# Gravitational production of massive scalars in the context of inflation

Urjit A. Yajnik<EMAIL_ADDRESS>Tata Institute of Fundamental Research,
Homi Bhabha Road, Bombay 400005, India

###### Abstract

We set up a formalism for calculating the energy density generated in a
quantized massive scalar field in the course of the drastic change in
spacetime geometry at the end of the inflationary era. The calculation relies
on the notion of adiabatic vacuum. The Bogolubov coefficients are computed by
employing the sudden approximation. After obtaining a general formula, we
calculate explicitly the energy density generated in a particle species with
$m/H\ll 1$, where $m$ is the particle mass and $H$ is the Hubble constant
during the inflationary epoch. We find the contribution of the long-wavelength
modes to be $\propto H^{5}/m$. If such particles are very weakly interacting,
they can come to dominate the total energy density in the Universe. Other
cosmological implications are also discussed.

> This preprint contains the calculations supporting the results published in
> Phys. Lett. B 234 (1990) 271-275. The unpublished preprint is now typeset in
> LaTeX 2εand submitted to the arXiv due to the renewed attention it has
> received.

††preprint: TIFR-TAP-11††preprint: July 1988

##  I Introduction

Quantum field theory in curved space1 2  is one of the most extensively
explored approaches to a full quantum theory of gravity. And yet it has some
very important issues that remain unresolved. One is that of defining the
vacuum state. Another is of computing the back reaction on the space in which
quantization is carried out. It is therefore important to test whatever
conceptual advances have been made so far by applying them to as many new
situations as possible. Inflationary universe3  is a very interesting proposal
whose correctness has not yet been established. While a search is on for
particle physics models within which it may be realized, or for variants of
the scenario in order to avoid pitfalls, it is also worth pursuing physical
effects other than those the scenario was originally invented to produce (or
to pre-empt). It is in this spirit that we carry out the present
investigation.

The most dramatic effect of quantizing a field in a curved space is that in
the generic case, quanta of the field must be spontaneously produced in the
vacuum. This effect of course can occur for a field coupled to any classical
background field. In the case of gravity, the situation is further complicated
because the very concept of a particle, or equivalently, that of the vacuum
state of the quantum field theory becomes ill-defined. Instead of worrying
about particle number, one might construe the problem as that of exchange of
energy between the quantized field and gravity. One would then try to compute
the expectation value of the energy-momentum stress tensor for the field. But
this attempt suffers from the dual handicap of the nonuniqueness of the ground
state in which to compute such an expectation value, and the divergences
typical of quantum field theory. So far, many techniques have been developed
to deal with these problems. In this paper we use the adiabatic procedurea4
for defining the vacuum and a simple normal ordering prescription to calculate
finite $\langle T_{0}^{0}\rangle$. The latter prescription, to be discussed in
II, is satisfactory for Friedmann- Robertson-Walker (FRW) universes but may
not be satisfactorily generalizable to other cases.

Our interest in carrying out this calculation is guided by the fact that the
production of particles is larger, the greater the rate of change of geometry
is6 . Since the generic inflationary scenario involves a change, over a
relatively short time scale, from a de Sitter like or inflationary phase to a
radiation dominated Friedmann phase, we expect it to be accompanied by copious
production of particles. The case of graviton production has been studied
extensively in the past7 8 . On the other hand, production of massive
particles has not received adequate attention.

While the present calculation was in progress, Turner and Widow9  have
calculated the same effect, using a different approach. Their particles
originate as quantum fluctuations during the inflationary phase. The evolution
of these fluctuations is then traced as the wavelengths of the modes cross
various critical physical scales. We shall compare their results with ours in
the Conclusion.

Our formalism suggests two cases of interest: $m\ll H$, and $m\sim H$. We are
able to display explicitly the dependence of the magnitude of the energy
density on the mass only for the small mass case. In the case of $m\sim H$, we
obtain the spectrum and the energy density for a representative value of $m$.
Our main results are that particles of small mass contribute a large amount to
the total energy density, which is comparable to the background energy density
if, for instance, inflation is driven by $\rho_{0}\simeq
M_{GUT}^{4}\sim(10^{14}GeV)^{4}$ and if $m\lesssim 0.01$eV. For particles of
larger mass, and which interact rapidly with other particles, we get a
nontrivial contribution to the total energy density, but we may get no
distinctly observable signature today. On the other hand, if these particles
are very weakly interacting, they can come to dominate the total energy
density at a later stage and cause a conflict with the nucleosynthesis data.

In deriving the above results however, we have not taken into account the
effects of nonzero temperature. This may be important for the particles of
standard model which acquire mass through spontaneous symmetry breaking, and
which remain effectively massless at high temperature. Thus it is unclear
whether the results obtained here are of direct validity for the known
particles, but the results are suggestive enough to warrant an independent
study for the case of spontaneous symmetry breaking. Some more remarks are
included in sec. V.

In what follows, in sec. II we describe the formalism to be used and set up
the problem. In sec. III we present the calculation and the general result. In
section IV we use the formula to obtain the energy density spectrum $d\rho/dk$
and the total energy density of the produced particles. In section V, we
consider the cosmological consequences of the results. Section VI contains
concluding remarks.

##  II The Formalism

Consider the field theory of a real massive scalar field in a spatially flat
FRW universe. We shall write the line element for the spacetime in the
conformal form

$ds^{2}=\Omega^{2}(\eta)\left(-d\eta^{2}+|d\vec{x}|^{2}\right)$ (1)

Given this coordinatization, the theory of a massive scalar field is given by
the action

$A=\frac{1}{2}\int d\eta
d^{3}x\Omega^{4}\left\\{\frac{1}{\Omega^{2}}\left(\frac{\partial\phi}{\partial\eta}\right)^{2}-\frac{1}{\Omega^{2}}|\vec{\nabla}\phi|^{2}-m^{2}\phi^{2}\right\\}$
(2)

We have chosen minimal coupling to scalar curvature. There is no experimental
evidence to the contrary, nor is there any theoretical motivation in the form
of higher symmetry, (such as exists, for instance, in the massless case), to
justify any nontrivial coupling.

In the quantum theory, we expand the field operator $\phi$ in terms of mode
functions in the following form

$\phi(\vec{x},\eta)=\int\frac{d^{3}k}{(2\pi)^{3/2}}\left\\{a_{\vec{k}}\frac{\chi_{k}(\eta)}{\Omega(\eta)}e^{-i\vec{k}\cdot\vec{x}}+a_{\vec{k}}{}^{\dagger}\frac{\chi_{k}^{*}(\eta)}{\Omega(\eta)}e^{i\vec{k}\cdot\vec{x}}\right\\}$
(3)

With the given scaling, the $\chi_{k}$ satisfy (with prime denoting
differentiation with respect to $\eta$)

$\chi_{k}^{\prime\prime}-\frac{\Omega^{\prime\prime}}{\Omega}\chi_{k}+|\vec{k}|^{2}\chi_{k}+m^{2}\Omega^{2}\chi_{k}=0$
(4)

The system is canonically quantized with

$\left[a_{\vec{k}},a_{\vec{k}^{\prime}}^{\dagger}\right]=\delta^{3}\left(\vec{k},\vec{k}^{\prime}\right)$
(5)

provided the Wronskian of the $\chi$-functions is normalized to

$\chi_{k}\chi_{k}^{*\prime}-\chi_{k}^{*}\chi_{k}^{\prime}=i$ (6)

For this system we shall compute

$\left\langle\psi\left|T^{0}{}_{0}\right|\psi\right\rangle=\left\langle\psi\left|\frac{1}{2\Omega^{2}}\left\\{\left|\phi^{\prime}\right|^{2}+|\vec{\nabla}\phi|^{2}+\Omega^{2}m^{2}|\phi|^{2}\right\\}\right|\psi\right\rangle$
(7)

The modulus sign refers to extracting real numbers from complex numbers and
vectors in the standard way, and also the hermitian combination according to
$|A|^{2}=A^{\dagger}A$ for operators. Here $|\psi\rangle$ is meant to be any
appropriately chosen ground state.

