the integration
numerically. We put an ad-hoc value $\Omega=0.0001$ for IR cut-off. Further we
put ${\cal H}=1$ for the plot which is shown in Figure 2. We again see the
behavior closely analogous to the (1+1) dimensional case reported previously
Modak:2018usa . It is easy to check that if one considers the outgoing modes
(34) and (48) to calculate Bogolyubov coefficient $\beta_{\Omega\omega}^{out}$
and then use it to calculate the particle content for outgoing sector in sub-
Hubble region, we once again obtain a result identical to the incoming sector
and the plot 2 still applies.

Notice that this result is obtained from the contribution of sub-Hubble field
modes $U^{\text{sub}}_{\Omega}$, and we can also look for the result in the
super-Hubble sector. If we now do the same calculation for the super-Hubble
modes $U^{\text{sup}}_{\Omega}$ in (60) and (34), together with the
corresponding transformation (10) we can realize that we obtain the same
integral for the relevant Bogolyubov coefficient, and therefore, we shall
obtain the same result, as depicted in Fig. 2 for the incoming sector in
super-Hubble region. The same is true for the outgoing $v-$modes in the super-
Hubble region. Therefore, we conclude that the plot 2 is generic and applies
for both the sub and super Hubble regions, and for the incoming and the
outgoing sectors.

###  V.2 Particle number density beyond the s-wave approximation

Let us now compute the particle number density for the most general case with
$\ell>0$. For this case we don’t have the luxury to calculate the Bogolyubov
coefficients or the particle content with all generalities. The only
situations where mode decomposition are possible in the $(T,R)$ frame, are
away from the Hubble radius, that is for both the deep sub-Hubble and the deep
super-Hubble modes ($R<<1/H$ and $R>>1/H$). These field expansions are shown
in (52) and (62). On the other hand, in the cosmological coordinates, mode
decomposition is valid for all sub Hubble and super Hubble scale as appear in
(39). Considering first, the incoming sector ($u-$modes) and deep sub-Hubble
region, we can express the Bogolyubov coefficient as

$\beta^{in}_{\omega\Omega\ell\ell^{\prime}mm^{\prime}}=i\int
a^{2}r^{2}dr\sin\theta d\theta d\phi(u_{\omega\ell
m}\partial_{\eta}U_{\Omega\ell^{\prime}m^{\prime}}-U_{\Omega\ell^{\prime}m^{\prime}}\partial_{\eta}u_{\omega\ell
m}).$ (77)

An explicit calculation of the integral (77) is lengthy and it is performed in
the appendix B. It is nice to see a closed form analytical result which is
given by

$|\beta^{in}_{\omega\Omega\ell}|^{2}=\frac{\eta\omega^{2}}{4\mathcal{H}e\Omega^{2}}J^{2}_{\ell-1/2}(\eta\omega)-\frac{2\ell\omega}{4\mathcal{H}e\Omega^{2}}J_{\ell-1/2}(\eta\omega)J_{\ell+1/2}(\eta\omega)+\frac{\ell^{2}+\eta^{2}\omega^{2}}{4\mathcal{H}e\Omega^{2}\eta}J^{2}_{\ell+1/2}(\eta\omega).$
(78)

The average number density of particles created by the $T-$vacuum is therefore
given by

$n_{\omega\ell}=\int{|\beta^{in}_{\omega\Omega\ell}|^{2}d\Omega}.$ (79)

Considering (78) it is straightforward to integrate and obtain once again a
close form expression for the particle content, given by

$\displaystyle\langle 0|n_{\omega\ell}|0\rangle_{T}$
$\displaystyle=\frac{1}{4{\cal
H}e\Omega_{0}\eta}\left(\eta^{2}\omega^{2}J^{2}_{\ell-1/2}(\eta\omega)-{2\ell\omega\eta}J_{\ell-1/2}(\eta\omega)J_{\ell+1/2}(\eta\omega)\right.$
(80)
$\displaystyle\left.+(\ell^{2}+\eta^{2}\omega^{2}){J^{2}_{\ell+1/2}(\eta\omega)}\right),$

where $\Omega_{0}$ is again the IR cutoff introduced in the integration limit
in equation (79). There is no ultra-violet divergence here which is
reassuring. Finally, we want to use the scale factor as the time parameter by
using its relationship with conformal time parameter given by $a(\eta)={\cal
H}e\eta$ for the radiation stage, and then the expression (80) takes the
following form

