# Particle physics processes in cosmology through an effective Minkowski space
formulation and the limitations of the method

Recai Erdem<EMAIL_ADDRESS>Kemal Gültekin<EMAIL_ADDRESS>Department of Physics
İzmir Institute of Technology
Gülbahçe, Urla 35430, İzmir, Turkey

###### Abstract

We introduce a method where particle physics processes in cosmology may be
calculated by the usual perturbative flat space quantum field theory through
an effective Minkowski space description at small time intervals provided that
the running of the effective particle masses are sufficiently slow. We discuss
the necessary conditions for the applicability of this method and illustrate
the method through a simple example. This method has the advantage of avoiding
the effects of gravitational particle creation in the calculation of rates and
cross sections i.e. giving directly the rates and the cross sections due to
the scatterings or the decay processes.

Quantum field theory in curved space, cosmology

##  I introduction

The current standard model of cosmology $\Lambda$CDM Weinberg  although being
quite successful at cosmological scales has some observational difficulties at
smaller scales and some theoretical problems CDM ; ccp . One of the most
popular and preferred alternatives to $\Lambda$CDM are those that employ
scalar fields (such as quintessence) or vector or fermionic fields for dark
energy or dark matter or for both DM-scalar-vector ; DM-fermionic ; DE-scalar-
vector ; DE-fermionic . The scalar, vector, or fermionic fields are also
employed for cosmic inflation DE-fermionic ; inflation-vector . For the study
of the evolution of these fields at quantum level the use of quantum field
theory in curved space is needed QFTC1 ; QFTC .

In quantum field theory in curved space one needs to determine the mode
function for a given cosmological background to calculate the cross section or
the rate corresponding to a particle physics process. The mode function must
be determined for each cosmological background separately, and this may be a
complicated process in general that makes the results more complicated and
less clear when compared to the usual Minkowski space perturbative quantum
field theory calculations. Moreover, in the calculations in the context of
quantum field theory in curved space, the rates and the cross sections involve
contributions due to (spontaneous) gravitational particle production in
addition to those due to interaction Lagrangians. This makes the problem of
identifying the results of the calculation even more difficult. The
contribution of gravitational production may be decreased by using adiabatic
approximation Bunch ; QFTC1 . In this way one may also make use of WKB type
solutions for mode function, and assume the space be approximately Minkowskian
for the modes with sufficiently short wavelengths. However, in general, the
quantum field theory in curved space calculations may have significant
departures from the results of S-matrix formulation of the usual Minkowski
space quantum field theory even when one may employ adiabatic approximation
Boyanovsky-new . The source of the problem is that, although, in principle,
Minkowski space is a good approximation for sufficiently small wavelengths,
adiabatic approximation by itself is not sufficient to justify the use of the
usual Minkowski space quantum field theory as a sufficiently good
approximation. One needs a more concrete and more general condition to justify
the use of approximate Minkowski space quantum field theory calculations in
cosmology. In this study we introduce a new rigorous condition (in addition to
adiabatic condition) that guarantees the use of Minkowski space quantum field
theory calculations in cosmology as a good approximation. Moreover, the
studies in literature in that context Boyanovsky1 ; Boyanovsky2 ; Vilja
usually employ the simpler case of scalar fields conformally coupled to
gravity or to the cases effectively equivalent to conformally coupled case.
Even in that case the results of the calculations are rather complicated. In
this paper, we consider scalars minimally coupled to gravity by using a method
that employs an effective Minkowski space formulation. We also discuss the
limitations of the applicability of this condition.

The effective Minkowski space formulation studied here was employed in the
context of a specific model for formation of Bose-Einstein condensation in
cosmology Erdem-Gultekin . The aim of this paper is to provide a more general,
formal and detailed treatment of this formulation i.e. of the effective
Minkowski space formulation of quantum field theory (in the background of a
Robertson-Walker metric). We employ the simple, yet elaborate enough Lagragian
density of Erdem-Gultekin , in particular, the
$\chi\chi\,\rightarrow\,\phi\phi$ processes in this framework, to see the
basic implications of the formulation in a simple setting. We show that the
corresponding calculations may be done by using the tools of the usual
perturbative Minkowski space quantum field theory in sufficiently small time
intervals provided that the parameters of the model satisfy some conditions.
We also show that there is a considerable parameter space that satisfies those
conditions.

##  II the basic conditions for effective Minkowski space formulation

Spacetime at cosmological scales is described by

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

For simplicity we take $k=0$ which is in agreement with observations PDG . We
consider the following action in this space

$\displaystyle S$
$\displaystyle\,=\,\int\sqrt{-g}\;d^{4}x\,\frac{1}{2}\\{-g^{\mu\nu}\left[\partial_{\mu}\phi\partial_{\nu}\phi\,+\,\partial_{\mu}\chi\partial_{\nu}\chi\right]\,-\,m_{\phi}^{2}\phi^{2}\,-\,m_{\chi}^{2}\chi^{2}\,-\,\mu\,\phi^{2}\chi\\}$
(2) $\displaystyle=$
$\displaystyle\int\,d^{3}x\,d\eta\,\frac{1}{2}\\{\,\tilde{\phi}^{\prime\;2}-(\vec{\nabla}\tilde{\phi})^{2}+\tilde{\chi}^{\prime\;2}-(\vec{\nabla}\tilde{\chi})^{2}-\tilde{m}_{\phi}^{2}\tilde{\phi}^{2}-\,\tilde{m}_{\chi}^{2}\tilde{\chi}^{2}-\tilde{\mu}\,\tilde{\phi}^{2}\tilde{\chi}\,\\}\;,$
(3)

where prime denotes derivative with respect to conformal time $\eta$ QFTC  and

$\displaystyle
d\eta\,=\,\frac{dt}{a(t)}~{},~{}\tilde{\phi}\,=\,a(\eta)\phi~{},~{}\tilde{\chi}\,=\,a(\eta)\chi~{},~{}a^{\prime}\,=\,\frac{da}{d\eta}~{},~{}\dot{a}\,=\,\frac{da}{dt}~{},~{}a^{\prime\prime}\,=\,\frac{d^{2}a}{d\eta^{2}}~{},~{}\ddot{a}\,=\,\frac{d^{2}a}{dt^{2}}$
$\displaystyle\tilde{m}_{i}^{2}\,=\,m_{i}^{2}a^{2}-\frac{a^{\prime\prime}}{a}\,=\,a^{2}\left(m_{i}^{2}-\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}\right)~{}~{}i\,=\,\phi,\chi~{},~{}~{}\tilde{\mu}\,=\,a\,\mu\;.$
(4)

We will use the above action to study the essential aspects of a formulation
(that we call effective Minkoski space formulation) that may be used to
calculate cross sections and rates of particle physics processes in the
background of a Robertson-Walker metric. Note that, the set-up given above
only serves as a simple and instructive toy model to study basics of this
formulation while, in principle, this formulation is applicable to any model
provided that the conditions introduced below are satisfied.

We assume that the rate of the decays and the scatterings are much larger than
the Hubble parameter so that one may take the masses $\tilde{m}_{\chi}$,
$\tilde{m}_{\phi}$ constant during times smaller than the average time between
two individual processes (e.g. between two individual
$\chi\chi\,\rightarrow\,\phi\phi$ processes) while the time dependence is
observed only at cosmological scales. This condition may be expressed as the
variation in
$\tilde{m}_{\chi}^{2}=a^{2}\left(m_{\chi}^{2}-\dot{H}-2H^{2}\right)$ and
$\tilde{m}_{\phi}^{2}=a^{2}\left(m_{\phi}^{2}-\dot{H}-2H^{2}\right)$ during
times $\Delta\,t$ smaller than the average time (e.g.
$\frac{1}{n_{\chi}\beta\sigma\,v}$ for $\chi\chi\,\rightarrow\,\phi\phi$)
between two individual processes (where $\beta$ is the effective penetration
depth of the incoming beam to the target, $n_{\chi}$ is the number density of
the target particles, $\sigma$ is the cross section of the process, $v$ is the
relative velocity of the incoming and the target particles) should be very
small i.e.

$\left|\frac{\Delta\,\tilde{m}^{2}}{\tilde{m}^{2}}\right|\,=\,\left|\frac{\Delta\,t\,\left(\frac{d\,a^{2}\left(m^{2}-\dot{H}-2H^{2}\right)}{dt}\right)}{a^{2}\left(m^{2}-\dot{H}-2H^{2}\right)}\right|\,\ll\,1~{},~{}~{}\mbox{where}~{}~{}\Delta\,t\,\leq\,\frac{1}{n_{\chi}\beta\sigma\,v}~{}~{}\mbox{for}~{}~{}\chi\chi\,\rightarrow\,\phi\phi\;.$
(5)

Here $m$ denotes either of $m_{\chi}$ or $m_{\phi}$,
$\dot{H}=\frac{dH}{dt}=\frac{\ddot{a}}{a}-H^{2}$. The upper bound in (5) on
$\Delta\,t$ is imposed because we take $\Delta\,t$ as the time interval where
the particles may be considered as free particles.

If we let

$H\,=\,\xi\,a^{-s}\;,$ (6)

which includes all simple interesting cases e.g. radiation, matter, stiff
matter, cosmological constant dominated universes, then (5) becomes

$\left|\frac{2\Delta\,t\,a^{2}H\left[m_{\chi}^{2}-(s^{2}-3s+2)H^{2}\right]}{a^{2}\left(m_{\chi}^{2}-(2-s)H^{2}\right)}\right|\,=\,\left|2H\Delta\,t\,\left(1-\frac{s(s-2)H^{2}}{m^{2}+(s-2)H^{2}}\right)\right|\ll\,1\;.$
(7)

Another basic condition for applicability of an effective Minkowski space
formulation is that the variation of the effective coupling constant
$\tilde{\mu}$ for a sufficiently small time interval $\Delta\,t$ in each
interval $\eta_{i}\,<\,\eta\,<\,\eta_{i+1}$ i should be negligible i.e.

$\frac{\Delta\,\tilde{\mu}}{\tilde{\mu}}\,=\,\frac{\frac{d\tilde{\mu}}{dt}\Delta\,t}{\tilde{\mu}}\,=\,H\,\Delta\,t\,\ll\,1~{},~{}~{}\mbox{where}~{}~{}\Delta\,t\,\leq\,\frac{1}{n_{\chi}\beta\sigma\,v}\;.$
(8)

Note that the condition (8) guarantees the condition (7) provided that
$\left|\left(1-\frac{s(s-2)H^{2}}{m^{2}+(s-2)H^{2}}\right)\right|$ is not very
large. $\left|\left(1-\frac{s(s-2)H^{2}}{m^{2}+(s-2)H^{2}}\right)\right|$ is
very large only if $s$ is very large or if $\frac{m_{\chi}^{2}}{H^{2}}$ is
very close to $2-s$. Thus, (8) is enough to guarantee (7) essentially in all
realistic cases. Moreover, (8) implies that $\Delta\,t$ is much smaller than
the Hubble time $\frac{1}{H}$. Therefore, in the following we will simply
suffice to impose the condition (8) rather than (7).

