# Fluctuations in the stress energy tensor of spinor fields evolving in
general FRW spacetimes

Ankit Dhanuka<EMAIL_ADDRESS><EMAIL_ADDRESS>Department
of Physical Sciences, IISER Mohali,
Sector 81, SAS Nagar, Manauli PO 140306, Punjab, India

###### Abstract

In this work, we study quantum fluctuations in the stress energy tensor of
spinor fields evolving in general FRW spacetimes. We quantify these
fluctuations by the noise kernel of spinor fields. For the particular case of
de Sitter spacetime, we place the spinor field in what is called the fermionic
Bunch Davies vacuum and study the variation of the noise kernel with the mass
of the field. We then make use of the conformal invariance of massless spinor
fields and employ an equivalence that relates a massless spinor field in any
given FRW spacetime with a corresponding massless spinor field in de Sitter
spacetime. Using this equivalence, we study the behaviour of the noise kernel
of massless spinor fields in general FRW spacetimes while placing the fields
in the Bunch Davies vacuum of the corresponding massless spinor field of the
de Sitter spacetime. We extend this analysis to the coordinate invariant
quantity of the energy energy correlator. We compare these results for spinor
fields in considered spacetimes with the analogous results for scalar fields.
This study helps us better understand whether the spin $1/2$ fermionic matter
remains strongly correlated in considered spacetimes leading to considerable
backreaction or not.

##  I Introduction

With the ever so precise cosmological data Aghanim  _et al._ (2020); Akrami
_et al._ (2020); Ade  _et al._ (2014) becoming available to us, we have
increasing density of empirical references against which we can test the
validity of our theories, arrived at either through intuition or to explain
observations obtained in other astrophysical or cosmological contexts, and
gain new insights into the origin of the universe and its evolution history.
For example, the anisotropies in the cosmic microwave background (CMB) are
believed to be related to the power spectrum of the quantum fluctuations of
the inflaton field Weinberg (2008); Baumann (2011) and therefore, corrections
to this quantity coming from higher order effects can show up themselves in
the finer details of the observed anisotropies Bartolo  _et al._ (2004). In
scenarios where fluctuations over the background quantities can be considered
small, we see that the matter fluctuations (over the background matter fields)
can couple to the gravitational fluctuations (over the background metric)
through their stress energy tensors111This can be easily seen by noticing that
$\displaystyle S_{total}$ $\displaystyle=$ $\displaystyle
S_{gravity}(\tilde{g})+S_{matter}(\tilde{g},\tilde{\phi})$ $\displaystyle=$
$\displaystyle
S_{gravity}(g_{bg}+h)+S_{matter}(g_{bg}+h,\phi_{bg}+\delta\phi)$
$\displaystyle=$
$\displaystyle\Big{(}S^{(0)}_{gravity}(g_{bg})+S^{(0)}_{matter}(g_{bg},\phi_{bg})\Big{)}+S^{(2)}_{matter}(g_{bg},\delta\phi)+S^{(2)}_{gravity}(h)$
$\displaystyle\hskip 56.9055pt+\int
d^{4}x\sqrt{-g_{bg}}\Big{(}-\frac{1}{2}h_{ab}T^{ab}_{\delta\phi}\Big{)}+....$
where $g_{bg}$ and $\phi_{bg}$ are the background metric and the background
matter field respectively and similarly $h$ and $\delta\phi$ are the
respective fluctuations over these background quantities. We have dropped the
terms which are first order in $h$ or $\delta\phi$ as they contain expressions
which correspond to the equations of motion for the background quantities and,
hence, are zero. Here the terms $S^{(2)}$ are second order in $h$ and
$\delta\phi$ and provide the free dynamics of the matter and the gravity
fluctuations. We see that one of the interaction terms between the gravity
fluctuations and the matter field fluctuations is proportional to the stress
energy tensor of the matter field fluctuations i.e.,
$T^{ab}_{\delta\phi}=-\frac{2}{\sqrt{-g_{bg}}}\frac{\delta
S_{matter}(\delta\phi)}{\delta g_{ab}(bg)}$.. Therefore, if we quantize these
fluctuations, then such an interaction term can be seen as providing a
perturbed Hamiltonian term to the free dynamics of these fluctuations and as
such the propagators of these fluctuations get corrections from this term
Weinberg (2005). From the structure of the interaction term, it is clear that
it would inevitably involve n-point correlators of the matter field stress
energy operators. Calculating n-point correlators of stress-energy operators
is, in general, a very difficult task. However, in order to study the 2nd
order effect (after taking into account the first order semiclassical effects
of the 1-point quantum averages of stress energy operators) coming from this
interaction term, one only needs the knowledge of two point correlators of
quantum field stress energy operators. A number of previous investigations
have undertaken the task of calculating and studying the behaviour of these
two point correlations of quantum field stress energy operators (mainly, that
of scalar fields). These works have considered the dynamics of quantum fields
over different types of background spacetimes ranging from the Minkowski
spacetime Ford and Roman (2005); Fröb (2013); Bates (2013) to the de Sitter
spacetime Perez-Nadal  _et al._ (2010); Dhanuka and Lochan (2020) to the more
general conformally flat spacetimes Bates  _et al._ (2013) among others
Eftekharzadeh  _et al._ (2012); Satin  _et al._ (2013); Wu  _et al._ (2021).
These studies have been performed by placing quantum fields in different types
of vacua, for example, Perez-Nadal  _et al._ (2010); Dhanuka and Lochan (2020)
have carried out the analysis by placing the scalar field in the Bunch Davies
vacuum Bunch and Davies (1978); Allen (1985). A number of works have also
focussed attention on how the fluctuations in quantum stress energy tensors
affect the geometry of spacetimes, for example Borgman and Ford (2004); Ford
_et al._ (2010) have considered how congruences of geodesics are affected by
these fluctuations and similarly, Wu  _et al._ (2007, 2011); Hsiang  _et al._
(2017) have considered how these fluctuations manifest themselves in the power
spectra of primordial gravity waves and hence provide possible observable
effects of these fluctuations. These fluctuations in stress energy operators,
or more appropriately the noise kernel (which is the vacuum expectation of
symmetrized stress energy bi-tensors, see below), play a fundamental role in
the stochastic gravity Hu and Verdaguer (2008) framework where they feed the
stochastic fluctuations appearing in the Einstein Langevin equations.
Therefore, in any attempt to go beyond the first order semiclassical analyses,
we would always end up requiring to calculate the behaviour of the noise
kernel of matter fields.
Since the present-day universe contains both bosonic and fermionic types of
particles, it is natural to assume that (during their journey through the
evolution of the universe) they could couple with gravitational waves with the
above types of interaction terms. Though majority of the above cited works
have considered only the effects of bosonic fields such as scalar and
electromagnetic fields, there are some works that have also considered the
effects of fermionic fields on gravitational waves. For example, assuming that
the fermions were produced after inflation, in the reheating era Peloso and
Sorbo (2000); Giudice  _et al._ (1999), Figueroa and Meriniemi (2013) has
considered the backreaction of fermionic fields on gravitational waves.
Similarly, Feng  _et al._ (2012); Feng and Piao (2015) have looked at the
modification in gravitational wave spectra caused by spinor fields during
inflation. These works have used the spinor mode sums in momentum space and
have arrived at the fermionic field corrections to the gravitational wave
spectra. In this work, we aim to determine the coordinate space expression of
the noise kernel for spinor fields in general FRW spacetimes. In order to
carry out this task for the particular case of spinor fields in de Sitter
spacetime, we consider that the fields are placed in the fermionic Bunch
Davies vacuum Collins (2005); Koksma and Prokopec (2009) which is defined by
choosing the fermionic mode functions in such a way that, in the asymptotic
past, the leading order behaviour of these mode functions is like that of the
Minkowski spacetime positive and negative frequency mode functions. Then, we
deal with massless spinor fields in general FRW spacetimes and for these
cases, we employ a mapping that relates a massless spinor field in any FRW
spacetime to another massless spinor field in any other FRW spacetime and also
relates the corresponding Wightman functions. Therefore, using this mapping,
we place a massless spinor field in a given FRW spacetime in the Bunch-Davies
vacuum of the corresponding massless spinor field in de Sitter spacetime (or
equivalently, in terms of the Poincare vacuum of the Minkowski spacetime). We
also compare the behaviour of the noise kernel of spinor fields in the above
spacetimes with the analogous studies for scalar fields in the same spacetimes
Dhanuka and Lochan (2020). Since different epochs of the universe are
approximately FRW type spacetimes, the analysis of massless spinor fields in
FRW spacetimes is applicable to these different epochs of the universe, in
particular, to the present day dark energy driven universe. As the surveys
Aghanim  _et al._ (2020) put the dark-energy equation of state parameter to be
$-1.03\pm 0.03$, having knowledge of the behaviour of the noise kernel for
spinor fields in both the quintessence regime ($-1<w<-\frac{1}{3}$) and the
phantom regime ($w<-1$) may be useful Johri (2004). Therefore, our analysis
considers spinor fields not only for quintessence but also for phantom
cosmologies and in fact, for all types of FRW spacetimes. Carrying out this
analysis for phantom cosmologies becomes physically more sound and, hence, all
the more appealing when we can evade the unwanted big rip feature (i.e., the
scaling factor and the energy density going to infinity in a finite interval
of time) of the phantom models Johri (2004); Caldwell (2002) with a dynamical
equation of state parameter generated by properly chosen phantom field
potentials Singh  _et al._ (2003). These types of potential allow for the
possibility of having a phantom spacetime phase for some time and then exiting
this phantom phase before big rip occurs.
The rest of the paper is organized as follows. In section $2$, we give a brief
review of the dynamics of spinor fields in curved spacetimes and also provide
an outline of a calculation to arrive at the expression of the Wightman
function corresponding to the fermionic Bunch Davies vacuum of the de Sitter
spacetime. We then employ a mapping between massless spinor fields in de
Sitter and general FRW spacetimes and, using this mapping, we relate the
expression of the Wightman function of a massless spinor field in de Sitter
spacetime with the corresponding expressions for massless spinor fields in
general FRW spacetimes. In section $3$, we use the point splitting technique
to write the noise kernel for spinor fields as an expression involving some
derivative operators acting on a product of Wightman functions. In section
$4$, we arrive at the expression of the noise kernel for a spinor field
(placed in the Bunch Davies vacuum) in de Sitter spacetime and analyse how the
noise kernel behaves as a function of the mass of the field. In section $5$,
we perform a similar analysis for massless fermions in general FRW spacetimes.
In section $6$, we summarize our findings and discuss their implications. We
use the $(-,+,+,+)$, convention for the spacetime metric.

