•

# Gravitational Fermion Creation During An Anisotropic Phase Of Cosmological
Expansion

Amit Bhoonah<EMAIL_ADDRESS>The McDonald Institute and Department of
Physics, Engineering Physics, and Astronomy, Queen’s University, Kingston,
Ontario, K7L 2S8, Canada  Eidgenössische Technische Hochschule Zürich,
Rämistrasse 101, 8092 Zürich, Switzerland

###### Abstract

The free Dirac equation is solved in a Bianchi Type I space-time, which
represents a homogeneous but anisotropic universe, to show the creation of
fermionic particles. It is found that unlike in the isotropic case, massless
fermion production is possible. An estimate of the energy density of massless
particles created during an early anisotropic phase of cosmological expansion
is shown to cause substantial back-reaction on the gravitational field. The
potential relevance to dark matter particle production, primordial
magnetogenesis, and early universe cosmology is discussed briefly.

##  I Introduction

The quantisation of matter fields in curved space-time is a rich subject that
has been treated extensively since the late 1960s. In semi-classical gravity,
or Quantum Field Theory (QFT) in curved space-time as it is commonly called,
the gravitational field is classical and treated using General Relativity
while fermionic, gauge, and scalar fields (collectively called matter fields
in this work) are quantized and treated as a QFT. The curved background gives
rise to a rich variety of phenomenology not present in flat space-time, such
as the creation of particles by the gravitational field. This is a consequence
of energy not being conserved in the most general gravitational field. There
is no globally defined vacuum for the entire space-time in the manner one is
familiar with from the treatment of free fields in QFT: a state of lowest
energy containing no particles. Two different vacuua at two different space-
time points will only be empty with respect to each other in a limited number
of scenarios, representing a major difficulty in the mathematical setup of the
problem. A typical approach in the literature has been to assume that the
space-time under consideration is asymptotically flat and define vacuum states
in those regions. These $in$ and $out$ vacuua are inequivalent and each one
contains particles relative to the other - the gravitational field has created
particles.

This phenomenon of particle creation by gravitational fields was first shown
by Parker in parkerparticlecreation , parkerspin0 , and parkerspin12 . He and
several others afterwards (see for example tagirov ) studied the quantisation
of matter fields in isotropic and homogeneously expanding universes described
by the Friedmann-Le Maître-Robertson-Walker (FLRW) metric. It was found, as
might seem intuitive, that the number density of created particles decreases
at higher masses. One therefore expects that low or zero mass particle
production occurs at a rate quite high compared to the more massive case,
motivating the study of gravitational particle creation in the massless limit.

However, a major result in FLRW universes is that massless fermions or vector
bosons - such as photons - are not created gravitationally. The reason has to
do with conformal invariance. For particles of non-zero spin the mass term is
the only one that breaks conformal symmetry. Therefore, in the massless
fermion or vector boson limit a conformal transformation can be used to recast
the equations of motion in an FLRW space-time into their Minkowskian forms, in
which there is no gravitational particle creation.

The situation is slightly more complicated for spin-0 particles. The most
general Lagrangian that can be written on account of renormalizability
includes a coupling between the Ricci scalar $\mathcal{R}\left(x\right)$ and
the scalar field $\phi$:

$\mathcal{L}=\frac{1}{2}\sqrt{-g}\left[g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{m^{2}}{2}\phi^{2}-\xi\mathcal{R}\phi^{2}\right],$
(1)

where $g^{\mu\nu}$ is the metric with determinant $g$, $\xi$ is a
dimensionless parameter, and a quartic interaction has been omitted since only
terms quadratic in the scalar field are being considered. Two values of $\xi$
are of particular interest: the minimally coupled case, $\xi=0$, and the
conformally coupled case, $\xi=\frac{1}{6}$, for which (1) is conformally
invariant when the scalar is massless parkerqftcurvedspacetime . Therefore, if
$\xi=\frac{1}{6}$ massless scalars are not gravitationally created in an FLRW
universe, as was shown explicitly in tagirov . However, there is no a priori
reason for setting $\xi=\frac{1}{6}$, and for complete generality one should
take $\xi$ to be an arbitrary dimensionless parameter. Of course this comes at
the expense of conformal invariance for the Lagrangian, meaning that massless
scalars will be gravitationally produced.

There is a subtlety in the above discussion of conformal invariance that
deserves further mention. That one can perform a conformal transformation
which reduces the equations of motion for massless fermions, vector bosons, or
conformally coupled scalars into their flat space-time forms is a property
particular to the FLRW metric. The same will not be true of other metrics
which are not conformally flat; even if the equations of motion are
conformally invariant, it will not be possible to reduce them to their
Minkowskian forms with a conformal transformation. This was illustrated by
Zeldovich and Starobinsky for conformally coupled scalar fields in a Bianchi
Type I universe, which is spatially homogeneous but not isotropic
bianchi1spin0 . Anisotropy is achieved by allowing the scale factors of the
three spatial directions to differ and homogeneity is preserved by restricting
the metric elements to functions of time but not space.

The central result of bianchi1spin0  is that conformally coupled massless
scalars are created in Bianchi Type I universe but that this stops in the
isotropic limit. In this current work I extend this result to
spin-$\frac{1}{2}$ fields. It is shown that massless fermions are
gravitationally created in a Bianchi Type I space-time but that this stops in
the isotropic limit, as one might anticipate from the preceeding discussion of
conformal invariance.

Before discussing the motivations for studying the creation of massless
fermions in an anisotropically expanding space-time, a brief cosmological
detour is apropos. Particle production in an FLRW space-time is particularly
relevant when the scale factor is increasing exponentially, for example in an
inflationary scenario. Massless minimally coupled scalars are abundantly
produced during inflation and considered to be the main source of scalar
perturbation while gravitons, produced because the Einstein-Hilbert action is
not conformally invariant, are the source of tensor perturbations (the
interested reader is invited to consult R. P. Woodard’s extensive review for a
deeper discussion of the subject woodard ). Clearly, standard cosmology, which
is well established experimentally, offers useful mechanisms for the
gravitational creation of massless particles and one does not have to go to
non-standard scenarios like an anisotropic phase of expansion to obtain such
results. Any massless fermions or vector bosons could be produced by the
subsequent decay of the gravitationally produced minimally coupled scalars.
However, this would only apply to particles that couple to the scalar and even
then such a process might be highly suppressed.

The main motivation behind this work is that the gravitational creation of
massless fermions or vector bosons during an anisotropic phase of cosmological
expansion offers an additional way of producing these particles. This paper
starts with a brief review of the mathematical setup of semi-classical gravity
in Section (II), describing in particular the conditions that determine the
restricted class of space-times for which it is possible to define a vacuum
state globally. Particular attention is paid to the role of conformal
invariance. In Section (III), it is argued that the equivalence principle
allows the definition of local vacuua around any arbitrary point in the
gravitational field provided some reasonable requirements are met. This has
the benefit of doing away with the asymptotic flatness assumption and being
applicable to more general situations of physical interest. The main result of
the paper - the creation of massless fermions in a Bianchi Type I universe -
is shown in Section (IV). Since any cosmological model must explain the
observed homogeneity and isotropy of the universe, it is tempting to believe
that the effects of particle production during a very early anisotropic phase
of the expansion will have negligible influence on the evolution of the
universe. However, in Section (V) a coarse estimate of the energy density of
created massless fermions is found to be substantial for times close to but
not at or before the Planck time. This is the same conclusion reached for
scalar fields in bianchi1spin0 , although I do not attempt a full computation
of the energy momentum tensor. Three examples where massless particle creation
during an early anisotropic phase might be relevant are also mentioned as
concluding remarks. The first two are that such a phase provides a mechanism
for dark matter production and primordial magnetic field generation. For the
third example, it is argued that the substantial amount of created massless
fermions around the Planck time deserves further investigation as it may alter
the form of collapse to a singularity of a hypothetical cosmological
contraction phase without stopping it. This is similar to what was found in
bianchi1spin0  for scalar fields.

##  II Quantisation In Curved Space-Time

###  II.1 Killing Vector Fields and Energy

Quantisation in the presence of a classical gravitational field is severely
plagued in one major way: in curved space-time Poincaré symmetry is broken.
This is problematic because when constructing QFTs, mostly gauge theories
motivated by dark matter and other phenomena that the Standard Model of
Particle Physics cannot explain, Poincaré invariance - a term often used
interchangeably with Lorentz invariance in the literature - is assumed de
facto. This allows for particles and anti-particles to be fundamental concepts
in QFT. A particle or an anti-particle is an excitation of the quantised
matter field that produces a localised quanta of energy, particles having
positive energy and anti-particles having negative energy. What enables this
unambiguous definition of particles and anti-particles is the fact that in
flat space-time energy is a well defined and conserved quantity. The quantised
fields can be written as plane waves of negative (particles) and positive
(anti-particles) frequencies representing positive and negative energies
respectively. This is in turn related to the fact that the Poincaré group
admits time translation as a symmetry - energy being the associated conserved
quantity. Intuitively, one may infer that a space-time - curved or flat - that
is invariant under time translations (meaning energy is conserved) will have a
fundamental notion of particles and anti-particles, even if the isometries of
the space-time in question do not correspond to the full Poincaré group. The
problem at hand can hence be formulated as a two part question: under what
conditions is a space-time invariant under time translations and how does one
construct a QFT if such conditions are not met?

