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\begin{document}
\title{ SELF-CONSISTENT SOLUTIONS OF THE SEMICLASSICAL\\ EINSTEIN-DIRAC
EQUATIONS WITH\\ COSMOLOGICAL CONSTANT }
\author{Marcelo G. Al\'{e}\\
{\it Departamento de F\'{\i}sica, Facultad de Ciencias Exactas y Naturales, }\\
{\it Universidad de Buenos Aires, Ciudad Universitaria, Pabell\'{o}n I, }\\
{\it1428 Buenos Aires, Argentina.}\\
\\
and \\
\\
Luis P. Chimento and Alejandro S. Jakubi \\
{\it Departamento de F\'{\i}sica, Facultad de Ciencias Exactas y Naturales, }\\
{\it Universidad de Buenos Aires, Ciudad Universitaria, Pabell\'{o}n I, }\\
{\it1428 Buenos Aires, Argentina.}}
\maketitle
\begin{abstract}
The general solution to the semiclassical backreaction equation is found for
conformally invariant free quantum fields in spatially flat homogeneous and
isotropic spacetimes containing a classical Dirac field, with or without
cosmological constant, when the ratio of the renormalisation parameters
$\beta/\alpha$ is $9/4$. It contains a two-parameter family of bouncing
solutions that avoid the singularity. There are several one-parameter
families which do not have particle horizons. A large set of solutions have
a de Sitter regime that satisfies the condition required for inflationary
models ($H\Delta t > 70$).
The stability of the de Sitter solutions is investigated and it is found that
they are stable when $\alpha$ and $\lambda$ have different signs. However,
when both parameters have the same sign the set of stable solutions is
restricted by the condition $ 0 < \lambda < 1/9$.
%\dag Fellow of the Consejo Nacional de Investigaciones Cient\'{\i}ficas
%y T\'{e}cnicas.
\end{abstract}
\section{Introduction}
The Standard Cosmological Model is founded in the classical theory of
General Relativity, and describes the universe as a spatially homogeneous and
isotropic spacetime stemming from a singularity in the remote past. This
model gave rise to the problems related with the existence of particle
horizons and the initial conditions. These difficulties could be overcame if
the universe underwent an inflationary expansion during the early stages of
its evolution. This inflation can be driven by a dynamical slowly-rolling
field \cite{1}${}^-$\cite{3}. Also another approach is possible considering
the contributions of higher order terms in the gravitational action,
originated by quantum corrections to the theory\cite{4}${}^-$\cite{12}. In
general, these quantum effects can violate the energy conditions and give
rise to bouncing solutions. This framework could solve the problem of the
initial singularity \cite{7}${}^-$\cite{12}. Hence
we are motivated to include a classical Dirac field source (as classical
matter) together with a cosmological constant and study their interaction with
the vacuum polarization terms. From the viewpoint of the inflationary
scenario we shall seek analytical solutions of the semiclassical Einstein
equations and we shall show that the contribution of the vacuum polarisation,
originated by the renormalisation of the vacuum expectation value of the
stress-energy tensor of the free quantum fields in conformally flat
spacetimes, can produce stable solutions with a final de Sitter stage and
singularity-free universes.
In section 2 we introduce the backreaction
equation and the necessary formalism to obtain its general solution.
In sections 3 and 4 we analyse its behaviour. In section 5 the conclusions are
stated.
