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\centerline{\normalsize\bf INFLUENCE OF A PERFECT FLUID }
\centerline{\normalsize\bf ON SCALAR FIELD COSMOLOGIES}
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\centerline{\footnotesize Luis P. Chimento }
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\centerline{\footnotesize\it Departamento de F\'{\i}sica,
Facultad de Ciencias Exactas y Naturales, }
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\centerline{\footnotesize\it Universidad de Buenos Aires, Ciudad
Universitaria, Pabell\'{o}n I, }
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\centerline{\footnotesize\it 1428 Buenos Aires, Argentina. }
\baselineskip=13pt\centerline{\footnotesize E-mail: chimento@df.uba.ar}
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\centerline{\footnotesize and}
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\centerline{\footnotesize Alejandro S. Jakubi}
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\centerline{\footnotesize\it Departamento de F\'{\i}sica,
Facultad de Ciencias Exactas y Naturales, }
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\centerline{\footnotesize\it Universidad de Buenos Aires, Ciudad
Universitaria, Pabell\'{o}n I, }
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\centerline{\footnotesize\it 1428 Buenos Aires, Argentina. }
\baselineskip=13pt\centerline{\footnotesize E-mail: jakubi@df.uba.ar}
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\abstracts{We solve isotropic, homogeneous cosmological models containing a
self-interacting scalar field and a perfect fluid source.
We find new exact asymptotically stable solutions for several
potentials of interest in inflationary models.
Two-stage power-law solutions arise, with potential-dominated
and perfect fluid dominated regimes.
}
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\section{Introduction}
A self-interacting scalar field has been introduced in cosmological models as
a matter source to the Einstein equations because, when dominated by the
potential energy, it violates the strong energy condition and drives the
universe into an inflationary period.~\cite{Lin}$^{,}$ \cite{Kolb} Besides, other
matter sources may also have been present in the early Universe, most probably
in the form of a radiation fluid. Thus, we are led to study the Einstein
equations in a Robertson-Walker metric with a minimally coupled
self-interacting scalar field plus a perfect fluid source. Few exact solutions
are known to this problem ,~\cite{Rat}$^{,}$ \cite{Bar93} and none of them with
spatial curvature . In this paper we show a procedure to reduce to quadratures
the set of equations for an arbitrary potential, a perfect fluid source and a
cosmological constant. This allows us to address the issue of whether the
scalar field is always the dominating force in driving the evolution of the
universe or the fluid and curvature terms may also play a role.
\section{The Model}
We wish to investigate the evolution of a universe filled with a perfect fluid
and a scalar field $ \phi $ which has a self-interaction potential $V(\phi )$
and is minimally coupled to gravity
\begin{equation}
\label{1}
\Box \phi +{\frac{dV}{d\phi }}=0
\end{equation}
The perfect fluid has four-velocity $u_i$ and its pressure $p$ and energy
density $\rho $ are related by the equation of state $p=(\gamma-1)\rho$
with a constant adiabatic index $0\le \gamma \le 2$. Thus, we must
solve Eq. (\ref{1}) together with the Einstein equations
\begin{equation}
\label{3}
R_{ik}-\frac 12g_{ik}R+\Lambda g_{ik}=T_{ik}^\phi +T_{ik}^f
\end{equation}
We are using units such that $c=8\pi G=1$, $\Lambda $ is the cosmological
constant and
\begin{equation}
\label{4a}
T_{ik}^\phi =\phi _{;i}\phi _{;k}-g_{ik}\left[ {\frac 12}\phi
_{;m}\phi ^{;m}-V(\phi )\right]
\end{equation}
\begin{equation}
\label{4b}
T_{ik}^f=(\rho +p)u_iu_k-pg_{ik}
\end{equation}
\noindent
are the stress-energy tensors of the field and the fluid.
In the Robertson-Walker metric
\begin{equation}
\label{5}
ds^2=dt^2-a^2(t)\left [{\frac{dr^2}{1-kr^2}}+r^2(d\theta ^2+\sin
{}^2\theta d\phi ^2)\right]
\end{equation}
\noindent with scale factor $a(t)$ and curvature parameter $k=0,\pm 1$,
Eqs. (\ref{1}) and (\ref{3}) become
\begin{equation}
\label{6}
\ddot \phi +3H\dot \phi +{\frac{dV}{d\phi }}=0
\end{equation}
\begin{equation}
\label{7}
3H^2=\frac {1}{2}{\dot \phi }^2+V(\phi )+\rho -3{\frac k{a^2}}+\Lambda
\end{equation}
\noindent where the dot means $d/dt$, $H=\dot a/a$ and $\phi =\phi (t)$. Also,
from the conservation of (\ref{4b}), $\rho
=\rho _0/a^{3\gamma }$ where $\rho _0\ge 0$ is a constant.