##  III implication of the basic conditions and the adiabatic condition for
mode functions

The fields $\tilde{\phi}$ and $\tilde{\chi}$ may be expanded in their Fourier
modes; for example,

$\tilde{\chi}(\vec{r},\eta)\,=\,\frac{1}{\sqrt{2}}\int\,\frac{d^{3}\tilde{p}}{(2\pi)^{\frac{3}{2}}}\left[a_{p}^{-}\,v_{p}^{*}(\eta)e^{i\vec{\tilde{p}}.\vec{r}}\,+\,a_{p}^{+}\,v_{p}(\eta)e^{-i\vec{\tilde{p}}.\vec{r}}\right]\;,$
(9)

where $\vec{r}=(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3})$, $a_{p}$ are the
expansion coefficients that are identified by annihilation operators after
quantization, $v_{p}$ are the basic normalized solutions of the equation of
motion (i.e. mode functions) of $\tilde{\chi}$. In the time interval between
two particle physics processes (such as those in Figure 1) one may consider
$\tilde{\chi}$ particles to be free particles, so they satisfy

$v_{p}^{\prime\prime}\,+\,\omega_{p}^{2}(\eta)\,v_{p}\,=\,0~{}~{}~{}\mbox{where}~{}~{}\omega_{p}\,=\,\sqrt{|\vec{\tilde{p}}|^{2}+\tilde{m}_{\chi}^{2}}~{}~{},~{}~{}~{}v_{p}^{\prime}\,v_{p}^{*}-v_{p}\,v_{p}^{*\,\prime}\,=\,2i\;.$
(10)

where $\tilde{}$’s in (10) refer to the action (3) while the quantities
without $\tilde{}$ will refer to (2).

We consider WKB-type solutions for (10) QFTC

$v_{p}(\eta)\,=\,\frac{1}{\sqrt{W_{p}(\eta)}}exp{\left(i\int_{\eta_{i}}^{\eta}\,W_{p}(\eta)\,d\eta\right)}\;,$
(11)

where $W_{p}$, by (10), satisfies

$W_{p}^{2}\,=\,\omega_{p}^{2}-\frac{1}{2}\left[\frac{W_{p}^{\prime\prime}}{W_{p}}-\frac{3}{2}\left(\frac{W_{p}^{\prime}}{W_{p}}\right)^{2}\right]\;.$
(12)

If we let $|W_{p}^{2}-\omega_{p}^{2}|$ in (12) be small with respect to
$\omega_{p}^{2}$, then we may take

${}^{(0)}W_{p}=\omega_{p}$ (13)

as an approximation for $W_{p}$. Further, if we let the variation of
$\omega_{p}^{2}$ with $\eta$ be small (i.e. if the contribution of
gravitational particle production to the particle physics processes is small
QFTC1 ) then we may take

${}^{(2)}W_{p}=\omega_{p}\left(1-\frac{\omega_{p}^{\prime\prime}}{4\omega_{p}^{3}}+\frac{3\omega_{p}^{\prime\;2}}{8\omega_{p}^{4}}\right)~{},~{}~{}\mbox{etc.}$
(14)

as higher order (better) approximations. These approximations are known as
adiabatic approximations of order zero, order two etc. Bunch ; QFTC1 . Note
that ${}^{(0)}W_{p}\,=\,\omega_{p}$ = constant corresponds to Minkowski case,
and ${}^{(2)}W_{p}$ is obtained by substituting ${}^{(0)}W_{p}$ with time
dependent $\omega_{p}$ on the right hand side of (12) and then Taylor
expanding the square root for slowly varying $\omega_{p}$.

The requirement of slow variation of $\omega_{p}^{2}$ with time introduced
above may be expressed in a formal way as

$\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}\,<\,1.$ (15)

As $\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}$ gets smaller and smaller, the
WKB approximate solutions in (14) get closer and closer to the solution in
(13) which, in turn, gets closer and closer to the exact solution at time
$\eta$. The above condition guarantees the more intuitive identification of
adiabatic approximation as slow variation of energy during Hubble time since
$\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}$$\,<\,$$\frac{\omega_{p}^{\prime}}{H\omega_{p}}$
for physically relevant modes inside horizon (i.e. for $\omega_{p}\,>\,H$). In
the following, first we will show that the condition (5) ensures adiabatic
approximation for most of the physically relevant parameter space, then we
will show that the condition (5) makes ${}^{(0)}W_{p}$ a perfect approximation
in each interval $\Delta\,t$.

The requirement of slow variation of $\omega_{p}^{2}$ with time in (15) is
similar to the condition (5)

$\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}\,=\,a\frac{\left(\frac{1}{\omega_{p}}\right)\frac{d\tilde{m}_{\chi}^{2}}{dt}}{2\omega_{p}^{2}}\,<\,a\frac{\left(\frac{1}{\omega_{p}}\right)\frac{d\tilde{m}_{\chi}^{2}}{dt}}{2\tilde{m}_{\chi}^{2}}\,<\,a\frac{\left(\frac{1}{\tilde{m}_{\chi}}\right)\frac{d\tilde{m}_{\chi}^{2}}{dt}}{2\tilde{m}_{\chi}^{2}}\,=\,a\frac{\left(\frac{1}{(\Delta\,t)\tilde{m}_{\chi}}\right)\,(\Delta\,t)\frac{d\tilde{m}_{\chi}^{2}}{dt}}{2\tilde{m}_{\chi}^{2}}\;.$
(16)

We see that the main differences between (15) and (5) are the replacements of
$\Delta\,t\,\leq\,\frac{1}{n_{\chi}\beta\sigma\,v}$ by $\frac{1}{\omega_{p}}$
or $\frac{1}{\tilde{m}_{\chi}}$ and of $\ll\,1$ by $<\,1$. We notice that (5)
ensures (15) if $\frac{1}{\Delta\,t\;\tilde{m}_{\chi}}$ is not much greater
than 1 i.e. if $\Delta\,t\;\tilde{m}_{\chi}$ is not much smaller than 1. To
see the range of the applicability of this condition, we take
$\Delta\,t\,=\,\frac{1}{n_{\chi}\beta\sigma\,v}$ and let
$\frac{\tilde{m}_{\chi}}{n_{\chi}\beta\sigma\,v}\,>\,{\cal O}(1)$ i.e.
$\tilde{m}_{\chi}\,>\,n_{\chi}\beta\sigma\,v$. This, for example, implies
$\tilde{m}_{\chi}\,>\,\hbar\,n_{\chi}\beta\sigma\,v\,=\,2\times\,10^{-32}\;eV$
for $\hbar\simeq\,6.6\times\,10^{-16}\,eV.second$,
$n_{\chi}=10^{8}\,(meter)^{-3}$ (i.e. at the order of the photon number
density in the universe), $\sigma=10^{-33}\,(meter)^{2}$ (i.e. in the order of
the cross section of electromagnetic interactions), $\beta=1$,
$v=c=3\times\,10^{8}\,meters/second$. Therefore, it is safe to say that (5)
ensures the applicability of (15) for all reasonable values of parameters
unless we do not take $\Delta\,t$ much smaller than
$\frac{1}{n_{\chi}\beta\sigma\,v}$. In fact, it is evident from the above
argument that in the case of
$\frac{1}{n_{\chi}\beta\sigma\,v}\tilde{m}_{\chi}\,=\,{\cal O}(1)$, (5) also
ensures that

$\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}\,\ll\,1\;.$ (17)

This implies that, in most of the cases, (5) ensures the adiabatic
approximation. Another, even more important, result is derived below.

By using the expression for $\omega_{p}$ in (10) we find that

$\left|\frac{\Delta\,\omega_{p}^{2}}{\omega_{p}^{2}}\right|\,=\,\left|\frac{\left(\frac{d\omega_{p}^{2}}{dt}\right)\,\Delta\,t}{\omega_{p}^{2}}\right|\,=\,\left|\frac{\left(\frac{d\tilde{m}_{\chi}^{2}}{dt}\right)\,\Delta\,t}{\omega_{p}^{2}}\right|\,<\,\left|\frac{\Delta\,\tilde{m}^{2}}{\tilde{m}^{2}}\right|.$
(18)

This implies that, if (5) holds in a time interval $\Delta\,t$, then
$\left|\frac{\Delta\,\omega_{p}^{2}}{\omega_{p}^{2}}\right|\,\ll\,1$ is
ensured in the same interval i.e. (5) ensures that $\omega_{p}$ may be taken
to be almost constant in that interval. Note that this argument is true for
any time interval $\Delta\,t$. This, in turn, implies that (13) is a very good
approximation to the exact solution and $\omega_{p}$ may be taken to be
constant for each time interval provided that (5) is satisfied in each
interval $\Delta\,t\,\leq\,\frac{1}{n_{\chi}\beta\sigma\,v}$. In other words,
the higher order approximations in (14) converge to (13) for reasonable values
of the parameters provided that (5) is satisfied. (A more explicit derivation
of this result may be found in A.) Hence we find that
$W_{p}\,\simeq\,^{(0)}W_{p}=\omega_{p}=\mbox{constant}$ may be taken as a good
approximation in each time interval. Therefore in each time interval
$\eta_{i}\,<\,\eta\,<\,\eta_{i+1}$ we may take $v_{p}\,\simeq\,v_{p}^{(i)}$
where

$v_{p}^{(i)}(\eta)\,=\,\frac{1}{\sqrt{\omega_{p}^{(i)}}}exp{\left(i\omega_{p}^{(i)}(\eta-\eta_{i})\right)}~{}~{}\mbox{where}~{}~{}\;\omega_{p}^{(i)}=\omega_{p}(\eta_{i})~{},~{}~{}\eta_{i}\,<\,\eta\,<\,\eta_{i+1}$
(19)

Hence (9) may be expressed as

$\displaystyle\tilde{\chi}^{(i)}(\vec{r},\eta)\,\simeq\,\int\,\frac{d^{3}\tilde{p}}{(2\pi)^{\frac{3}{2}}\sqrt{2\omega_{p}^{(i)}}}\left[a_{p}^{(i)\,-}\,e^{i\left(\vec{\tilde{p}}.\vec{r}-\omega_{p}^{(i)}(\eta-\eta_{i})\right)}\,+\,a_{p}^{(i)\,+}\,e^{i\left(-\vec{\tilde{p}}.\vec{r}+\omega_{p}^{(i)}(\eta-\eta_{i})\right)}\right]$
(20) $\displaystyle\eta_{i}\,<\,\eta\,<\,\eta_{i+1}\;,$

where (i) refers to the $i$th time interval between the $i$th and $(i+1)$th
processes and $\eta_{i+1}-\eta_{i}=\Delta\,t$.