We choose $|\psi\rangle$ by demanding

$a_{\vec{k}}\ |\psi\rangle\ =\ 0$ (8)

This choice of course is not unique. It was specified by assuming
$\chi_{k}(\eta)/\Omega(\eta)$ to be “positive frequency” mode functions. Due
to the general covariance of the classical theory, another choice
$\tilde{\chi}_{k}(\eta)=\,\alpha_{k}\chi_{k}(\eta)+\beta_{k}\chi_{k}^{*}(\eta)$,
(appropriately normalized), would have been just as good. The corresponding
$\tilde{a}_{\vec{k}}$ would yield a different vacuum than in Eq. (8). Although
a FRW space offers a natural choice for $3+1$ slicing into hypersurfaces of
constant $TrK$ (trace of the extrinsic curvature tensor),10  this favoured
choice of slicing is not available in arbitrary spacetime manifolds. In such
cases, each possibility for slicing introduces further possibilities for the
choice of a vacuum.

In the FRW case at hand, we shall make use of the natural choice it affords
for slicing (and which is implied in the expression Eq. (1) for the metric).
We are then left with having to decide on one particular linear combination
from the set of two fundamental solutions of Eq. (4), which should be called
the positive frequency mode function. In the language of complex vector
spaces, this amounts to singling out a particular complex structure on the
one-complex-dimensional space spanned by the two real solutions of (4). For
this purpose we shall use the notion of adiabatic vacuum developed by Parker11
(see also ref. 2  for other references). We shall not discuss the details of
this approach here, though the prescription will be stated briefly in sec. III
where it is used. Another natural way to define the vacuum in the conformally
flat case such as we have, is the conformal vacuum prescription2 . This will
not be used in the present work.

To continue, we note that in any vacuum $|\psi\rangle$ suitably identified,
Eq. (7) becomes

$\langle\psi|T^{0}{}_{0}|\psi\rangle=\int\\!\frac{d^{3}k}{(2\pi)^{3}}\,\frac{1}{2\Omega^{2}}\,\left\\{\left|\left(\frac{\chi_{\vec{k}}}{\Omega}\right)^{\prime}\right|^{2}+\left(\frac{|\vec{k}|^{2}}{\Omega^{2}}+m^{2}\right)|\chi_{\vec{k}}|^{2}\right\\}\left\langle\psi|a_{\vec{k}}^{\dagger}a_{\vec{k}}+a_{\vec{k}}a_{\vec{k}}^{\dagger}|\psi\right\rangle$
(9)

We have already dropped the off-diagonal terms that do not contribute to the
vacuum expectation value. The remaining two terms will be together interpreted
by normal ordering just like in flat space. That is, the operator expression
for $T^{0}_{0}$ will be taken with $a_{\vec{k}}$’s to the right of
$a^{\dagger}_{\vec{k}}$’s, and the $c$-number generated in this arrangement
will be discarded. Since this makes
$\langle\psi|T_{\phantom{\mu}\nu}^{\scriptscriptstyle\mu}|\psi\rangle$ vanish
for all $\mu$, $\nu$, the $c$-numbers constitute a tensor. Thus the normal
ordering prescription is covariant, with the proviso that we do not seek a new
complex structure for the quantum theory when we carry out a
reparameterization of the background manifold. For example, in the FRW case, a
natural slicing is already suggested, so that the only reparameterizations
affecting $\chi_{k}(\eta)$ are $\eta\rightarrow{\bar{\eta}}(\eta)$, which do
not warrant making a new choice for mode functions in which $\chi_{k}$ and
$\chi^{*}_{k}$ get mixed. The only change needed (since the $\chi_{k}$ are
scalar), is $\chi_{k}(\eta)\,\rightarrow\,\chi_{k}(\eta(\bar{\eta}))$, so that
$\langle\psi|T_{\phantom{\mu}0}^{0}|\psi\rangle$ will continue to be zero.
Finally, the natural $\chi_{k}$’s will be singled out for us by the adiabatic
procedure. The answer produced by the latter procedure should not get affected
by $\eta\rightarrow{\bar{\eta}}(\eta)$ transformations.

Thus, in the symmetric case of FRW, we utilise the freedom allowed by the
reparameterization gauge symmetry (viz., choose the preferred slicing) to
obtain a sensible answer. However, in a manifold lacking symmetry, no
preferred time direction will exist, and different observers choosing
different coordinate systems for their convenience will also decide to choose
different complex structures consistent with their time direction, and the
proviso on reparameterization stated in the preceding paragraph will appear
unjustifiable.

For the symmetric case of FRW, the above picture accords with our intuition.
The free (although coupled to a classical field) field theory has a stable
ground state of zero energy; a single particle state has energy

$w_{k}(\eta)\quad\equiv\quad\Omega\mathrm{~{}}\left\\{\left|\left(\frac{\chi_{k}}{\Omega}\right)^{\prime}\right|^{2}+\left(\frac{|\vec{k}|^{2}}{\Omega^{2}}+m^{2}\right)|\chi_{k}|^{2}\right\\}$
(10)

which changes with time $\eta$ due to interaction with the gravitational
field, conventionally known as gravitational red or blue shifting.

If a particular $|\psi\rangle$ is adhered to in a FRW universe as in the
preceding discussion, no “particle creation” ( or, in less picturesque
language, spontaneous generation of matter energy density, gravitational in
origin), can occur. However, the energy density as defined in Eq. (9) can
change spontaneously if physical considerations indicate choosing different
$|\psi\rangle$’s at different epochs. The simplest of such cases are the toy
models in which the conformal scale factor $\Omega$ evolves from one constant
value to another in a finite duration of time12 6 . One is led to define two
distinct vacua, one (say the “in”) during the beginning epoch of constant
$\Omega$ and another, (say the “out”) during the final epoch of constant
$\Omega$. Then the out vacuum turns out to be a state of infinite particle
number built on the in vacuum, and we would refer to such a phenomenon as
creation of particles. The amount of particles produced, and their spectrum
are determined to a large extent by the rate at which the scale factor changes
in the transition region.