##  II Preliminaries

The dynamics of spinor fields in curved spacetimes is best described using the
formalism of tetrad bases. The action for a minimally coupled spinor field in
curved spacetime is given by

$\displaystyle\hskip 62.59596ptS$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\big{[}i\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-m\bar{\psi}\psi\big{]}$
(1) $\displaystyle=$ $\displaystyle\int
d^{4}x\frac{\sqrt{-g}}{2}\big{[}i\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-i(\nabla_{\mu}\bar{\psi})\gamma^{\mu}\psi-2m\bar{\psi}\psi\big{]}$
$\displaystyle\hskip 56.9055pt+\int
d^{4}x\frac{\sqrt{-g}}{2}\nabla_{\mu}\big{(}i\bar{\psi}\gamma^{\mu}\psi\big{)}\,.$

where $\gamma^{\mu}=e^{\mu}_{a}\Gamma^{a}$,
$\\{\Gamma^{a},\Gamma^{b}\\}=-2\eta^{ab}$ and the four divergence term is a
boundary term. Here $e^{\mu}_{a}$ are the tetrad basis which satisfy
$e^{\mu}_{a}e^{\nu}_{b}\eta^{ab}=g^{\mu\nu}$. Also
$\nabla_{\mu}=\partial_{\mu}-\frac{1}{8}\omega^{ab}_{\,\
\mu}[\Gamma_{a},\Gamma_{b}]$ where $\omega^{ab}_{\,\
\mu}=e^{a}_{\lambda}e^{\tau b}\Gamma^{\lambda}_{\tau\mu}-e^{\tau
b}\partial_{\mu}e^{a}_{\tau}$ (See Parker and Toms (2009); Shapiro (2016)).
The equation of motion for the above action has the following form

$(i\gamma^{\lambda}\nabla_{\lambda}-m)\psi=0.$ (2)

Since we are ultimately interested in general FRW spacetimes for which one can
choose coordinate systems in which the metric can be cast into conformally
flat form Spradlin  _et al._ (2001), let us consider
$ds^{2}=a^{2}(\eta)(-d\eta^{2}+d\vec{x}^{2})$. For these spacetimes, we can
take the tetrad basis to be $e^{\mu}_{a}=\frac{1}{a}\delta^{\mu}_{a}$. Using
this tetrad basis and the fact that
$\Gamma^{\lambda}_{\mu\nu}=\frac{a^{\prime}}{a}(\delta^{\lambda}_{\nu}\delta^{0}_{\mu}+\delta^{\lambda}_{\mu}\delta^{0}_{\nu}-\eta_{\mu\nu}\eta^{\lambda
0})$, we can calculate $\omega_{\mu
ab}=\frac{a^{\prime}}{a}(\eta_{a\mu}\delta^{0}_{b}-\eta_{b\mu}\delta^{0}_{a})$
where ′ represents a derivative w.r.t. $\eta$. Using these expressions, we
find that

$i\gamma^{\lambda}\nabla_{\lambda}=\frac{i}{a}\Gamma^{\mu}\bigg{(}\partial_{\mu}-\frac{1}{8}\omega_{\mu
ab}[\Gamma^{a},\Gamma^{b}]\bigg{)}=\frac{i}{a}\bigg{(}\Gamma^{\mu}\partial_{\mu}+\frac{a^{\prime}}{4a}\eta_{\mu
a}\Gamma^{\mu}[\Gamma^{0},\Gamma^{a}]\bigg{)}=\frac{i}{a}\bigg{(}\Gamma^{\mu}\partial_{\mu}+\frac{3a^{\prime}}{2a}\Gamma^{0}\bigg{)}\,.$
(3)

Substituting this into the equation of motion, it modifies to

$\bigg{(}i\Gamma^{\mu}\partial_{\mu}+\frac{3ia^{\prime}}{2a}\Gamma^{0}-am\bigg{)}\psi=0\,.$
(4)

Going to the Fourier space i.e., writing
$\psi(\eta,\vec{x})=\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}e^{i\vec{k}.\vec{x}}\psi_{\vec{k}}(\eta)$,
we obtain that

$\bigg{(}i\Gamma^{0}\partial_{0}-\vec{k}.\vec{\Gamma}+\frac{3ia^{\prime}}{2a}\Gamma^{0}-am\bigg{)}\psi_{\vec{k}}(\eta)=0\,,$
(5)

and, for field redefinition
$\chi_{\vec{k}}(\eta)=a^{\frac{3}{2}}(\eta)\psi_{\vec{k}}(\eta)$, it reduces
further to the following form

$\bigg{(}i\Gamma^{0}\partial_{0}-\vec{k}.\vec{\Gamma}-am\bigg{)}\chi_{\vec{k}}(\eta)=0\,.$
(6)

Following Koksma and Prokopec (2009), let us now decompose the 4-column
spinors into chirality-helicity basis i.e.,

$\displaystyle\chi^{h}(\vec{k},\eta)\equiv\chi_{\vec{k}}(\eta)=\begin{bmatrix}\chi_{L,h}(\vec{k},\eta)\\\
\chi_{R,h}(\vec{k},\eta)\end{bmatrix}\otimes\xi_{h}$ (7)

where $\xi_{h}$ are 2-column eigenvectors of $(\hat{k}.\vec{\sigma})$ i.e.,
$(\hat{k}.\vec{\sigma})\xi_{h}=h\xi_{h}$ with eigenvalues $h$ being $\pm
1$222Using the form of the Pauli matrices and demanding that
$\xi^{\dagger}_{h}\xi_{h}=1$, we can take the helicity eigenvectors to be
$\displaystyle\xi_{+}=\frac{1}{\sqrt{2(1-\hat{k}_{z})}}\begin{bmatrix}\hat{k}_{x}-i\hat{k}_{y}\\\
1-\hat{k}_{z}\end{bmatrix},\,\xi_{-}=\frac{1}{\sqrt{2(1+\hat{k}_{z})}}\begin{bmatrix}-\hat{k}_{x}+i\hat{k}_{y}\\\
1+\hat{k}_{z}\end{bmatrix}\,,$ where
$\hat{\vec{k}}=\frac{\vec{k}}{||\vec{k}||}$.. The functions
$\chi_{L,h}(\vec{k},\eta)$ and $\chi_{R,h}(\vec{k},\eta)$ are 1-column
functions and are called left- and right-handed spinors of helicity $h$,
respectively. Here $\otimes$ is the tensor product symbol.
Using the Weyl representation for Gamma matrices i.e.,

$\displaystyle\Gamma^{0}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\otimes\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\,,\,\,\,\Gamma^{i}=\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}\otimes\sigma_{i}$ (8)

where $\otimes$ is, again, the tensor product symbol and
$\sigma_{i}^{\prime}s$ are $2*2$ Pauli matrices, the Dirac equation reduces to
the following set of coupled linear differential equations

$\displaystyle
i\partial_{0}\chi_{R,h}(\vec{k},\eta)-kh\chi_{R,h}(\vec{k},\eta)-am\chi_{L,h}(\vec{k},\eta)$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
i\partial_{0}\chi_{L,h}(\vec{k},\eta)+kh\chi_{L,h}(\vec{k},\eta)-am\chi_{R,h}(\vec{k},\eta)$
$\displaystyle=$ $\displaystyle 0\,.$ (9)

Defining $u_{\pm
h}(k,\eta)=\frac{\chi_{L,h}(\vec{k},\eta)\pm\chi_{R,h}(\vec{k},\eta)}{\sqrt{2}}$
and specializing to the de Sitter spacetime case i.e., taking
$a(\eta)=-\frac{1}{H\eta}$, we get the following equations

$u^{\prime\prime}_{\pm
h}+\Big{(}k^{2}+\frac{\frac{1}{4}-(\frac{1}{2}\mp\frac{im}{H})^{2}}{\eta^{2}}\Big{)}u_{\pm
h}=0\,,$ (10)

where ′ represents a derivative w.r.t. to $\eta$. These equations have the
form of Bessel’s equation and hence the most general solutions for $u_{\pm h}$
are given by Gradshteyn and Ryzhik (2014); DLMF

$u_{\pm h}(k,\eta)=\alpha^{h}_{\pm
k}\sqrt{-k\eta}H^{(1)}_{\nu_{\pm}}(-k\eta)+\beta^{h}_{\pm
k}\sqrt{-k\eta}H^{(2)}_{\nu_{\pm}}(-k\eta)$ (11)

where $\nu_{\pm}=\frac{1}{2}\mp\frac{im}{H}$ and $\alpha_{\pm k}^{h}$’s and
$\beta_{\pm k}^{h}$’s are arbitrary constants to be fixed by initial
condition. In order to proceed further with the analysis, we need to make a
choice of vacuum for the spinor field and in this work, we choose to work with
what is called the fermionic Bunch Davies vacuum.