To see the difference of this framework with the standard framework for the
study of quantum field theory in curved space, it may be useful to point out
the similarities and the differences between the formulation in Boyanovsky2
and in this paper regarding their domains of the applicability and the forms
of the expansion of $\tilde{\chi}$ in terms of mode functions. The condition
(5) may be expressed as

$\frac{\Delta\,\tilde{m}_{\chi}}{\tilde{m}_{\chi}}=\left(\frac{H\hbar}{E_{k}}\right)\left(\frac{E_{k}}{\tilde{m}_{\chi}}\right)\left(\frac{\Delta\tilde{m}_{\chi}}{H\hbar}\right)\,\ll\,1\;,$
(21)

where $E_{k}=\sqrt{\frac{|\vec{\tilde{p}}|^{2}}{a^{2}}+m_{\chi}^{2}}$. In
Boyanovsky2  the condition $\frac{H\hbar}{E_{k}}\,\ll\,1$ is imposed, which is
equivalent to (17) in the case of conformally coupled scalars where
$\tilde{m}=a\,m$. It is evident that the condition
$\frac{H\hbar}{E_{k}}\,\ll\,1$ does not automatically imply (21) and vice
versa. In most of the cases
$\left(\frac{E_{k}}{\tilde{m}_{\chi}}\right)\,>\,1$, and it is quite possible
that one may have $\left(\frac{E_{k}}{\tilde{m}_{\chi}}\right)\,\gg\,1$,
$\left(\frac{\Delta\tilde{m}_{\chi}}{H\hbar}\right)\,\gg\,1$ while
$\frac{H\hbar}{E_{k}}\,\ll\,1$ or one may have
$\left(\frac{E_{k}}{\tilde{m}_{\chi}}\right)\,\sim\,{\cal O}(1)$ while
$\frac{\Delta\,\tilde{m}_{\chi}}{\tilde{m}_{\chi}}\,\ll\,1$ since
$\left(\frac{E_{k}}{\tilde{m}_{\chi}}\right)$ or
$\left(\frac{\Delta\tilde{m}_{\chi}}{H\hbar}\right)$ or both may be smaller
than one. This difference results in different forms for the mode functions.
The condition $\frac{H\hbar}{E_{k}}\,\ll\,1$ makes $a^{\prime\prime}\simeq\,0$
so that $\tilde{m}_{\chi}\simeq\,a\,m_{\chi}$. Therefore, in the case of
Boyanovsky2  $\tilde{m}_{\chi}$ can not be taken to be constant, so the
frequency $\omega_{p}$ in (10) depends on $\eta$ in general in that case while
in this study the condition (5) (i.e. (21)) guarantees that one may take
$\tilde{m}_{\chi}$ constant in each $\Delta\,t$, so
$\omega_{p}=\omega_{p}^{(i)}$ is constant in each interval
$\eta_{i}\,<\,\eta\,<\,\eta_{i+1}$, hence the corresponding mode function is
just that of the Minkowski space, namely,
$e^{\pm\,i\omega_{p}^{(i)}(\eta-\eta_{i})}$ in that interval.

The form of (3) and the above analysis imply that in each interval
$\eta_{i}\,<\,\eta\,<\,\eta_{i+1}$ we have an effective Minkowski space given
by Parker

$d\tilde{s}^{2}\,=\,-d\eta^{2}\,+\,d\tilde{x}_{1}^{2}+d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2}\;,$
(22)

where the masses of the particles are time dependent, and $\tilde{x}_{i}$ are
related to (1) by
$d\tilde{x}_{1}^{2}+d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2}$=$dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})$.
This together with the condition (5) implies we may use the tools of the usual
perturbative quantum field theory for calculation of the rates and cross
sections in an effective Minkowski space given by (22) for each $\Delta\,t$ in
(5). For example, for the process given in Figure 1, one may take the masses
be constant during $\Delta\,t$ and use the usual formulas for the rates and
cross sections of the usual (Minkowski space) quantum field theory and then we
take masses of the particles during the next $\Delta\,t$ be another constant
and then calculate the rates and cross sections for that $\Delta\,t$, and so
on. Eq.(5), hence Eq.(17) ensures that a possible contribution to the particle
physics processes due to a change in effective masses and decay widths and
gravitational particle production is small since the change in the effective
mass of the particles is small QFTC ; Bunch .

##  IV Additional conditions to be satisfied for applicability of the method

Another constraint for this formulation is the cluster decomposition principle
cluster-decomposition-1 ; cluster-decomposition-2 ; Weinberg2  which may be
stated that as ”the outcome of a scattering event, in which two or several
particles come in close contact with each other is unaffected by the presence
of any number of particles very far away, or differently stated, that several
scattering events separated from each other by large distances are independent
of each other” cluster-decomposition-1 . In fact, once the particles
participating in the interactions may be expressed in terms of field
expansions as in (9) and then quantized, then the cluster-decomposition is
guaranteed Weinberg2 . However such an expansion for individual particles may
be inapplicable in some cases because the particles may tend to act
collectively due to long-range correlations or due to long-range interactions.
If the de Broglie wave-lengths of particles in a system of particles overlap
then there may be long-range order (e.g. a Bose-Einstein condensation BEC ) so
that the system of particles tend to move as a whole rather than acting
independently. In that case an expansion of the form of (9) can be done only
for the particles that are not in the condensate system. Therefore we should
impose the condition

$\frac{h}{|\vec{\tilde{p}}|}\,\ll\,\tilde{n}_{\chi}^{-\frac{1}{3}}$ (23)

for the Broglie wavelength of the particles where $\vec{\tilde{p}}$ and
$\tilde{n}_{\chi}$ are the momenta and the number density of the incoming
particles in a scattering process. Therefore this method is not applicable in
the case of very low momenta particles in the incoming beam. In the case of
long-range forces also there may be collective behavior. This is characterized
by the Compton wavelength of the particles mediating the interaction (since at
non-relativistic case the corresponding interaction potential may be expressed
as $V(r)\,\sim\,\frac{e^{-\alpha\,m_{\chi}\,r}}{r}$). In the case of the usual
scattering processes the long-ranged character of electromagnetic interactions
do not pose a significant problem since the atoms and molecules in the target
are electrically neutral as a whole and the interactions take place inside
each atom separately. However this an important issue for strong interactions
inside nucleons where all densely distributed colored particles may be
effected by the scattering, especially when the coupling constant may be large
CDP-QCD . Therefore we also impose that

$\frac{h}{\tilde{m}_{\chi}\,c}\,\ll\,\tilde{n}_{\phi}^{-\frac{1}{3}}\;.$ (24)

These conditions require that for the applicability of the method, the momenta
and the density of incoming particles should not violate (23), and the mass of
the intermediate particle $\tilde{m}_{\chi}$ and the density of the
interacting particles $\tilde{n}_{\phi}$ should not violate (24). These
conditions become important at the extremely low momenta of incoming
particles, and extremely low mass of intermediate particles (especially in the
case of appreciably strong interactions) for extremely high number density of
incoming particles.

After identifying the effective Minkowski spaces and the corresponding field
expansions in (20) the next step is to find the rates and the cross sections
corresponding to some process. To this end, first we must identify the
asymptotic states, the in and out states to be able to express the interaction
Lagrangian in terms of these states and then calculate the corresponding
scattering amplitudes QFT . In the usual Minkowski space quantum field theory
the out and in states are taken as the free particle states in the remote
future and remote past i.e. taken as the states for ${\it
t\,\rightarrow\,\infty}$ and ${\it t\,\rightarrow\,-\infty}$, respectively in
text books. However, in practice these measurements are done by detectors that
are located sufficiently far from the interaction region and the coming
particles come from sufficiently far locations. In other words, in practice
the out and in states are identified as those corresponding to ${\it
t\,\rightarrow\,T}$ and ${\it t\,\rightarrow\,-T}$, respectively, where T is
sufficiently large so that the in and out states may be identified as free
states. In this study we take T$=\frac{1}{2}\Delta\,t$ as the half of the time
between two interactions. By the conditions (18), (23), (24) we guarantee the
assumption of taking this time to be sufficiently large so that each incoming
and outgoing particle can be treated to be a free particle, so one may use the
tools of the usual perturbative Minkowski space quantum field theory. This
point is especially important for applicability of S-matrix formulation of
Minkowski space quantum field theory in this framework. Adiabatic condition
(15), even in the form of
$\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}\,\ll\,1$, is not enough to ensure
the applicability of S-matrix formulation of Minkowski space quantum field
theory. In fact, $\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}\,\ll\,1$ only
ensures (20) in a time interval $\frac{1}{\omega_{p}}$ which should be much
smaller than $T\,\sim\,\frac{1}{n_{\chi}\beta\sigma\,v}$ since, otherwise, one
cannot identify free asymptotic states. Therefore, imposing (5) or (7) (or
(8)) is essential for applicability of S-matrix formulation of Minkowski space
quantum field theory in cosmology.

It is evident that (7) can be satisfied in most of the times by taking
$\Delta\,t$ sufficiently small. The only lower bound on $\Delta\,t$ comes from
(23) and (24) which are satisfied unless one takes momentum of the incoming
particles extremely small and/or the masses of the intermediate particles and
the number density of the final state particles violate (24). In fact, in many
cases, (7) is satisfied even in the case where $\Delta\,t$ saturates the upper
bound i.e. when $\Delta\,t=\frac{1}{n_{\chi}\beta\sigma\,v}$. In that case
(where $m=m_{\chi}$ and $\Delta\,t=\frac{1}{n_{\chi}\beta\sigma\,v}$), (7)
becomes

$\left|\frac{2a^{2}H\left[m_{\chi}^{2}-(s^{2}-3s+2)H^{2}\right]}{n_{\chi}\beta\sigma\,v\,a^{2}\left(m_{\chi}^{2}-(2-s)H^{2}\right)}\right|\,=\,\left|\frac{2H}{n_{\chi}\beta\sigma\,v}\left(1-\frac{s(s-2)H^{2}}{m_{\chi}^{2}+(s-2)H^{2}}\right)\right|\ll\,1\;.$
(25)

Eq.(25) implies that either of $|\frac{2H}{n_{\chi}\beta\sigma\,v}|$,
$|1-\frac{s(s-2)H^{2}}{m_{\chi}^{2}+(s-2)H^{2}}|$ is much smaller than one,
and the other is at most at the order of one. In other words there are two
extreme cases for (25) to be satisfied, ${\bf i}$\-
$|\left(1-\frac{s(s-2)H^{2}}{m_{\chi}^{2}+(s-2)H^{2}}\right)|$ has a value of
at most,
$|\left(1-\frac{s(s-2)H^{2}}{m_{\chi}^{2}+(s-2)H^{2}}\right)|\,\sim\,\mathcal{O}(1)$
provided that $s\simeq\,0$ or $s\simeq\,2$ or $\frac{m_{\chi}^{2}}{H^{2}}$ is
not close to $2-s$. In that case it is enough to let
$|\frac{2H}{n_{\chi}\beta\sigma\,v}|\,\ll\,1$ to satisfy (25). Note that
$|\frac{H}{n_{\chi}\beta\sigma\,v}|\,<\,1$ should already be satisfied to
enable the process to take place. ${\bf ii}$\-
$|\frac{2H}{n_{\chi}\beta\sigma\,v}|$ has a value of at most,
$|\frac{2H}{n_{\chi}\beta\sigma\,v}|\,\sim\,\mathcal{O}(1)$. In that case it
is enough to let
$|\left(1-\frac{s(s-2)H^{2}}{m_{\chi}^{2}+(s-2)H^{2}}\right)|\,\ll\,1$ i.e. to
set $\frac{m_{\chi}^{2}}{H^{2}}\,\simeq\,(s-1)(s-2)$ which also implies that
$s\,>\,2$ or $s\,<\,1$ (in addition to
$|\frac{m_{\chi}^{2}}{H^{2}}|\,\simeq\,|(s-1)(s-2)|$). Eq.(25) is satisfied
for a considerable range of parameters. For example, the case i) may be
realized in the radiation dominated era ($s=2$) well after its start (to make
$|\frac{2H}{n_{\chi}\beta\sigma\,v}|\,\ll\,1$ applicable) independent of the
value of $m_{\chi}$. The case i) is also satisfied for the current epoch of
accelerated cosmic expansion (where $s\,\simeq\,0$) independent of the value
of $m_{\chi}$ provided that
$\frac{H_{0}}{n_{0}\beta\sigma_{0}\,v}\simeq\,\frac{H_{0}}{n_{0}\sigma_{0}\,v}\simeq\,\frac{10^{-26}m^{-1}}{n_{0}\sigma_{0}(v/c)}\,\ll\,1$.
The case ii) may be satisfied in a possible stiff matter dominated era ($s=3$)
after inflation or in the current accelerated expansion era, $s\neq\,0\sim\,0$
provided that $m_{\chi}^{2}\sim\,2H_{0}^{2}$.