We expect inflation to lead to production of particles for similar reasons.
The universe is supposed to have undergone a change from a de Sitter like
phase to radiation dominated phase over a relatively short period of time. Any
reasonable vacuum defined in the de Sitter like phase can turn out to be a
state of nonzero particle number built on a vacuum defined in the radiation
dominated phase. The main difference from the toy models is that we do not
have flat space in the past or in the future of the transition region. For
this purpose we rely on the notion of adiabatic vacuum. We define adiabatic
vacuum $|\psi^{I}\rangle$ in the de Sitter like or the inflationary phase, and
$|\psi^{R}\rangle$ in the radiation dominated phase. We then need the
Bogolubov transformations13  relating the two vacua, i.e., a set of
transformations that relate the creation and destruction operators of the two
phases

$a_{\vec{k}}^{R}=\alpha_{k}a_{\vec{k}}^{I}+\beta_{k}a_{\vec{k}}^{I}$ (11)

In practice, this requires knowing the mode functions valid throughout the
history of the universe, including the region of transition. Calculation of
the Bogolubov coefficients can then be carried out as in the toy models
mentioned above. Unfortunately, this is too difficult in the present case, as
will be discussed in the concluding section. Here we shall adopt what may be
called the sudden approximation14 . We assume that the Universe goes over from
the inflationary phase to the radiation dominated phase abruptly. We then
match the mode functions at the juncture of the two phases:

$\frac{\chi_{k}^{I}}{\Omega}=\alpha_{k}\frac{\chi_{k}^{R}}{\Omega}+\beta_{k}\frac{\chi_{k}^{R*}}{\Omega}\qquad\mathrm{(at\
the\ juncture)}$ (12)

and, because the mode functions satisfy a second order differential equation,

$\left(\frac{\chi_{k}^{I}}{\Omega}\right)^{\prime}=\alpha_{k}\left(\frac{\chi_{k}^{R}}{\Omega}\right)^{\prime}+\beta_{k}\left(\frac{\chi_{k}^{R*}}{\Omega}\right)^{\prime}\qquad\mathrm{(at\
the\ juncture)}$ (13)

Due to the field expansion (3), these are the coefficients needed in (11).
Since the mode functions continue to be normalised according to (6), the
$\alpha$ and $\beta$ satisfy

$|\alpha_{k}|^{2}-|\beta_{k}|^{2}=1$ (14)

The number $\textstyle\sum_{k}|\beta_{k}|^{2}$ can be interpreted as the
number of particles created in the course of the phase change6 . In the
present case, we evaluate (9) with $|\psi\rangle=|\psi^{I}\rangle$ but with
the normal ordering of the radiation dominated phase. We then get

$\langle\psi^{I}|T_{\;\;0}^{0}\,|\psi^{I}\rangle=\frac{1}{\Omega^{3}}\int\\!\frac{d^{3}k}{(2\pi)^{3}}w_{k}|\beta_{k}|^{2}$
(15)

This is the contribution to the energy density in the radiation dominated
phase, from particles gravitationally generated due to ending of inflation.

The procedure of mode matching for obtaining the Bogolubov coefficients has
been used previously15 9  in the context of graviton production. The
limitation of the approximation is that we lose the information about the
modes with wavelengths comparable to or smaller than the (temporal) length of
the transition region. However, due to technical difficulties to be explained
in the conclusion, this is the best we have been able to do. It should be
noted however, that in any reasonable formalism (such as the exact adiabatic
treatment of the entire evolution), the value of $|\beta_{k}|^{2}$ becomes
exponentially damped for wavelengths smaller than the typical time scale
involved in the transition. As such, the wavelengths we have foregone the
information about are not of much interest either.

##  III Adiabatic vacua and mode matching

Let $\eta_{1}$ and $\eta_{2}$ denote the times when the inflationary phase
begins and ends, respectively. In most inflationary universe models, the epoch
between the Planck time, (before which we do no know the physics), and the
time $\eta_{1}$, the Universe is radiation dominated. In the models with
primordial inflation, the time n in fact runs into the Planck time. In either
case, since inflation must enhance the scale factor by the stupendous factor
of $10^{27}$ or more (see Guth3 ), any relics of the time h and earlier must
become completely irrelevant to physics. In particular, this allows us to
assume that at the time $\eta_{2}$, our scalar field of Eq. (2) is in its
ground state111The validity of this assumption is not at all obvious,
especially because the dilution that occurs is of energy density but not, for
instance, of the number of occupied modes per causal horizon volume. We adopt
this assumption here primarily because it simplifies the calculation without
sacrificing essential physics. Also, it appears reasonable that at least in
the range of wavelengths that we shall be concerned with at time $\eta_{2}$
there will be no relics, because at time $\eta_{1}$ these wavelengths were
inordinately small compared to the natural physical scale of that epoch. I
thank T. Padmanabhan for pointing this out.. We shall not have much occasion
to refer to $\eta_{1}$ in the following.

In addition, we shall not be concerned in the present work with the subsequent
major changes in the course of the evolution of the Universe, such as
transition to a matter dominated epoch.

We model inflation by taking the scale factor to be 9

$\displaystyle\Omega$ $\displaystyle=\frac{\Omega_{2}}{2-\eta/\eta_{2}}$
$\displaystyle\eta_{1}<\eta<\eta_{2}$ (16)
$\displaystyle=\Omega_{2}\frac{\eta}{\eta_{2}}$ $\displaystyle\eta>\eta_{2}$

The constants have been chosen to ensure that the scale factor itself as well
as its first derivative are continuous across $\eta_{2}$. The continuity of
the first derivative is required by Einstein’s equations which imply
$\left(\Omega^{\prime}/\Omega^{2}\right)^{2}\propto\rho$, which in turn may
reasonably be assumed to be continuous. Assuming any greater degree of
smoothness goes beyond the requirements of the approximation we are making.
With the above choice of constants, the Hubble constant during the
inflationary phase is given by

$H^{2}\equiv\frac{1}{\left(\Omega_{2}\eta_{2}\right)^{2}}=8\pi G\rho_{0}$ (17)

where $\rho_{0}$ is the constant energy density that drives inflation. In the
new inflation it has the order of magnitude
$\left(10^{14}\mathrm{GeV}\right)^{4}$ to
$\left(10^{16}\mathrm{GeV}\right)^{4}$ derived from Grand Unified Theories.

We now proceed to finding the mode functions and the adiabatic vacuua. In the
de Sitter like phase, the mode functions satisfy

$\chi_{k}^{\prime\prime}+\left\\{-\frac{2}{\left(2\eta_{2}-\eta\right)^{2}}+k^{2}+\frac{m^{2}\Omega^{2}\eta_{2}^{2}}{\left(2\eta_{2}-\eta\right)^{2}}\right\\}\chi_{k}=0$
(18)

The two independent solutions of this equation are found to be

$\displaystyle\chi_{k}$
$\displaystyle\sim\left(2\eta_{2}-\eta\right)^{1/2}H_{\lambda}^{(1)}\left(k\left(2\eta_{2}-\eta\right)\right)$
(19)
$\displaystyle\sim\left(2\eta_{2}-\eta\right)^{1/2}H_{\lambda}^{(2)}\left(k\left(2\eta_{2}-\eta\right)\right)$

with

$\lambda^{2}=\frac{9}{4}-m^{2}\Omega_{2}^{2}\eta_{2}^{2}$ (20)

Here $H_{\lambda}^{(1)},H_{\lambda}^{(2)}$ are Hankel functions of order
$\lambda$. The adiabatic prescription requires us to identify that combination
(and normalisation) of the fundamental set (19) to be the positive frequency
mode which behaves as

$\chi_{k}\sim\frac{1}{\sqrt{2\bar{\omega}(\eta)}}\exp\left(-i\int^{\prime}\bar{\omega}\left(\eta^{\prime}\right)d\eta^{\prime}\right)$
(21)

in some limiting value of some parameter such as $m\Omega_{2}\eta_{2}$ or $k$.
The $\bar{\omega}$ appearing in the ansatz is in turn to be obtained as the
limit of

$\omega^{2}\equiv-\frac{2}{\left(2\eta_{2}-\eta\right)^{2}}+k^{2}+\frac{m^{2}\Omega^{2}\eta_{2}^{2}}{\left(2\eta_{2}-\eta\right)^{2}}$
(22)

in the same limit of the same parameter. For the case at hand, the answer
appears in 2 , sec.5.4, which one can check222Note that in 2 , the argument of
the Hankel function $k\eta=-(k/H)e^{-Ht}$ increases as the Universe expands,
whereas our argument $k\left(2\eta_{2}-\eta\right)$ decreases. Hence our
positive frequency function is $H^{(1)}$..