The answer to the first part of this question is of course well known and the
material presented in this section is intended as a brief summary to motivate
the approach taken in this work to address the second part. The material
presented here draws heavily from the classic texts dewitt , birrell , and
parkerqftcurvedspacetime  and the interested reader is invited to consult
these references for a more detailed treatment of the subject, or blureview
for a more recent review.

A space-time with a metric $g_{\mu\nu}$ and covariant derivative
$\nabla_{\mu}=\partial_{\mu}+\Gamma_{\mu\nu}^{\rho}$ ($\Gamma_{\mu\nu}^{\rho}$
are the Christoffel symbols of the metric, with space-time labels suppressed
for clarity - this will be done throughout this text) is invariant with
respect to time translations if it possesses a time-like Killing vector, i.e.,
a vector $K$ satisfying

$\displaystyle\nabla_{\mu}K_{\nu}+\nabla_{\nu}K_{\mu}=0$ (2) $\displaystyle
g^{\mu\nu}K_{\mu}K_{\nu}>0,$ (3)

where (2) is the condition that the Lie derivative of the metric with respect
to the vector $K$ vanishes. When such a time-like Killing vector exists, there
will also be a time-like coordinate, denoted by $t$ here, and it will be
possible to choose a coordinate system in which the metric is independent of
that coordinate. When setting up a coordinate system for the corresponding
gravitational field, $t$ is therefore a natural choice to represent time. It
is important to highlight here that conditions (2) and (3) are completely
general in the sense that no assumption has been made about whether the space-
time is flat or curved. In the case of flat space-time, the covariant
derivatives simply reduce to partial derivatives as the connections vanish.

If the theory is conformally invariant, condition (2) can be relaxed to that
of $K$ being a conformal killing vector,

$\nabla_{\mu}K_{\nu}+\nabla_{\nu}K_{\mu}=\lambda\left(x\right)g_{\mu\nu},$ (4)

with $\lambda\left(x\right)$ a scalar function of space-time. For non-
interacting fields the mass term is the only source of conformal symmetry
breaking, with the exception of scalar fields with a non-conformal coupling to
the Ricci scalar parkerspin0 . A space-time containaing only massless non-
interacting fields will therefore be invariant under time-translations if it
possesses a time-like conformal Killing vector.

The Killing vector $K$ defines energy as the conserved quantity $E$,

$E=\int_{\Sigma}K_{\mu}T^{\mu\nu}d\Sigma_{\nu}$ (5)

where repeated indicies are summed over, $T^{\mu\nu}$ is the Energy Momentum
Tensor, $\Sigma$ is a Cauchy hypersurface, and $E$ is independent of the
choice of $\Sigma$.

###  II.2 Field Decomposition, Natural Bases, and the G Vacuum

Having adopted a coordinate system with the natural time and found a globally
acceptable definition of energy, one now proceeds with the quantisation of
matter fields existing on that background space-time. The matter fields,
denoted generically by $\varphi\left(x\right)$ here, obey the Euler-Lagrange
equations of motion obtained from the Lagrangian, and, in seeking solutions to
these equations, one considers a basis for the decomposition of the solutions
into negative and positive frequencies, corresponding to particles with
positive energies and and anti-particles with negative energies respectively.
There is no unique way of doing this. However, with the time-like Killing
vector $K$ and coordinate system with the metric independent of $t$, there is
natural eigenfunction basis $\mathit{e}_{j}$,

$\begin{split}\varphi_{j}\left(x\right)&\propto
a_{\mathbf{\omega_{j}}}\mathit{e}_{j}+a_{\mathbf{\omega_{j}}}^{\star}\mathit{e}^{\star}_{j}\\\
\mathcal{L}_{K}e_{j}&=-i\omega_{j}e_{j}\\\
\mathcal{L}_{K}e^{\star}_{j}&=+i\omega_{j}e^{\star}_{j}\end{split}$

where $j$ is a generic label for the quantum numbers used to describe the
field and the full solution is a super-position of eigenfunctions
$\varphi_{j}$. $\mathcal{L}_{K}$ is the Lie derivative of the field with
respect to the vector $K$ and $\omega_{j}$ is the energy of the state $e_{j}$.
In the natural basis for the fields and with $K=\left(1,0,0,0\right)$ \- $K$
is written using the basis $V_{\mu}=V^{\mu\nu}\partial_{\nu}$ for vectors -
this reduces to

$\frac{\partial e_{j}}{\partial t}=-i\omega_{j}e_{j},$

with $\mathit{e}^{\star}_{j}$ satisfying the same equation but with a $+i$
sign. It should again be emphasized that this discussion is completely general
and includes the flat space-time case. The solutions to the above equation are
the exponentials, $\varphi_{j}\left(x\right)\propto\exp\left(\mp\
i\omega_{j}t\right)$, one is familiar with from QFT.

After imposing canonical quantisation on the fields, the coefficients
$a_{\mathbf{\omega_{j}}}$ and $a_{\mathbf{\omega_{j}}}^{\star}$ of the basis
functions $e_{j}$ and their conjugates $e_{j}^{\star}$ respectively will be
interpreted as the creation and annihilation operators of quantas of energy
$\pm\ \omega_{j}$. The energy $E$ being a conserved quantity, a vacuum state
can now be defined

$E\ |\ 0\ >\ =\ 0$ (6)

In anticipation of the next section, I will call this vacuum a global, or $G$,
vacuum as it is unique on the entire manifold up to isometries of the space-
time. It is a true vacuum in the sense that it can be unambiguously stated
that it contains no particles. The Fock space is constructed by repeated
application of creation operators on the vacuum just like in QFT.

##  III The General Case

The discussion until now assumed that the space-time under consideration
possesses a global time-like Killing vector. This is a rather restrictive. A
general space-time need not possess any Killing vectors at all, let alone a
time-like one. This is a serious challenge to the foundation upon which QFTs
are built and, before proceeding further, it is helpful to understand the full
range of scenarios that one may encounter. DeWitt gave a comprehensive account
of these in dewitt :

  1. 1.

There may be no global Killing vector at all, time-like or otherwise. In the
literature, this has been typically dealt with by assuming that the space-time
is asymptotically flat. Only in those limits can a meaningful definition be
assigned to particles and anti-particles. The set up includes two asymptotic
and inequivalent IN and OUT vacuua with particle creation between the two.
This may not be very applicable to most physical situations of interest -
especially in the cosmological context - and one of the chief aims of this
paper is to show that more general assumptions are possible.

  2. 2.

Even if there exists a time-like Killing vector, it may not be time-like
everywhere on the manifold. A typical example is the Schwarzchild metric, for
which $K=\left(1,0,0,0\right)$ is time-like outside the event horizon but
space-like inside it.

  3. 3.

The space-time may admit a time-like Killing vector only within a limited
region. In this case, there are several vacua - one for every region in which
there exists a time-like Killing vector. DeWitt argued in favour of a
principle of relativity by which all of these vacua must be equivalent dewitt
. This is the principle that will be used in this work for the quantisation of
matter fields in curved space-time. Using the nomenclature of Chernikov and
Tagirov tagirov , these will be called quasi, or $Q_{X}$, vacua, with $X$ the
region of space-time in which the vacua are defined.

The quasi vacuum in is in fact guaranteed for any arbitrary metric by virtue
of the equivalence principle. At an arbitrary point in a gravitational field,
a locally inertial coordinate system may be constructed for a small
neighbourhood around that point. One must however be careful to avoid
situations where the small neighbourhood includes a singularity or a horizon.
To steer clear of such situations, I will consider only ordinary points, by
which I mean that the small neighbourhood around them contains neither a
singularity nor a horizon. Within that small neighbourhood, a QFT may be set
up.

Naturally, one now needs to precisely define the small neighbourhood within
which the QFT is set up. To that end, it is helpful to think of the
Christoffel symbols as spurions that break Poincaré symmetry. Returning to the
Killing equation and writing it out explicitly,

$\partial_{\mu}K_{\nu}+\partial_{\nu}K_{\mu}+2\Gamma_{\mu\nu}^{\rho}K_{\rho}=0.$

In the limit of vanishing Christoffel symbols, full Poincaré symmetry is
recovered. The magnitude of $\Gamma_{\mu\nu}^{\rho}\left(x\right)$ therefore
indicates the degree to which Poincaré symmetry is broken. On the scale of
quantum processes in weak gravitational fields, this can be safely ignored.

The size of the time translation spurion is of particular interest. To obtain
this, one first fixes an ordinary point $X$ as the reference point and
considers a nearby point $x$. Using the notation of Appendix (A), the tetrads
$e^{\bar{\mu}}_{\nu}\left(X\right)$ can be used to define a local orthonormal
basis
$\tilde{\partial}_{\bar{\mu}}=e^{\nu}_{\bar{\mu}}\left(X\right)\partial_{\nu}$
for vectors. Applying the Killing condition to the vector
$K_{X}=\left(e^{\nu}_{\bar{0}}\left(X\right)\partial_{\nu},0,0,0\right)$ and
using the expansion in terms of local inertial coordinates described in
Appendix (A),

$\nabla_{\mu}K_{\nu}+\nabla_{\nu}K_{\mu}=-\frac{2}{3}\left(\tilde{R}^{0}_{\mu\nu\rho}+\tilde{R}^{0}_{\nu\mu\rho}\right)\Delta
X^{\rho}+\mathcal{O}\left(\Delta X^{3}\right)$

where $\Delta X^{\rho}=\left(x^{\rho}-X^{\rho}\right)$. The condition for
$K_{X}$ to be a local time-like Killing vector is

$\left|\left(\tilde{R}^{0}_{\mu\nu\rho}+\tilde{R}^{0}_{\nu\mu\rho}\right)\Delta
X^{\rho}\right|\ll 1$ (7)

In a region where (7) is satisfied, a local energy, $E_{X}$, and a $Q_{X}$
vacuum can be defined as well as annililation and creation operators,
$a_{\mathbf{\omega_{j}}}\left(X\right)$ and
$a_{\mathbf{\omega_{j}}}^{\star}\left(X\right)$, respectively

$\begin{split}E_{X}|\ 0_{X}\ >\ &=\ 0\\\
a_{\mathbf{\omega_{j}}}\left(X\right)|\ 0_{X}\ >&=0.\end{split}$ (8)

The corresponding Fock space, $\mathcal{F}_{X}$, is constructed by repeated
application of $a_{\mathbf{p}}^{\star}\left(X\right)$.