$\displaystyle\langle 0|_{T}n_{\omega\ell}|0\rangle_{T}$
$\displaystyle=\frac{1}{4a\Omega_{0}}\left((a\omega/{\cal
H}e)^{2}J^{2}_{\ell-1/2}(a\omega/{\cal H}e)-\frac{2\ell a\omega}{{\cal
H}e}J_{\ell-1/2}(a\omega/{\cal H}e)J_{\ell+1/2}(a\omega/{\cal H}e)\right.$
(81) $\displaystyle\left.+(\ell^{2}+a^{2}\omega^{2}/{\cal
H}^{2}e^{2}){J^{2}_{\ell+1/2}(a\omega/{\cal H}e)}\right).$

A few comments are in order regarding our final result (81) – first and
foremost, the particle content is manifestly anisotropic and now known for any
value of the angular momentum quantum number $\ell>0$. This is a direct
consequence of the fact that the $T-$vacuum is intrinsically anisotropic as it
is a natural state for observers who envision the radiation dominated universe
as such. Of course, these observers, who are the successors of static de
Sitter observers, do not see any particles as the quantum state is in vacuum
configuration for them. However, the cosmological observers with proper time
$\eta$ are bombarded with particles due its peculiar motion with respect to
$(T,R)$ frame. These particles are anisotropically distributed in the sky and
its distribution is given by (81). Evidently, this is not a blackbody
distribution and it is not at all a surprise since the relationship between
the cosmological and $(T,R)$ frames do not invoke an exponential redshift
relative to each other – rather it is a power law and quadratic in nature.
Furthermore, (81) is time dependent and it has an oscillatory behavior with
the scale factor which we depict in Figure (3).

Figure 3: Plot of the particle number density with the scale factor as appear
in equation (81). Particle number oscillates with the scale factor during the
expansion of the universe with a diminishing amplitude of oscillation and
finally approaching a saturation towards the end of the radiation epoch.

Notice again that if we consider the outgoing $v-$modes in (34) and (54) we
shall end up with the same result as (81). In addition, the result for the
deep super-Hubble modes (63) and (64) lead us to an identical result (81).
Thus we can conclude that (81) and figure (3) are generic behavior of the
anisotropic particle creation by the $T-$vacuum in appropriate limits.

##  VI Discussions and Outlook

We have made a significant step forward in introducing a physically appealing
anisotropic quantum vacuum state in the early universe. To our knowledge this
is the only quantum vacuum state either in the pre/post inflationary era which
manifestly breaks homogeneity and isotropy, even for cosmological observers.
Our results are however valid in the deep sub/super Hubble regions. In the
absence of $T-$vacuum there is nothing that may provide an observational
angular variance for an cosmological observer in the radiation dominated
epoch. But if the field is in $T-$vacuum then such observers naturally detect
anisotropic radiation.

This particular nature of $T-$vacuum raises a few interesting theoretical
implications and it is worth investigating further on it. One natural ambition
might be calculating some definite imprints of these anisotropies originating
from the $T-$vacuum on the CMB map. This prospect is borne by the fact that
the CMB map depicts the early universe in the matter dominated stage which is
after the radiation stage and therefore a suitable signature in the radiation
stage, such as in (81), may have some residual imprint on the CMB map itself.
It provides exact number of particles (such as photons) corresponding to a
given frequency and direction in the sky. This prediction is extremely precise
and not to be confused with the statistical average, which is usually done
while analysing CMB data, in the lack of a precise physical theory that might
be producing the CMB radiation. Of course, we do not claim anything about the
observed CMB anisotropy with the anisotropy calculated in (81) yet, but it is
curious that such an expression can be calculated from such a first principle
theory where a anisotropic photon background might be related with the
symmetry of quantum vacuum itself. We plan to report on these curious
observations in a future publication ams .

## Acknowledgements

Research of SKM is supported by CONAHCyT Grants CB/2017-18/A1-S-33440 and
CF/140630, Mexico. DSA acknowledges CONAHCyT Grant CB/2017-18/A1-S-33440 for
financial support.