###  II.1 The fermionic Bunch Davies vacuum

The Bunch Davies vacuum for spinor fields in de Sitter spacetime is defined
analogously to how one defines it for scalar fields. For this, we fix
$\alpha_{\pm k}^{h}$’s and $\beta_{\pm k}^{h}$’s in such a way that the mode
functions $u_{\pm h}(k,\eta)$, considered in the asymptotic past, behave like
flat spacetime mode functions. To make it more precise, we notice that in the
asymptotic past i.e., $\eta\to-\infty$ limit, equation (10) reduces to
$u^{\prime\prime}_{\pm h}+k^{2}u_{\pm h}=0$. This implies that there exists
positive and negative frequency modes in the asymptotic past i.e., $e^{\pm
ik\eta}$. Using the following large argument expansions of Hankel functions
DLMF  i.e.,

$\displaystyle H^{(1)}_{\nu}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\pi
z}}e^{i(z-\frac{\pi}{2}\nu-\frac{\pi}{4})}\Big{[}1+O\Big{(}\frac{1}{z}\Big{)}\Big{]}\,,$
(12) $\displaystyle H^{(2)}_{\nu}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\pi
z}}e^{-i(z-\frac{\pi}{2}\nu-\frac{\pi}{4})}\Big{[}1+O\Big{(}\frac{1}{z}\Big{)}\Big{]}\,,$
(14) $\displaystyle\hskip
56.9055pt\,\,\,for\,\,|z|\to\infty,\,\,\,Re(\nu)>-\frac{1}{2}\,\,\,and\,\,|arg(z)|<\pi,$

we see that, for non-zero $\alpha_{\pm k}^{h}$’s and vanishing $\beta_{\pm
k}^{h}$’s, the mode functions, $u_{\pm h}(k,\eta)$, indeed behave like
positive frequncy (or particle) modes in the $\eta\to-\infty$ limit. In fact,
by substituting $u_{\pm h}(k,\eta)=\alpha^{h}_{\pm
k}\sqrt{-k\eta}H^{(1)}_{\nu_{\pm}}(-k\eta)$ in equations of motion and using
certain properties of Hankel functions, we find that
$\alpha^{h}_{-k}=ih\alpha^{h}_{+k}e^{i\pi\nu_{-}}.$ Demanding that
$\lim_{\eta\to-\infty}u_{+h}(k,\eta)=\frac{e^{-ik\eta}}{\sqrt{2}}$, we should
take $\alpha^{h}_{+k}=\sqrt{\frac{\pi}{4}}e^{i\frac{\pi}{2}(\nu_{+}+1/2)}$ and
therefore, we have, for particle modes,

$\displaystyle u_{+h}(k,\eta)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{-\pi
k\eta}{4}}e^{i\frac{\pi}{2}(\nu_{+}+1/2)}H^{(1)}_{\nu_{+}}(-k\eta)\equiv
f(k,\eta)$ (15) $\displaystyle u_{-h}(k,\eta)$ $\displaystyle=$
$\displaystyle-h\sqrt{\frac{-\pi
k\eta}{4}}e^{i\frac{\pi}{2}(\nu_{-}+1/2)}H^{(1)}_{\nu_{-}}(-k\eta)\equiv-
hg^{*}(k,\eta)\,.$ (16)

This implies that

$\displaystyle\chi_{L,h}(k,\eta)$ $\displaystyle=$
$\displaystyle\frac{f-hg^{*}}{\sqrt{2}}$ (17)
$\displaystyle\chi_{R,h}(k,\eta)$ $\displaystyle=$
$\displaystyle\frac{f+hg^{*}}{\sqrt{2}}\,.$ (18)

Similarly, we can determine the anti-particle modes. For anti-particle modes,
we denote the Fourier coefficient of $e^{-i\vec{k}.\vec{x}}$ to be
$\nu_{\vec{k}}(\eta)$ and hence, in the helicity-chirality basis, the
differential equations have $-k$ in place of $k$. Therefore, by flipping the
definitions of the left and right handed fermions i.e., taking

$\displaystyle\nu^{h}(\vec{k},\eta)\equiv\nu_{\vec{k}}(\eta)=\begin{bmatrix}\nu_{R,h}(\vec{k},\eta)\\\
\nu_{L,h}(\vec{k},\eta)\end{bmatrix}\otimes\xi_{h}\,,$ (19)

we end up with the same set of differential equations for $\nu_{R,h}$ and
$\nu_{L,h}$ as (II). Now demanding that the corresponding $u_{\pm h}$’s go to
$\frac{e^{ik\eta}}{\sqrt{2}}$ in the asymptotic past, we conclude that

$\displaystyle\nu_{L,h}(k,\eta)$ $\displaystyle=$
$\displaystyle\frac{hf^{*}+g}{\sqrt{2}}$ (20) $\displaystyle\nu_{R,h}(k,\eta)$
$\displaystyle=$ $\displaystyle\frac{-hf^{*}+g}{\sqrt{2}}\,.$ (21)

Using the above considered Bunch Davies modes, we can write the field operator
and its conjugate as follows333For the rest of this subsection, $a(\eta)$
represents $-\frac{1}{H\eta}$.

$\hat{\psi}(\eta,\vec{x})=a^{-\frac{3}{2}}(\eta)\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\sum_{h}\Big{[}\hat{a}_{\vec{k},h}\chi^{h}(\vec{k},\eta)e^{i\vec{k}.\vec{x}}+\hat{b}^{\dagger}_{\vec{k},h}\nu^{h}(\vec{k},\eta)e^{-i\vec{k}.\vec{x}}\Big{]}$
(22)

and

$\hat{\bar{\psi}}(\eta,\vec{x})=a^{-\frac{3}{2}}(\eta)\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\sum_{h}\Big{[}\hat{a}^{\dagger}_{\vec{k},h}\bar{\chi}^{h}(\vec{k},\eta)e^{-i\vec{k}.\vec{x}}+\hat{b}_{\vec{k},h}\bar{\nu}^{h}(\vec{k},\eta)e^{i\vec{k}.\vec{x}}\Big{]}\,.$
(23)

Here $\hat{a}$’s and $\hat{b}$’s are the annihilation operators while
$\hat{a}^{\dagger}$’s and $\hat{b}^{\dagger}$’s are the creation operators
corresponding to the above described Bunch Davies modes. We now define the
Bunch Davies vacuum, $\ket{0}$, to be the state which is annihilated by all
$\hat{a}_{\vec{k},h}$’s and $\hat{b}_{\vec{k},h}$’s i.e.,
$\hat{a}_{\vec{k},h}\ket{0}=0$ and $\hat{b}_{\vec{k},h}\ket{0}=0$.
From the expressions for the field operator and its adjoint, we find that the
Wightman functions are given by

$S_{ij}(x,x^{\prime})=\langle\hat{\psi}_{i}(x)\hat{\bar{\psi}}_{j}(x^{\prime})\rangle=a^{-\frac{3}{2}}(\eta)a^{-\frac{3}{2}}(\eta^{\prime})\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\sum_{h}\chi^{h}_{i}(\vec{k},\eta)\bar{\chi}^{h}_{j}(\vec{k},\eta^{\prime})e^{i\vec{k}.(\vec{x}-\vec{x}^{\prime})}$
(24)

and

$R_{ji}(x^{\prime},x)=\langle\hat{\bar{\psi}}_{j}(x^{\prime})\hat{\psi}_{i}(x)\rangle=a^{-\frac{3}{2}}(\eta)a^{-\frac{3}{2}}(\eta^{\prime})\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\sum_{h}\nu^{h}_{i}(-\vec{k},\eta)\bar{\nu}^{h}_{j}(-\vec{k},\eta^{\prime})e^{i\vec{k}.(\vec{x}-\vec{x}^{\prime})}\\\
=-S_{ij}(x,x^{\prime})\hskip 227.62204pt$ (25)

After some manipulations, one can show that (See Koksma and Prokopec (2009);
Collins (2005))

$S_{ij}(x,x^{\prime})=\langle\hat{\psi}_{i}(x)\hat{\bar{\psi}}_{j}(x^{\prime})\rangle=a(\eta_{x})\Big{[}i\gamma^{\lambda}\overrightarrow{\nabla}^{x}_{\lambda}+m\Big{]}\frac{H^{2}}{\sqrt{a(\eta_{x})a(\eta_{x^{\prime}})}}\Bigg{[}S_{+}(x,x^{\prime})\frac{1+\Gamma^{0}}{2}+S_{-}(x,x^{\prime})\frac{1-\Gamma^{0}}{2}\Bigg{]}$
(26)

where

$S_{\pm}(x,x^{\prime})=\frac{\Gamma\big{(}2\pm
i\frac{m}{H}\big{)}\Gamma\big{(}1\mp
i\frac{m}{H}\big{)}}{(4\pi)^{2}}{}_{2}F_{1}\Big{(}2\pm i\frac{m}{H},1\mp
i\frac{m}{H},2,Z(x,x^{\prime})\Big{)}$ (27)

and
$Z(x,x^{\prime})=1+\frac{(\eta-\eta^{\prime})^{2}-(\Delta\vec{x})^{2}}{4\eta\eta^{\prime}}$.
Using that
$\overrightarrow{\nabla}^{x}_{\lambda}=\partial_{\lambda}-\frac{1}{8}\omega_{ab\lambda}[\Gamma^{a},\Gamma^{b}]=\partial_{\lambda}+\frac{a^{\prime}}{4a}\eta_{\lambda
a}[\Gamma^{0},\Gamma^{a}]$ and
$\gamma^{\lambda}\overrightarrow{\nabla}^{x}_{\lambda}=\frac{1}{a}\big{(}\Gamma^{\lambda}\partial_{\lambda}+\frac{3a^{\prime}}{2a}\Gamma^{0}\big{)}$,
the above expression can be written as

$S_{ij}(x,x^{\prime})=\Big{[}i\Gamma^{\lambda}\partial^{x}_{\lambda}+\frac{3ia^{\prime}(\eta_{x})}{2a(\eta_{x})}\Gamma^{0}+a(\eta_{x})m\Big{]}\frac{H^{2}}{\sqrt{a(\eta_{x})a(\eta_{x^{\prime}})}}\Bigg{[}S_{+}(x,x^{\prime})\frac{1+\Gamma^{0}}{2}+S_{-}(x,x^{\prime})\frac{1-\Gamma^{0}}{2}\Bigg{]}\\\
=\frac{H^{2}}{\sqrt{a(\eta_{x})a(\eta_{x^{\prime}})}}\Big{[}i\Gamma^{\lambda}\partial^{x}_{\lambda}+i\frac{a^{\prime}(\eta_{x})}{a(\eta_{x})}\Gamma^{0}+a(\eta_{x})m\Big{]}\Bigg{[}S_{+}(x,x^{\prime})\frac{1+\Gamma^{0}}{2}+S_{-}(x,x^{\prime})\frac{1-\Gamma^{0}}{2}\Bigg{]}\,.$
(28)

We need the forms of the Wightman functions, $S_{ij}(x,x^{\prime})$ and
$R_{ji}(x^{\prime},x)$, later on in this work when we study the behaviour of
the fluctuations in the stress energy operator which, as we shall see, can be
expressed as certain derivatives acting on products of Wightman functions.