The same procedure can of course be applied at another ordinary point
$X^{\prime}$. In this case, energy eigenfunctions and creation and
annihilation operators will be related by Bogoliubov transformations

$a_{\mathbf{\omega_{j}}}\left(X^{\prime}\right)=D_{1}a_{\mathbf{\omega_{j}}}\left(X\right)+D_{-1}a_{\mathbf{\omega_{j}}}^{\star}\left(X\right)$
(9)

where $D_{\pm 1}$ are the Bogoliubov matrix elements with their space-time
dependence suppressed and $a_{\mathbf{\omega_{j}}}\left(X^{\prime}\right)$
satisfies the conjugate equation. The $Q_{X^{\prime}}$ vacuum is the state of
lowest energy with respect to $E_{X^{\prime}}$ and is annihilated by
$a_{\mathbf{\omega_{j}}}\left(X^{\prime}\right)$

$\begin{split}E_{X^{\prime}}|\ 0_{X^{\prime}}\ >&=\ 0\\\
a_{\mathbf{\omega_{j}}}\left(X^{\prime}\right)|\ 0_{X^{\prime}}\
>&=0\end{split}$

However, the $Q_{X}$ vacuum is not the state of lowest energy with respect to
$E_{X}$,

$E_{X}|\ 0_{X^{\prime}}\ >\ \neq\ 0$

This can be seen by calculating the number density of $Q_{X^{\prime}}$ with
respect to $X$

$N_{X^{\prime}}=\
<0_{X}\left|a^{\star}_{\mathbf{\omega_{j}}}\left(X^{\prime}\right)a_{\mathbf{\omega_{j}}}\left(X^{\prime}\right)\right|0_{X}>=\left|D_{-1}\right|^{2}$
(10)

$N_{X^{\prime}}$ will only be zero if the space-time has a $G$ vacuum. If not,
the $Q_{X^{\prime}}$ vacuum is not empty with respect to the $Q_{X}$ vacuum -
the gravitational field has created particles.

##  IV A Cosmological Example

###  IV.1 Bianchi I Universes

I now apply the result presented in the previous section to a Bianchi Type I
metric of the form

$ds^{2}=dt^{2}-\sum_{j=1}^{3}r^{2}_{j}\left(t\right)dx_{j}^{2},$ (11)

where $r_{j}\left(t\right)$ are arbitrary functions of time. This describes a
spatially homogeneous but non-isotropic space-time, with isotropy recovered in
the limit $r_{j}\left(t\right)\rightarrow r\left(t\right)\ \forall j$ for an
arbitrary function $r\left(t\right)$. The isotropic limit is of course the
FLRW metric used in cosmology.

Time translation is obviously not a symmetry in this case so gravitational
particle creation is expected. It is however useful to start by considering
the time-like vector $K=\left(1,0,0,0\right)$ and seeing if there are any non-
trivial conditions under which it is a conformal Killing vector. For
illustrative purposes, it is useful to fix a $k$ and perform the coordinate
transformation $dt^{2}=r_{k}^{2}\left(\eta\right)d\eta$ so that the line
element becomes

$ds^{2}=r_{k}\left(\eta\right)^{2}d\eta^{2}-\sum_{j=1}^{3}r^{2}_{j}\left(\eta\right)dx_{j}^{2}.$

The Lie derivative of the new metric with respect to $K$ is non-zero and does
not satisfy the condition for being a conformal Killing vector. However, in
the isotropic - FLRW - limit one finds

$\begin{split}\nabla_{\mu}K_{\nu}+\nabla_{\nu}K_{\mu}=-\frac{\dot{r}}{r}r^{2}=\frac{\dot{r}}{r}g_{\mu\nu}\end{split}$

where a dot denotes derivative with respect to $\eta$. A conformal Killing
vector therefore exists in the FLRW limit, meaning that a $G$ vacuum can be
defined if the QFT is conformally invariant. In other words, massless
particles cannot be created in an isotropic and homogeneously expanding
universe.

After using the coordinate transformation above to show the existence of a
conformal Killing vector in the FLRW limit, I now revert to the original
metric (11) for the remainder of this work. The general case has no time-like
Killing vector, so a $Q$ vacuum must be defined. Fixing an arbitrary ordinary
point $X=\left(T,0,0,0\right)$ on the manifold, one constructs a locally
inertial frame around it. The spatial coordinates have been set to zero since
the manifold admits spatial translational invariance as a symmetry. To linear
order in $|X|$, (62) gives the Riemann normal coordinates

$\begin{split}\xi_{X}\left(x\right)&=\left(t,r^{-1}_{j}\left(T\right)x^{j}\right)\end{split}$
(12)

for a neighbouring point $x=\left(t+T,x,y,z\right)$. Here, local translational
invariance was used to set $\xi^{\mu}_{X}\left(X\right)$ as the origin.
Condition (7) for metric (11) is

$t^{2}\left|\left(r^{-1}_{j}\left(T\right)\right)^{2}+r_{j}\left(T\right)r_{j}^{\prime\prime}\left(T\right)\right|^{2}\ll
1,$ (13)

The quantum scale associated to a particle of mass $m$ is its Compton
wavelength, below which quantum effects can no longer be ignored. One can
therefore define the reduced Compton time $t_{C}$ as the typical time scale
associated with quantum processes of that particle,

$t_{C}\equiv\frac{\hbar}{mc^{2}},$ (14)

where the reduced Planck’s constant, $\hbar$, and the speed of light $c$ have
been made explicit. Substituting $t_{C}$ into (13) and calling
$\mathcal{R\left(T\right)}=\left|\left(r^{-1}_{j}\left(T\right)\right)^{2}+r_{j}\left(T\right)r_{j}^{\prime\prime}\left(T\right)\right|$
for clarity (a scalar product over $j$ is implied in the absolute value),

$\frac{\hbar^{2}}{m^{2}c^{4}}\left|\mathcal{R\left(T\right)}\right|^{2}\ll 1$
(15)

From this, two things are immediately apparent:

  1. 1.

For massive particles, condition (13) is violated when $\mathcal{R}\gg m$, or
the radius of curvature of the space-time, $\mathcal{R}^{-1}$, is comparable
to or much smaller than the Compton wavelength of the particle. In the
cosmological context, this implies that quantum effects due to semi-classical
gravity are relevant in the very early universe near the Planck time.

  2. 2.

Gravitational particle creation is more relevant for lighter particles. In the
massless limit, quantum effects cannot be ignored as (15) is badly violated.
Massless particles are therefore most relevant for gravitational particle
creation.

Finally, it should be mentioned that this semi-classical treatment of gravity
is at best valid up until the Planck scale. At those energies, quantum
gravitational effects can no longer be ignored. In the cosmological context
studied here, this means that the theory is only valid for times $t$ larger
than the Planck time, $t_{P}$, which can be thought of as the Compton time of
the Planck mass. The validity of the semi-classical treatment can be
parameterised by a dimensionless factor $\gamma\geq 1$,

$t\ \geq\ \gamma\ t_{P}=\gamma\ \sqrt{\frac{\hbar
G}{c^{5}}}\approx\gamma\times 10^{-44}\ sec$ (16)

where $\gamma=1$ is reserved for the most optimistic of theorists.

###  IV.2 Fermion Creation

I now proceed to explicitly show an important property mentioned above using
spin-$\frac{1}{2}$ particles as an example: massless particle creation in the
anisotropic case but not in the isotropic case. From Appendix (B), the Dirac
equation for fermionic fields in space-time (11) is

$\begin{split}&\left[i\gamma_{0}\frac{\partial}{\partial
t}+i\sum_{j=1}^{3}r^{-1}_{j}\gamma_{j}\frac{\partial}{\partial
x_{j}}+\frac{i}{2}\sum_{j=1}^{3}\frac{r^{\prime}_{j}}{r_{j}}\gamma_{0}-m\right]\psi=0.\end{split}$
(17)

where ′ denotes derivative with respect to time. I use the Dirac
representation in which the $\gamma$ matrices are given by

$\gamma_{0}=\begin{pmatrix}I_{2}&0\\\ 0&-I_{2}\end{pmatrix}\ \ \
\gamma_{j}=\begin{pmatrix}0&\sigma_{j}\\\ -\sigma_{j}&0\end{pmatrix}\ \ $ (18)

with j = 1,2,3. $I_{2}$ is the 2$\times$2 identity matrix and the
$\sigma_{j}$s are the Pauli matrices.