\section{Solution of the back-reaction equations}
An spatially flat homogeneous and isotropic spacetime is described by
the Robertson-Walker metric
\begin{equation}
ds^2 = dt^2 - a^2 (t)\left( dx^2_1 + dx^2_2 + dx^2_3\right)
\label{1}
\end{equation}
where a(t) is the scale factor. The backreaction equation arises from the
one-loop approximation to quantum gravity \cite{13}
\begin{equation}
G_{ik} +\Lambda g_{ik} = -8\pi G\left(T^{cl}_{ik} + \right)
\label{2}
\end{equation}
where $G_{ik}$ is the Einstein tensor, $\Lambda$ the cosmological constant,
$T^{cl}_{ik}$ the classical source and $$ the renormalised
vacuum expectation value of the quantum fields stress-energy tensor
operator. For conformally invariant free quantum fields $$ contains
only local terms
\begin{equation}
=\frac{1}{3}\alpha\> ^{(1)}H_{ik}+ \beta\>
^{(3)}H_{ik}
\end{equation}
where the constant $\alpha$ and $\beta$ are determined by the gravitational
conformal anomaly. So, the 00 component of Eq. \ref{2} becomes:
\begin{equation}
2H\ddot{H}-\dot{H}^2 + 6H^2\dot{H} + \frac{\beta}{\alpha} H^4 =
\left[H^2 - \lambda - |\alpha| l^4_p \rho\right] \rm{sgn}\alpha
\label{4}
\end{equation}
where $H=\dot{a}/a$ is the Hubble expansion rate, $l_p =(16\pi G)^{1/2}$ the
Planck length , $\lambda = \frac{1}{6}\Lambda|\alpha|l^2_p$ the dimensionless
cosmological constant, $\rho$ is the energy density of the classical source,
and the dot means differentiation with respect to the
dimensionless time $\tau=(\frac{1}{2}|\alpha|l^2_p)^{-1/2}t$. Also $\hbar=c=1$
are assumed throughout this paper. The classical Dirac field equations in
curved spacetime is \cite{13}
\begin{equation}
\left(\Gamma^i \nabla_i - m\right)\psi = 0
\label{5}
\end{equation}
where $\Gamma^i$ are the generalized Dirac matrices, and $\nabla_i$ are
the spinorial covariant derivatives. In the metric Eq. \ref{1}, the general solution
of Eq. \ref{5} can be found and the only nonvanishing component of the Dirac
energy-momentum tensor is \cite{11}
\begin{equation}
T^D_{00} = \frac{m}{a^3}\left( |b_1|^2 + |b_2|^2 - |d_1|^2 - |d_2|^2 \right)=\frac{\rho_D}{a^3}
\end{equation}
where the complex coefficients $b_1 , b_2 , d_1$ and $d_2$ are arbitrary.
So, $\rho$ is non-positive definite and behaves as a classical dust
for $\rho_D >0$. The classical Dirac field will allow us to extend the
analysis for negative values of the energy density.
We shall find the general solution of Eq. \ref{4} when $\beta/\alpha=9/4$.
In this case many exact solutions of Eq. \ref{2} were found
\cite{8}${}^,$\cite{9}${}^,$\cite{11}${}^,$\cite{12}.
This fact could suggest that this particular
value had an interesting physical meaning. Then, inserting
$H = \frac{2\dot{s}}{3s}$ in Eq. \ref{4} and differentiating
it, we get the following fourth order linear homogeneous equation
with constant coefficients:
\begin{equation}
s^{(IV)} - \rm{sgn}\alpha\> \ddot{s} +\frac{9}{4}\lambda\>
\rm{sgn}\alpha \> s = 0
\label{7}
\end{equation}
The $ \dot{s}=0$ solution corresponds to the Minkowski spacetime. The
general solution of Eq. \ref{7} which satisfies (2.4) is:
a) $\lambda \rm{sgn}\alpha \not= \frac{1}{9}$
\begin{equation}
a(\tau) = a_0\left[\cosh \left(\omega_+ \tau + \phi_1 \right) +
\frac{\omega_+}{\omega_-}\sqrt{1-\frac{\rho_D \rm{sgn}\alpha}{a^3_0\omega_+^2
\left(\omega_+^2 - \omega_-^2 \right)}}\cosh \left(\omega_- \tau + \phi_2
\right)\right]^{2/3}
\end{equation}
\begin{equation}
a(\tau) = a_0\left[\exp^{\pm \omega^{+}\tau} +
\sqrt{1-\frac{\rho_D \rm{sgn}\alpha}{a^3_0\omega_+^2
\left(\omega_+^2 - \omega_-^2 \right)}}\cosh \left(\omega_- \tau + \phi
\right)\right]^{2/3}
\end{equation}
\begin{equation}
a(\tau) = a_0\left[\exp^{\pm \omega^{+}\tau} -
\sqrt{1-\frac{\rho_D \rm{sgn}\alpha}{a^3_0\omega_-^2
\left(\omega_+^2 - \omega_-^2 \right)}}\sinh \left(\omega_- \tau + \phi
\right) \right]^{2/3}
\end{equation}
\begin{equation}
a(\tau) = \left[
\sqrt{1-\frac{\rho_D \rm{sgn}\alpha}{a^3_0\omega_-^2
\left(\omega_+^2 - \omega_-^2 \right)}}\sinh \left(\omega_- \tau + \phi
\right)\right]^{2/3}
\end{equation}
b) $\lambda \rm{sgn}\alpha=\frac{1}{9}$
\begin{equation}
a(\tau) = a_0\left[\left[1+\frac{\rho_D \rm{sgn}\alpha}{a_0^3}\right]
\sinh\left(\omega\tau + \phi_1\right) - \left[\omega\tau + \phi_2\right]
\cosh\left(\omega\tau + \phi_1\right)\right]^{2/3}
\end{equation}
\begin{equation}
a(\tau) = \left[\left[\tau + \frac{\rho_D}{4\omega} + \phi_2\right]
\sinh\left(\omega\tau + \phi_1\right) - \left[\tau - \frac{\rho_D}{4\omega} +
\phi_2\right]\cosh\left(\omega\tau + \phi_1\right)\right]^{2/3}
\end{equation}
\begin{equation}
a(\tau) = a_0 \left[\omega\tau\frac{\rho_D\rm{sgn}\alpha}{a_0^3}
\exp^{\pm\left(\omega\tau + \phi\right)} \pm
\cosh\left(\omega\tau + \phi\right)\right]^{2/3}
\end{equation}
\begin{equation}
a(\tau) = \left(\frac{\rho_D}{2}\right)^{1/3}
\left[\tau\sinh\left(\omega\tau + \phi\right)\right]^{2/3}
\end{equation}
where $\omega_{\pm}$ and $\omega$ are the characteristic roots of
Eq. \ref{7} and $a_0 , \phi, \phi_1$ and $\phi_2$ are arbitrary
integration constants. In addition to the above solutions we have also their
time reversal.