###  II.2 Equivalence between massless spinor fields in FRW spacetimes and de
Sitter spacetime

It has been established, in Lochan  _et al._ (2018), that massless scalar
fields in FRW spacetimes can be mapped to massive scalar fields in de Sitter
spacetime and hence one can make use of this equivalence to study the dynamics
in one setup in terms of the another. In this subsection, we make use of the
conformal invariance of massless spinor fields and relate the Wightman
functions of massless spinor fields in general FRW spacetimes with the
Wightman function of a massless spinor field in de Sitter spacetime. In fact,
we start with a massive spinor field in an arbitrary FRW spacetime and try to
relate it to another spinor field in some other arbitrary FRW spacetime. For
this, consider the action of a spinor field in a spacetime with
$ds^{2}=c^{2}(\eta)(-d\eta^{2}+d\vec{x}^{2})$ i.e.,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\big{[}i\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-m\bar{\psi}\psi\big{]}$
(29) $\displaystyle=$ $\displaystyle\int
d^{4}xc^{3}\bar{\psi}\Big{[}i\Gamma^{\mu}\partial_{\mu}+i\frac{3c^{\prime}}{2c}\Gamma^{0}-cm\Big{]}\psi\,.$

Considering the transformation $\psi=F(\eta)\Omega$, we observe that the above
action transforms to

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}xc^{3}F^{2}\bar{\Omega}\Big{[}i\Gamma^{\mu}\partial_{\mu}+i\frac{3c^{\prime}}{2c}\Gamma^{0}+i\frac{F^{\prime}}{F}\Gamma^{0}-cm\Big{]}\Omega$
(30)

and now demanding that $c^{3}F^{2}=b^{3}$ and
$\Big{[}i\frac{3c^{\prime}}{2c}\Gamma^{0}+i\frac{F^{\prime}}{F}\Gamma^{0}-cm\Big{]}=\Big{[}i\frac{3b^{\prime}}{2b}\Gamma^{0}-bm^{\prime}\Big{]}$,
the above action becomes

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}xb^{3}\bar{\Omega}\Big{[}i\Gamma^{\mu}\partial_{\mu}+i\frac{3b^{\prime}}{2b}\Gamma^{0}-bm^{\prime}\Big{]}\Omega$
(31)

which is the action of a spinor field, $\Omega$, in a spacetime with
$ds^{2}=b^{2}(\eta)(-d\eta^{2}+d\vec{x}^{2})$.

We notice that the condition that $c^{3}F^{2}=b^{3}$ automatically gives

$\frac{3c^{\prime}}{2c}+\frac{F^{\prime}}{F}=\frac{3b^{\prime}}{2b}\,.$ (32)

Therefore, from the condition that

$\Big{[}i\frac{3c^{\prime}}{2c}\Gamma^{0}+i\frac{F^{\prime}}{F}\Gamma^{0}-cm\Big{]}=\Big{[}i\frac{3b^{\prime}}{2b}\Gamma^{0}-bm^{\prime}\Big{]}\,,$
(33)

we obtain that $m^{\prime}=\frac{c}{b}m$. A number of interesting conclusions
can be drawn from this analysis. But for our purposes, all we need to observe
is that a massless spinor field in an FRW spacetime can always be mapped to
another massless spinor field in any other FRW spacetime, in particular we can
map a massless spinor field in any FRW spacetime to a massless spinor field in
de Sitter or flat spacetime. We make use of this equivalence and determine the
desired results in general FRW spacetimes in terms of the de Sitter
quantities444As we have seen that massless spinor fields in FRW spacetimes are
equivalent to massless spinor fields in flat spacetime as well, we could have
equally well carried out the FRW analyses in terms of the flat space-time
quantities. But since we are going to analyse the de Sitter noise kernel
before going to the general FRW cases, it seems more seamless with the flow of
the paper to map FRW analyses to the de Sitter case. i.e., we take
$b(\eta)=a(\eta)=-\frac{1}{H\eta}$. In particular, we observe that the
Wightmann function for a massless spinor field in an FRW spacetime (with scale
factor being $c(\eta)$) is given by

$S^{FRW}_{ij}(x,x^{\prime})=\langle\psi_{i}(x)\bar{\psi}_{j}(x^{\prime})\rangle=(F(\eta)F(\eta^{\prime}))\langle\Omega_{i}(x)\bar{\Omega}_{j}(x^{\prime})\rangle=\Big{(}\frac{a(\eta)}{c(\eta)}\Big{)}^{\frac{3}{2}}\Big{(}\frac{a(\eta^{\prime})}{c(\eta^{\prime})}\Big{)}^{\frac{3}{2}}S^{dS}_{ij}(x,x^{\prime})\\\
=\frac{1}{{c^{\frac{3}{2}}(\eta_{x})c^{\frac{3}{2}}(\eta_{x^{\prime}})}}S^{flat}_{ij}(x,x^{\prime})\,.$
(34)

Similarly, we have

$R^{FRW}_{ji}(x^{\prime},x)=\langle\bar{\psi}_{j}(x^{\prime})\psi_{i}(x)\rangle=(F(\eta)F(\eta^{\prime}))\langle\bar{\Omega}_{j}(x^{\prime})\Omega_{i}(x)\rangle=\Big{(}\frac{a(\eta)}{c(\eta)}\Big{)}^{\frac{3}{2}}\Big{(}\frac{a(\eta^{\prime})}{c(\eta^{\prime})}\Big{)}^{\frac{3}{2}}R^{dS}_{ji}(x^{\prime},x)\\\
=\frac{1}{{c^{\frac{3}{2}}(\eta_{x})c^{\frac{3}{2}}(\eta_{x^{\prime}})}}R^{flat}_{ji}(x^{\prime},x)\,.$
(35)

We make use of these relations later on in this work and find that the noise
kernel for massless spinor fields in general FRW spacetimes is related to the
de Sitter noise kernel again by conformal factors.

##  III Noise kernel and point splitting

Using the expression that

$T_{\mu\nu}(x)=-\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}(x)}\,,$
(36)

we find that, for spinor fields in curved spacetimes, the stress energy tensor
Parker and Toms (2009); Shapiro (2016) is given by

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}g_{\mu\nu}\big{[}\bar{\psi}\gamma^{\lambda}\overrightarrow{\nabla}_{\lambda}\psi-\bar{\psi}\overleftarrow{\nabla}_{\lambda}\gamma^{\lambda}\psi\big{]}+\frac{i}{2}\big{[}\bar{\psi}\gamma_{(\mu}\overrightarrow{\nabla}_{\nu)}\psi-\bar{\psi}\overleftarrow{\nabla}_{(\nu}\gamma_{\mu)}\psi\big{]}+m\bar{\psi}\psi
g_{\mu\nu}$ (37) $\displaystyle=$
$\displaystyle-\frac{g_{\mu\nu}}{2}\bar{\psi}\Big{[}(i\gamma^{\lambda}\overrightarrow{\nabla}_{\lambda}-m)-(i\overleftarrow{\nabla}_{\lambda}\gamma^{\lambda}+m)\Big{]}\psi+\frac{i}{2}\bar{\psi}\big{[}\gamma_{(\mu}\overrightarrow{\nabla}_{\nu)}-\overleftarrow{\nabla}_{(\nu}\gamma_{\mu)}\big{]}\psi\,.$

Using the point splitting technique Bunch and Davies (1978), this can
succinctly be written as

$T_{\mu\nu}(x)=\lim_{x^{\prime}\to x}P_{\mu\nu
ij}(x,x^{\prime})\bar{\psi}_{i}(x)\psi_{j}(x^{\prime})\,,$ (38)

where summation over $i$ and $j$ is understood and

$P_{\mu\nu
ij}(x,x^{\prime})=-\frac{g_{\mu\nu}}{2}\Big{[}(i\gamma^{\lambda}\overrightarrow{\nabla}^{x^{\prime}}_{\lambda}-m)-(i\overleftarrow{\nabla}^{x}_{\lambda}\gamma^{\lambda}+m)\Big{]}_{ij}+\frac{i}{2}\big{[}\gamma_{(\mu}\overrightarrow{\nabla}^{x^{\prime}}_{\nu)}-\overleftarrow{\nabla}^{x}_{(\nu}\gamma_{\mu)}\big{]}_{ij}\,.$
(39)

To conclude whether the semiclassical analyses (based only on the expectation
values of the stress energy operators) alone are sufficient or not, we must
have knowledge of the behaviour of the quantum fluctuations in stress energy
tensors. In stochastic gravity paradigm Hu and Verdaguer (2008), these are
characterised by the noise kernel which is given by

$N_{abcd}(x,y)=\langle\hat{T}_{ab}(x)\hat{T}_{cd}(y)\rangle-\langle\hat{T}_{ab}(x)\rangle\langle\hat{T}_{cd}(y)\rangle\,,$
(40)

where, as usual, the quantum stress energy operator is obtained from the
classical expression by replacing the classical fields by their corresponding
operator quantities.
Using the above notations, we find that the noise kernel can be written as

$N_{abcd}(x,y)=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}P_{abij}(x,x^{\prime})P_{cdkl}(y,y^{\prime})\Big{[}\langle\hat{\bar{\psi}}_{i}(x)\hat{\psi}_{j}(x^{\prime})\hat{\bar{\psi}}_{k}(y)\hat{\psi}_{l}(y^{\prime})\rangle\\\
-\langle\hat{\bar{\psi}}_{i}(x)\hat{\psi}_{j}(x^{\prime})\rangle\langle\hat{\bar{\psi}}_{k}(y)\hat{\psi}_{l}(y^{\prime})\rangle\Big{]}\,.$
(41)

Applying Wick’s theorem to the first term, we see that the noise kernel is now
given by

$N_{abcd}(x,y)=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}P_{abij}(x,x^{\prime})P_{cdkl}(y,y^{\prime})\Big{[}\langle\hat{\bar{\psi}}_{i}(x)\hat{\psi}_{l}(y^{\prime})\rangle\langle\hat{\psi}_{j}(x^{\prime})\hat{\bar{\psi}}_{k}(y)\rangle\Big{]}\,.$
(42)