Since the metric is diagonal, (17) contains no mixed derivative terms and
separable solutions are appropriate. The space-time is also spatially
homogeneous, which means that momentum is a conserved quantity. For a fixed
momentum $\mathbf{p}$, one therefore considers solutions of the form

$\psi^{\alpha}_{\mathbf{p}}=\chi^{\alpha}_{b}\left(\mathbf{p},t\right)\exp\left(i\alpha\mathbf{p}\cdot\mathbf{x}\right)$
(19)

where $b$ labels a spinor index and $\alpha=\pm 1$ represent negative and
positive frequency modes respectively. Substituting (19) into (17),

$\begin{split}&i\gamma_{0}\frac{\partial\chi^{\alpha}_{b}}{\partial
t}=\left[\sum_{j=1}^{3}\alpha
p_{j}r^{-1}_{j}\gamma_{j}-\frac{i}{2}V^{-1}V^{\prime}\gamma_{0}+m\right]\chi^{\alpha}_{b}.\end{split}$
(20)

The spin connection term can be absorbed into a redefinition of
$\chi\left(t\right)$ by substituting solutions of the form

$\chi\left(t\right)=\frac{1}{\sqrt{r1\left(t\right)r2\left(t\right)r3\left(t\right)}}\tilde{\chi}\left(t\right)$

into (20). As this results in an overall factor that drops out of the
equation, the tilde will be suppressed for convenience. From this substitution
one obtains

$\begin{split}&i\gamma_{0}\frac{\partial\chi^{\alpha}_{b}}{\partial
t}=\left[\sum_{j=1}^{3}\alpha
p_{j}r^{-1}_{j}\gamma_{j}+m\right]\chi^{\alpha}_{b}.\end{split}$ (21)

In order to solve (21), initial conditions are required. Taking these to be at
the time $T=\gamma t_{p}$ at which a local inertial frame was set up in the
previous section, one considers a linear superposition

$\begin{split}&\sum_{\alpha=-1}^{1}a^{\mathbf{p}}_{\alpha
s}\left(T\right)\chi^{\alpha}_{b}\left(T\right)\end{split}$

where

$\displaystyle\chi^{\alpha}_{b}\left(t\right)$
$\displaystyle\equiv\exp\left[-i\alpha\int_{T}^{t}dt^{\prime}\
\omega_{\mathbf{p}}\left(t^{\prime}\right)\right]u_{\alpha
s}\left(\mathbf{p},t\right)$ (22)
$\displaystyle\omega^{2}_{\mathbf{p}}\left(t\right)$
$\displaystyle=\sum_{j=1}^{3}\left(\frac{p_{j}}{r_{j}\left(t\right)}\right)^{2}+m^{2}$
(23)

Some comments about the form of the expression in (22) are in order. An exact
analytic solution is available only for a limited number of functional forms
of the scale factors $r_{j}\left(t\right)$. The exponential phase with an
integral is obtained by “squaring” the Dirac equation (21), which yields the
corresponding second order Klein-Gordon equation (hereafter referred to as KG)
satisfied by each component of the spinor and writing a solution of the latter
in terms of adiabatic mode functions, a WKB-like solution first described in
parkerspin0  and also used in bianchi1spin0 , parkerspin12  and wkb . For
practical applications where an analytic solution of the KG equation is not
available for $W_{\mathbf{p}}\equiv\int_{T}^{t}dt^{\prime}\
\omega_{\mathbf{p}}\left(t^{\prime}\right)$, a WKB solution such as the one
described in wkb  is useful. One expands the solution as an asymptotic series
in derivatives of the metric and truncates it at a given order.

The coefficients $a^{\mathbf{p}}_{\alpha s}$ and their adjoints are to be
interpreted as annihilation and creation operators in the local inertial frame
at $t=T$ after canonical quantisation. $u_{\alpha s}\left(\mathbf{p},t\right)$
carry spinor indices that have been suppressed for clarity and a label $s$ has
been used in anticipation of the spin projection that will be defined in the
next section. A general solution to (20) can be written as

$\chi^{\alpha}_{b}\left(t\right)=\sum_{\alpha^{\prime}=-1}^{1}D^{\alpha}_{\alpha^{\prime}}\left(\mathbf{p},t\right)A^{\mathbf{p}}_{\alpha^{\prime}s}\exp\left[-i\alpha
W_{\mathbf{p}}\right]u_{\alpha s}$

$D_{\alpha\alpha^{\prime}}\left(\mathbf{p},t\right)$ are elements of the
Bogoliubov matrix describing the mixing of positive and negative frequencies.
The complete to (17) can now be written as a superposition of plane waves

$\psi_{\alpha
s}=\int\sqrt{-g}\frac{d^{3}\mathbf{p}}{\left(2\pi\right)^{3}}\sum_{\alpha^{\prime}}D_{\alpha\alpha^{\prime}}A_{\alpha^{\prime}s}^{\mathbf{p}}u_{\alpha^{\prime}s}\exp\left(-i\alpha^{\prime}W_{\mathbf{p}}+i\alpha\mathbf{p}\cdot\mathbf{x}\right)$
(24)

where the full solution involves sums over $\alpha$ and $s$. The
$A_{\alpha^{\prime}s}^{\mathbf{p}}$s and their adjoints are the coefficients
written as a column vector

$\vec{A}_{\alpha^{\prime}s^{\prime}}^{\mathbf{p}}=\begin{pmatrix}a_{s^{\prime}}^{\mathbf{p}}\left(T\right)\\\
b_{s^{\prime}}^{\dagger\mathbf{p}}\left(T\right)\end{pmatrix}$ (25)

To quantisatise the fermionic field, one imposes the anti-commutation
conditions on them which results in the following relations between the
coefficients

$\left[a_{s}^{\mathbf{p}},a_{s^{\prime}}^{\mathbf{\dagger
p^{\prime}}}\right]_{+}=\left[b_{s}^{\mathbf{p}},b_{s^{\prime}}^{\dagger\mathbf{p^{\prime}}}\right]_{+}=\delta_{ss^{\prime}}\delta_{pp^{\prime}}.$
(26)

where the $T$ labels have been suppressed for clarity. The
$a_{s^{\prime}}^{\mathbf{p}}\left(T\right)$ and
$b_{s^{\prime}}^{\dagger\mathbf{p}}\left(T\right)$ can now be interpreted as
annihilation and creation operators for a fermion and an anti-fermion of
momentum $\mathbf{p}$ and spin $s^{\prime}$ respectively defined for the
locally inertial frame at $t=T$. They define a $Q_{T}$ vacuum

$\begin{split}a_{s^{\prime}}^{\mathbf{p}}\left(T\right)|\ 0_{T}\ >\ &=\ 0\\\
b_{s^{\prime}}^{\mathbf{p}}\left(T\right)|\ 0_{T}\ >\ &=\ 0\\\ \end{split}$
(27)

which is empty with respect the local fermion and anti-fermion number
operators
$a_{s^{\prime}}^{\dagger\mathbf{p}}\left(T\right)a_{s^{\prime}}^{\mathbf{p}}\left(T\right)$
and
$b_{s^{\prime}}^{\dagger\mathbf{p}}\left(T\right)b_{s^{\prime}}^{\mathbf{p}}\left(T\right)$
respectively. By applying the creation operators to the $Q_{T}$ vacuum, one
generates a Fock space of locally defined particles and anti-particles as long
as their wavelengths are not larger than the size of the local inertial frame
in question. For longer wavelengths this approximate definition of particles
and anti-particles is no longer valid. One therefore imposes a lower bound on
the momentum, $\mathbf{p_{\mathbf{>}}}$, determined by requiring that the de
Broglie wavelength of the particle, $\lambda_{B}=\frac{h}{\mathbf{p_{>}}}$,
not be larger than the size of the region defined by setting equality in (13).
Writing factors of $\hbar$ and $c$ explicitly,

$\mathbf{p_{>}}=\frac{\hbar}{c|\mathcal{R}\left(T\right)|}.$ (28)

The creation and annihilation operators at an arbitrary time $t$ are

$\vec{a}_{s}^{\mathbf{p}}\left(t\right)=\mathbf{D}\left(\mathbf{p},t\right)\vec{A}_{s}^{\mathbf{p}}$
(29)

where $\mathbf{D}\left(\mathbf{p},t\right)$ is the Bogoliubov matrix.
$\vec{a}_{s}^{\mathbf{p}}\left(t\right)$ must satisfy anti-commutation
relations (26) at all times, which is another way of saying that the matrix
$\mathbf{D}\left(\mathbf{p},t\right)$ must describe canonical transformations.
Direct substitution of (29) into (26) shows that the following conditions must
hold

$\displaystyle|D_{\alpha\alpha}|^{2}+|D_{\alpha-\alpha}|^{2}$
$\displaystyle=1$ (30) $\displaystyle
D_{\alpha\alpha}D^{\dagger}_{-\alpha\alpha}+D_{\alpha-\alpha}D_{-\alpha-\alpha}^{\dagger}$
$\displaystyle=0$ (31)

Condition (30) can be implemented by normalising the Bogoliubov coefficients
with the factor

$\displaystyle\mathcal{N}=\frac{1}{\sqrt{|D_{\alpha\alpha}|^{2}+|D_{\alpha-\alpha}|^{2}}}.$

Ultimately, one is interested in determining the number density of created
(anti-)particles, $|D_{\alpha-\alpha}|^{2}$ . To that end, a suitable form for
$u_{\alpha^{\prime}s}\left(\mathbf{p},t\right)$ must be determined and then
used them to find expressions for
$D_{\alpha\alpha^{\prime}}\left(\mathbf{p},t\right)$.