$\chi_{k}^{I(+)}(\eta)=\frac{1}{2}\left(\pi\left(2\eta_{2}-\eta\right)\right)^{1/2}H_{\lambda}^{(1)}\left(k\left(2\eta_{2}-\eta\right)\right)$
(23)

and similarly the negative frequency function in terms of $H^{(2)}$.

In the radiation dominated phase one finds

$\chi_{k}^{\prime\prime}+\left(k^{2}+\frac{m^{2}\Omega_{2}^{2}}{\eta_{2}^{2}}\eta^{2}\right)\chi_{k}=0$
(24)

The fundamental solutions for this are the parabolic cylinder functions of
Weber16

$\displaystyle\chi_{k}$ $\displaystyle\sim
D_{\nu}\left(e^{i\pi/4}\sigma\eta\right)$ (25) $\displaystyle\sim
D_{\nu}\left(e^{-i\pi/4}\sigma\eta\right)$

with

$\sigma=\sqrt{\frac{2m\Omega_{2}}{\eta_{2}}};\quad\nu=-i\frac{k^{2}}{\sigma^{2}}-\frac{1}{2}$
(26)

We now note that the limit of the quantity in the brackets in (24), as either
$\eta$ or $\sigma$ go to infinity is $\frac{1}{4}\sigma^{4}\eta^{2}$, so that
we must identify positive frequency modes to be that combination of (25) which
behaves like
$(\sigma\sqrt{\eta})^{-1}\exp\left(-i\frac{1}{4}\sigma^{2}\eta^{2}\right)$ in
the same limit. Using the fact that 16

$D_{n}(z)\xrightarrow{|x|\rightarrow\infty}e^{-\frac{1}{4}z^{2}}z^{n}\quad\text{
for }|\arg z|<\frac{3\pi}{4}$ (27)

and using the Wronskian 17

$\mathcal{W}\left\\{D_{n}(z),D_{-n-1}(-iz)\right\\}=e^{i\pi(1+n)/2},$ (28)

we identify the radiation dominated era mode functions to be

$\chi_{k}^{R(+)}(\eta)=\frac{e^{-\pi
k^{2}/4\sigma^{2}}}{\sqrt{\sigma}}D_{\nu}\left(e^{i\pi/4}\sigma\eta\right)$
(29)

and similarly the negative frequency function in terms of the complex
conjugate function.

We now match the mode functions at $\eta=\eta_{2}$. We get

$\frac{1}{2}\sqrt{\pi\eta_{2}}H_{\lambda}^{(1)}\left(k\eta_{2}\right)=\frac{1}{\sqrt{\sigma}}\exp\left(-\pi
k^{2}/4\sigma^{2}\right)\left\\{\alpha_{k}D_{\nu}\left(e^{i\pi/4}\sigma\eta_{2}\right)+\beta_{k}(\text{
c.c. })\right\\}$ (30)

and

$\displaystyle\frac{\sqrt{\pi}}{2}\left\\{\frac{-1}{\Omega_{2}\eta_{2}}\right.$
$\displaystyle\left.\sqrt{\eta_{2}}H_{\lambda}^{(1)}\left(k\eta_{2}\right)+\frac{1}{2\Omega_{2}\sqrt{\eta_{2}}}H_{\lambda}^{(1)}\left(k\eta_{2}\right)+\frac{k\sqrt{\eta_{2}}}{\Omega_{2}}H_{\lambda}^{(1)\prime}\left(k\eta_{2}\right)\right\\}$
(31) $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\sigma}}\exp\left(-\pi
k^{2}/4\sigma^{2}\right)\left[\alpha_{k}\left\\{-\frac{1}{\Omega_{2}\eta_{2}}D_{\nu}\left(e^{i\pi/4}\sigma\eta_{2}\right)\right.\right.$
$\displaystyle\left.\left.+\frac{e^{i\pi/4}\sigma}{\Omega_{2}}D_{\nu}^{\prime}\left(e^{i\pi/4}\sigma\eta_{2}\right)\right\\}+\beta_{k}\\{\text{
c.c. }\\}\right]$

In both these equations (c.c.) means the complex conjugate of the coefficient
of the preceding $\alpha_{k}$, and the prime on the special functions denotes
derivative with respect to their own argument, not simply $\eta$. Now denote

$\displaystyle E$ $\displaystyle\equiv
D_{\nu}\left(e^{i\pi/4}\sigma\eta_{2}\right),$ (32) $\displaystyle F$
$\displaystyle\equiv D_{\nu}^{\prime}\left(e^{i\pi/4}\sigma\eta_{2}\right),$
$\displaystyle X$ $\displaystyle\equiv
H_{\lambda}^{(1)}\left(k\eta_{2}\right)$ $\displaystyle Y$
$\displaystyle\equiv H_{\lambda}^{(1)\prime}\left(k\eta_{2}\right)$

and the complex conjugates by $\bar{E},\bar{F}$ etc. After some
simplification, and using the fact that due to the normalisation (6),

$\exp\left(-\pi
k^{2}/2\sigma^{2}\right)\left(e^{-i\pi/4}E\bar{F}-e^{i\pi/4}\bar{E}\boldsymbol{F}\right)=i$
(33)

we get a master formula

$\displaystyle\left|\beta_{k}\right|^{2}=$
$\displaystyle\frac{\pi}{4}\frac{\exp\left(-\pi
k^{2}/2\sigma^{2}\right)}{\sigma\eta_{2}}\left[\sigma^{2}\eta_{2}^{2}F\bar{F}X\bar{X}+E\bar{E}\left|\frac{1}{2}X+k\eta_{2}Y\right|^{2}\right.$
(34)
$\displaystyle\left.+2\operatorname{Re}e^{i\pi/4}\sigma\eta_{2}F\bar{E}X\left(\frac{1}{2}\bar{X}+k\eta_{2}\bar{Y}\right)\right]$

##  IV Calculation of energy density

###  IV.1 General discussion

Formula (34) gives the desired results in terms of standard special functions.
It can be used to numerically plot $\left|\beta_{k}\right|^{2}$ as a function
of $k$ given any value of $m$ (in the units of $H$ ). Instead, here we shall
try to obtain analytic estimates of the dependence of
$\left|\beta_{k}\right|^{2}$ on $k$ as well as on $m$. Hence we begin by
identifying the different cases of interest.

One important scale in the problem is $H$; for two reasons. Firstly we recall
that our approximation restricts us to values of $k_{phy}\equiv k/\Omega<H$.
Secondly, on physical grounds we know that at $k_{phy}>H$, the spectrum of
produced particles gets exponentially cut off. The other scale to compare
$k_{phy}$ to is provided by $m$. We expect qualitatively different behaviours
when $k_{phy}$ is greater than or less than $m$ (i.e., when
$\lambda_{phy}\equiv 2\pi/k_{phy}$ is less than or greater than the Compton
wavelength). This is at least true of flat space physics, where field
theoretic phenomena set in when the system is probed at wavelengths small
compared to the Compton wavelength. Considering both the above mentioned
length scales together, we see that three cases arise. i) When $k_{phy}$
crosses $m$ when it is yet smaller than $H$, ii) when it crosses $m$ when it
is roughly $H$, and iii)when it does so beyond $H$. In other words, i)
$m/H<1$, ii) $m/H\simeq 1$ and iii) $m/H>1$. Case ii) is distinguished from
iii) only by the fact that the interesting range $k_{phy}\simeq m$ is not lost
in the exponential tail of the spectrum. Unfortunately, due to our
approximation becoming unreliable precisely at $k_{phy}\simeq H$, the only
information we can possibly get from (34), both for cases ii) and iii) is
restricted to the case $k_{phy}\ll m$. But even this is difficult in the
general case for technical reasons, i.e., it has to be done numerically. It
turns out however, that we can get analytic estimate of the $k$ dependence
even in this case for a particular value of $m$. This we shall do later. At
first we take up the case $m\ll H$.