An important comment is in order at this point. In the case
$m_{i}^{2}a^{2}\,<\,\frac{a^{\prime\prime}}{a}$ in (4) (which corresponds to
$m_{i}^{2}\,<\,(2-s)H^{2}$ in the case of (6)) the effective mass
$\tilde{m}_{i}$ becomes tachyonic. However $\frac{a^{\prime\prime}}{a}$ gets
sufficiently small by time so that the particle masses become real after some
time for all physically relevant cases (e.g. as given in (6)) except in the
case where strictly $s=0$. Therefore this is not a true problem in general for
the physically interesting cases because either the mass becomes real after a
finite time for $s\,>\,0$ or it can not interact with other particles (so,
making the tachyonic state harmless)for $s\,\leq\,0$ due to fast expansion
rate. However a tachyonic state can not be dealt within this formulation
because the would-be ground state (e.g. $\chi=0$, $\phi=0$) will not be the
ground state anymore, making the perturbation expansion about the ground state
inapplicable. In other words this formulation is not applicable to the case,
$m_{i}^{2}a^{2}\,<\,\frac{a^{\prime\prime}}{a}$ (which corresponds to
$m_{i}^{2}\,<\,(2-s)H^{2}$ in the case of (6)). Therefore the case of
$s\,\geq\,2$ (e.g. of radiation and stiff matter) is safe in this regard
while, in the the case of $s\,<\,2$ (e.g. for cosmological constant and dust)
$H^{2}$ should be sufficiently small compared to $m_{\chi}^{2}$ so that the
problem of tachyons do not emerge. After combining this constraint with those
discussed after (25) one notices that there is still a significant relevant
available parameter space left. The conclusions obtained after (25) remain
intact for radiation and stiff matter dominated eras, and the conclusions
obtained for the current accelerated expansion era still hold provided that
$m_{\chi}^{2}$ is not equal to $(2-s)\,H_{0}^{2}\hbar^{2}\,\sim\,10^{-66}$
eV2.

##  V The allowed range of parameters

Let us summarize what we have done up to this point: we have introduced three
conditions, namely, (5) (which in the case of (6) reduces to (7)), (23), (24)
for applicability of the method of approximating the Robertson-Walker space by
Minkowski space in sufficiently small time intervals. One may also impose (8)
for a standard setup. Note that, (8) guarantees (7). Therefore, in physically
relavant cases the essential conditions to be satisfied (for applicability of
the effective Minkowski formulation introduced in this study) may be taken to
be (23), (24), (8). Another constraint we had imposed is the exclusion of
tachyonic states that is automatically guaranteed in a radiation dominated
universe and for cosmological constant and matter dominated universe at
current time is guaranteed for
$m_{\chi}\,>\,\sim\,H_{0}\hbar\,\sim\,10^{-33}\,eV$ as we have mention in the
preceding paragraph. We will assume that either of these conditions is
satisfied i.e. no tachyonic states emerge. Moreover, to guarantee the
applicibality of (47) it is useful to impose
$\left(\frac{a}{\omega_{p}}\right)(\Delta\,t)^{-1}\sim\left(\frac{a\,n_{\chi}\beta\sigma\,v}{\omega_{p}}\right)\,<\,{\cal
O}(1)$ as we have mentioned after Eq.(47) although it is not a condition
independent of (7).

In other words, we have to impose the following conditions for the
applicability of the effective Minkowski space formulation introduced in this
study;

$\displaystyle\left(\frac{\tilde{m}_{\chi}\,c^{2}}{\hbar}\right)^{3}\,=\,\left(\frac{\tilde{m}_{\chi}\,c^{2}}{eV}\right)^{3}\left(\frac{eV}{\hbar}\right)^{3}\,\simeq\,1.3\,\times\,10^{20}m^{-3}\left(\frac{\tilde{m}_{\chi}\,c^{2}}{eV}\right)^{3}\gg\,\tilde{n}_{\phi}\;,$
(26)
$\displaystyle\left(\frac{|\vec{\tilde{p}}|}{\hbar}\right)^{3}\,\simeq\,1.3\,\times\,10^{20}m^{-3}\left(\frac{|\vec{\tilde{p}}|\,c}{eV}\right)^{3}\,\gg\,\tilde{n}_{\chi}\;,$
(27)
$\displaystyle(H\,\Delta\,t)^{-1}\,\gg\,1~{}~{}\mbox{where}~{}~{}\Delta\,t\,\leq\,\frac{1}{n_{\chi}\beta\sigma\,v}\;,$
(28)
$\displaystyle\left(\frac{a}{\omega_{p}}\right)(\Delta\,t)^{-1}\sim\left(\frac{a\,n_{\chi}\beta\sigma\,v}{\omega_{p}}\right)\,<\,{\cal
O}(1)\;,$ (29)

provided that $m_{\chi}\,>\,\sim\,H_{0}\hbar\,\sim\,10^{-33}\,eV$. Note that
the condition (26) is imposed for the processes where the intermediate
particle in the process is $\chi$ such as in the tree level
$\phi\phi\,\rightarrow\,\phi\phi$. In the cases where the intermediate
particle is $\phi$, such as in the tree level
$\chi\chi\,\rightarrow\,\phi\phi$ processes, $\tilde{m}_{\chi}$ in (26) should
be replaced by $\tilde{m}_{\phi}$. It is evident that, once (26) is satisfied,
the corresponding expression where $\tilde{m}_{\chi}$ in (26) being replaced
by $\tilde{m}_{\phi}$ will be automatically satisfied.

Eq.(26) tells us that one can not use the scattering theory sufficiently well
when the range of the forces are too long for a successful particle
description for the individual particles in the system. This implies that,
when the scattering of a system of charged particles is considered, the
scattering cross section calculated in this formulation becomes less and less
reliable when the mass of the intermediate particles goes to zero, especially
when one considers high number densities of the scattering particles. However,
this is not a big problem when one considers scatterers like atoms that are
neutral up to very small distances. In fact, the bound (26) may be easily
satisfied even for charged scatterers for reasonable choices of the number
densities, and for sufficiently weak interactions as in Erdem-Gultekin . For
example, for $\tilde{m}_{\chi}\,c^{2}\,=\,1eV$ we have
$\left(\frac{\tilde{m}_{\chi}\,c}{\hbar}\right)^{3}\,\sim\,10^{20}\,(meter)^{-3}$.
Note that the number densities of nucleons, photons, and cold dark matter
particles at present time are in the order of $\sim\,10^{-1}\,(meter)^{-3}$,
$>\,10^{8}\,(meter)^{-3}$, and
$\left(\frac{3\,\times\,10^{-3}eV}{\tilde{m}_{\chi}\,c^{2}}\right)$$\left(\frac{3\,\times\,10^{-3}eV}{\hbar\,c}\right)^{3}\,\simeq\,10^{13}\,\left(\frac{eV}{m_{\chi}\,c^{2}}\right)(meter)^{-3}$;
respectively.

The second condition (27) tell us that one can not use the scattering theory
when the number density of the particles in the initial state becomes
sufficiently high (or if the momentum is sufficiently low) so that Bose-
Einstein correlation takes place and the system acts like a single quantity.
For most of the parameter space this bound is not saturated. For example, we
have
$\left(\frac{|\vec{\tilde{p}}|}{\hbar}\right)^{3}\,\sim\,10^{9}\,(meter)^{-3}$
for $|\vec{\tilde{p}}|c\,\sim\,2\,\times\,10^{-4}\,eV$. This condition, for
example, in Erdem-Gultekin  is guaranteed by taking the momenta of the
particles in the initial state being high.

The third condition (28) is the essential condition for validity of the
effective Minkowski space formulation in this paper. It, in principle, can be
always satisfied by taking a smaller $\Delta\,t$ (provided that the
interaction can be still taken to localized). However taking a smaller
$\Delta\,t$ implies the excluding modes whose wavelengths greater than
$\frac{|\vec{\tilde{p}}|}{\tilde{E}}\Delta\,t$. Therefore the optimum choice
is to take $\Delta\,t\,=\,\frac{1}{n_{\chi}\beta\sigma\,v}$. In this case, one
may put constraint on the possible values of $H$, $n_{\chi}$ and $\sigma$. For
example, if one lets $H\,=\,H_{0}\,\sim\,2\,\times\,10^{-18}\,sec^{-1}$,
$\sigma\,=\,10^{-40}\,(meter)^{2}$ (i.e. in the order of weak interaction
cross sections), $\beta\sim\,1$, $v\sim\,c$, then one finds
$n_{\chi}\,\gg\,\frac{H_{0}}{\beta\sigma\,v}\,\sim\,10^{14}\,(meter)^{-3}$
which may be only satisfied by cold dark matter particles of masses less than
meV range. However, instead if one takes $\sigma\,=\,10^{-33}\,(meter)^{2}$
(i.e. in the order of electromagnetic interaction cross sections) and the same
values of $H$, $\beta$, then one finds
$n_{\chi}\,\gg\,\frac{H_{0}}{\beta\sigma\,v}\,\sim\,10^{7}\,(meter)^{-3}$
which is satisfied if one takes $n_{\chi}$ in the order of
$10^{8}(meter)^{-3}$ i.e. in the order of the number density of nucleons in
the universe. At earlier times this condition is usually satisfied more easily
since $n\,\propto\,a^{-3}$ while $H\,\propto\,a^{-2}$ for a radiation
dominated universe and $H\,\propto\,a^{-\frac{3}{2}}$ for a matter dominated
universe.

The last condition (29) may be satisfied provided that we consider the
particles with momenta satisfying
$\omega_{p}\,>\,a(t)\,n_{\chi}\beta\sigma\,v$. For example, for
$n_{\chi}\,\sim\,10^{8}\,(meter)^{-3}$, $\beta\,\sim\,1$,
$\sigma\,\sim\,10^{-32}\,(meter)^{2}$, $v\,\sim\,c$ at present we must have
$\hbar\omega_{p}\,>\,10^{-31}\,eV$ to be able to use this method.

There is a considerable parameter space that satisfies all conditions in
(26-29) simultaneously. For example, for
$\Delta\,t\,=\,\frac{1}{n_{\chi}\beta\sigma\,v}$ the conditions (27) and (28)
may be combined into

$10^{20}m^{-3}\,\left(\frac{|\vec{\tilde{p}}|\,c}{eV}\right)^{3}\,\gg\,\tilde{n}_{\chi}\,\gg\,\frac{a^{3}\,H}{\beta\sigma\,v}\,\;,$
(30)

(where we have used $\tilde{n}_{\chi}=a^{3}\,n_{\chi}$) which, for example;
for $|\vec{\tilde{p}}|\,c\,>\,1\,eV$, $\sigma\,=\,10^{-33}\,m^{2}$,
$\beta\,\sim\,1$, $v\sim\,c$,
$H\,=\,H_{0}\,\sim\,2\,\times\,10^{-18}\,sec^{-1}$ implies that
$10^{20}m^{-3}\,\gg\,\tilde{n}_{\chi}\,\gg\,10^{7}\,m^{-3}$. The equation (26)
remains as an independent equation that restricts the number density of $\phi$
particles for a given $\tilde{m}_{\chi}$, or vice versa. The last equation
(29) gives a lower bound for $\omega_{p}$ for a given interval of the values
of $\tilde{n}_{\chi}$ as the one given in (30). For example, for the values
given above it results in $\omega_{p}\hbar\,>\,10^{-29}\,eV$.