##  Appendix A The normalization constant

Let us first express the normalization constant for the cosmological
coordinates. Applying the inner product to the field modes

$\displaystyle(u_{\omega 1},u_{\omega 2})$
$\displaystyle=-i\mathcal{N}^{2}\int
d\Sigma\frac{\sqrt{-g{{}_{{}_{\Sigma}}}}}{a}\sum_{\ell,m,\ell^{\prime},m^{\prime}}[\frac{e^{-i\omega_{1}\eta}}{\omega_{1}\eta}\frac{J_{\nu}(\omega_{1}r)}{\sqrt{r}}Y_{\ell
m}(\theta,\phi)(i\omega_{2}-\frac{1}{\eta})\frac{e^{i\omega_{2}\eta}}{\omega_{2}\eta}\frac{J_{\nu^{\prime}}(\omega_{2}r)}{\sqrt{r}}$
$\displaystyle\times
Y^{*}_{\ell^{\prime}m^{\prime}}(\theta,\phi)-\frac{e^{i\omega_{2}\eta}}{\omega_{2}\eta}\frac{J_{\nu^{\prime}}(\omega_{2}r)}{\sqrt{r}}Y^{*}_{\ell^{\prime}m^{\prime}}(\theta,\phi)(-i\omega_{1}-\frac{1}{\eta})\frac{e^{-i\omega_{1}\eta}}{\omega_{1}\eta}\frac{J_{\nu}(\omega_{1}r)}{\sqrt{r}}Y_{\ell
m}(\theta,\phi)]$
$\displaystyle=\mathcal{N}^{2}\int\sum_{\ell,m,\ell^{\prime},m^{\prime}}(\mathcal{H}e\eta)^{2}r^{2}\sin\theta
drd\theta
d\phi\frac{e^{-i(\omega_{1}-\omega_{2})\eta}}{\omega_{1}\omega_{2}\eta^{2}}$
$\displaystyle\qquad\times\frac{J_{\nu}(\omega_{1}r)J_{\nu^{\prime}}(\omega_{2}r)}{r}Y_{\ell
m}(\theta,\phi)Y^{*}_{\ell^{\prime}m^{\prime}}(\theta,\phi)\frac{(\omega_{1}+\omega_{2})}{\omega_{1}\omega_{2}}$
$\displaystyle=\mathcal{N}^{2}\int\sum_{\ell,m,\ell^{\prime},m^{\prime}}(\mathcal{H}e)^{2}r^{2}dre^{-i(\omega_{1}-\omega_{2})\eta}\frac{J_{\nu}(\omega_{1}r)J_{\nu^{\prime}}(\omega_{2}r)}{r}(\omega_{1}+\omega_{2})\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}$
$\displaystyle=\mathcal{N}^{2}\mathcal{H}^{2}e^{2}e^{-i(\omega_{1}-\omega_{2})\eta}\frac{(\omega_{1}+\omega_{2})}{\omega_{1}\omega_{2}}\int
rdrJ_{\nu}(\omega_{1}r)J_{\nu}(\omega_{2}r)$
$\displaystyle=\mathcal{N}^{2}\mathcal{H}^{2}e^{2}e^{-i(\omega_{1}-\omega_{2})\eta}\frac{(\omega_{1}+\omega_{2})}{\omega_{1}\omega_{2}}\times\frac{1}{\omega_{1}}\delta(\omega_{1}-\omega_{2})$
$\displaystyle\implies\mathcal{N}$
$\displaystyle=\frac{\omega}{\sqrt{2}\mathcal{H}e}.$

Similar calculation can be performed to evaluate the normalization constant
for the field expansion made in $T,R$ coordinate system. Once again, first
notice $\sqrt{-\gamma}d\Sigma^{\mu}=R^{2}\sin\theta dRd\theta d\phi$ and
therefore the inner product