#### Pseudo-Helicity Operator

A first step in attempting the two tasks mentioned above is to substitute the
solution for a single wave of momentum $\mathbf{p}$ into (17),

$\begin{split}&\sum_{\alpha^{\prime}}\gamma_{0}\frac{dD_{\alpha\alpha^{\prime}}}{dt}u_{\alpha^{\prime}}=\sum_{\beta}D_{\alpha\beta}\exp\left(-i\left(\alpha^{\prime}-\beta\right)W_{\mathbf{p}}\right)\times\\\
&\left[-\gamma_{0}\frac{d}{dt}u_{\beta}\
+-i\left(\beta\omega_{\mathbf{p}}\gamma_{0}-\alpha\sum_{j=1}^{3}\frac{p_{j}}{r_{j}}\gamma_{j}-m\right)u_{\beta}\right]\end{split}$
(32)

where spin labels have been suppressed for clarity.

Within any locally inertial frame, one should recover the Dirac equation in
flat space-time. Inspection of (32) shows that in the limit
$D_{\alpha\beta}\left(\mathbf{p},t\right)=\delta_{\alpha\beta}$ and
$u\left(\mathbf{p},t\right)$ independent of time - which are the appropriate
limits for a locally inertial frame at an arbitrary time $t$ \- the only
surviving term is

$\begin{split}\left(\alpha\omega_{\mathbf{p}}\gamma_{0}-\alpha\sum_{j=1}^{3}\frac{p_{j}}{r_{j}}\gamma_{j}-m\right)u_{\alpha}=0\end{split}$
(33)

This is the Dirac equation in flat space-time in terms of locally inertial
four momentum - $u_{\alpha s}\left(\mathbf{p},t\right)$ must therefore be the
Dirac spinor. However, a difficulty arises immediately. In flat space-time,
the spinor is conveniently expressed in terms of two component helicity
eigenstates. Helicity, which represents the component of the particle’s spin
along its direction of motion and is a conserved quantity in flat space-time,
is defined by the operator

$\Sigma_{\mathbf{p}}=\frac{1}{|\mathbf{p}|}\begin{pmatrix}\mathbf{\sigma}\cdot\mathbf{p}&0\\\
0&\mathbf{\sigma}\cdot\mathbf{p}\end{pmatrix}$ (34)

where the $\sigma_{j}$s are the Pauli matrices. In the most general curved
space-time, neither the Hamiltonian nor the helicity operators represent
conserved quantities. How then does one make sense of spin and helicity in
curved space-time?

For the FLRW spacetime considered in parkerspin12 , the helicity operator
commutes with the Hamiltonian as a consequence of the spatial isotropy of the
metric. Inspection of (21) in the limit $r_{j}\rightarrow r\ \forall j$ shows
that the commutation of the helicity operator with the Hamiltonian implies
that the spinor remains an eigenvector of the helicity operator at all times.

For spacetime (11), the Hamiltonian operator is

$\mathcal{H}=\sum_{j=1}^{3}\alpha\frac{p_{j}}{r_{j}}\gamma_{0}\gamma_{j}-\frac{i}{2}V^{-1}V^{\prime}+m\gamma_{0}$
(35)

It is easily verified that the helicity operator (34) does not commute with
the Hamiltonian. However, the operator

$\begin{split}\Sigma^{P}_{\mathbf{p}}&=\frac{1}{\tilde{p}}\begin{pmatrix}\sum_{j=1}^{3}\sigma_{j}\frac{p_{j}}{r_{j}}&0\\\
0&\sum_{j=1}^{3}\sigma_{j}\frac{p_{j}}{r_{j}}\end{pmatrix}\end{split}$ (36)

does. Here
$\tilde{p}\equiv\sqrt{\sum_{j=1}^{3}\left(\frac{p_{j}}{r_{j}}\right)^{2}}$.

#### Dirac spinors in the pseudo-Helicity basis

Eigenvectors of operator (36) are therefore particularly suitable for
representing the Dirac spinor. I will call $\Sigma^{P}_{\mathbf{p}}$ the
pseudo-Helicity operator as it is the flat space-time helicity operator in any
locally inertial frame. In the FLRW limit, the pseudo-Helicity operator
reduces to the helicity operator (34) as the inverse scale factor
$r^{-1}\left(t\right)$ drops out of the matrix in (36) and is canceled by
$\tilde{p}\left(t\right)=r\left(t\right)p$ , with $p$ the magnitude of the
(three) momentum vector. One can therefore have a meaningful notion of
helicity over the entire manifold in FLRW space-times while in Bianchi Type I
metrics of the form (11), helicity can only be well defined locally.

In choosing Dirac spinors for a general curved space-time, one therefore seeks
solutions to (33) with $u_{\alpha s}\left(\mathbf{p},t\right)$ eigenvectors of
the pseudo-Helicty operator. These are

$\begin{split}u_{\tiny{1}s}=N\begin{pmatrix}\xi_{s}\\\
\frac{s\tilde{p}}{\omega_{\mathbf{p}}+m}\xi_{s}\end{pmatrix},\ \ \ \ \
u_{\tiny{-1}s}=N\begin{pmatrix}\frac{s\tilde{p}}{\omega_{\mathbf{p}}+m}\xi_{s}\\\
\xi_{s}\end{pmatrix}\end{split}$ (37)

where ant-fermion spinors are understood to have negative momentum, $N$ is a
normalisation factor for the spinors, and

$\begin{split}\xi^{s}=\begin{pmatrix}1\\\ f^{s}\left(p\right)\end{pmatrix},\ \
\ \ \
f^{s}\left(p\right)\equiv\frac{\frac{p_{1}}{r_{1}}+i\frac{p_{2}}{r_{2}}}{\frac{p_{3}}{r_{3}}+s\tilde{p}}\end{split}$
(38)

When the sign of the momentum is switched, $\xi_{s}\rightarrow\xi_{-s}$.
Therefore, an anti-particle of spin $s$ should be understood as an anti-spinor
with $\xi_{-s}\left(-p\right)$. As this does not affect calculations, it will
not be made explicit to avoid cumbersome notation. It will be convenient to
pick

$\begin{split}N=\sqrt{\frac{\omega_{\mathbf{p}}+m}{2m\ 2s\ \xi^{\dagger
s}\xi^{s^{\prime}}}},\ \ \ \ \ \xi^{\dagger
s}\xi^{s^{\prime}}&=\frac{2s\tilde{p}}{\frac{p_{3}}{r_{3}}+s\tilde{p}}\
\delta_{ss^{\prime}}\end{split}$

for later computations. With this normalisation, the spinors satisfy

$\bar{u}_{\alpha^{\prime}s^{\prime}}u_{\alpha
s}=\delta_{\alpha^{\prime}\alpha}\delta_{ss^{\prime}}$ (39)

where $\bar{u}_{\alpha^{\prime}s}=u^{\dagger}_{\alpha^{\prime}s}\gamma_{0}$.

###  IV.3 Bogoliubov Matrix Elements

With this choice of spinors, equation (32) for the Bogoliubov matrix elements
now simplifies to

$\begin{split}&\sum_{\alpha^{\prime}}\gamma_{0}\frac{dD_{\alpha\alpha^{\prime}}}{dt}u_{\alpha^{\prime}}=\sum_{\beta}D_{\alpha\beta}\exp\left(-i\left(\alpha^{\prime}-\beta\right)W_{\mathbf{p}}\right)\left(-\gamma_{0}\frac{d}{dt}u_{\beta}\right)\end{split}$
(40)

The orthogonality relation (39) can be used to isolate each component of the
Bogoliubov matrix. Multiplying (40) from the right by
$u^{\dagger}_{\alpha^{\prime}s}$,

$\begin{split}\frac{dD_{\alpha\alpha^{\prime}}}{dt}=\sum_{\beta}D_{\alpha\beta}\theta_{\alpha\alpha^{\prime}\beta}\left(t\right)\end{split}$
(41)

where

$\begin{split}&\theta_{\alpha\alpha^{\prime}\beta}\left(t\right)\equiv-\alpha^{\prime}\exp\left(-i\left(\alpha^{\prime}-\beta\right)W_{\mathbf{p}}\right)\left(\bar{u}_{\left(\alpha^{\prime},s\right)}\frac{d}{dt}u_{\left(\beta,s\right)}\right)\end{split}$

Integrating equation (41), one obtains a system of coupled linear Volterra
integral equations of the second kind with kernel
$\theta_{\alpha\alpha^{\prime}\beta}\left(t\right)$

$\begin{split}&D_{\alpha\alpha^{\prime}}\left(t\right)=D_{\alpha\alpha^{\prime}}\left(T\right)-\sum_{\beta}\int_{T}^{t}dt^{\prime}\
D_{\alpha\beta}\left(t^{\prime}\right)\theta_{\alpha\alpha^{\prime}\beta}\left(t^{\prime}\right)\end{split}$
(42)

In general, equation (42) will not be solvable analytically. However, for
times small compared to a characteristic value $\tau$ associated with the
scale factors $r_{j}\left(t\right)$, a Neumann series solution can be written
by using the inverse of $\tau$ as a small parameter. To see this, one
considers a rescaled time $t\rightarrow\tau t$, which weighs each derivative
by a factor of $\frac{1}{\tau}\equiv\lambda$. Since the kernel (see (45)
below) contains only one time derivative, an iterative solution to (42) can be
written as a power series in $\lambda$:

$\begin{split}D_{\alpha\alpha^{\prime}}\left(\eta\right)=D_{\alpha\alpha^{\prime}}^{0}\sum_{n=0}^{\infty}\lambda^{n}\left(\int_{T}^{t}\
dt^{\prime}\ \theta_{\alpha\alpha^{\prime}}^{\alpha^{\prime}}\right)^{n}\\\
+D_{\alpha-\alpha^{\prime}}^{0}\sum_{n=1}^{\infty}\lambda^{n}\left(\int_{T}^{t}\
dt^{\prime}\
\theta_{\alpha\alpha^{\prime}}^{-\alpha^{\prime}}\right)^{n}\end{split}$ (43)

where

$\begin{split}&\left(\int_{T}^{t}\ dt^{\prime}\
\theta_{\alpha^{\prime}}^{\beta}\left(t^{\prime}\right)\right)^{0}=1\\\
&\left(\int_{T}^{t}\ dt^{\prime}\
\theta_{\alpha^{\prime}}^{\beta}\left(t^{\prime}\right)\right)^{n}=\int_{T}^{t_{n-1}}dt_{1}...dt_{n}\
\theta_{\alpha^{\prime}}^{\gamma_{1}}\theta_{\gamma_{1}}^{\gamma_{2}}...\theta_{\gamma_{n-1}}^{\beta},\end{split}$

and $\alpha$ labels have been suppressed for clarity. For the remainder of
this work I will suppress powers of $\lambda$ to lighten the notation but it
should be understood that the results are only valid for times equal to or
smaller than a characteristic time associated with the scale factors
$r_{j}\left(t\right)$.

$D_{\alpha\alpha^{\prime}}^{0}\equiv D_{\alpha\alpha^{\prime}}\left(T\right)$
are the initial value of the Bogoliubov matrix elements. By construction, at
the initial time $t=T$, the vacuum is empty and only pure states are present,
implying

$D_{\alpha\alpha^{\prime}}^{0}=\delta_{\alpha\alpha^{\prime}}$ (44)

Using the explicit form of the spinors given in (37), the kernel elements
$\theta_{\alpha\ \alpha^{\prime}\beta}\left(t\right)$ are

$\begin{split}\theta_{\alpha-\alpha\alpha}\left(t\right)&=\frac{\alpha}{\omega_{\mathbf{p}}}\
\frac{1}{2\tilde{p}}\sum_{j=1}^{3}\left(\frac{p_{j}}{r_{j}}\right)^{2}\frac{r_{j}^{\prime}}{r_{j}}\exp\left(i2\alpha
W_{\mathbf{p}}\right)\\\ \\\
\theta_{\alpha\alpha\alpha}\left(t\right)&=i\frac{\frac{p_{1}}{r_{1}}\frac{p_{2}}{r_{2}}}{\frac{p_{3}}{r_{3}}+s\tilde{p}}\frac{1}{2\tilde{p}}\left(\frac{r_{1}^{\prime}}{r_{1}}-\frac{r_{2}^{\prime}}{r_{2}}\right)\end{split}$
(45)

It can additionally be verified that

$\begin{split}\theta_{\alpha-\alpha\alpha}\left(t\right)&=-\
\theta_{\alpha\alpha-\alpha}\left(t\right)\\\
\theta_{\alpha\alpha\alpha}\left(t\right)&=-\
\theta_{\alpha-\alpha-\alpha}\left(t\right)\end{split}$ (46)

Having obtained expressions for kernels of the Bogoliubov matrix elements, one
naturally proceeds to find a closed form integral expression for them. It is
found after some tedious algebra that the easiest way to make these Bogoliubov
transformations canonical - having (29) obeying (26) - is to impose
$\theta_{\alpha\alpha\alpha}\left(t\right)=0$, or
$r_{1}\left(t\right)=r_{2}\left(t\right)$. There are less constraining ways of
achieving canonical transformations, most notably by normalising the
coefficients with the factor $\mathcal{N}$ mentioned above. However, as this
diverts from the main subject treated in this work, I shall not consider them
here and will henceforth set $r_{1}\left(t\right)=r_{2}\left(t\right)$, which
result in the Bogoliubov matrix elements being simple trigonometric functions
that automatically satisfy normalisation conditions (30) and (31).

Using (46) and imposing time-ordering as is familiar from Quantum Field
Theory, the Volterra integral solution for the Bogoliubov matrix elements are

$\begin{split}D_{\alpha\alpha}\left(t\right)&=\cos\left(\Theta_{\alpha-\alpha\alpha}\left(t\right)\right)\\\
D_{\alpha-\alpha}\left(t\right)&=\sin\left(\Theta_{\alpha-\alpha\alpha}\left(t\right)\right)\end{split}$
(47)

where
$\Theta_{\alpha-\alpha\alpha}\left(t\right)=\int_{T}^{t}\theta_{\alpha-\alpha\alpha}\left(t\right)$.
The trigonometric form of the solution is what one expects for fermionic
Bogoliubov transformations. Two other properties are immediately verified

$\begin{split}D^{\dagger}_{\alpha\alpha^{\prime}}\left(t\right)&=D_{\alpha\alpha^{\prime}}\left(t\right)\\\
D_{\alpha\alpha^{\prime}}\left(t\right)&=\left(-1\right)^{\alpha^{\prime}-\alpha}D_{-\alpha-\alpha^{\prime}}\left(t\right)\end{split}$
(48)

and, from these, it can be seen that (30) and (31) are satisfied, making the
Bogoliubov transformations canonical.

Finally, the time-dependent number density - number of particles per unit
volume at a time $t$ \- of created fermions is

$\begin{split}N\left(t\right)&=\sum_{s}\
<0_{T}\left|a_{s^{\prime}}^{\dagger\mathbf{p}}\left(t\right)a_{s^{\prime}}^{\mathbf{p}}\left(t\right)\right|0_{T}>\\\
&=2\sin^{2}\left(\Theta_{\alpha-\alpha\alpha}\left(t\right)\right)\end{split}$
(49)

where the factor of 2 comes from the sum over spins. The number density of
created anti-fermions is the same as the theory has no source of global
$U\left(1\right)$ symmetry breaking.

###  IV.4 Massless and Infinite Mass Limits

I now come to the explicit proof that massless particles creation is possible
in an anisotropic background space-time but vanishes when the isotropic limit
is taken. As mentioned previously, several authors (see for example
parkerspin0 , parkerspin12 , and tagirov ) have shown that no massless
particles of non-zero spin - or spin zero particles but conformally coupled to
the Ricci scalar - are created in an FLRW universe. Bronnikov and Tagirov, in
particular, discussed the vacuum and the construction of Fock spaces for an
FLRW space-time extensively in tagirov . Their work considers only scalar
fields conformally coupled to the Ricci scalar but the main argument applies
to particles of non-zero spins as well: in a FLRW space-time and in the
absence of a (conformal symmetry breaking) mass term a conformal time-like
Killing vector and, by extension a $G$ vacuum, exists. To see this, it is best
to first define two terms for a general metric. A conformal transformation is
a local rescaling of the line element defined by

$g_{\mu\nu}\left(x\right)dx^{\mu}dx^{\nu}\rightarrow\exp\left[-2\Omega\left(x\right)\right]g_{\mu\nu}\left(x\right)dx^{\mu}dx^{\nu}$
(50)

with $\Omega\left(x\right)$ a continuous, non-vanishing, finite, real
function. A manifold $M$ with a metric $g_{\mu\nu}\left(x\right)$ is said to
be conformally flat if there exists a space-time transformation which allows
the line element to be recast in the form

$g_{\mu\nu}\left(x\right)dx^{\mu}dx^{\nu}=H^{2}\left(x\right)\eta_{\mu\nu}dx^{\mu}dx^{\nu}$
(51)

where $H\left(x\right)$ is an arbitrary function of space-time. Conformally
flat metrics belong to the same equivalence class as the Minkowski metric and
the former can be transformed into the latter through a conformal
transformation.

From this, it is evident that the FLRW metric is conformally flat. The
coordinate transformation $dt^{2}=r^{2}\left(t\right)d\eta^{2}$ that was used
in Section (IV.1) to show the presence of a conformal Killing vector casts the
FLRW metric into a conformally flat form. This can now be better understood:
the FLRW metric belonging to the same equivalence class as the Minkowski
metric, the two are equivalent and have the same symmetries up to a conformal
transformation. When the theory is conformally invariant, time translation is
therefore a symmetry and a $G$ vacuum can be defined.

For the gravitational creation of massless fermions, one is concerned with how
independent components of the anti-symmetric spin connection,
$\omega_{\mu}^{\nu\rho}$, transform under a conformal transformation. For
space-time (11), the spin connection is just the time derivative of the scale
factor in a particular direction. Its non-zero independent components
transform as

$\omega_{j}^{j0}=r_{j}^{\prime}\left(t\right)\rightarrow
r_{j}^{\prime}\left(t\right)-\Omega^{\prime}\left(t\right)$ (52)

where $j=1,2,3$ and $\Omega\left(x\right)$ is chosen to be a function of time
only. Clearly, by picking

$\Omega\left(t\right)=r_{j}\left(t\right)$

for a fixed $j$ at least one component of the spin-connection for space-time
(11) can be eliminated. More than one component can be eliminated if the
$r_{j}\left(t\right)$s are not all distinct. In particular, in the FLRW limit
all three independent components are eliminated.