##  VI Basic quantum field theory calculations in this framework

It is evident from (3) that the cosmological time evolution of physical
quantities may be considered mainly to be due to the time evolutions of the
effective masses and the coupling constants. We have shown that, in each time
interval $\Delta\,t\,=\,\frac{1}{n_{\chi}\beta\sigma\,v}$, one may consider
the space approximately as a Minkowski space and use the S-matrix formulation
of the flat space quantum field theory provided that the condition (5) (or
(8)) and the conditions studied in Section IV are satisfied in that interval.
Thus, one may let $\tilde{m}_{\chi}^{2}=\tilde{m}_{\chi}^{2}(\eta_{r})$,
$\tilde{m}_{\phi}^{2}=\tilde{m}_{\phi}^{2}(\eta_{r})$,
$\tilde{\mu}=\tilde{\mu}(\eta_{r})$ in the $r$’th interval
$\eta_{r}\,<\,\eta\,<\,\eta_{r}\,+\Delta\,\eta$ (where $\Delta\,\eta$ is the
conformal time interval corresponding to $\Delta\,t$), and then do the cross
section or rate calculations by using the S-matrix formulation of the flat
space space quantum field theory in that interval in the space described by
(22). After the calculation one may identify the
$\tilde{m}_{\chi}^{2}(\eta_{r})$, $\tilde{m}_{\phi}^{2}(\eta_{r})$,
$\tilde{\mu}(\eta_{r})$ terms in the result and replace them by
$\tilde{m}_{\chi}^{2}(\eta)$, $\tilde{m}_{\phi}^{2}(\eta)$,
$\tilde{\mu}(\eta)$, and convert the momenta $\vec{\tilde{k}}$ and the
energies $\tilde{E}$ for (22) to the corresponding quantities for (1) by using
$\vec{\tilde{k}}=a(t)\,\vec{k}$ (as discussed in Appendix B) to determine the
final result with cosmic evolution. This method can be used for any particle
physics process. In the following paragraphs, first we illustrate the method
through a simple example, namely, calculation of the total cross section for
the Feynman diagrams in Figure 1, then we present the general framework for
this type of calculations.

To illustrate how to apply this method we consider a simple example. We
consider two types of particles $\chi$ and $\phi$ with significant
cosmological number densities and making random collisions, and assume that
all conditions for the applicability of this method hold. We also assume that
the coupling constant $\mu$ is small enough so that one may use perturbative
quantum field theory e.g. we let
$\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\,\ll\,1$ for Figure 1. The corresponding
cross section in the effective Minkowski space is given by

$\tilde{\sigma}\,=\,\frac{(2\pi)^{4}}{4\sqrt{(\tilde{p}_{1}.\tilde{p}_{2})^{2}-\tilde{m}_{\chi}^{4}}}\int\int\,\delta^{(4)}(\tilde{p}_{1}+\tilde{p}_{2}-\tilde{p}_{3}-\tilde{p}_{4})\,|\tilde{M}|^{2}\frac{d^{3}\vec{\tilde{p}}_{3}}{\tilde{E}_{3}}\frac{d^{3}\vec{\tilde{p}}_{4}}{\tilde{E}_{4}}\;,$
(31)

where $\tilde{m}_{\chi}$, $\tilde{m}_{\phi}$ are assumed to be constant during
a time interval much smaller than the average collision time between two
particles, $\tilde{M}$ corresponding to Figure 1 is given by

$\tilde{M}\,=\,\tilde{\mu}^{2}\left[\frac{1}{(\tilde{p}_{1}-\tilde{p}_{3})^{2}+\tilde{m}_{\phi}^{2}}\,+\,\frac{1}{(\tilde{p}_{1}-\tilde{p}_{4})^{2}+\tilde{m}_{\phi}^{2}}\right]\;.$
(32)

As we have mentioned before, the quantities with $\tilde{}$ correspond to the
metric (22) while those without $\tilde{}$ correspond to the metric (1). For
example, $E$ corresponds to (1) while $\tilde{E}$ corresponds to the same
quantity for (22) i.e.

$E_{i}^{2}\,=\,a^{2}\left(m_{i}^{2}-\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}\right)+a^{2}\vec{p}_{i}^{2}~{}~{}~{}\mbox{while}~{}~{}~{}\tilde{E}_{i}^{2}\,=\,a^{2}m_{i}^{2}-\frac{a^{\prime\prime}}{a}+\vec{\tilde{p}}^{2}\;.$
(33)

where we have used $\vec{\tilde{p}}=a\,\vec{p}$ (see Appendix B). In the
center of mass frame conservation of energy amounts to

$\vec{\tilde{p}}^{2}+\tilde{m}_{\chi}^{2}=\vec{\tilde{k}}^{2}+\tilde{m}_{\phi}^{2}~{}~{}~{}\mbox{i.e.}~{}~{}~{}\vec{\tilde{p}}^{2}-\vec{\tilde{k}}^{2}=\tilde{m}_{\phi}^{2}-\tilde{m}_{\chi}^{2}=a^{2}\left(m_{\phi}^{2}-m_{\chi}^{2}\right)\;,$
(34)

where $\vec{\tilde{k}}$ and $\vec{\tilde{p}}$ are $\vec{\tilde{p}}_{3}$ and
$\vec{\tilde{p}}_{1}$ in the center of mass of the system, respectively. Note
that $\vec{\tilde{p}}$, $\vec{\tilde{k}}$ do not depend on time, so (34) is
satisfied (provided that (5) is satisfied), in general, only at the moment of
each transition $\chi\chi\,\rightarrow\,\phi\phi$. It is evident from the
right-hand side of (34) that it may not hold at later times. Therefore one may
use it in the evaluation of the cross sections and rates while it should not
be used for studying the cosmological evolution of the relation between a
particular $\vec{\tilde{p}}$, $\vec{\tilde{k}}$ pair (while it may be used for
cosmological evolution of a generic $\vec{\tilde{k}}$ provided that
cosmological evolution of a generic $\vec{\tilde{p}}$ is given). Another point
is that if the time interval $\Delta\,t$ is taken extremely small to satisfy
(5) then the uncertainty introduced to
$\tilde{E}=\sqrt{\vec{\tilde{p}}^{2}+\tilde{m}_{\chi}^{2}}$ by
$\delta\,E=\frac{1}{\delta\,t}$ Boyanovsky2  may be comparable to or larger
than $E$ itself, so making (34) inapplicable. In fact, such a situation would
also make (31) inapplicable since the energy conservation given in (34) is
also used in the temporal part of the four dimensional Dirac delta function in
(31). However, we observe that $\delta\,E$ is small for most of the
phenomenologically relevant set of parameters. This may be seen as follows.
For $\delta\,t\sim\,\Delta\,t\sim\,\frac{1}{n_{\chi}\beta\sigma\,v}$, Eq.(30)
may be reexpressed as

$10^{20}m^{-3}\,\left(\frac{|\vec{\tilde{p}}|\,c}{eV}\right)^{3}\hbar\beta\sigma\,v\,\gg\,\delta\,E\,\sim\,\frac{\hbar}{\Delta\,t}\,\sim\,\hbar\tilde{n}_{\chi}\,\beta\sigma\,v\,\gg\,\hbar\,a^{3}\,H.$
(35)

For example, for the values after (30), namely, for
$|\vec{\tilde{p}}|\,c\,>\,1\,eV$, $\sigma\,=\,10^{-33}\,m^{2}$,
$\beta\,\sim\,1$, $v\sim\,c$, $H\,=\,H_{0}\,\sim\,10^{-18}\,sec^{-1}$, (35) at
present time ($a=1$) implies
$2\times\,10^{-20}\,eV$$\,\gg\,\delta\,E\,\sim\,\frac{\hbar}{\Delta\,t}\,\gg\,$$\,10^{-33}\,eV$.
This, in turn, implies that $\frac{\delta\,E}{E}$ is negligible for these
values of the parameters, provided that $E$ is greater than
$2\times\,10^{-20}\,eV$. It is evident from (35) that the upper bound on
$\delta\,E$ may be increased by increasing the value of
$|\vec{\tilde{p}}|\,\sigma\,\beta\,v$. For example, if
$|\vec{\tilde{p}}|\,\sigma\,\beta\,v$ is increased $10^{4}$ times, then the
upper bound on $\delta\,E$ changes as
$2\times\,10^{-16}\,eV$$\,\gg\,\delta\,E$. In that case, taking $E$ be greater
than $2\times\,10^{-16}\,eV$ ensures $\frac{\delta\,E}{E}$ be negligible while
the conditions necessary for the applicability of effective Minkowski space
remain intact. In other words, there is a considerable parameter space where
one may use the Minkowski space quantum field theory formula (31) in each time
interval $\Delta\,t$ for particle physics processes in cosmological
backgrounds that satisfy the conditions discussed in the preceding section as
a good approximation to an exact result that would follow from a standard
quantum field theory in curved space calculation.

If we take $\tilde{m}_{\phi}\,\gg\,\tilde{m}_{\chi}$,
$\vec{\tilde{p}}^{2}\,\gg\,\tilde{m}_{\chi}^{2}$,
$\vec{\tilde{k}}^{2}\,\ll\,\tilde{m}_{\phi}^{2}$, then (32) may be
approximated by

$\tilde{M}\,\simeq\,\left(\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\right)^{2}\frac{1}{1-\frac{\vec{\tilde{k}}^{2}}{\tilde{m}_{\phi}^{2}}\cos^{2}{\theta}}\;.$
(36)

In this case the cross section corresponding to Figure 1 is

$\tilde{\sigma}\,\simeq\,\left(\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\right)^{4}\frac{|\vec{\tilde{k}}|}{64\pi\,|\vec{\tilde{p}}|^{2}\tilde{m}_{\phi}}\;.$
(37)

Next we relate $\tilde{\sigma}$ (which is expressed in terms of the
coordinates in (22)) to the physically observed cross section $\sigma$ (which
is expressed in terms of the coordinates in (1)). We note that

$\tilde{n}(\eta)\,=\,\frac{d\tilde{N}(\eta)}{d^{3}\tilde{x}}\,=\,\frac{dN(t)}{a^{-3}d^{3}x}\,=\,a^{3}\,n(t)~{}~{}~{}\mbox{and}~{}~{}a^{4}\left(\dot{n}_{4}+3H\,n_{4}\right)\,=\,\frac{d\,\tilde{n}_{4}(\eta)}{d\eta}\,=\,\beta\tilde{n}_{1}\tilde{n}_{2}\tilde{\sigma}\,\tilde{v}$
(38) $\dot{n}_{4}+3H\,n_{4}\,=\,\beta\,n_{1}n_{2}\sigma\,v\;.$ (39)

The equations (38) and (39) and
$\frac{d|\vec{x}|}{dt}\,=\,v\,=\,\tilde{v}=\frac{d|\vec{\tilde{x}}|}{d\eta}$
together imply that $\sigma\,=\,a^{2}\,\tilde{\sigma}$. Therefore the cross
section corresponding to Figure 1 is

$\sigma\,\simeq\,\frac{\mu^{4}}{\left(m_{\phi}^{2}-\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}\right)^{\frac{5}{2}}}\left(\frac{a^{2}|\vec{\tilde{k}}|}{64\pi\,|\vec{\tilde{p}}|^{2}}\right)\;.$
(40)

Hence once we know the scale factor one may determine the cosmological
evolution of the cross section. Note that, even when
$\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\,\sim\,0$, the physical cross
section $\sigma$ has a dependence on the scale factor. For example, in a
radiation dominated universe we have
$\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\,=\,0$ while $a(t)$ varies with
time considerably. Moreover, even when the variation of $a(t)$ in a time
interval $\Delta\,t$ may be small, it may vary considerably during a Hubble
time. To see the situation better, let
$\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\,\sim\,0$ in (40). In that case,
by (8), we get $\left|\frac{\Delta\,\sigma}{\sigma}\right|\,\ll\,1$ in a time
interval $\Delta\,t\,=\,\frac{1}{n_{\chi}\beta\sigma\,v}$ as expected while
$\left|\frac{\Delta\,\sigma}{\sigma}\right|$ in a Hubble time $\frac{1}{H}$
becomes of the order of one.