###  IV.2 The case of small mass

We take up this case first, partly because we will be able to do full justice
to it by analytical methods, but also because it is the most interesting. One
can learn the behaviour of $\left|\beta_{k}\right|$ for both the ranges
$k_{phy}\ll m$, and $k_{phy}\gg m$ within the validity of our approximation,
i.e., from (34). Let us define spectral distribution of the energy density,
$d\rho/dk$ through

$\displaystyle\left\langle\psi^{R}\left|T^{0}_{\phantom{\mu}0}\right|\psi^{R}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\Omega_{2}^{3}}\int\frac{d^{3}k}{(2\pi)^{3}}w_{k}\left|\beta_{k}\right|^{2}$
(35) $\displaystyle\equiv$ $\displaystyle\int dk_{phy}\frac{d\rho}{dk_{phy}}$
(36)

The quantities $\left|\beta_{k}\right|^{2}$ and $w_{k}$ can be calculated as
shown in Appendix A. We find that immediately at the end of inflation,

$\displaystyle\frac{d\rho^{(1)}}{dk_{phy}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}H^{4}\left(\frac{1}{k_{phy}}+\frac{H^{2}}{2k_{phy}^{3}}\right)\qquad
k_{phy}^{2}\gg mH$ (37) $\displaystyle=$ $\displaystyle
0.04\frac{H^{5}}{m}\frac{1}{k_{phy}}\qquad\qquad\qquad\qquad k_{phy}^{2}\ll
mH$ (38)

The superscript (1) is meant to label the case $m\ll H$. We find that the
large $k$ case becomes independent of the mass. We then expect it to be the
same as that, for instance, for gravitons 9 . Indeed Appendix A shows that our
$\left|\beta_{k}\right|^{2}$ is the same as that obtained by Allen. However,
our definition of the single-particle energy $w_{k}$, derived from our
covariant definition of $\left\langle T^{0}{}_{0}\right\rangle$ differs from
that used by Allen, which is simply $k_{phy}$.

The low-mode-number end of (38) shows a curious dependence on mass. As a
result, for small mass, we can get a large contribution to the total energy
density, coming from the extremely long wavelengths. We can estimate this
contribution by integrating (38). We need to put reasonable limits. We may
take the upper limit to be $m$.

To determine the lower limit, recall the comment made at the beginning of sec.
III, along with the footnote. At the epoch $\eta_{1}$ when inflation begins,
the Hubble value is $H$ which sets the physical scale. Our entire calculation
applies only to those modes that have $k_{phy}\left(\eta_{1}\right)<H$. The
reason is that the modes $k_{phy}\left(\eta_{1}\right)\gtrsim H$ carry the
information regarding the preceding (possibly radiation dominated) phase, and
it is inappropriate to define the de Sitter adiabatic vacuum for them. Only
for $k_{\text{phy }}\left(\eta_{1}\right)\ll H$, i.e., modes which do not
explore the global geometry at $\eta_{1}$ can we safely pretend that the
Universe has been truly de Sitter (with infinite past) and also that these
modes are unoccupied at $\eta_{2}$. Hence at $\eta_{2}$, we take the lower
limit on $k_{phy}$ as $\Omega_{1}H/\Omega_{2}$. We then get

$\rho_{\text{low
}}^{(1)}=0.04\frac{H^{5}}{m}\ln\left(\frac{m}{H}\frac{\Omega_{2}}{\Omega_{1}}\right)$
(39)

Recall that inflation requires $\Omega_{2}/\Omega_{1}\sim 10^{27}$. This means
that the low-mode-number contribution vanishes if $m$ (or more generally, the
upper limit of validity of Eq. (38)) becomes as small as $10^{-27}H$. Thus the
lower limit on $k_{phy}$ values dictated by the validity of the approximation
scheme also sets a lower limit on the mass value for which the contribution
Eq. (39) can be taken seriously.

We may similarly evaluate the contribution of high-mode-number modes. Here we
take the upper limit to be $H$ and the lower limit, for the sake of argument,
$\sqrt{mH}$. Then

$\rho_{high}^{(1)}=\frac{H^{4}}{8\pi^{2}}\left\\{\frac{1}{2}\ln\frac{H}{m}+\frac{H}{4m}-\frac{1}{4}\right\\}$
(40)

The dominant contribution appears to be $\sim H^{5}/m$ as in the low-mode-
number case, due to our rather liberal lower limit. If we ignore this in Eq.
(40), we get a contribution essentially $\sim H^{4}$.

The case of massless scalars was considered by Ford 18  and the related case
of gravitons by Allen 8 . It is interesting to check whether the zero-mass
limit of our results agrees with their results. Eq. (37), aside from its
validity range indicated there for the small-mass case, is also the spectrum
for the massless case. (Eq. (38) refers to a range of $k$ values not possible
for the massless case). Our $\left|\beta_{k}\right|^{2}$ for this case (67) is
the same as that obtained by above authors, and the first term of eq. (37) is
the answer obtained by them. We additionally get the second term, due to our
definition of $\rho$ which we have identified with $\left\langle
T^{0}{}_{0}\right\rangle$. Ford, Allen as well as Starobinsky compute $\rho$
by assigning to each mode of mode-number $k$ a single-particle energy
$\hbar\omega_{k}\equiv\hbar k_{phy}$. By contrast, our choice amounts to (see
eq.s (10), (68) and (75))

$\hbar w_{k}=\hbar\left(k_{phy}+\frac{H^{2}}{2k_{phy}}\right)$ (41)

The definition of energy density in curved space is intrinsically ambiguous.
We have here tried a choice that is justified by being a component of a
covariant object. Further comments are included in the Conclusion.

###  IV.3  Spectrum for one $m\sim H$ example

In this subsection we briefly mention the results for a particular value of
$m$ close to $H$. We see that if
$m^{2}\Omega_{2}^{2}\eta_{2}^{2}=\frac{1}{4}\sigma^{4}\eta_{2}^{2}=2$, the
order $\lambda$ of the Hankel functions of (3.4) becomes $1/2$, in which case
$H_{1/2}(z)\sim z^{-1/2}e^{iz}$ and the problem simplifies somewhat. The
details are given in Appendix B.

$\frac{d\rho^{(2)}}{dk_{phy}}=3.86H^{2}k_{phy}\quad m=\sqrt{2}H;\quad
k_{phy}^{2}\ll mH$ (42)

The superscript (2) denotes the case $m\sim H$. Note that the case
$k^{2}/\sigma^{2}\gg 1$ is inappropriate here because with $m\sim H,\quad
k_{phy}^{2}\gg mH$ lends us far out of the domain of validity of our
approximation. The so called low end $\left(k^{2}/\sigma^{2}\ll 1\right)$ may
now be safely taken to be the representative of the entire allowed range
$\left(k_{pky}<H\right)$ of $k$.

We may take the above spectrum as representative of the case ii), i.e., $m\sim
H$. A different value of mass will only change the numerical constant. We now
integrate this spectrum, with upper limit $fH,f$ being some number less than
one, since assuming this spectrum to be valid all the way to $\mathrm{H}$ is
incorrect.