Since $\hat{\psi}$ is a linear combination of the solutions of the Dirac
equation and $\hat{\bar{\psi}}$ is a linear combination of the solutions of
the adjoint of the Dirac equation, we see that, in the stress energy operator,
the terms which correspond to the Dirac equation and its adjoint drop out.
Therefore, we are left with the following expression for the noise kernel

$N_{abcd}(x,y)=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{1}{4}\big{[}\gamma_{(a}\overrightarrow{\nabla}^{x^{\prime}}_{b)}-\overleftarrow{\nabla}^{x}_{(a}\gamma_{b)}\big{]}_{ij}\big{[}\gamma_{(c}\overrightarrow{\nabla}^{y^{\prime}}_{d)}-\overleftarrow{\nabla}^{y}_{(c}\gamma_{d)}\big{]}_{kl}S_{jk}(x^{\prime},y)S_{li}(y^{\prime},x)\\\
=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{1}{4}\Big{[}Tr\big{(}\gamma_{(a}\overrightarrow{\nabla}^{x^{\prime}}_{b)}S(x^{\prime},y)\gamma_{(c}\overrightarrow{\nabla}^{y^{\prime}}_{d)}S(y^{\prime},x)\big{)}-Tr\big{(}\gamma_{(a}\overrightarrow{\nabla}^{x^{\prime}}_{b)}S(x^{\prime},y)\overleftarrow{\nabla}^{y}_{(c}\gamma_{d)}S(y^{\prime},x)\big{)}\\\
-Tr\big{(}S(x^{\prime},y)\gamma_{(c}\overrightarrow{\nabla}^{y^{\prime}}_{d)}S(y^{\prime},x)\overleftarrow{\nabla}^{x}_{(a}\gamma_{b)}\big{)}+Tr\big{(}S(x^{\prime},y)\overleftarrow{\nabla}^{y}_{(c}\gamma_{d)}S(y^{\prime},x)\overleftarrow{\nabla}^{x}_{(a}\gamma_{b)}\big{)}\Big{]}\,.$
(43)

The analysis done in this section is independent of the spacetime that we
consider and as such holds for all types of spacetimes. In the next section,
we calculate the behaviour of the noise kernel in de Sitter spacetime while
placing the quantum spinor field in the Bunch Davies vacuum and therefore we
use the Wightmann function for this state given in the previous section. In
particular, we calculate the $(a=b=c=d=0)$ component of the noise kernel as
this can be related to an invariant quantity called the energy-energy
correlator as discussed later on in this work.

##  IV Noise kernel in de Sitter spacetime

In this section, we specialize to the case of a spinor field evolving in the
de Sitter spacetime. As mentioned in the previous section, we are interested
in calculating the $(a=b=c=d=0)$ component of the noise kernel i.e.,
$N_{0000}(x,y)$. Since we have expressed the noise kernel as a sum of
derivatives acting on a product of Wightman functions, we take the above given
expression of the Wightman function for a spinor field (in de Sitter
spacetime) placing it in the Bunch Davies vacuum. Using that
$\overrightarrow{\nabla}^{x}_{0}=\partial^{x}_{0}$, we see that555From this
section onwards, we always take $a_{x}\equiv a(\eta_{x})=-\frac{1}{H\eta_{x}}$

$N_{0000}(x,y)=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{a(\eta_{x})a(\eta_{y})}{4}\Bigg{[}\partial^{x^{\prime}}_{0}\partial^{y^{\prime}}_{0}+\partial^{x}_{0}\partial^{y}_{0}-\partial^{x^{\prime}}_{0}\partial^{y}_{0}-\partial^{x}_{0}\partial^{y^{\prime}}_{0}\Bigg{]}Tr\big{(}\Gamma_{0}S(x^{\prime},y)\Gamma_{0}S(y^{\prime},x)\big{)}\\\
=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{a_{x}a_{y}}{4}\Bigg{[}\partial^{x^{\prime}}_{0}\partial^{y^{\prime}}_{0}+\partial^{x}_{0}\partial^{y}_{0}-\partial^{x^{\prime}}_{0}\partial^{y}_{0}-\partial^{x}_{0}\partial^{y^{\prime}}_{0}\Bigg{]}\frac{H^{4}}{\sqrt{a_{x}a_{x^{\prime}}a_{y}a_{y^{\prime}}}}\\\
Tr\Bigg{(}\Gamma_{0}\Big{[}i\Gamma^{\lambda}\partial^{x^{\prime}}_{\lambda}+i\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}\Gamma^{0}+a_{x^{\prime}}m\Big{]}\Big{[}\sum_{\epsilon=\pm}S_{\epsilon}(x^{\prime},y)\frac{1+\epsilon\Gamma^{0}}{2}\Big{]}\\\
\Gamma_{0}\Big{[}i\Gamma^{\sigma}\partial^{y^{\prime}}_{\sigma}+i\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}\Gamma^{0}+a_{y^{\prime}}m\Big{]}\Big{[}\sum_{\epsilon=\pm}S_{\epsilon}(y^{\prime},x)\frac{1+\epsilon\Gamma^{0}}{2}\Big{]}\Bigg{)}\,.$
(44)

It can be easily shown that the factor
$\frac{H^{4}}{\sqrt{a_{x}a_{x^{\prime}}a_{y}a_{y^{\prime}}}}$, after having
been operated by all the derivative operators, simply come in front of the
bigger square brackets (which contain the derivative operators) as it is and
in the above limits cancel out the already present $a_{x}a_{y}$ factor.
Finally, we obtain

$N_{0000}(x,y)=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{H^{4}}{4}\Bigg{[}\partial^{x^{\prime}}_{0}\partial^{y^{\prime}}_{0}+\partial^{x}_{0}\partial^{y}_{0}-\partial^{x^{\prime}}_{0}\partial^{y}_{0}-\partial^{x}_{0}\partial^{y^{\prime}}_{0}\Bigg{]}\\\
Tr\Bigg{(}\Gamma_{0}\Big{[}i\Gamma^{\lambda}\partial^{x^{\prime}}_{\lambda}+i\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}\Gamma^{0}+a_{x^{\prime}}m\Big{]}\Big{[}\sum_{\epsilon=\pm}S_{\epsilon}(x^{\prime},y)\frac{1+\epsilon\Gamma^{0}}{2}\Big{]}\\\
\Gamma_{0}\Big{[}i\Gamma^{\sigma}\partial^{y^{\prime}}_{\sigma}+i\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}\Gamma^{0}+a_{y^{\prime}}m\Big{]}\Big{[}\sum_{\epsilon=\pm}S_{\epsilon}(y^{\prime},x)\frac{1+\epsilon\Gamma^{0}}{2}\Big{]}\Bigg{)}\,.$
(45)

Thus, we see that, in the expression of the noise kernel, we have traces of
the Dirac matrices. Therefore, we can use the well known properties Parker and
Toms (2009) of these traces to proceed further with our calculations.

###  IV.1 “Gammatronics”

Using the cyclic property of traces, we can write the noise kernel expression
in the following expanded form

$N_{0000}(x,y)=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{H^{4}}{4}\Bigg{[}\partial^{x^{\prime}}_{0}\partial^{y^{\prime}}_{0}+\partial^{x}_{0}\partial^{y}_{0}-\partial^{x^{\prime}}_{0}\partial^{y}_{0}-\partial^{x}_{0}\partial^{y^{\prime}}_{0}\Bigg{]}\\\
\sum_{\epsilon=\pm}\sum_{\delta=\pm}Tr\Bigg{(}\Gamma_{0}\Big{[}i\Gamma^{\lambda}\partial^{x^{\prime}}_{\lambda}+i\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}\Gamma^{0}+a_{x^{\prime}}m\Big{]}\Big{[}\frac{1+\epsilon\Gamma^{0}}{2}\Big{]}\\\
\Gamma_{0}\Big{[}i\Gamma^{\sigma}\partial^{y^{\prime}}_{\sigma}+i\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}\Gamma^{0}+a_{y^{\prime}}m\Big{]}\Big{[}\frac{1+\delta\Gamma^{0}}{2}\Big{]}\Bigg{)}S_{\epsilon}(x^{\prime},y)S_{\delta}(y^{\prime},x)\\\
=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{H^{4}}{4}\Bigg{[}\partial^{x^{\prime}}_{0}\partial^{y^{\prime}}_{0}+\partial^{x}_{0}\partial^{y}_{0}-\partial^{x^{\prime}}_{0}\partial^{y}_{0}-\partial^{x}_{0}\partial^{y^{\prime}}_{0}\Bigg{]}\sum_{\epsilon=\pm}\sum_{\delta=\pm}Tr\Bigg{(}i^{2}\Gamma^{\lambda}M\Gamma^{\sigma}P\partial_{\lambda}^{x^{\prime}}\partial_{\sigma}^{y^{\prime}}\\\
+i^{2}\Gamma^{\lambda}M\Gamma^{0}P\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}\partial^{x^{\prime}}_{\lambda}+ia_{y^{\prime}}m\Gamma^{\lambda}MP\partial^{x^{\prime}}_{\lambda}+i^{2}\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}\Gamma^{0}M\Gamma^{\sigma}P\partial^{y^{\prime}}_{\sigma}+i^{2}\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}\Gamma^{0}M\Gamma^{0}P\\\
+im\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}a_{y^{\prime}}\Gamma^{0}MP+ia_{x^{\prime}}mM\Gamma^{\sigma}P\partial^{y^{\prime}}_{\sigma}+ia_{x^{\prime}}m\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}M\Gamma^{0}P+a_{x^{\prime}}a_{y^{\prime}}m^{2}MP\Bigg{)}S_{\epsilon}(x^{\prime},y)S_{\delta}(y^{\prime},x)$
(46)

where $M=\Big{[}\frac{\Gamma^{0}+\epsilon}{2}\Big{]}$ and
$P=\Big{[}\frac{\Gamma^{0}+\delta}{2}\Big{]}$.
Employing the known properties of the traces of the Gamma matrices, we find
that the above expression reduces to