The relation $\sigma\,=\,a^{2}\,\tilde{\sigma}$ that is obtained above for
Figure 1 may be seen more clearly by using a more formal consideration. The
scattering amplitude from an in state (with $\,{}^{1}n$ particles with momenta
${\bf p}_{1}$, $\,{}^{2}n$ particles with momenta ${\bf p}_{2}$, etc.) to an
out state (with $\,{}^{1}m$ particles with momenta ${\bf k}_{1}$, $\,{}^{2}m$
particles with momenta ${\bf k}_{2}$, etc.) may be expressed as Birrell-Taylor
; QFTC1

$\displaystyle<\,out;\;^{r}\bar{m}_{k_{r}},\ldots,\,^{2}\bar{m}_{k_{2}},\,^{1}\bar{m}_{k_{1}}\,|\,^{1}n_{p_{1}},\,^{2}n_{p_{2}},\ldots\,^{s}n_{p_{s}}\;;in\,>$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{\infty}\frac{1}{h!}\sum_{q_{1}\ldots\,q_{h}}\,<\,out;\;^{r}\bar{m}_{k_{r}},\ldots,\,^{2}\bar{m}_{k_{2}},\,^{1}\bar{m}_{k_{1}}\,|I_{q_{1}},I_{q_{2}},\ldots,I_{q_{h}};out>$
(41)
$\displaystyle\times\,<out;I_{q_{h}},\ldots,I_{q_{2}},I_{q_{1}}|\,^{1}n_{p_{1}}\,^{2}n_{p_{2}},\ldots\,^{s}n_{p_{s}}\;;in\,>\;,$

where the bars over the indices signify that they are expressed in terms of
the creation operators of the out states $\bar{b}_{k_{i}}^{\dagger}$ (with
$\bar{b}_{k_{i}}|0,out>\,=\,0$) while those without bar are expressed in terms
of creation operators of the in states $a_{p_{i}}^{\dagger}$ (with
$a_{p_{i}}|0,in>\,=\,0$); and $|I_{q_{1}},I_{q_{2}},\ldots,I_{q_{h}};out>$
denotes the complete set of all possible intermediate states in the final
state that are expressed in terms of the creation operators of the in fields.
Therefore, the terms
$<\,out;\;^{r}\bar{m}_{k_{r}},\ldots,\,^{2}\bar{m}_{k_{2}},\,^{1}\bar{m}_{k_{1}}\,|I_{q_{1}},I_{q_{2}},\ldots,I_{q_{h}};out>$
do not contain information about the effect of the particle physics
interactions, they only contain information about the effect of the curved
space and are expressed in terms of Bogolyubov transformation matrices (see
Appendix C). Note that the in and the out states in (41) are the same as the
in and the out states in the effective Minkowski space formulation discussed
in this study in the paragraph after Eq.(24). The in and the out states in
(41) are the usual in and the out states for $t\,\rightarrow\,-\infty$ and
$t\,\rightarrow\,\infty$, respectively while the in and the out states in the
paragraph after (24) correspond to $t\,\rightarrow\,-L$ and
$t\,\rightarrow\,L$, respectively, where $L$ is taken to be much larger than
the range of the interactions so that may taken to be sufficiently good
approximation to $t\,\rightarrow\,-\infty$ and $t\,\rightarrow\,\infty$. We
remark that identification of in and out states in a standard quantum field
theory in curved space analysis in general may be difficult. Although such an
identification can be rigorously may be done for cosmological backgrounds that
are localized in time, for general cosmological backgrounds such an
identification is a highly non-trivial issue Wald . On the other hand, the
issue of the identification in and out states is not a problem in the present
effective Minkowski space formulation since we take the space to be
approximately Minkowskian in each time interval $\Delta\,t=2T$ where we
calculate the cross sections and rates, and we identify the in state as the
state at $t\,\rightarrow\,-L$ and the out state as the state at
$t\,\rightarrow\,L$. We use (41) and the formulas of the standard exact
quantum field theory in curved space analysis in the following paragraphs to
compare the result $\sigma\,=\,a^{2}\,\tilde{\sigma}$ obtained after (39) (by
using the effective Minkowski space formulation) with the result of a standard
exact quantum field theory in curved space analysis.

The transformation $\tilde{\phi}\,=\,a(\eta)\phi$ is an overall rescaling of
the field both for in an out field operators, so it does not change the
Bogolyubov transformation matrices. Therefore, the only effect of the field
rescaling is through the second term that may be expressed as

$\displaystyle\frac{<out;I_{q_{h}},\ldots,I_{q_{2}},I_{q_{1}}|\,^{1}n_{p_{1}}\,^{2}n_{p_{2}},\ldots\,^{s}n_{p_{s}}\;;in\,>}{<out,0|0,in>}\,=\,i^{h+s}\,\prod_{i=1}^{s}\int\,d^{4}x_{i}\left[-g(x_{i})\right]^{\frac{1}{2}}$
$\displaystyle\times\,\prod_{j=1}^{h}\int\,d^{4}y_{j}\left[-g(y_{j})\right]^{\frac{1}{2}}\,g_{p_{i}}(x_{i})\,f_{q_{j}}^{*}(y_{j})\;\left[-\nabla_{x_{i}\,\mu}\nabla_{x_{i}}^{\mu}+m^{2}+\xi\,R_{x_{i}}\right]\left[-\nabla_{y_{j}\,\mu}\nabla_{y_{j}}^{\mu}+m^{2}+\xi\,R_{y_{j}}\right]$
$\displaystyle\times\,\tau\left(y_{h},\ldots,y_{1},x_{1},\ldots,x_{s}\right),$
(42)

where $g_{p_{i}}(x_{i})$, $f_{q_{j}}(y_{j})$ are the mode functions defined in
(51) and (52). We take $\xi=0$ in this study. The explicit form of the Green’s
function in (42) (for $s$ $\chi$ particles in the initial state and $h$ $\phi$
particles in the final state) is

$\tau\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)\,=\,\frac{<out;0|T\left(\phi(y_{1})\ldots\,\phi(y_{h})\,\chi(x_{1})\ldots\,\chi(x_{s})\right)|0;in>}{<out,0|0,in>}\;.$
(43)

The corresponding expression in terms of $\tilde{\chi}$ and $\tilde{\phi}$ is
given by

$\tilde{\tau}\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)\,=\,\frac{<out;0|T\left(\tilde{\phi}(y_{1})\ldots\,\tilde{\phi}(y_{h})\,\tilde{\chi}(x_{1})\ldots\,\tilde{\chi}(x_{s})\right)|0;in>}{<out,0|0,in>}\;.$
(44)

It is evident that $\tilde{\tau}$, $\tau$ are related by
$\tilde{\tau}\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)$ =
$a^{h+s}\,\tau\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)$ since
$\tilde{\chi}=a\,\chi$, $\tilde{\phi}=a\,\phi$ (see Appnedix D for a more
formal and rigorous derivation).

Next, let us analyze the behaviour of the terms before the Green’s function
$\tau$ in (42) under the rescaling $\tilde{\chi}\,=\,a(\eta)\chi$,
$\tilde{\phi}\,=\,a(\eta)\phi$. By (4), (2), (3) we have
$\sqrt{-\tilde{g}}\,d^{4}\tilde{x}\,$=$\,d\eta\,d^{3}x$,
$\sqrt{-g}\,d^{4}x\,$=$\,a^{3}dt\,d^{3}x$, so
$\sqrt{-\tilde{g}}\,d^{4}\tilde{x}\,$=$\,a^{-4}\sqrt{-g}\,d^{4}x$. The mode
functions $g_{p_{i}}(x_{i})$, $f_{q_{j}}(y_{j})$ remain the same under the
rescaling because the equations of motion for $\tilde{\phi}$ and $\phi$ are
related by

$\left[-\tilde{\nabla}_{y_{j}\,\mu}\tilde{\nabla}_{y_{j}}^{\mu}+\tilde{m}^{2}\right]\tilde{\phi}\,=\,a^{3}(t)\,\left[-\nabla_{y_{j}\,\mu}\nabla_{y_{j}}^{\mu}+m^{2}\right]\phi\,=\,0$
(45)

so the corresponding solutions, namely mode functions, are the same. Moreover,
(45) implies that when each of
$\left[-\nabla_{y_{j}\,\mu}\nabla_{y_{j}}^{\mu}+m^{2}\right]$ and
$\left[-\nabla_{x_{i}\,\mu}\nabla_{x_{i}}^{\mu}+m^{2}\right]$ in (42) acts to
the corresponding $\chi$ and $\phi$ in (43), then each term results in a
factor of $a^{-3}$ after the rescaling $\chi\,\rightarrow\,\tilde{\chi}$,
$\phi\,\rightarrow\,\tilde{\phi}$. Hence, after combining these factors, one
obtains

$\displaystyle<\,out;\;^{r}\bar{m}_{k_{r}},\ldots,\,^{2}\bar{m}_{k_{2}},\,^{1}\bar{m}_{k_{1}}\,|\,^{1}n_{p_{1}},\,^{2}n_{p_{2}},\ldots\,^{s}n_{p_{s}}\;;in\,>$
$\displaystyle=$ $\displaystyle
a^{h+s}\,<\,out;\;^{r}\bar{\tilde{m}}_{k_{r}},\ldots,\,^{2}\bar{\tilde{m}}_{k_{2}},\,^{1}\bar{\tilde{m}}_{k_{1}}\,|\,^{1}\tilde{n}_{p_{1}},\,^{2}\tilde{n}_{p_{2}},\ldots\,^{s}\tilde{n}_{p_{s}}\;;in\,>\;.$
(46)

i.e. it has the same scaling as the Green’s function.

For Figure 1 we have $h=s=2$ in (46), so, the amplitude after the scaling is
$a^{-4}(t)$ times the one before rescaling. Note that the square of the
absolute value of the amplitude in (41) when expressed in in Minkowski space
in terms of the rescaled fields is equal to $|\tilde{M}|^{2}$ in (31) up to
the delta function in (31). Taking also the fact that
$\vec{\tilde{p}}_{i}=a\,\vec{p}_{i}$, and $\tilde{E}_{i}=E_{i}$ by (33) into
account we find that we should have $\tilde{\sigma}\,=\,a^{-2}\,\sigma$ which
confirms the relation after Eq.(39) that is obtained in a less formal way.