$\rho^{(2)}=\mathrm{O}(1)\cdot f^{2}H^{4}$ (43)

##  V Implications for the Universe

The expressions (37), (38) and (43) give the energy density added to the
background energy density immediately after inflation. Before considering the
effects of this in the early Universe, we must make two caveats. The first
concerns the fact that we have considered only a scalar field. whereas no
fundamental massive scalar is known to date. This brings us to the second
point. In the standard model. the known light particles acquire mass only
after the temperature has dropped below the Weinberg-Salam symmetry breaking
scale. It is unclear how this temperature dependent effect is to be taken
account of in our formalism. One might suspect however, that this
gravitational phenomenon occurring over length scales $H^{-1}$ may not be
seriously affected by the temperature, which is $\sim M_{GUT}\gg H$.

The first thing we would like to check is whether the energy density of the
produced particles is comparable to the background energy density. If it is,
then we cannot really trust the answer because we have not taken account of
the back reaction of this newly generated energy density on the background
geometry. This problem is generic, and there is no known extension of the
formalism to take this into account. Under the circumstances, in the case in
which we find the gravitationally generated energy density to be comparable to
the background density, we shall conclude that something nontrivial is
occurring but which needs to be investigated using techniques yet to be
invented.

In the following, while making order of magnitude estimates, we take the GUT
mass scale $M_{GVT}\simeq\rho_{0}^{1/4}$ (see (17)) to be
$10^{14}\mathrm{GeV},G^{-1/2}\equiv M_{Pl}\simeq 10^{19}\mathrm{GeV}$, and
$H^{2}\sim$ $\rho_{0}/M_{Pl}^{2}=\left(10^{9}\mathrm{GeV}\right)^{2}$. First
consider the terms in $\rho^{(1)}$ and $\rho^{(2)}$ that are $\sim H^{4}$.
Substituting the above numbers, we see that these are certainly small compared
to $\rho_{0}$. The interesting term is in $\rho_{\text{low }}^{(1)}$, which,
aside from a logarithm is $\sim H^{5}/\mathrm{m}$. We find

$\frac{\rho^{(1)}}{\rho_{0}}\sim\frac{H^{3}}{mM_{Pl}^{2}}=\frac{10^{-11}GeV}{m}$
(44)

This means that for $m\lesssim 0.01\mathrm{eV}$, (and $m/H\lesssim 10^{-20}$)
the formalism of quantum field theory in curved spacetime (in the form we have
used here) breaks down. In fact if $\rho_{0}$ were as large as
$\left(10^{16}\mathrm{GeV}\right)^{4}$ (so that $H\sim 10^{13}\mathrm{GeV}$ ),
as in some models of inflation, this value would be $m\sim 100\mathrm{eV}$.

Let us assume next that $m$ is larger than the limits found in the preceding
paragraph, so that our formalism is still valid. First consider the case in
which these particles are freely interacting with other species. Then they
will quickly reach thermal equilibrium with the radiation constituting the
background energy density. In that case their subsequent evolution is the same
as that of a massive species in the standard cosmology. Despite the fact that
they contribute significant fraction of the energy density at that epoch,
there will be no observable signature left. It is possible that the produced
particles decay into lighter particles. Then their evolution has to be traced
as has been done in some dark matter scenarios 19 .

The other possibility is that these particles are stable and interact very
weakly with the rest of matter. Further, due to their low number density,
their self-interaction may also be ignorable. In this case, the spectral
distribution of this energy-density will remain non-thermal. We then expect
that the long-wavelength part $\left(k_{phy}<m\right)$ can be treated as
pressureless dust even as early as the epoch $\eta_{2}$. Then these particles
would come to dominate the total energy density too quickly, as follows

$\displaystyle\frac{\rho^{(1)}}{\rho_{\gamma}}(\eta)$ $\displaystyle\simeq$
$\displaystyle\frac{H^{5}}{m}\frac{T^{3}(\eta)}{T_{2}^{3}}\frac{1}{H^{2}M_{Pl}^{2}}\frac{T_{2}^{4}}{T^{4}(\eta)}$
(45) $\displaystyle\simeq$ $\displaystyle\frac{T_{2}}{mT(\eta)}\cdot
10^{-11}\mathrm{GeV}$

If we take the reheating temperature $T_{2}$ to be $10^{14}\mathrm{GeV}$, for
any $m\lesssim 10^{7}\mathrm{GeV}$, the Universe would become matter dominated
as early as the epoch of nucleosynthesis and affect that process adversely.
(we took $T_{nucleosynthesis}=0.1\mathrm{MeV}$ ). Thus the known abundance of
primordial light nuclei places a lower limit on the mass of non-interacting
particles that may be produced by this mechanism. As before, the bound is more
stringent if $M_{GUT}\sim 10^{15}-$ $10^{16}\mathrm{GeV}$. In a more detailed
treatment, we must remember that the actual contribution to the long-
wavelength part of the spectrum is continuously increasing due to cosmological
red-shifting.

It is clear that one needs to investigate the above possibilities in specific
models of inflation and particle physics in greater detail. This will be
reported in a separate publication.

##  VI Conclusion

We considered a quantised scalar field in an inflationary universe. Using some
standard techniques, we calculated the energy density appearing in the form of
quanta of this field due to the rather abrupt change from a de Sitter like
phase to a radiation dominated phase. We employed the sudden approximation to
do this calculation. This restricts the validity of our results to the case of
modes with wavelengths long compared to the interval over which the transition
from one phase to the other occurs. This is not as great a disadvantage as it
might appear at first sight, since from other tractable problems we know that
the spectrum of produced particles must taper off exponentially at large wave-
numbers. The interesting case lost is that of $m\sim H$ for its modes with
$k_{phy}\sim m$. For small mass case we got reasonable results even for
$k_{phy}\sim H$. viz.. our $\left|\beta_{k}\right|^{2}$ is the same as that
for the massless rase, calculated independently by Allen.

One may wonder whether a better treatment avoiding the sudden approximation
was possible. I shall give here the problems encountered in the process.
Ideally one would like to determine the mode functions that would be valid
throughout spacetime, including through the transition region. One can then
compare them with the adiabatic modes of the inflationary and the radiation
dominated phases to obtain the required Bogolubov coefficients. To do this, we
have to first model inflation using a smooth function that changes over from
the $1/(1-\eta)$ form to a linear function. With any function that does this,
(that I could find), the differential equations satisfied by the mode
functions no longer remain within the hyperbolic scheme. Although solutions
may be still possible to find, one no longer has the advantage of the
systematic approach possible with the hyperbolic scheme. Since the price paid
in making the approximation is not great we have opted to rely on the sudden
approximation.

The surprising result that comes out of this analysis is that the very long
wavelength modes of the small mass particles give a large contribution to the
energy density, proportional to $\mathrm{H}^{5}/\mathrm{m}$. We note that two
factors contribute to this. One is (62)

$\left|\beta_{k}\right|^{2}\simeq
0.74\sqrt{\frac{H}{2m}}\left(\frac{H}{k_{phy}}\right)^{3}\quad k_{phy}^{2}\ll
mH$ (46)

The other is (72)

$u_{k}\simeq 1.5\sqrt{\frac{H}{2m}}H\quad k_{phy}^{2}\ll mH$ (47)

The appearance of $(H/m)^{1/2}$ in these expressions is puzzling. It has to do
with how the modes of a massive scalar have to be normalised in a radiation
dominated universe. Note the appearance of $\sigma^{-1/2}\sim m^{-1/4}$ in eq.
(29). This normlisation has appeared in the literature before. for instance,
in ref. 12 . As for the singular $k$ dependence of
$\left|\beta_{k}\right|^{2}$. a more physical reasoning may exist, having to
do with global properties of the two spaces concerned. Such a reasoning has
been given by Allen in the massless case.