For the creation of massless particle creation in space-time (11) assuming
$r_{1}=r_{2}$, it is convenient to write the different scale factors as

$\begin{split}r_{1}\left(t\right)&=r_{2}\left(t\right)=r\left(t\right)\\\
r_{3}\left(t\right)&=\sqrt{\delta^{2}\left(t\right)+r^{2}\left(t\right)}\end{split}$
(53)

where $\delta\left(t\right)$ is a measure of the anisotropy, which need not be
small. Under a conformal transformation with
$\Omega\left(t\right)=r\left(t\right)$, equation (45) for the non-zero kernel
element is

$\begin{split}\theta_{\alpha-\alpha\alpha}\left(t\right)&=\frac{1}{2\tilde{p}^{2}}\frac{p_{3}^{2}}{1+\left(\frac{\delta}{r}\right)^{2}}\frac{\delta^{\prime}\left(t\right)\delta\left(t\right)}{\sqrt{1+\left(\frac{\delta}{r}\right)^{2}}}\exp\left(i2\alpha
W_{\mathbf{p}}\right),\end{split}$ (54)

where it is assumed that $\tilde{p}$ and $W_{\mathbf{p}}$ are appropriately
transformed. In the FLRW limit $\delta\left(t\right)=0$ and
$\Theta_{\alpha-\alpha\alpha}\left(t\right)=0$ \- there is no massless fermion
production, as anticipated.

Finally, it is easily shown, as a consistency check, that fermion creation
stops in the infinite mass limit: $\theta_{\alpha-\alpha\alpha}\left(t\right)$
vanishes due to dependence on the inverse mass through the
$\frac{1}{\omega_{p}}$ factor in (45). This verifies the claim made previously
that gravitational particle creation is more substantial for lighter particles
- in the infinite mass limit, there is no particle creation, as one might
expect intuitively.

##  V Some Cosmological Implications

In this final section I mention some areas where massless gravitational
particle creation might be consequential. Since the latter is not possible in
isotropic universes, this is of particular interest for cosmological models
with an early anisotropic phase. For a full computation one would need to
evaluate the energy momentum tensor, most likely numerically using specific
models of anisotropic gravitational fields such as the Krasner metric used in
bianchi1spin0 . This formidable undertaking is left as a future work.
Nonetheless, one can get a flavour of things to come by considering various
approximations.

To estimate $\Theta_{\alpha-\alpha\alpha}\left(t\right)$, the exponential
phase factor in (45) can be neglected since, from the Euler formula, this
introduces an order one correction at most. In the massless limit and assuming
$r_{1}\left(t\right)=r_{2}\left(t\right)$,

$\Theta_{\alpha-\alpha\alpha}\left(t\right)\sim\frac{\alpha}{2}\log\left[\frac{\tilde{p}\left(t\right)}{\tilde{p}\left(T\right)}\right]$
(55)

where
$\tilde{p}\left(t\right)=\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\left(\frac{r_{3}\left(t\right)}{r\left(t\right)}\right)^{2}}$.
Here, conformal symmetry has been used to absorb
$r_{1}\left(t\right)=r_{2}\left(t\right)=r\left(t\right)$ into a redefinition
of the scale factors. In the FLRW limit,
$r_{3}\left(t\right)=r\left(t\right)$, the right-hand side of (55) is the
logarithm of unity, which is zero - massless gravitational fermion creation is
not possible. An estimate for energy density, $\epsilon$, of created fermions
is

$\epsilon\sim\int d^{3}p\ \tilde{p}\left(t\right)\
\sin^{2}\left[\frac{\alpha}{2}\log\left(\frac{\tilde{p}\left(t\right)}{\tilde{p}\left(T\right)}\right)\right]$
(56)

As before, I define
$\delta^{2}\left(t\right)=r^{2}_{3}\left(t\right)-r^{2}\left(t\right)$ for
convenience. Using the double angle formula and neglecting the resulting
cosine term, which is at most an order one correction to the $\sim\Lambda^{4}$
dependence of the energy density ($\Lambda$ is a momentum cutoff),

$\epsilon\sim\ 2\pi\frac{\Lambda^{4}}{\left(\hbar
c\right)^{3}}\left[\sqrt{\delta^{2}\left(t\right)+r^{2}\left(t\right)}+\frac{r\left(t\right)}{\delta\left(t\right)}\sinh^{-1}\left[\frac{\delta\left(t\right)}{r\left(t\right)}\right]\right]$

where the $\hbar$ factor is made explicit. Clearly, for large anisotropies,

$\frac{\delta\left(t\right)}{r\left(t\right)}\gg 1,$

the energy density of the gravitationally produced massless particles is
substantial. This is of interest in situations where a large anisotropy
results in a substantial backreaction on the metric, and for a quantitative
estimate one must compare this energy density to that of the background
gravitational field, $\epsilon_{G}$, obtained from the $00$ component of the
Einstein tensor (without a cosmological constant term). Writing factors of $c$
and $G$ explicitly,

$\epsilon_{G}=\frac{c^{4}}{8\pi
G}\left[\left(\frac{r^{\prime}\left(t\right)}{r\left(t\right)}\right)^{2}+2\
\frac{r^{\prime}\left(t\right)r_{3}^{\prime}\left(t\right)}{r\left(t\right)r_{3}\left(t\right)}\right]$

For a measure of the back-reaction on the metric the ratio of the energy
densities $\epsilon$ and $\epsilon_{G}$ must be found. To that end, one writes
the momentum cutoff in terms of the dimensionless parameter $\gamma\geq 1$
defined previously as $T=\gamma t_{P}$ and obtains $\Lambda=\hbar T^{-1}$. The
resulting ratio is

$\frac{\epsilon}{\epsilon_{G}}\sim\
8\pi^{2}\times\left(\frac{t_{P}}{T}\right)^{2}\times\frac{\delta^{2}\left(t\right)r\left(t\right)}{r^{\prime}\left(t\right)\delta^{\prime}\left(t\right)}.$
(57)

At late initial times, $T\gg t_{P}$, the backreaction of the created fermions
on the metric is small. However, as $T$ approaches $t_{P}$, the energy density
is substantial and may cause backreaction that substantially alters the form
of the metric. Even if one assumes $T\approx\left(10^{3}\right)t_{P}$, this
suppression in (57) can be compensated with large anisotropies from the
$\delta$ dependence of the same expression. Therefore, an early anisotropic
phase and the resulting massless fermion creation may find applications to
several problems of interest to Particle Physics and Cosmology, three of which
I mention briefly here.

Dark Matter Production: Gravitational particle creation as a mechanism for
generating dark matter was discussed in superheavydm  and superheavydm2 ,
where the authors consider the production of very heavy particles in an FLRW
universe. As shown in this work, this effect is less substantial for high mass
particles, and massless particles cannot be created in FLRW universes. Hence,
in FLRW universes one is left with the gravitational production of very
massive particles, a process that is not efficient.

A better mechanism suggests itself if one considers that these masses are
generated through a scalar acquiring a vacuum expectation value (VEV). The
main result of this work is that in a cosmological model with a very early
anisotropic phase a substantial amount of massless particles will be produced.
These can acquire a mass later as the scalar acquires a VEV. In addition to
being more efficient, this is also less restrictive as there is no imposition
on when the VEV has to appear. In the FLRW case, since the particles have to
be massive to be produced and production has to happen in the very early
universe to be substantial, the phase transition for the appearance of the VEV
is restricted to near Planck times.

Primordial Magnetic Fields: Astrophysical observations indicate a lack of GeV
range photon detection from astrophysical sources known to reach TeV level
energies. The simplest explanation for this is the presence of magnetic fields
which deflect the photons accelerated by inverse Compton scattering primbfield
. A compelling explanation that has received significant attention in the
literature is that some physical process generates a seed field which is then
amplified by a dynamo mechanism - see primbfieldlarry  for a review. If one is
focused on early universe solutions to the problem, the main difficulty to
overcome is that photons, being massless, are not produced in an FLRW
universe. The reason for this is exactly the same as for fermions discussed in
this work: the presence of a time-like conformal Killing vector results in a
$G$ vacuum and there is no gravitational photon creation. The approach most
commonly taken in the literature to circumvent this problem has been to
introduce conformal symmetry breaking interactions that enable the generation
of magnetic fields. However, the conclusion of this section applies to photons
as well: an early anisotropic phase would produce a potentially substantial
amount of photons that may provide a seed magnetic field that is later
amplified. The viability of an early anisotropic phase as a mechanism for
primordial magnetogenesis is left as a future work.

Anisotropic Cosmological Collapse: Massless particle creation in Bianchi Type
I space-times might be of interest in the study of cosmological collapse.
Classically, it is well established that if the Hawking-Penrose singularity
theorems apply, any cosmological contracting phase ultimately results in the
formation of a singularity. Zeldovich and Starobinsky discussed the subject
briefly in bianchi1spin0 , arguing that the substantial scalar particle
creation as the cosmological singularity is approach is unlikely to halt
cosmological collapse or change the dynamics from a contracting phase to an
expanding one. It is doubtful that this conclusion changes with fermions, even
though the large energy density of created particles found in this work as the
Planck time is approached will result in a similarly large degeneracy
pressure. However, a quantitative study of anisotropic versus isotropic
collapse in the presence of spin-$\frac{1}{2}$ fields is well motivated on
account of the gravitational creation of massless fermions in the former case
but not the latter.