##  VII Conclusion

We have demonstrated that, if the variation of the effective masses defined by
(4) are very small compared to the average collision time between the
particles in the universe, then the spacetime may be taken to be approximately
Minkowskian in these time intervals. Moreover, if the conditions of cluster
decomposition principle are satisfied in these time intervals, and if the
momenta of the incoming particles are sufficiently large to prevent a
significant amount of Bose-Einstein correlation, and if the range of the
interactions between the particles are small or are effectively short ranged,
then one may use the tools of the usual Minkowski space quantum field theory
to determine cross sections and the rates of the interactions between cosmic
particles. We have shown that there is a considerable parameter space that
satisfies these conditions. In the cases where this method is applicable it
may be a simple tool to study the rates and the cross sections in cosmology.

###### Acknowledgements.

We would like to thank Professor Daniel Boyanovsky for reading the draft
version of this manuscript and for his valuable comments. This paper is
financially supported by The Scientific and Technical Research Council of
Turkey (TÜBITAK) under the project 117F296 in the context of the COST action
CA 16104 ”GWverse”

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##  Appendix A Explicit derivation of the convergence of higher order
approximations to the zeroth in the presence of (5) for phenomenologically
relevant set of parameters

One may show that (5) guarantees
$\frac{\omega_{p}^{\prime\prime}}{4\omega_{p}^{3}}\,\simeq\,0$ and
$\frac{3\omega_{p}^{\prime\;2}}{8\omega_{p}^{4}}\,\simeq\,0$ for reasonable
values of the parameters, so one may take $W_{p}\,\simeq\,\omega_{p}$. This
may be seen as follows:
$\omega_{p}\,=\,\sqrt{|\vec{\tilde{p}}|^{2}+\tilde{m}_{\chi}^{2}}$ implies
that

$\frac{\omega_{p}^{\prime}}{\omega_{p}^{2}}\,=\,\frac{a}{\omega_{p}^{2}}\frac{d\omega_{p}}{dt}\,=\,\left(\frac{a}{2\omega_{p}^{3}}\right)\frac{d\tilde{m}_{\chi}^{2}}{dt}\,=\,\left(\frac{a}{2\omega_{p}}\right)\left(\frac{\tilde{m}_{\chi}}{\omega_{p}}\right)^{2}(\Delta\,t)^{-1}\left[\frac{\left(\frac{d\tilde{m}_{\chi}^{2}}{dt}\right)(\Delta\,t)}{\tilde{m}_{\chi}^{2}}\right]\,\simeq\,0\;,$
(47)

where $\Delta\,t\,\sim\,\left(\frac{1}{n_{\chi}\beta\sigma\,v}\right)$. By
(5), the last term in the square brackets in (47) is much smaller than one, so
in order to show that (47) is almost zero the remaining terms in (47) must not
be much larger than one. For reasonable values of the parameters this is
really the case. $\left(\frac{\tilde{m}_{\chi}}{\omega_{p}}\right)^{2}\,<\,1$,
and
$\left(\frac{a}{\omega_{p}}\right)(\Delta\,t)^{-1}\sim\left(\frac{a\,n_{\chi}\beta\sigma\,v}{\omega_{p}}\right)\,<\,{\cal
O}(1)$ provided we take $\omega_{p}$ sufficiently large. This point will be
discussed after (29). Next we show that one may take
$\frac{\omega_{p}^{\prime\prime}}{4\omega_{p}^{3}}\,\simeq\,0$ for reasonable
values of the parameters. After using (6) and
$\frac{d^{2}\omega_{p}^{2}}{dt^{2}}\,=\,\frac{d^{2}\tilde{m}_{\chi}^{2}}{dt^{2}}$,
and
$\frac{d^{2}\omega_{p}^{2}}{dt^{2}}\,=\,2\left(\frac{d\omega_{p}}{dt}\right)^{2}+2\omega\frac{d^{2}\omega_{p}}{dt^{2}}$,
and
$\frac{d^{2}\tilde{m}_{\chi}^{2}}{dt^{2}}\,=\,2\left(\frac{d\tilde{m}_{\chi}}{dt}\right)^{2}+2\tilde{m}_{\chi}\frac{d^{2}\tilde{m}_{\chi}}{dt^{2}}$
we observe that

$\displaystyle\frac{d\tilde{m}_{\chi}}{dt}\,=\,\frac{1}{2\tilde{m}_{\chi}}\frac{d\tilde{m}_{\chi}^{2}}{dt}\,=\,H\tilde{m}_{\chi}\,+\,\frac{a^{2}s(2-s)H^{3}}{\tilde{m}_{\chi}}$
$\displaystyle\frac{d~{}^{2}\tilde{m}_{\chi}}{dt^{2}}\,=\,(1-s)\,H^{2}\tilde{m}_{\chi}\,+\,\frac{a^{2}s(2-s)(2-3s)H^{4}}{\tilde{m}_{\chi}}\,-\,\frac{a^{4}s^{2}(2-s)^{2}H^{6}}{\tilde{m}_{\chi}^{3}}$
(48)
$\displaystyle\mbox{so},~{}~{}~{}\frac{\omega_{p}^{\prime\prime}}{4\omega_{p}^{3}}\,=\,\frac{a}{4\omega_{p}^{3}}\frac{d\left(a\frac{d\omega_{p}}{dt}\right)}{dt}\,=\,\frac{a^{2}H\tilde{m}_{\chi}}{4\omega_{p}^{4}}\frac{d\tilde{m}_{\chi}}{dt}\,+\,\frac{a^{2}}{4\omega_{p}^{4}}\left(1-\frac{\tilde{m}_{\chi}^{2}}{\omega_{p}^{2}}\right)\left(\frac{d\tilde{m}_{\chi}}{dt}\right)^{2}$
$\displaystyle+\,\frac{a^{2}\tilde{m}_{\chi}}{4\omega_{p}^{4}}\left[(1-s)\,H^{2}\tilde{m}_{\chi}\,+\,\frac{a^{2}s(2-s)(2-3s)H^{4}}{\tilde{m}_{\chi}}\,-\,\frac{a^{4}s^{2}(2-s)^{2}H^{6}}{\tilde{m}_{\chi}^{3}}\right]$
$\displaystyle=\,\left(\frac{a}{\omega_{p}}(\Delta\,t)^{-1}\right)^{2}\left(\frac{\tilde{m}_{\chi}}{\omega_{p}}\right)^{2}\\{\frac{1}{8}H\Delta\,t\,\left[\frac{\left(\frac{d\tilde{m}_{\chi}^{2}}{dt}\right)(\Delta\,t)}{\tilde{m}_{\chi}^{2}}\right]\,+\,\frac{1}{16}\left(1-\frac{\tilde{m}_{\chi}^{2}}{\omega_{p}^{2}}\right)\left[\frac{\left(\frac{d\tilde{m}_{\chi}^{2}}{dt}\right)(\Delta\,t)}{\tilde{m}_{\chi}^{2}}\right]^{2}$
$\displaystyle+\,\frac{1}{4}(H\Delta\,t)^{2}\left[(1-s)\,\,+\,a^{2}s(2-s)(2-3s)\frac{H^{2}}{\tilde{m}_{\chi}^{2}}\,-\,a^{4}s^{2}(2-s)^{2}\left(\frac{H}{\tilde{m}_{\chi}}\right)^{4}\right]\\}\,\simeq\,0\;,$
(49)

provided that (5) is satisfied. After using (47) and (48) one observes that
(49) too gives negligible contribution to (14) for reasonable values of the
parameters provided that either $H\Delta\,t$ and $\frac{H}{\tilde{m}_{\chi}}$
are not much larger than one. Hence we have shown that
$W_{p}\,\simeq\,\omega_{p}$ in this case. In fact a similar result is obtained
in Boyanovsky2  where the condition $\frac{H\hbar}{E_{k}}\,\ll\,1$
($E_{k}=\sqrt{\frac{|\vec{\tilde{p}}|^{2}}{a^{2}}+m_{\chi}^{2}}$) is imposed.
In that study $\omega_{p}$ is time dependent in general since the condition
$\frac{H\hbar}{E_{k}}\,\ll\,1$ can not impose effective Minkowski spaces in
small time intervals while the condition (5) in this study makes
$W_{p}\,\simeq\,^{(0)}W_{p}=\omega_{p}=\mbox{constant}$ in each time interval.

##  Appendix B Relation between the momenta with tilde and without tilde

The relation between the momenta with tilde and without tilde in this paper is
the same as the momenta in the geodesic equations for the metrics (1) and (22)
in a given time interval. Therefore, the relation between the momenta with
$\tilde{}$’s and those without $\tilde{}$’s is

$|\vec{\tilde{p}}|\,=\,\tilde{m}_{\chi}\sqrt{\tilde{g}_{ij}\frac{d\tilde{x}^{i}}{|d\tilde{s}|}\frac{d\tilde{x}^{j}}{|d\tilde{s}|}}\,=\,a\,m_{\chi}\sqrt{g_{ij}\frac{dx^{i}}{|ds|}\frac{dx^{j}}{|ds|}}\,=\,a|\vec{p}|\;.$
(50)

The equality in (50) follows from extremization of the action for a free
particle of mass $m$ i.e. $\int\,mds$ where $m=\tilde{m}_{\chi}$,
$ds=d\tilde{s}$ for the metric (22) and $m=m_{\chi}$, $ds=ds$ for the metric
(1). In the case of (1) it reduces to the extremization of $\int\,ds$ since
$m=m_{\chi}$ is constant. The resulting geodesic equations are
$\frac{d\tilde{x}^{i}}{|d\tilde{s}|}\,=\,\frac{\sqrt{m_{\chi}\,v_{i}}}{\tilde{m}_{\chi}}$
and $\frac{dx^{i}}{|ds|}\,=\,\frac{\sqrt{v_{i}}}{a^{2}}$ where $v_{i}$ is some
constant, so (50) follows. $|\vec{p}|\propto\,\frac{1}{a}$ is the physical
momentum while $|\vec{\tilde{p}}|$ does not depend on redshift. We require
that the variations in $\tilde{m}_{\chi}^{2}$ and $\tilde{m}_{\phi}^{2}$ are
small (that are already ensured by (5)). This, in turn, implies that the
variation in $\omega_{p}^{2}$ of (10) with time is small since
$\vec{\tilde{p}}$ does not depend on time.