$\displaystyle(U_{\Omega 1},U_{\Omega 2})$
$\displaystyle=-i\mathcal{N}^{2}\int
d\Sigma\sum_{\ell,m,\ell^{\prime},m^{\prime}}\frac{1}{R}[e^{-i\Omega_{1}T}J_{\nu}(\Omega_{1}R)Y_{\ell
m}(\theta,\phi)(i\Omega_{2})e^{i\Omega_{2}T}J_{\nu^{\prime}}(\Omega_{2}R)$
$\displaystyle\times
Y^{*}_{\ell^{\prime}m^{\prime}}(\theta,\phi)-e^{i\omega_{2}\eta}J_{\ell^{\prime}}(\omega_{2}r)Y^{*}_{\ell^{\prime}m^{\prime}}(\theta,\phi)(-i\Omega_{1})e^{-i\Omega_{1}T}J_{\nu}(\Omega_{1}R)Y_{\ell
m}(\theta,\phi)]$
$\displaystyle=\mathcal{N}^{2}\int\sum_{\ell,m,\ell^{\prime},m^{\prime}}R\sin\theta
dRd\theta d\phi e^{-i(\omega_{1}-\omega_{2})\eta}$ $\displaystyle\qquad\times
J_{\nu}(\Omega_{1}R)J_{\nu^{\prime}}(\Omega_{2}R)Y_{\ell
m}(\theta,\phi)Y^{*}_{\ell^{\prime}m^{\prime}}(\theta,\phi)(\Omega_{1}+\Omega_{2})$
$\displaystyle=\mathcal{N}^{2}\int\sum_{\ell,m,\ell^{\prime},m^{\prime}}RdRe^{-i(\Omega_{1}-\Omega_{2})T}J_{\nu}(\Omega_{1}R)J_{\nu^{\prime}}(\Omega_{2}R)(\Omega_{1}+\Omega_{2})\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}$
$\displaystyle=\mathcal{N}^{2}e^{-i(\omega_{1}-\omega_{2})T}(\Omega_{1}+\Omega_{2})\int
dRRJ_{\nu}(\Omega_{1}R)J_{\nu}(\Omega_{2}R)$ (82)
$\displaystyle=\mathcal{N}^{2}e^{-i(\omega_{1}-\omega_{2})T}(\Omega_{1}+\Omega_{2})\times\frac{1}{\Omega_{1}}\delta(\Omega_{1}-\Omega_{2})$
$\displaystyle\implies\mathcal{N}$ $\displaystyle=\frac{1}{\sqrt{2}}$

##  Appendix B Bogolyubov coefficient for $\ell>0$

The calculation of the Bogolyubov coefficient by taking a spacelike
hypersurface at $\eta=\text{constant}$. It should be recalled that the scale
factor is related with the conformal time by a linear relationship
$a(\eta)={\cal H}e\eta$. We start from the basic definition (77)