The singular $m$ dependence of the above equations need not worry us about the
massless case because they are valid only for physical wavelengths larger than
the Compton wavelength. and not appropriate for studying the massless limit.
In this context. note that the mode functions we have found in eq.s (23), (29)
reduce to those for the massless case as found, for instance, by Allen. The
limit of the de Sitter era mode functions is obtained easily. For the mode
functions of the radiation dominated era, one needs to consider $D_{n}(z)$ in
the limit of $z\rightarrow 0$, and $n\rightarrow\infty$, but with
$z^{2}n\rightarrow k^{2}\eta^{2}$. Such limits however, can not be taken after
mode-matching has been performed at a specific epoch.

The massless limit can be found in our answer from the $k_{phy}\sim H$ case.
which turns out to be $O\left(m^{0}\right)$. In this case we find for $w_{k}$
(See eq. (75))

$w_{k}=k_{phy}+\frac{H^{2}}{2k_{phy}}\quad k_{phy}\lesssim H$ (48)

The second term here happens to be the same term from the complete expression
for $w_{k}$ (eq. (68) )that gave the leading contribution to the $k_{p\text{
hy }}^{2}\ll mH$ case above. As pointed out below eq. (40), this term,
although dictated by covariance, is not included in other works the author is
aware of (viz., ref.s 7 , 8 , 18 ), in their definition of the effective
energy of each mode. Energy-momentum tensor for gravitons is a subtle issue,
but for massless non-conformal scalars the energy density must contain the
above term, and to that extent we have corrected the result of ref. 18 . Note
that the above term becomes more important than the preceding one when
$k_{phy}<H$, but this is precisely the range in which modes are produced by
the gravitational mechanism, and hence is not ignorable. The term is absent
for conformally coupled fields. We plan to discuss this issue in greater depth
in a separate publication dealing with the massless case.

We must also note that the present calculation has been carried out for scalar
particles. The results however, are interesting enough that it seems worth
carrying out a similar exercise for fermions. Otherwise, in absence of
fundamental scalars, our results remain only of instructional value.
Similarly, it is unclear what modifications, if any, the results will undergo
if the finite temperature symmetry restoration effect of spontaneously broken
gauge theories is taken into account.

The techniques of quantum field theory in curved space that we have used here
are not well established. They are not unambiguous generalisations of the flat
space field theory, and there has been no experimental conformation for them,
for instance the notion of adiabatic vacuum. This calculation shows one
example where the results may be testable against astrophysical observations.
Since our procedure is not unambiguous, it is interesting to compare our
results with those obtained by Turner and Widrow using a more intuitive
picture and heuristic arguments. Restated in the notation used here, their
results for minimally coupled scalar massive particles are as follows. If we
assume the values of $\rho_{0}$ and $T_{2}$ as above, their answer for the
ratio of the energy density in these particles today to today’s closure
density is

$\Omega_{\phi}\simeq 3.9\times
10^{17}\cdot\left(\frac{H}{M_{PI}}\right)^{2}\cdot\left(\frac{m}{GeV}\right)^{\frac{1}{2}}$
(49)

in the range of $m$ values that coincides with that permitted by our
formalism. If we take $m\simeq 10^{6}\mathrm{GeV}$ (which case we found
conflicts with nucleosynthesis in our calculation if the particles are very
weakly interacting), we get $\Omega_{\phi}\sim O(1)$, i.e.capable of seriously
affecting the observed Universe as in our case. However, the dependence on $H$
and $m$ are completely different and we have no way of reconciling the two
answers.

###### Acknowledgements.

I thank Prof. J. V. Narlikar, Prof. S. M. Chitre, T. R. Seshadri, T. P. Singh
and Ashoke Sen for useful conversations in the course of this work, and T.
Padmanabhan for many useful discussions. I also wish to record here my
indebtedness to Prof. P. Candelas who taught me mathematical physics. Note
added in 2024 : This conversion of the original preprint has benefited from
use of the tools at mathpix.com

## Appendix A

The most important simplification that occurs on assuming $m/H\ll 1$, that is,
$m\Omega_{2}\eta_{2}=\frac{1}{2}\sigma^{2}\eta_{2}^{2}\ll 1$ is that the order
$\lambda$ of the Hankel functions of eq. (23) becomes $3/2$, in which case

$\displaystyle X=H_{3/2}^{(1)}\left(k\eta_{2}\right)=\sqrt{\frac{2}{\pi
k\eta_{2}}}\left(-1-\frac{i}{k\eta_{2}}\right)e^{ik\eta_{2}}$ (50)
$\displaystyle Y=H_{3/2}^{(1)\prime}\left(k\eta_{2}\right)=\sqrt{\frac{2}{\pi
k\eta_{2}}}\left\\{\frac{3}{2k\eta_{2}}-i+\frac{3i}{2k^{2}\eta_{2}^{2}}\right\\}e^{ik\eta_{2}}$
(51)

Further, the argument $\sigma\eta_{2}$ of the parabolic cylinder functions
goes to zero, in which case

$D_{\nu}(0)=\sqrt{\pi}\frac{2^{\nu/2}}{\Gamma\left(\frac{1-\nu}{2}\right)}$
(52)
$D_{\nu}^{\prime}(0)=-\sqrt{\pi}\frac{2^{(\nu+1)/2}}{\Gamma\left(-\frac{\nu}{2}\right)}$
(53)

It follows that

$F\bar{F}X\bar{X}\sigma^{2}\eta_{2}^{2}=2\sqrt{2}\frac{\sigma^{2}}{k}\eta_{2}\left(1+\frac{1}{k^{2}\eta_{2}^{2}}\right)\frac{1}{\left|\Gamma\left(\frac{1}{4}+\frac{ik^{2}}{2\sigma^{2}}\right)\right|^{2}}$
(54)
$E\bar{E}\left|\frac{1}{2}X+k\eta_{2}Y\right|^{2}=\frac{\sqrt{2}}{k\eta_{2}}\left(1+\left(k\eta_{2}-\frac{1}{k\eta_{2}}\right)^{2}\right)\frac{1}{\left|\Gamma\left(\frac{3}{4}+\frac{ik^{2}}{2\sigma^{2}}\right)\right|^{2}}$
(55) $\displaystyle
2\sigma\eta_{2}\operatorname{Re}e^{i\pi/4}F\bar{E}X\left(\frac{1}{2}\bar{X}+k\eta_{2}\bar{Y}\right)$
(56) $\displaystyle=\frac{2\sigma}{\pi
k}\left(\frac{1}{k^{2}\eta_{2}^{2}}\exp\left(-\pi
k^{2}/2\sigma^{2}\right)-k\eta_{2}\exp\left(\pi
k^{2}/2\sigma^{2}\right)\right)$

We now estimate the contribution of the $\Gamma$-functions. As indicated in
the text, the two interesting ranges of values are $k/\sigma\ll 1$ and
$k/\sigma\gg 1$. For $k/\sigma\ll 1$, we use 16

$|\Gamma(x+iy)|^{2}=\Gamma^{2}(x)\prod_{n=0}^{\infty}\left[\frac{1}{1+y^{2}/(x+n)^{2}}\right]$
(57)

From which we learn that

$\frac{1}{\left|\Gamma\left(\frac{1}{4}+\frac{ik^{2}}{2\sigma^{2}}\right)\right|^{2}}=\frac{1}{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}\left[1+O\left(\frac{k^{4}}{\sigma^{4}}\right)\right]$
(58)

Since $k/\sigma\ll 1$ also means $k\eta_{2}<1$, we find the dominant
contribution of (54) to be