As a final note I would like to mention that even though this work has
considered only the Bianchi Type I space-time, the main conclusion carries
over to other cosmological scenarios such as large inhomogeneities in early
times or FLRW space-times with inhomogeneous perturbations. As long as the
cosmological model considered is not conformally flat massless particles will
be created, with all the interesting consequences mentioned here.

##  Appendix A Tetrad Formulation Of General Relativity

The tetrad formulation of General Relativity has been used extensively in this
work for the setting up of the quantisation method and subsequently in
treating spin-$\frac{1}{2}$ fields in curved space-time. As this is not such a
common topic, I will introduce it briefly in this Appendix. The interested
reader is invited to see weinberg  for a thorough review. The following
conventions are used: bars represent Lorentz indices, raised and lowered with
the Minkowski metric $\eta_{\mu\nu}$, unbarred indices represent generally
covariant indices, raised and lowered with $g_{\mu\nu}\left(x\right)$, and
repeated indices of the same kind are summed over.

The prescription for incorporating gravity into a physical system is to
replace all Lorentz tensors with objects that are covariant under general
coordinate transformations. Mathematically, this means that in General
Relativity the Poincaré symmetry group of Special Relativity is extended to
the GL$\left(\mathbb{R},4\right)$ group. This approach works for Lorentz
vectors and tensors but not for spinors as there are no representations of
GL$\left(\mathbb{R},4\right)$ that transform like spinors under the Lorentz
subgroup. To introduce spinors in curved space-time, one must use the tetrad
formulation of General Relativity.

Under a coordinate transformation $x^{\mu}\rightarrow\xi^{\mu}\left(x\right)$,
the line element transforms as

$\begin{split}g_{\mu\nu}\left(x\right)dx^{\mu}dx^{\nu}\rightarrow
g_{\mu\nu}\left(\xi\right)\frac{\partial\xi^{\rho}}{\partial
x^{\mu}}\frac{\partial\xi^{\sigma}}{\partial
x^{\nu}}dx^{\rho}dx^{\sigma}=\tilde{g}_{\mu\nu}\left(\xi\right)d\xi^{\mu}d\xi^{\nu}\end{split}$
(58)

Invoking the equivalence principle, one can choose, at an arbitrary point $X$,
a set of locally inertial coordinates $\xi^{\bar{\rho}}_{X}$ such that
$\tilde{g}_{\mu\nu}\left(X\right)=\eta_{\mu\nu}$. In other words,

$\begin{split}g_{\mu\nu}\left(X\right)dx^{\mu}dx^{\nu}&=\frac{\partial\xi^{\bar{\rho}}_{X}}{\partial
x^{\mu}}\frac{\partial\xi^{\bar{\sigma}}_{X}}{\partial x^{\nu}}|_{x=X}\
\eta_{\bar{\rho}\bar{\sigma}}dx^{\bar{\rho}}dx^{\bar{\sigma}}\\\
\frac{\partial\xi^{\bar{\sigma}}_{X}}{\partial x^{\nu}}|_{x=X}&\equiv
e^{\bar{\rho}}_{\mu}\left(X\right)\end{split}$ (59)

The $e^{\bar{\rho}}_{\mu}\left(x\right)$ are called tetrads (or vierbiens) and
can be raised and lowered as follows

$e^{\bar{\rho}}_{\mu}\left(x\right)=\eta^{\bar{\rho}\bar{\sigma}}g_{\mu\nu}e^{\nu}_{\bar{\sigma}}\left(x\right)$
(60)

Tetrads provide a local orthonormal basis for vectors. One can write any
vector field $V^{\mu}\left(x\right)$ in terms of the coordinate system
$\xi^{\bar{\rho}}_{X}\left(x\right)$ locally inertial at $x=X$,

$V_{X}^{\bar{\mu}}\left({x}\right)\equiv
e^{\bar{\mu}}_{\nu}\left({x}\right)V^{\nu}\left({x}\right)$ (61)

where the subscript $X$ is used to indicate the fact that one has used a basis
for the vectors that is locally orthonormal at the point $X$.

The local inertial coordinates can be expanded in powers of $X$ blau :

$\begin{split}\xi^{\bar{\mu}}_{X}\left(x\right)&=\xi^{\bar{\mu}}_{X}\left(X\right)\
+\ \left(x^{\rho}-X^{\rho}\right)e^{\bar{\mu}}_{\rho}\left(X\right)\\\
&+\left(x^{\rho}-X^{\rho}\right)\left(x^{\sigma}-X^{\sigma}\right)\Gamma_{\rho\sigma}^{\lambda}\left(X\right)e^{\bar{\mu}}_{\lambda}\left(X\right)+\mathcal{O}\left(X^{3}\right)\end{split}$
(62)

The transformed metric, $\tilde{g}_{\mu\nu}\left(\xi\right)$, and Christoffel
symbols, $\tilde{\Gamma}_{\mu\nu}^{\lambda}\left(\xi\right)$, in that local
inertial frame are

$\begin{split}\tilde{g}_{\mu\nu}\left(\xi\right)&=\eta_{\mu\nu}-\frac{1}{3}\tilde{R}_{\mu\rho\nu\sigma}\left(\xi_{0}\right)\left(\xi^{\rho}-\xi_{0}^{\rho}\right)\left(\xi^{\sigma}-\xi_{0}^{\sigma}\right)+\mathcal{O}\left(\xi^{3}\right)\\\
\tilde{\Gamma}_{\mu\nu}^{\lambda}\left(\xi\right)&=-\frac{1}{3}\left(\xi^{\rho}-\xi_{0}^{\rho}\right)\left(\tilde{R}^{\lambda}_{\mu\nu\rho}\left(\xi_{0}\right)+\tilde{R}^{\lambda}_{\nu\mu\rho}\left(\xi_{0}\right)\right)+\mathcal{O}\left(\xi^{3}\right)\end{split}$
(63)

where $x^{\mu}\left(\xi_{0}\right)=X^{\mu}$ and
$\tilde{R}^{\lambda}_{\mu\nu\rho}\left(\xi_{0}\right)$ is the transformed
Riemann tensor

$\tilde{R}_{\mu\rho\nu\sigma}\left(\xi_{0}\right)=\frac{\partial^{2}g_{\mu\sigma}}{\partial
x_{\rho}\partial
x_{\nu}}|_{\xi=\xi_{0}}-\frac{\partial^{2}g_{\mu\nu}}{\partial
x_{\rho}\partial x_{\sigma}}|_{\xi=\xi_{0}}+\mathcal{O}\left(\xi^{3}\right)$
(64)

##  Appendix B Spinors In Curved Space-time

Spin-$\frac{1}{2}$ particles in a background gravitational field are described
by the covariant generalization of free Dirac equation, with the covariant
derivative that includes a spin connection. A spinor
$\psi\left(\mathbf{x},t\right)$ existing on a curved background space-time
satisfies

$i\left[\ e^{\mu}_{\bar{\nu}}\gamma^{\bar{\nu}}D_{\mu}-m\
\right]\psi\left(\mathbf{x},t\right)=0,$ (65)

with $D_{\mu}$ is the covariant derivative for spinor fields

$\begin{split}D_{\mu}&=\partial_{\mu}-\frac{i}{4}\omega^{\bar{\rho}\bar{\lambda}}_{\mu}\sigma_{\bar{\rho}\bar{\lambda}}\\\
\sigma_{\bar{\rho}\bar{\lambda}}&=\frac{i}{2}\left[\gamma_{\bar{\rho}},\gamma_{\bar{\lambda}}\right]_{-}\end{split}$
(66)

The 4$\times$4 $\gamma$ matrices satisfy the Clifford Algebra

$\left[\gamma^{\mu},\gamma^{\nu}\right]_{+}\equiv\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\eta^{\mu\nu}$
(67)

Defining $\tilde{\gamma}^{\mu}\equiv e^{\mu}_{\bar{\nu}}\gamma_{\bar{\nu}}$,

$\left[\tilde{\gamma^{\mu}},\tilde{\gamma^{\nu}}\right]_{+}=2g^{\mu\nu}$ (68)

The $\sigma_{\bar{\rho}\bar{\lambda}}$ furnish a local spinor representation
of the Lorentz group. The $\omega^{\bar{\rho}\bar{\lambda}}_{\mu}$ are the
spin connection, anti-symmetric in $\bar{\rho}\bar{\lambda}$ exchange, defined
in fujikawa  as:

$\begin{split}\omega^{\bar{\rho}\bar{\lambda}}_{\mu}&=\frac{1}{2}e^{\bar{\rho}\alpha}e^{\bar{\lambda}\beta}\left(C_{\alpha\beta\mu}-C_{\beta\alpha\mu}-C_{\mu\alpha\beta}\right)\\\
C_{\alpha\beta\mu}&\equiv
e^{\bar{\kappa}}_{\alpha}\left(\partial_{\beta}e_{\bar{\kappa}\mu}-\partial_{\mu}e_{\bar{\kappa}\beta}\right)\end{split}$
(69)

###### Acknowledgements.

I am especially grateful to Joseph Bramante for useful comments after
proofreading this manuscript and for fruitful discussions about the
applications of gravitational particle creation to dark matter freeze in and
cosmology. I would also like to thank the two anonymous referees who provided
thoughtful comments and useful references that have significantly improved
this manuscript. This research was supported in part by Perimeter Institute
for Theoretical Physics. Research at Perimeter Institute is supported by the
Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Economic Development & Innovation.

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