\section{Analysis of the Solutions}
\subsection{$\alpha<0$}
In this case all the solutions have singularities. For $\rho_D>0$,
the initial behavior is $\left(\Delta\tau\right)^{2/3}$, while for
$\rho_D<0$, there exist a one-parameter family of solutions without
particle horizons that begin as $\left(\Delta\tau\right)^{4/3}$ .
The solutions with $\lambda \rm{sgn}\alpha<0$ oscillate at near the Planck
frequency. This oscillations (scalarons) represent the effect of
quantum corrections and could lead to abundant particle production
providing the necessary reheating for baryogenesis \cite{14}. Those
solutions with a
final de Sitter stage are stable in the far future as it was shown in \cite{10}.
For $\lambda <0$ the solutions have a finite time span.
\subsection{$\alpha>0$}
We found that the singular solutions begin as $\left(\Delta\tau\right)^{2/3}$
or $\left(\Delta\tau\right)^{4/3}$ ,
whereas the bouncing solutions end into a de Sitter phase
displaying an oscillatory behavior in the $\lambda \rm{sgn}\alpha<0$ case.
Also, there is a two-parameter family of bouncing
solutions that start in the remote past with a contracting
de Sitter era, reach a minimum and reexpand into a stable de Sitter phase.
There are singular
bouncing solutions that begin as $\left(\Delta\tau\right)^{2/3}$
then contract and
reexpand again, having a final de Sitter stage.
For a wide range of initial conditions, there exist solutions
that enter into a de Sitter expansion phase, reach a maximum and
then collapse in a second singularity. The approximate Hubble
parameter is
\begin{equation}
H_0=\frac{2}{3}\omega^-
\end{equation}
and the period of exponential expansion may
satisfy $H\Delta t>70$ as required by the inflationary scenario,
and is
easily attainable without fine tuning of the initial conditions.
\section{Conclusions}
We have found the general solution of Einstein semiclassical
equation with cosmological constant in the presence of conformally
invariant fields and a classical Dirac field, for an spatially
flat Robertson-Walker model when $\beta/\alpha$ = $9/4$.
For $\alpha<0$ the quantum effects do not remove the singularity.
For both signs of $\rho_D$ there exist a two-parameter family of solutions
that begin as $\left(\Delta\tau\right)^{2/3}$ . For $\rho_D<0$ and $\lambda>0$
one-parameter families exist without particle horizons.
We find two-parameter families of solutions that avoid the
singularity for both signs of $\rho_D$ when $\alpha>0$. They start and end
with a de Sitter phase. We find several two-parameter families of
solutions that begin as if the universe were matter dominated and
have a finite time span or expand with a final de Sitter
behavior. Also, for $\rho_D>0$ there are one-parameter solutions
without particle horizons.
For both signs of $\rho_D$ we find two-parameter families
of solutions that within a wide range of initial conditions yield
the expansion factor
$\left(H\Delta t>70\right)$ required by inflation.
These results show that the quantum contribution of the
vacuum polarization terms, have profound importance in the
solution of cosmological puzzles such as the particle horizon and
initial singularity. On the other hand, they allow for an
alternative formulation of the inflationary scenario.
\section{References}
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\leftmargin 2.5em
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%\bibitem{10}
%P.~ Spindel
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\end{thebibliography}
\end{document}