$F\bar{F}\text{ term
}:\frac{2\sqrt{2}}{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}\sigma^{2}\eta_{2}^{2}\frac{1}{k^{3}\eta_{2}^{3}}\left(1+O\left(k^{2}\eta_{2}^{2}\right)+O\left(\frac{k^{4}}{\sigma^{4}}\right)\right)$
(59)

Similarly, for (55) we find

$E\bar{E}\text{ term
}:\frac{\sqrt{2}}{\left(\Gamma\left(\frac{3}{4}\right)\right)^{2}}\frac{1}{k^{3}\eta_{2}^{3}}\left(1+O\left(k^{2}\eta_{2}^{2}\right)+O\left(\frac{k^{4}}{\sigma^{4}}\right)\right)$
(60)

Finally, for the cross term (56) we get

$F\bar{E}\text{ term
}:\frac{2}{\pi}\sigma\eta_{2}\frac{1}{k^{3}\eta_{2}^{3}}\left(1+O\left(k^{3}\eta_{2}^{3}\right)+O\left(\frac{k^{2}}{\sigma^{2}}\right)\right)$
(61)

Comparing the three terms, we see that the $E\bar{E}$ term is the leading term
of the expansion in $m/H$. Hence,

$\left|\beta_{k}\right|^{2}\simeq\frac{0.74}{\sqrt{2m/H}\left(k_{phy}/H\right)^{3}}\quad
k_{phy}^{2}\ll mH$ (62)

where $k_{phy}\equiv k/\Omega_{2}$. In the limit $k/\sigma\gg 1$, we use
Stirling’s approximation for the $\Gamma$-functions to find

$|\Gamma(x+iy)|^{2}\longrightarrow 2\pi
e^{-\pi|y|}|y|^{2x-1}\quad(|y|\rightarrow\infty,|x|\ll|y|)$ (63)

Now the $\Gamma$-functions contribute important multiplicative factors
$k/\sigma$, so that

$F\bar{F}\text{ term
}:\frac{1}{\pi}\sigma\eta_{2}\left(1+\frac{1}{k^{2}\eta_{2}^{2}}\right)\exp\left(\frac{\pi
k^{2}}{2\sigma^{2}}\right)$ (64) $\displaystyle E\bar{E}\text{ term :
}\frac{1}{\pi}\sigma\eta_{2}\left(1-\frac{1}{k^{2}\eta_{2}^{2}}+\frac{1}{k^{4}\eta_{2}^{4}}\right)\exp\left(\frac{\pi
k^{2}}{2\sigma^{2}}\right)$ (65) $\displaystyle F\bar{E}\text{ term :
}\quad-\frac{2}{\pi}\sigma\eta_{2}\exp\left(\frac{\pi
k^{2}}{2\sigma^{2}}\right)$ (66)

Since this range of $k$ corresponds to $k_{phy}\sim H$, we have retained all
the powers of $k\eta_{2}=$ $k_{\text{phy }}/H$. Combining these, we find

$\left|\beta_{k}\right|^{2}=\frac{1}{4}\left(\frac{H}{k_{phy}}\right)^{4}\quad
k_{phy}\leq H$ (67)

We now estimate $w_{k}$ of (10) for both these cases.

$\displaystyle w_{k}=$
$\displaystyle\frac{1}{\Omega}\left|\chi_{k}^{\prime}\right|^{2}-2\frac{\Omega^{\prime}}{\Omega^{2}}\operatorname{Re}\chi_{k}^{\prime}\chi_{k}^{*}$
(68)
$\displaystyle+\Omega\left(\frac{|\vec{k}|^{2}}{\Omega^{2}}+m^{2}+\left(\frac{\Omega^{\prime}}{\Omega^{2}}\right)^{2}\right)\left|\chi_{k}\right|^{2}$

Since we evaluate this immediately at the end of inflation, we shall take
$\Omega^{\prime}/\Omega^{2}=H$. Then for $k^{2}/\sigma^{2}\ll 1$,

$\displaystyle\left|\chi_{k}\right|^{2}$
$\displaystyle=\frac{1}{\sigma}\exp\left(-\pi
k^{2}/2\sigma^{2}\right)\frac{\pi}{\sqrt{2}}\frac{1}{\left(\Gamma\left(\frac{3}{4}\right)\right)^{2}}$
(69) $\displaystyle\left|\chi_{k}^{\prime}\right|^{2}$
$\displaystyle=\sigma\exp\left(-\pi
k^{2}/2\sigma^{2}\right)\pi\sqrt{2}\frac{1}{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}$
(70) $\displaystyle\operatorname{Re}\chi_{k}^{\prime}\chi_{k}^{*}$
$\displaystyle=-\frac{1}{2}\exp\left(-\pi k^{2}/\sigma^{2}\right)$ (71)

On comparing we find $H^{2}\left|\chi_{k}\right|^{2}$ making the dominant
contribution, so that

$w_{k}\simeq
1.5\frac{H}{\sqrt{2m/H}}+O\left(\left(\frac{m}{H}\right)^{0},\left(\frac{k}{\sigma}\right)^{2}\right)\quad
k_{ph_{y}}^{2}\ll mH$ (72)

For $k^{2}/\sigma^{2}\gg 1$, we use (63) to find that

$\left|\chi_{k}\right|^{2}\rightarrow\frac{1}{2k}$ (73)
$\left|\chi_{k}^{\prime}\right|^{2}\rightarrow\frac{k}{2}$ (74)

and the cross term is exponentially suppressed. Hence

$w_{k}=k_{phy}+\frac{H^{2}}{2k_{phy}}+O\left(\left(\frac{\sigma}{k}\right)^{2}\right)\quad\quad
k_{phy}\lesssim H$ (75)

## Appendix B

Here we calculate the spectrum for the case $m/H=\sqrt{2}$. This makes the
order of the Hankel functions $\lambda=1/2$. So

$\displaystyle X$ $\displaystyle=-i\sqrt{\frac{2}{\pi
k\eta_{2}}}e^{ik\eta_{2}}$ (76) $\displaystyle Y$
$\displaystyle=-i\sqrt{\frac{2}{\pi
k\eta_{2}}}\left\\{ik\eta_{2}-\frac{1}{2k^{2}\eta_{2}^{2}}\right\\}e^{ik\eta_{2}}$

We have $\sigma\eta_{2}=\sqrt{2m/H}=2^{3/4}$. Since we want the case
$k_{phy}^{2}/\sigma^{2}\ll 1$, we shall evaluate
$D_{-\frac{1}{2}}\left(e^{i\pi/4}\sigma\eta_{2}\right)$. Ideally we would like
to expand in $k_{phy}^{2}/\sigma^{2}$ but this is not readily available, and
as we shall see, not needed. We find the required values from the tables of
Kireyeva and Karpov 20  and using standard recursion relations for the Weber
functions.

$\displaystyle E=0.3960-i0.6244$ (77) $\displaystyle F=-0.0147+i0.5565$

The $F\bar{F}$ term turns out to be the leading term in $k/\sigma$.

$\left|\beta_{k}\right|^{2}=0.155\frac{\sigma}{k}+0\left(\frac{k^{2}}{\sigma^{2}}\right)$
(78)

Next we calculate $w_{k}$ using

$\left|\chi_{k}\right|^{2}=0.546\frac{1}{\sigma};\quad\left|\chi_{k}^{\prime}\right|^{2}=0.31\sigma$
(79) $\mathrm{Re}\chi_{k}^{\prime}\chi_{k}^{*}\ =\
\mathrm{Re}e^{i\pi/4}F\bar{E}\ =\ -0.4$ (80) $w_{k}=2.3H$ (81)
$\frac{d\rho}{dk_{phy}}~{}=~{}3.86~{}H^{2}k_{phy}$ (82)

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