##  Appendix C Derivation of the equations (41) and (42)

In general, the in and out field operators may be expressed as

$\displaystyle\chi_{in}(x)\,=\,\sum_{n}\left[a_{n}^{in\,-}f_{n}(x)\,+\,a_{n}^{in\,+}f_{n}^{*}(x)\right]\,=\,\sum_{n}\left[b_{n}^{in\,-}g_{n}(x)\,+\,b_{n}^{in\,+}g_{n}^{*}(x)\right]$
(51)
$\displaystyle\phi_{out}(x)\,=\,\sum_{n}\left[\bar{a}_{n}^{out\,-}f_{n}(x)\,+\,\bar{a}_{n}^{out\,+}f_{n}^{*}(x)\right]\,=\,\sum_{n}\left[\bar{b}_{n}^{out\,-}g_{n}(x)\,+\,\bar{b}_{n}^{out\,+}g_{n}^{*}(x)\right]\;,$
(52)

where the creation, annihilation operators and the mode functions in the first
and the second equations of (51) and (52) are related by Bogolyubov
transformations. We have put an additional bar over the annihilation and the
creation operators in (52) to simplify the notation in the following lines.
The initial and the final vacuum states being identified by
$a_{n}^{in\,-}|0,in>\,=\,0$, $\bar{b}_{n}^{out\,-}|0,out>\,=\,0$ for all $n$.
Consider the scattering amplitude from an in state
$|^{1}n_{p_{1}}\,^{2}n_{p_{2}}.....\,^{s}n_{p_{s}}\;;in>$ to an out state
$|^{1}\bar{m}_{k_{1}}\,^{2}\bar{m}_{k_{2}}.....\,^{r}\bar{m}_{k_{r}}\;;out>$,
namely, Birrell-Taylor ; QFTC1

$<out,\bar{m}^{(s)}|n^{(s)}>\,=\,<\,out;\;^{r}\bar{m}_{k_{r}}\,.....\,^{2}\bar{m}_{k_{2}}\,^{1}\bar{m}_{k_{1}}\,|\,^{1}n_{p_{1}}\,^{2}n_{p_{2}}.....\,^{s}n_{p_{s}}\;;in\,>\;,$
(53)

where $\,{}^{n}l_{k_{j}}$ stands for $\,l^{n}$ particles with momentum
$k_{j}$, and

$\displaystyle|^{1}n_{p(1)}\,^{2}n_{p(2)}.....\,^{s}n_{p(s)}\;;in\,>\,=\,\left({}^{1}n!\,^{2}n!\,......\,^{s}n!\right)^{-\frac{1}{2}}\,\left(a_{p(1)}^{\dagger}\right)^{\,{}^{1}n}\left(a_{p(2)}^{\dagger}\right)^{\,{}^{2}n},\ldots,\left(a_{p(s)}^{\dagger}\right)^{\,{}^{s}n}\,|0,in>\;,$
$\displaystyle|^{1}\bar{m}_{k(1)}\,^{2}\bar{m}_{k(2)},\ldots\,^{r}\bar{m}_{k(r)}\;;out>\,=\,\left({}^{1}m!\,^{2}m!\ldots\,^{r}m!\right)^{-\frac{1}{2}}\left(\bar{b}_{k(1)}^{\dagger}\right)^{\,{}^{1}m}\left(\bar{b}_{k(2)}^{\dagger}\right)^{\,{}^{2}m},\ldots,\left(\bar{b}_{k(r)}^{\dagger}\right)^{\,{}^{r}m}\,|0,out>\;.$
(54)

Then the corresponding amplitude $<out,\bar{m}^{(r)}|n^{(s)},in>$ may be found
to be Birrell-Taylor

$\displaystyle<out,\bar{m}^{(r)}|n^{(s)},in>\,=\,\sum_{\rho}\sum_{\sigma}\,i^{\frac{(s-q)}{2}}i^{\frac{(r-q-w+k)}{2}}\\{\,O_{z\rho(1)}\ldots\,O_{z\rho(k)}\,\,\alpha^{-1}_{\rho(k+1),\rho(k+1)}\ldots\alpha^{-1}_{\rho(q),\rho(q)}\;$
$\displaystyle\times\,\Lambda_{\rho(q+1),\rho(q+2)}\ldots\Lambda_{\rho(s-1),\rho(s)}\,Q_{z\sigma(1)}\ldots\,Q_{z\sigma(w)}\;V_{\sigma(w+1),\sigma(w+2)}\ldots\,V_{\sigma(r-q+k-1),\sigma(s-q+k-q)}\,$
(55)
$\displaystyle\times\,<out,0|\,T\left(\phi(z_{\sigma(1)})\ldots\,\phi(z_{\sigma(w)})\,\phi(z_{\rho(1)})\ldots\,\phi(z_{\rho(k)})\right)\,|0,in>\\}\;,$

where the subscripts $z\rho(i)$ imply $z\rho(i)=z_{\rho(i)}$, and
$\sum_{\rho}\sum_{\sigma}$ stands for the summations over $0\leq\,k\,\leq\,s$,
$0\leq\,q\,\leq\,s-1$, $0\leq\,w\,\leq\,r$. Here the subindex $\rho$ stands
for all possible combinations between $O_{z\rho()}$,
$\alpha^{-1}_{\rho(),\rho()}$, $\Lambda_{\rho(),\rho()}$; and the subindex
$\sigma$ stands for all possible combinations between $O_{z\sigma()}$ and
$V_{\sigma(),\sigma()}$, where

$\displaystyle
O_{z(i)}=i\sum_{p}\,\alpha^{-1}_{ip}\,\int\,\sqrt{-g}\,d^{4}z_{i}\,g_{p}(z_{i})\,\left(-\nabla_{z_{i}\,\mu}\nabla_{z_{i}}^{\mu}+m^{2}+\xi\,R_{z(i)}\right)$
$\displaystyle
Q_{z(i)}=i\sum_{p}\,\alpha^{-1}_{pi}\,\int\,\sqrt{-g}\,d^{4}z_{i}\,f_{p}(z_{i})\,\left(-\nabla_{z_{i}\,\mu}\nabla_{z_{i}}^{\mu}+m^{2}+\xi\,R_{z(i)}\right)$
$\displaystyle\Lambda_{ij}\,=\,-i\sum_{p}\beta_{pi}\alpha^{-1}_{jp}~{},~{}~{}V_{ij}\,=\,i\sum_{p}\beta_{jp}\alpha^{-1}_{pi}$

with $m$ being the mass of $\phi$, $R$ is the curvature scalar, and $\xi=0$ in
this case (i.e. for minimal coupling between matter and curvature). It is
evident that (55) has the form given in Eq.(41) QFTC1 , and after excluding
the terms containing Bogolyubov coefficients $\alpha_{ij}$, $\beta_{ij}$ it
corresponds to Eq.(42).

##  Appendix D Transformation of
$\frac{<out;0|T\left(\phi(y_{1})\ldots\,\phi(y_{h})\,\chi(x_{1})\ldots\,\chi(x_{s})\right)|0;in>}{<out,0|0,in>}\,$
under $\chi\,\rightarrow\,\tilde{\chi}$, $\phi\,\rightarrow\,\tilde{\phi}$

In this appendix we study the effect of $\chi\,\rightarrow\,\tilde{\chi}$,
$\phi\,\rightarrow\,\tilde{\phi}$ on
$\tau\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)$ in (43). We apply a
careful path integral analysis since a naive analysis could be misleading.
$\tau\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)$ in terms of
functional derivatives may be expressed as QFTC1

$\tau\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)\,=\,i^{-(h+s)}\left[\frac{\delta^{h+s}\ln{Z[J]}}{\delta\,J_{\phi}(y_{h})\ldots\,\delta\,J_{\phi}(y_{1})\,\delta\,J_{\chi}(x_{1})\ldots\,\delta\,J_{\chi}(x_{s})}\right]_{J=0},$
(56)

The Green function generating functional for this study before the rescaling
is

$Z[J]\,=\,\int\,{\cal D}[\phi]\,{\cal
D}[\chi]\;\exp{\left[i\,S\,+\,i\int\,\sqrt{-g}\,d^{4}x\,\left(J_{\phi}\,\phi+J_{\chi}\,\chi\right)\,\right]}\;.$
(57)

Here, path-integral1 ; path-integral2

$\int\,{\cal
D}[\phi]\,=\,lim_{L\rightarrow\,\infty}lim_{N\rightarrow\,\infty}\,{}_{N\epsilon=L}\int\prod_{m}^{N}\,d\phi_{m}~{},~{}~{}~{}\int\,{\cal
D}[\chi]\,=\,lim_{L\rightarrow\,\infty}lim_{N\rightarrow\,\infty}\,{}_{N\epsilon=L}\int\prod_{m}d\chi_{m}\;,$
(58)

where the 4-dimensional volume $L^{4}$ is divided into $N^{4}$ small cubes of
sides $\epsilon$, the subindex $m$ denotes the $m$th cube.

The explicit form of $S$ in (57) is given in (2), and The $S$ before and after
the rescaling are the same by equality of (2) and (3) by construction. The
$J_{\phi}\,\phi$ and $J_{\chi}\,\chi$ terms may be made to be invariant under
rescaling by defining

$\tilde{J}_{\phi}\,=\,a^{3}\,J_{\phi}\,,~{}~{}\tilde{J}_{\chi}\,=\,a^{3}\,J_{\chi}\;,$
(59)

where we have used $\tilde{\chi}=a\,\chi$, $\tilde{\phi}=a\,\phi$,
$\sqrt{-\tilde{g}}\,d^{4}\tilde{x}\,$=$\,d\eta\,d^{3}x$ as mentioned before
(45).

On the other hand, there could be some nontrivial effect of rescaling on (57)
through the ${\cal D}[\phi]$ and ${\cal D}[\chi]$ terms. Another potential
effect could be through the intermediate states since (57) may be also
expressed as

$\displaystyle Z[J]\,=$
$\displaystyle\exp{\\{i\int\,d^{4}x\;\mu\left(\frac{1}{i}\frac{\delta}{\delta\,J_{\phi}}\right)^{2}\left(\frac{1}{i}\frac{\delta}{\delta\,J_{\chi}}\right)\\}}$
(60) $\displaystyle\times\,\int{\cal D}[\phi]{\cal
D}[\chi]\,\exp{\left[i\,S_{0}+i\int\sqrt{-g}\,d^{4}x\,\left(J_{\phi}\phi+J_{\chi}\chi\right)\right]}\;.$

Hence a factor of $a^{9}$ will be introduced through the rescaling of
$J_{\phi}$ and $J_{\chi}$ in the functional derivatives in (60) after using
(59). However this rescaling does not effect (56) because of the presence of
$\ln{Z}$ rather than $Z$. Because of the same reason, the effects of rescaling
on ${\cal D}[\phi]$ and ${\cal D}[\chi]$ terms in (58) cancel in (56).

In other words, the only effect of the rescaling
$\chi\,\rightarrow\,\tilde{\chi}$, $\phi\,\rightarrow\,\tilde{\phi}$ on
$\tau\left(y_{1},\ldots,y_{h},x_{1},\ldots,x_{s}\right)$ is through the
functional derivatives in (56) i.e. only through the rescaling of the fields
$\phi$ and $\chi$ in (43) as expected.

$\tilde{p}_{1}$$\chi$$\tilde{p}_{2}$$\chi$$\tilde{p}_{3}$$\phi$$\tilde{p}_{4}$$\phi$$\tilde{q}_{t}$$\phi$(a)$\tilde{p}_{1}$$\chi$$\tilde{p}_{2}$$\chi$$\tilde{p}_{3}$$\phi$$\tilde{p}_{4}$$\phi$$\tilde{q}_{u}$$\phi$(b)

Figure 1: The leading order Feynman diagrams that contribute to
$\chi\chi\,\rightarrow\,\phi\phi$. Here
$\tilde{q}_{t}\,=\,\tilde{p}_{1}-\tilde{p}_{3}=\tilde{p}_{4}-\tilde{p}_{2}$,
$\tilde{q}_{u}\,=\,\tilde{p}_{1}-\tilde{p}_{4}\,=\tilde{p}_{3}-\tilde{p}_{2}$
are the 4-momenta carried in the internal lines. Note that the s-channel is
forbidden by kinematics in Minkowski space. In a curved space, in general, one
expects the s-channel to be opened due to the violation of energy/momentum
conservation Boyanovsky2 . In this case where the space is taken to be nearly
Minkowski in the intervals corresponding to average time for each process, one
expects a small contribution to the scattering amplitudes due to the opened
s-channel processes. However this contribution may be neglected since we take
the space to be almost Minkowski in these time intervals by the condition (5).