$N_{0000}(x,y)=-\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{2H^{4}}{4}\Bigg{[}\partial^{x^{\prime}}_{0}\partial^{y^{\prime}}_{0}+\partial^{x}_{0}\partial^{y}_{0}-\partial^{x^{\prime}}_{0}\partial^{y}_{0}-\partial^{x}_{0}\partial^{y^{\prime}}_{0}\Bigg{]}\\\
\Bigg{[}\sum_{\epsilon=\pm}\Bigg{(}\partial_{0}^{x^{\prime}}\partial_{0}^{y^{\prime}}+\Big{(}\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}-\epsilon
ima_{x^{\prime}}\Big{)}\partial^{y^{\prime}}_{0}+\Big{(}\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}-\epsilon
ima_{y^{\prime}}\Big{)}\partial^{x^{\prime}}_{0}+\Big{(}\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}-\epsilon
ima_{y^{\prime}}\Big{)}\Big{(}\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}-\epsilon
ima_{x^{\prime}}\Big{)}\Bigg{)}S_{\epsilon}(x^{\prime},y)S_{\epsilon}(y^{\prime},x)\\\
+\delta^{kl}\partial^{x^{\prime}}_{k}\partial^{y^{\prime}}_{l}\Big{(}S_{+}(x^{\prime},y)S_{-}(y^{\prime},x)+S_{-}(x^{\prime},y)S_{+}(y^{\prime},x)\Big{)}\Bigg{]}\,.$
(47)

###  IV.2 Behaviour on equal time sheets

In order to study how the energy densities for spatially separated points are
correlated with each other, we specialize to the case where
$\eta_{x}=\eta_{y}$. But before we consider this case, we have to perform all
the derivatives above. While performing these calculations, we make use of the
fact that $S_{\pm}(x,x^{\prime})=S_{\pm}(Z(x,x^{\prime}))$ i.e., it is a
function of the invariant distance $Z$. We should take extra care to make sure
that we take all the limits only after performing all the derivatives. Using
the formulae given in Appendix A, we find that, on equal time sheets, we have

$-\frac{2}{H^{4}}N_{0000}(x,y)=\big{(}S^{\prime\prime}_{+}S^{\prime}_{-}+(-\leftrightarrow+)\big{)}\Big{[}-\frac{(\Delta\vec{x})^{4}}{8\eta^{8}}-\frac{(\Delta\vec{x})^{2}}{4\eta^{6}}\Big{]}+\big{(}S^{\prime\prime}_{+}S^{\prime\prime}_{-})\Big{[}-\frac{(\Delta\vec{x})^{6}}{16\eta^{10}}\Big{]}\\\
+\big{(}S^{\prime\prime\prime}_{+}S^{\prime}_{-}+(-\leftrightarrow+)\big{)}\Big{[}\frac{(\Delta\vec{x})^{6}}{32\eta^{10}}\Big{]}+\sum_{\epsilon=\pm}\Big{(}1-\frac{\epsilon
im}{H}\Big{)}\Bigg{[}\frac{(S^{\prime}_{\epsilon})^{2}}{4}\Big{[}\frac{(\Delta\vec{x})^{4}}{2\eta^{8}}-\frac{(\Delta\vec{x})^{2}}{\eta^{6}}\Big{]}+\frac{2}{\eta^{4}}S_{\epsilon}S^{\prime}_{\epsilon}+S_{\epsilon}S^{\prime\prime}_{\epsilon}\Big{[}-\frac{3(\Delta\vec{x})^{4}}{8\eta^{8}}-\frac{(\Delta\vec{x})^{2}}{4\eta^{6}}\Big{]}\\\
+(S_{\epsilon}S^{\prime\prime\prime}_{\epsilon}-S^{\prime}_{\epsilon}S^{\prime\prime}_{\epsilon})\frac{(\Delta\vec{x})^{6}}{32\eta^{10}}\Bigg{]}+\Big{(}1-\frac{\epsilon
im}{H}\Big{)}^{2}\Bigg{[}\frac{(S^{\prime}_{\epsilon})^{2}}{4}\Big{[}\frac{(\Delta\vec{x})^{4}}{2\eta^{8}}\Big{]}+\frac{S_{\epsilon}S_{\epsilon}}{\eta^{4}}+S_{\epsilon}S^{\prime}_{\epsilon}\Big{[}\frac{(\Delta\vec{x})^{2}}{2\eta^{6}}+\frac{1}{\eta^{4}}\Big{]}-S_{\epsilon}S^{\prime\prime}_{\epsilon}\frac{(\Delta\vec{x})^{4}}{8\eta^{8}}\Bigg{]}\\\
+\Bigg{[}\frac{(S^{\prime}_{\epsilon})^{2}}{4}\Big{[}\frac{(\Delta\vec{x})^{4}}{4\eta^{8}}-2\frac{(\Delta\vec{x})^{2}}{\eta^{6}}+\frac{2}{\eta^{4}}\Big{]}+\frac{S^{\prime}_{\epsilon}S^{\prime\prime}_{\epsilon}}{8}\frac{(\Delta\vec{x})^{4}}{4\eta^{6}}\Big{[}\frac{2}{\eta^{2}}+\frac{(\Delta\vec{x})^{2}}{\eta^{4}}\Big{]}+\frac{S^{\prime\prime}_{\epsilon}S^{\prime\prime}_{\epsilon}}{16}\frac{(\Delta\vec{x})^{8}}{8\eta^{12}}-\frac{S^{\prime}_{\epsilon}S^{\prime\prime\prime}_{\epsilon}}{16}\frac{(\Delta\vec{x})^{8}}{8\eta^{12}}\Bigg{]}\,.$
(48)

Now we are interested in seeing the behaviour of this noise kernel for late
time limits i.e., $\eta\to 0$ limit. Recalling that the
$S_{\epsilon}^{\prime}s$ are just Hypergeometric functions (see section $2$)
and using the fact that their derivatives are again Hypergeometric functions
DLMF  and also making use of their asymptotic behaviour, we find that the
leading order behaviour of the noise kernel is given by

$N_{0000}(x,y)=\frac{2H^{4}\eta^{2}}{\pi^{3}(\Delta\vec{x})^{6}}\frac{(1+\frac{m^{2}}{H^{2}})(\frac{m^{3}}{H^{3}})}{\mathrm{sinh}(\frac{2\pi
m}{H})}+O(\eta^{4}).$ (49)

Thus, from the above expression, we see that, irrespective of how massive or
light the fermionic fields are, the considered component of the noise kernel
vanishes in the late time limit i.e., $\eta\to 0$ limit. In that sense, the
decay of the noise kernel component for a spinor field in de Sitter spacetime
is a universal phenomenon independent of the mass of the field. This is in
contrast to the behaviour of the noise kernel for quantum scalar fields in de
Sitter spacetime where the considered noise kernel component shows a
transition from vanishing to divergent behaviour as mass is varied from
$[0,\frac{3}{2}]$ with ‘critical mass’ value being $\frac{m^{2}}{H^{2}}=2$
(Refer Dhanuka and Lochan (2020)). This implies that, if we start with two
spatially separated points on a constant time sheet in the past, then the
corresponding local stress energy tensors (physically speaking, energy
momentum content) don’t develop any correlations in the far future. In fact,
more appropriately, the correlations between the quantum stress energy tensor
at different points get washed away as the de Sitter spacetime evolves to late
time limit. To present this conclusion in a more illuminating manner, we
notice that at the start of the inflation i.e., in the $\eta\to-\infty$ limit,
all points, on this very early constant time slice of the de Sitter spacetime,
are separated by zero physical distances and hence, the quantum field at
different coordinate distances (but with zero physical separations) have
maximum correlations. However, as the de Sitter spacetime evolves, the
physical distances between spatially separated points increase and
intuitively, we expect that the correlations between the corresponding local
stress energy tensors (for that matter, between the field operators themselves
or any other functions of the field operators) should decrease. The above
obtained results suggest that the quantum dynamics of the spinor fields in de
Sitter spacetime is not so that it can overcome the diminishing effect of
increasing physical distances and we have an overall vanishing of the noise
kernel in the late time limit. But we see (from the behaviour of the noise
kernel of scalar fields Dhanuka and Lochan (2020)) that, for quantum scalar
fields in de Sitter spacetime, there are mass ranges for which their dyanmics
can overcome the effect of the classical expansion of the de Sitter spacetime
and we obtain divergent noise kernel for these cases. Another important yet
very intuitive property of the noise kernel is that the vanishing of the noise
kernel is inversely proportional to the coordinate distances between spatially
separated points i.e., the lesser the coordinate distances are (between the
spatially separated points), the later (in the evolution history of the de
Sitter spacetime) the correlations decay.
As the noise kernel is a (bi-)tensor quantity and hence is expressed
differently in different coordinate systems, let us express the above obtained
results in terms of a coordinate invariant quantity i.e., the energy-energy
correlator (Refer Dhanuka and Lochan (2020)). To define this quantity, we
consider a comoving observer (i.e., one whose spatial coordinates are fixed)
for which the normalized tangent vector field along its trajectory is given by
$t^{\alpha}(\eta,\vec{x})=(\frac{1}{a(\eta)},0,0,0)$ (and hence
$g_{\alpha\beta}t^{\alpha}t^{\beta}=-1$). One then defines the energy density
at any point by the contraction $T_{\alpha\beta}(x)t^{\alpha}(x)t^{\beta}(x)$.
Using this, we find that the energy-energy correlator, between the spacetime
points $x$ and $y$, is given by

$\frac{N_{0000}(x,y)}{a^{2}(\eta_{x})a^{2}(\eta_{y})}\,.$ (50)

From the expression of the noise kernel on constant time slices, we find that
the energy-energy correlator, on late constant time sheets, has leading order
behaviour of the form
$\frac{2H^{8}\eta^{6}}{\pi^{3}(\Delta\vec{x})^{6}}\frac{(1+\frac{m^{2}}{H^{2}})(\frac{m^{3}}{H^{3}})}{\mathrm{sinh}(\frac{2\pi
m}{H})}$. Although there is an extra factor of $(H\eta)^{4}$ in front of the
energy correlator compared to the noise kernel and hence leading to an even
faster decay at late times, the qualitative remarks, made above for the noise
kernel, hold equally good for this invariant correlator as well. Let us now
move on to discuss similar studies but now in general FRW settings and for
massless spinor fields only.