$\displaystyle\beta_{\omega\Omega\ell\ell^{\prime}mm^{\prime}}$
$\displaystyle=i\int a^{2}r^{2}dr\sin\theta d\theta d\phi(u_{\omega\ell
m}\partial_{\eta}U_{\Omega\ell^{\prime}m^{\prime}}-U_{\Omega\ell^{\prime}m^{\prime}}\partial_{\eta}u_{\omega\ell
m})$
$\displaystyle=i\frac{\sqrt{\mathcal{H}e\eta}}{2}e^{-i(\omega\eta+\tfrac{1}{2}\mathcal{H}e\Omega\eta^{2})}\int^{\infty}_{0}dr\;re^{-i\mathcal{H}e\Omega
r^{2}/2}J_{\ell+1/2}(\omega r)$
$\displaystyle\times\left[2\eta\frac{d}{d\eta}J_{\ell^{\prime}+1/2}(\mathcal{H}e\Omega\eta
r)+(1+2i\eta(\omega-\mathcal{H}e\Omega\eta))J_{\ell^{\prime}+1/2}(\mathcal{H}e\Omega\eta
r)\right]$
$\displaystyle\times\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}d\theta
d\phi\sin\theta Y_{\ell
m}(\theta,\phi)*Y_{\ell^{\prime}m^{\prime}}(\theta,\phi)$
$\displaystyle\implies\beta_{\omega\Omega\ell m}$
$\displaystyle=i(-1)^{m}\frac{\sqrt{\mathcal{H}e\eta}}{4}e^{-i(\omega\eta+\tfrac{1}{2}\mathcal{H}e\Omega\eta^{2})}\left[2\eta\frac{d}{d\eta}+(1+i2\eta(\omega-\mathcal{H}e\Omega\eta))\right]$
$\displaystyle\times\int^{\infty}_{0}dr\;re^{-i\mathcal{H}e\Omega
r^{2}/2}J_{\ell+1/2}(\omega r)J_{\ell^{\prime}+1/2}(\mathcal{H}e\Omega\eta r)$
$\displaystyle=i(-1)^{m}\frac{\sqrt{\mathcal{H}e\eta}}{2}e^{-i(\omega\eta+\tfrac{1}{2}\mathcal{H}e\Omega\eta^{2})}\left[2\eta\frac{d}{d\eta}+(1+i2\eta(\omega-\mathcal{H}e\Omega\eta))\right]\frac{(\tfrac{1}{2}\omega\mathcal{H}e\Omega\eta)^{\ell+1/2}}{\Gamma(\ell+1)\Gamma(1/2)}$
$\displaystyle\times\int^{\infty}_{0}\int^{\pi}_{0}e^{-(i\mathcal{H}e\Omega/2)^{2}r^{2}}\frac{J_{\ell+1/2}(\alpha
r)}{\alpha^{\ell+1/2}}r\sin^{2{\ell+2}}\phi d\phi$
$\displaystyle=i(-1)^{m}\tfrac{\sqrt{\mathcal{H}e\eta}}{2}e^{-i(\omega\eta+\tfrac{1}{2}\mathcal{H}e\Omega\eta^{2})}\left[2\eta\frac{d}{d\eta}+(1+i2\eta(\omega-\mathcal{H}e\Omega\eta))\right]\frac{\Gamma({\ell+3/2})}{2\pi(i\mathcal{H}e\Omega/2)^{2}\Gamma(2\ell+2)}$
$\displaystyle\times\left(\frac{\omega\mathcal{H}e\Omega\eta}{(i\mathcal{H}e\Omega/2)^{2}}\right)^{\ell+\tfrac{1}{2}}\int^{\pi}_{0}e^{-\frac{\alpha^{2}}{4(i\mathcal{H}e\Omega/2)^{2}}}\sin^{2\ell+1}(\phi)d\phi$
$\displaystyle=i(-1)^{m}\frac{\sqrt{\mathcal{H}e\eta}}{2}e^{-i(\omega\eta+\tfrac{1}{2}\mathcal{H}e\Omega\eta^{2})}\left[2\eta\frac{d}{d\eta}+(1+i2\eta(\omega-\mathcal{H}e\Omega\eta))\right]\times$
$\displaystyle\times\frac{(\tfrac{1}{4}\omega\mathcal{H}e\Omega\eta/(i\mathcal{H}e\Omega/2)^{2})^{\ell+\tfrac{1}{2}}}{2(i\mathcal{H}e\Omega/2)^{2}\Gamma(\ell+1)\Gamma(1/2)}e^{-\frac{\omega^{2}+(\mathcal{H}e\Omega\eta)}{4(i\mathcal{H}e\Omega/2)^{2}}}\int^{\pi}_{0}e^{\frac{\omega\mathcal{H}e\Omega\eta\cos(\phi)}{2(i\mathcal{H}e\Omega/2)^{2}}}\sin^{2\ell+1}(\phi)d\phi$
$\displaystyle=i(-1)^{m}\frac{\sqrt{\mathcal{H}e\eta}}{2}e^{-i(\omega\eta+\tfrac{1}{2}\mathcal{H}e\Omega\eta^{2})}\left[2\eta\frac{d}{d\eta}+(1+i2\eta(\omega-\mathcal{H}e\Omega\eta))\right]\frac{-i}{\mathcal{H}e\Omega}e^{i\frac{\omega^{2}+\mathcal{H}^{2}e^{2}\Omega^{2}\eta^{2}}{2\mathcal{H}e\Omega}}I_{\ell+1/2}\left(-i\omega\eta\right)$
$\displaystyle=i(-1)^{m}\frac{\sqrt{\mathcal{H}e\eta}}{2}e^{-i(\omega\eta+\tfrac{1}{2}\mathcal{H}e\Omega\eta^{2})}\left[2\eta\frac{d}{d\eta}+(1+i2\eta(\omega-\mathcal{H}e\Omega\eta))\right]\frac{1}{i\mathcal{H}e\Omega}e^{i\frac{(\omega^{2}+\mathcal{H}^{2}e^{2}\Omega^{2}\eta^{2})}{2\mathcal{H}e\Omega}}I_{\ell+1/2}(-i\omega\eta)$
$\displaystyle=(-1)^{m}\frac{\sqrt{\mathcal{H}e\eta}}{2\mathcal{H}e\Omega}e^{-i\omega\eta}\left[2\eta\frac{d}{d\eta}+(1+i2\eta(\omega-\mathcal{H}e\Omega\eta))\right]e^{i\frac{(\omega^{2}+\mathcal{H}^{2}e^{2}\Omega^{2}\eta^{2})}{2\mathcal{H}e\Omega}}I_{\ell+1/2}(-i\omega\eta),$

where $I_{\ell+1/2}(\eta\omega)$ is modified Bessel function of first kind,
such that $I_{\ell+1/2}(z)=e^{-i(\ell+1/2)\pi/2}J_{\ell+1/2}(iz)$. From the
above expression we obtain (78).

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