
\documentstyle[12pt,wspro]{article}
\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} + <T^q_{ik}
>\right)
\label{2}
\end{equation}
where $G_{ik}$ is the Einstein tensor, $\Lambda$ the  cosmological  constant,
$T^{cl}_{ik}$ the  classical  source  and $<T^q_{ik}>$  the  renormalised
vacuum expectation value of  the  quantum  fields  stress-energy  tensor
operator. For conformally invariant free quantum  fields $<T^q_{ik}>$ contains
only local terms

\begin{equation}
                 <T^q_{ik} > =\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}

\begin{thebibliography} {9}
\leftmargin 2.5em

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%\bibitem{10}
%P.~ Spindel
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\bibitem{9}
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\bibitem{11}
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\end{thebibliography}

\end{document}