##  V Behaviour of the noise kernel and the energy-energy correlator for
massless spinors in general FRW spacetimes

In this section, we analyse the behaviour of the noise kernel and the energy-
energy correlator for massless spinor fields in general FRW spacetimes. To
carry out this task, we make use of the equivalence of massless spinors in FRW
spacetimes with massless spinors in de Sitter spacetime to relate the
corresponding noise kernels. This equivalence between massless spinors in FRW
and de Sitter spacetimes is analogous to the equivalence between massless
scalar fields in FRW spacetimes with massive scalar fields in de Sitter
spacetime established in Lochan  _et al._ (2018). Now recalling that a
massless spinor field, $\psi$, in an FRW spacetime (with scale factor,
$c(\eta)$) is related to a massless spinor field, $\Omega$, in de Sitter
spacetime by the relation $\psi(x)$ =
$\Big{(}\frac{a(\eta)}{c(\eta)}\Big{)}^{\frac{3}{2}}\Omega(x)$ (see subsection
$2.2$), we find that the Wightmann functions in the corresponding spacetimes
are related by

$S^{FRW}_{ij}(x,y)=\Big{(}\frac{a(\eta)}{c(\eta)}\Big{)}^{\frac{3}{2}}\Big{(}\frac{a(\eta^{\prime})}{c(\eta^{\prime})}\Big{)}^{\frac{3}{2}}S^{dS}_{ij}(x,y)$
(51)

and

$R^{FRW}_{ji}(y,x)=\Big{(}\frac{a(\eta)}{c(\eta)}\Big{)}^{\frac{3}{2}}\Big{(}\frac{a(\eta^{\prime})}{c(\eta^{\prime})}\Big{)}^{\frac{3}{2}}R^{dS}_{ji}(y,x)\,,$
(52)

where $a(\eta)=-\frac{1}{H\eta}.$ Using these expressions in equation (43), we
see that the considered component of the FRW noise kernel is given by666Here,
like in the case of de Sitter scale factor ($a_{x}$), $c_{x}$ stands for
$c(\eta_{x})$.

$N^{FRW}_{0000}(x,y)=\lim_{x^{\prime}\to x}\lim_{y^{\prime}\to
y}\frac{c_{x}c_{y}}{4}\Bigg{[}\partial^{x^{\prime}}_{0}\partial^{y^{\prime}}_{0}+\partial^{x}_{0}\partial^{y}_{0}-\partial^{x^{\prime}}_{0}\partial^{y}_{0}-\partial^{x}_{0}\partial^{y^{\prime}}_{0}\Bigg{]}H^{4}\frac{(a_{x}a_{x^{\prime}}a_{y}a_{y^{\prime}})}{(c_{x}c_{x^{\prime}}c_{y}c_{y^{\prime}})^{\frac{3}{2}}}\\\
Tr\Bigg{(}\Gamma_{0}\Big{[}i\Gamma^{\lambda}\partial^{x^{\prime}}_{\lambda}+i\frac{a^{\prime}_{x^{\prime}}}{a_{x^{\prime}}}\Gamma^{0}\Big{]}\Big{[}\sum_{\epsilon=\pm}S_{\epsilon}(x^{\prime},y)\frac{1+\epsilon\Gamma^{0}}{2}\Big{]}\Gamma_{0}\Big{[}i\Gamma^{\sigma}\partial^{y^{\prime}}_{\sigma}+i\frac{a^{\prime}_{y^{\prime}}}{a_{y^{\prime}}}\Gamma^{0}\Big{]}\Big{[}\sum_{\epsilon=\pm}S_{\epsilon}(y^{\prime},x)\frac{1+\epsilon\Gamma^{0}}{2}\Big{]}\Bigg{)}\,.$
(53)

After operating all the derivative operators (present in the bigger square
brackets) on the
$H^{4}\frac{(a_{x}a_{x^{\prime}}a_{y}a_{y^{\prime}})}{(c_{x}c_{x^{\prime}}c_{y}c_{y^{\prime}})^{\frac{3}{2}}}$
term, we find that this term comes in front of the bigger square brackets as
it is. Therefore, we can conclude that the massless spinor field noise kernel
in general FRW spacetimes is just $\frac{(a_{x}a_{y})^{2}}{(c_{x}c_{y})^{2}}$
factor multiplying the de Sitter noise kernel expression of the previous
section along with the fact that we also have $m=0$ and hence
$S_{+}(Z(x,y))=S_{-}(Z(x,y))=\frac{1}{16\pi^{2}(1-Z(x,y))}.$ Like in the
previous section, we consider the $(a=b=c=d=0)$ component of the noise kernel
on equal time sheets i.e., $\eta_{x}=\eta_{y}=\eta$, which, in this case, is
given by

$N^{FRW}_{0000}(x,y)=\frac{a_{x}^{4}}{c_{x}^{4}}\frac{3H^{4}\eta^{4}}{2\pi^{4}(\Delta\vec{x})^{8}}=\frac{3}{2\pi^{4}(\Delta\vec{x})^{8}}c_{x}^{-4}\,.$
(54)

Thus, we see that the considered component of the noise kernel of a massless
spinor field in an FRW spacetime (with scale factor being $c(\eta)$) behaves
in a manner that is opposite to the behaviour of the scale factor. This
implies that the correlations, between (massless) spinor matter located at
spatially separated points on constant time sheets, decay during the expanding
phases of the universe while they grow during the contracting phases of the
universe. We also notice that the behaviour of the noise kernel component is
monotonic (non-monotonic) if the scale factor changes monotonically (non-
monotonically). By monotonic growth (or decay) of the noise kernel, we simply
mean that the value of the noise kernel for spatially separated points on any
constant time sheet is always more (or less) than its values on earlier time-
sheets. We now specialize to the cosmologically interesting power-law type
expanding FRW spacetimes i.e., $c(\eta)\propto\eta^{-q}$. For these cases, we
have

$N^{FRW}_{0000}(x,y)=\frac{3}{2\pi^{4}(\Delta\vec{x})^{8}}(H\eta)^{4q}\,.$
(55)

Since $\eta\to 0$ limit is the late time limit for spacetimes with positive
values of q, we notice that, for these spacetimes, the considered component of
the noise kernel vanishes in this limit. For these spacetimes i.e.,
$q\in(0,\infty)$, the equation of state parameter, $w$, lies in the range
$(-\infty,-\frac{1}{3})$ and these are acceleratingly expanding spacetimes.
These spacetimes include both phantom cosmologies (see Johri (2004)) i.e.,
$w\in(-\infty,-1)$, as well as quintessence cosmologies i.e.,
$w\in(-1,-\frac{1}{3})$. In fact, the most accurate data till date puts the
present day dark-energy driven universe in this regime i.e., $\omega=-1.03\pm
0.03$ Aghanim  _et al._ (2020). For spacetimes with negative values of $q$,
the late time limit is the $\eta\to\infty$ limit and we see that, for these
spacetimes also, the considered component of the noise kernel vanishes in the
late time limit. For these spacetimes i.e., $q\in(-\infty,0)$, the equation of
state parameter, $w$, lies in the interval $(-\frac{1}{3},\infty)$. These
spacetimes are also expanding spacetimes but, for them, the rate of expansion
decreases with time i.e., the expansion is decelerating. These spacetimes
include cosmologically interesting epochs like radiation dominated and dust
dominated cases as well. These considerations imply that the vanishing of the
noise kernel component, in the scaling factor going to large value limit, is
universal for all power-law type expanding FRW cosmologies and we know that
this should be the case for expanding spacetimes as we have already seen that
the behaviour of the considered noise kernel component is opposite compared to
the behaviour of the scaling factor. As argued in the de Sitter case, this
implies that the quantum dynamics of massless spinor fields in power-law type
FRW spacetimes is not such that it can overcome the effects of increasing
physical distances between spatially separated points. Thus, we see that the
quantum correlations between local stress-energy operators (at different
points on constant time slices) are suppressed by the increasing physical
distances in these expanding spacetimes.
Like in the previous section, we see that the considered noise kernel
component may change its form under coordinate transformations. Therefore, we
look for the coordinate independent quantity of the previous section i.e., the
energy-energy correlator which is given by (on equal time sheets)

$\frac{N^{FRW}_{0000}(x,y)}{c^{2}(\eta_{x})c^{2}(\eta_{y})}\,=c_{x}^{-8}\frac{3}{2\pi^{4}(\Delta\vec{x})^{8}}.$
(56)

This implies that, like the considered component of the noise kernel, the
behaviour of the energy-energy correlator is also opposite to the behaviour of
the scale factor (of the FRW spacetime). In particular, it vanishes for all
power-law type expanding FRW spacetimes. All the remarks made above for the
$(a=b=c=d=0)$ component of the noise kernel hold for the energy-energy
correlator as well except for the fact that the energy-energy correlator
depends on even more negative power of the scale factor compared to the
considered noise kernel component.
For the special case of power-law type FRW spacetimes, one can compare these
results for massless spinor fields with the corresponding results for massless
scalar fields Dhanuka and Lochan (2020). Whereas in the case of massless
spinor fields in power-law FRW spacetimes, there are no 2nd order quantum
corrections, Dhanuka and Lochan (2020) shows that, in case of massless scalar
fields evolving in these FRW spacetimes, there are a number of these power-law
FRW spacetimes for which the 2nd order quantum corrections coming from the
noise kernel are significant e.g., for
$\omega\in(0,-\frac{1}{3})\cup(-\frac{1}{3},-1)$ universes there are large
quantum fluctuations. This implies that, for scenarios where we have both
massless spinor and massless scalar fields present in power-law FRW
spacetimes, the results, obtained by performing only the (first order)
semiclassical analysis, get corrections only from the scalar sector but not
from the spinor sector.

##  VI Summary and conclusions

In this work, we have tried to understand the dynamics of arbitrarily massive
spinor fields living on de Sitter spacetime. In particular, we have looked at
the correlations between local stress energy operators in the late time limit
while assuming the Bunch Davies initial conditions for the spinor fields at
early times. We have also considered the behaviour of these correlations for
masssless spinor fields in general FRW spacetimes. In order to carry out this
analysis, we employed a conformal mapping from massless spinor fields in a
general FRW spacetime to massless spinor fields in de Sitter spacetime. This
mapping helps us place massless spinor fields (in FRW spacetimes) in Bunch
Davies like vacua. Below, we give a brief summary of the results obtained in
this work:

  * •

Behaviour of the noise kernel for spinor fields in de Sitter spacetime: We
consider the $(a=b=c=d=0)$ component of the noise kernel for a spinor field in
the de Sitter spacetime. We find that, in the late time i.e., $\eta\to 0$
limit, the leading order behaviour tells us that the considered component of
the noise kernel (on constant time slices) decays. From this, we infer that
the quantum dynamics of the spinor fields is washed away by the classical
accelerated expansion of the de Sitter spacetime. This dominance of the
classical dynamics of the background over the quantum dynamics of the spinor
field is universal, in the sense that it occurs irrespective of the mass of
the field. This behaviour of the noise kernel for spinor fields is in stark
contrast to the behaviour of the noise kernel for scalar fields (in de Sitter
spacetime) Dhanuka and Lochan (2020) where the noise kernel, considered at
late times, decays for scalar fields with
($2<\frac{m^{2}}{H^{2}}<\frac{9}{4}$) but diverges for scalar fields with
($0<\frac{m^{2}}{H^{2}}<2$). For spinor fields, we observe that the invariant
energy-energy correlator also decays. In fact, the decay is faster for this
correlator. We also find that these correlators show the usual dependence on
the coordinate distance for points on equal time sheets i.e., that these
correlations become less and less important as the comoving distances
increase.

  * •

Behaviour of the noise kernel for spinor fields in FRW spacetimes: Using the
equivalence that exists between massless spinor fields in FRW spacetimes and
massless spinor fields in de Sitter spacetime, we consider the behaviour of
the $(a=b=c=d=0)$ component of the noise kernel for massless spinor fields in
FRW spacetimes, again on constant time slices. In this setting, we find that
the considered component of the noise kernel behaves in a manner that is
opposite to the behaviour of the scale factor i.e., it decays (or grows) with
expanding (or contracting) scale factors. This implies that the noise kernel
component always decays for power-law type expanding FRW spacetimes. These
results, for power-law type expanding FRW spacetimes, are in stark contrast to
the behaviour of the considered noise kernel component for quantum scalar
fields for which the noise kernel does not always decay Dhanuka and Lochan
(2020). Like the noise kernel component, the energy-energy correlator of the
massless spinor fields is also inversely related to the scale factor.

From the above results, one can conclude that, for systems involving massless
spinor fields in power-law type expanding FRW spacetimes and arbitrarily
massive spinor fields in de Sitter spacetime, the first order quantum
treatments (based solely upon considering quantum averages of the stress
energy operator) do not get significant corrections from the second order
quantum effects coming from the noise kernel and hence, inferences made from a
first order analysis will remain robust against the considered quantum
fluctuations. It is important to emphasize that these conclusions have been
arrived at for spinor fields by placing them in the Bunch Davies like vacua.
Since the conclusions of field theories in curved spacetimes are markedly
different for different vacua (Collins (2005); Goldstein and Lowe (2003);
Collins and Martin (2004)), it would be interesting to investigate how the
noise kernel for spinor fields behaves for other de Sitter spacetime vacua
like the fermionic alpha-vacua Collins (2005) etc. For general FRW spacetimes,
we can again use the conformal invariance of massless spinor fields and carry
out the analysis for the FRW vacua corresponding to other de Sitter space-time
vacua.

## Acknowledgments

AD would like to thank Kinjalk Lochan for carefully reading the manuscript of
this paper and giving useful suggestions. AD would also like to acknowledge
the financial support from University Grants Commission, Government of India,
in the form of Senior Research Fellowship (UGC-CSIR JRF/Dec2016/510944).

## Appendix A

Here we enlist some important formulae that are used in evaluating noise
kernel in the main portion of the draft

$\displaystyle Z(x^{\prime},y)$ $\displaystyle=$ $\displaystyle
1+\frac{(\eta_{x^{\prime}}-\eta_{y})^{2}-(\vec{x}^{\prime}-\vec{y})^{2}}{4\eta_{x^{\prime}}\eta_{y}}$
(57) $\displaystyle\frac{\partial Z(x^{\prime},y)}{\partial x^{\prime\mu}}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\Bigg{[}\frac{\Delta
s^{2}}{2\eta_{x^{\prime}}^{2}\eta_{y}}\delta_{\mu
0}-\frac{(x^{\prime}-y)_{\mu}}{\eta_{x^{\prime}}\eta_{y}}\Bigg{]}$ (58)
$\displaystyle\frac{\partial Z(x^{\prime},y)}{\partial y^{\nu}}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\Bigg{[}\frac{\Delta
s^{2}}{2\eta_{x^{\prime}}\eta_{y}^{2}}\delta_{\nu
0}+\frac{(x^{\prime}-y)_{\nu}}{\eta_{x^{\prime}}\eta_{y}}\Bigg{]}$ (59)
$\displaystyle\frac{\partial^{2}Z(x^{\prime},y)}{\partial
x^{\prime\nu}\partial x^{\prime\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}-\frac{\Delta
s^{2}}{\eta_{x^{\prime}}^{3}\eta_{y}}\delta_{\mu 0}\delta_{\nu
0}+\frac{(x^{\prime}-y)_{\nu}}{\eta_{x^{\prime}}^{2}\eta_{y}}\delta_{\mu
0}+\frac{(x^{\prime}-y)_{\mu}}{\eta_{x^{\prime}}^{2}\eta_{y}}\delta_{\nu
0}-\frac{\eta_{\mu\nu}}{\eta_{x^{\prime}}\eta_{y}}\Bigg{]}$ (60)
$\displaystyle\frac{\partial^{2}Z(x^{\prime},y)}{\partial y^{\nu}\partial
y^{\mu}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\Bigg{[}-\frac{\Delta
s^{2}}{\eta_{x^{\prime}}\eta_{y}^{3}}\delta_{\mu 0}\delta_{\nu
0}-\frac{(x^{\prime}-y)_{\nu}}{\eta_{x^{\prime}}\eta_{y}^{2}}\delta_{\mu
0}-\frac{(x^{\prime}-y)_{\mu}}{\eta_{x^{\prime}}\eta_{y}^{2}}\delta_{\nu
0}-\frac{\eta_{\mu\nu}}{\eta_{x^{\prime}}\eta_{y}}\Bigg{]}$ (61)
$\displaystyle\frac{\partial^{2}Z(x^{\prime},y)}{\partial y^{\nu}\partial
x^{\prime\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}-\frac{\Delta
s^{2}}{2\eta_{x^{\prime}}^{2}\eta_{y}^{2}}\delta_{\mu 0}\delta_{\nu
0}-\frac{(x^{\prime}-y)_{\nu}}{\eta_{x^{\prime}}^{2}\eta_{y}}\delta_{\mu
0}+\frac{(x^{\prime}-y)_{\mu}}{\eta_{x^{\prime}}\eta_{y}^{2}}\delta_{\nu
0}+\frac{\eta_{\mu\nu}}{\eta_{x^{\prime}}\eta_{y}}\Bigg{]}$ (62)
$\displaystyle\frac{\partial^{3}Z(x^{\prime},y)}{\partial y^{\rho}\partial
x^{\prime\nu}\partial x^{\prime\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}\frac{\Delta
s^{2}}{\eta_{x^{\prime}}^{3}\eta_{y}^{2}}\delta_{\mu 0}\delta_{\nu
0}\delta_{\rho
0}-\frac{(x^{\prime}-y)_{\nu}}{\eta_{x^{\prime}}^{2}\eta_{y}^{2}}\delta_{\mu
0}\delta_{\rho
0}-\frac{(x^{\prime}-y)_{\mu}}{\eta_{x^{\prime}}^{2}\eta_{y}^{2}}\delta_{\nu
0}\delta_{\rho 0}$ (64)
$\displaystyle+\frac{\eta_{\mu\nu}}{\eta_{x^{\prime}}\eta_{y}^{2}}\delta_{\rho
0}-\frac{\eta_{\mu\rho}}{\eta_{x^{\prime}}^{2}\eta_{y}}\delta_{\nu
0}-\frac{\eta_{\nu\rho}}{\eta_{x^{\prime}}^{2}\eta_{y}}\delta_{\mu
0}+2\frac{(x^{\prime}-y)_{\rho}}{\eta_{x^{\prime}}^{3}\eta_{y}}\delta_{\mu
0}\delta_{\nu 0}\Bigg{]}$
$\displaystyle\frac{\partial^{3}Z(x^{\prime},y)}{\partial
x^{\prime\rho}\partial y^{\nu}\partial y^{\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}\frac{\Delta
s^{2}}{\eta_{x^{\prime}}^{2}\eta_{y}^{3}}\delta_{\mu 0}\delta_{\nu
0}\delta_{\rho
0}+\frac{(x^{\prime}-y)_{\nu}}{\eta_{x^{\prime}}^{2}\eta_{y}^{2}}\delta_{\mu
0}\delta_{\rho
0}+\frac{(x^{\prime}-y)_{\mu}}{\eta_{x^{\prime}}^{2}\eta_{y}^{2}}\delta_{\nu
0}\delta_{\rho 0}$ (66)
$\displaystyle+\frac{\eta_{\mu\nu}}{\eta_{x^{\prime}}^{2}\eta_{y}}\delta_{\rho
0}-\frac{\eta_{\mu\rho}}{\eta_{x^{\prime}}\eta_{y}^{2}}\delta_{\nu
0}-\frac{\eta_{\nu\rho}}{\eta_{x^{\prime}}\eta_{y}^{2}}\delta_{\mu
0}-2\frac{(x^{\prime}-y)_{\rho}}{\eta_{x^{\prime}}\eta_{y}^{3}}\delta_{\mu
0}\delta_{\nu 0}\Bigg{]}$

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