It becomes convenient to use the scale factor as the independent variable
and write the potential in the following form:
\begin{equation}
\label{10}
V[\phi (a)] = {\frac{F(a)}{a^{6}}}
\end{equation}
\noindent with a suitable function $F(a)$. Thus we obtain a first integral
of Eq. \ref{6}
\begin{equation}\label{11}
{\frac 12\dot \phi }^2+V(\phi )-{\frac 6{a^6}\ \int }da{\frac Fa}=
{\ \frac C{a^6}}
\end{equation}
\noindent where $C$ is an arbitrary integration constant. Then, using Eqs.
\ref{7} and \ref{11}, we have reduced the problem to quadratures:
\begin{equation}\label{12}
\Delta t={\sqrt{3}\int \frac{da}a}\left[ {\frac 6{a^6}\ \int }da{
\frac Fa+\frac C{a^6}}+\rho -3{\frac k{a^2}}+\Lambda\right] ^{-1/2}
\end{equation}
\begin{equation}\label{13}
\Delta \phi ={\sqrt{6}\int \frac{da}a\left[ \frac{-F+6\int daF/a+C
}{6\int daF/a+C+\rho a^6-3k a^4+\Lambda a^6}\right] }^{1/2}
\end{equation}
\noindent where $\Delta t\equiv t-t_0$, $\Delta \phi \equiv \phi -\phi _0$
and $t_0$, $\phi _0$ are arbitrary integration constants.
\subsection{ Stability of the Solutions}
For models such that $V(\phi )$ has a local minimum at $\phi _m$ and $V(\phi
_m)+\Lambda \ge 0$, we can study the stability of solutions with
asymptotic behavior $\phi (t)\rightarrow \phi _m$. First we note that the
evolution $a(t)$ is monotonic. Then, we may use $a$ as the independent variable,
and restrict the analysis to the phase space $(\phi,\phi')$.
We find that the energy density of the field
is a suitable a Lyapunov function as
it satisfies $\rho_\phi(\phi,\phi',a)>-\Lambda$ and ${\rho_\phi}'=-3aH^2
\phi'^2<0$ in a neighborhood of $(\phi _m,0)$. So this point is an attractor
and any
solution such that $\phi \rightarrow \phi _m$ for $a\rightarrow \infty $
(equivalently $t\rightarrow \infty $) is asymptotically stable.~ \cite{Kra}
\section{ Examples}
We illustrate our procedure with
several new exact solutions for potentials of interest in inflationary models.
We consider first an exponential potential:
\begin{equation}\label{181}
V(\phi)=B \exp(-\sigma \Delta\phi) ,\qquad B>0
\end{equation}
\noindent
which yields a power-law solution
\begin{equation} \label{182}
a(t)=\left(K\Delta t\right)^\lambda\qquad
\Delta\phi=M\ln a
\end{equation}
\noindent with some constants $K$, $M$ and $\lambda$ which are functions of
the parameters of the system. We summarize known and new exact solutions for
this potential in the following table.~\cite{Luc}$^{,}$ \cite{Ell}$^{,}$
\cite{Rat}
\bigskip
\begin{tabular}{|c|c|c|}
\hline
&$\lambda$&$\sigma$\\
\hline
$\rho_0=k=0$,\quad$0~~0
\end{equation}
\noindent
we obtain asymptotically stable monotonic
solutions which begin at a singularity.
They have complicated expressions in terms
of the hypergeometric function, but
their most remarkable feature
is that a time-scale appears when a switch between two kinds of regimes
occurs. These regimes are characterized by their asymptotical power-laws:
$a(t)\sim\Delta t^{\lambda_1}$ for $\Delta t\to 0$ and $a(t)\sim
t^{\lambda_2}$ for $t\to\infty$. We find always that $\lambda_1<\lambda_2$.
Besides, we find
\begin{equation} \label{33}
\Delta\phi=-\frac{1}{\sigma}{\rm arccoth} \left(1+\omega
a^r\right)^{1/2}
\end{equation}
\noindent
with some constants $\sigma$, $\omega$ and $r$ which are functions of the
parameters of the system.
All these are new exact solutions for this potential, and we summarize our
results in the following table.
\bigskip
\begin{tabular}{|c|c|c|c|}
\hline
&$\lambda_1$&$\lambda_2$&sgn $q$\\
\hline
$k=-1$,\quad$0<\gamma<2/3$&$1$&$2/(3\gamma)$&$-$\\
$k=-1$,\quad$2/3<\gamma<2$&$2/(3\gamma)$&$1$&$+$\\
\hline
$k=0$,\quad$0<3\gamma<6-s<6$&$2/(6-s)$&$2/(3\gamma)$&$+$\\
$k=0$,\quad$0<6-s<3\gamma<6$&$2/(3\gamma)$&$2/(6-s)$&$-$\\
\hline
$k=0,\pm 1$,\quad$\gamma<2/3$&$1$&$2/(3\gamma)$&$+$\\
$k=0,\pm 1$,\quad$\gamma>2/3$&$2/(3\gamma)$&$1$&$-$\\
\hline
$k=-1$,\,$\rho_0=0$,\quad$0~~~~2/3$. We have shown asymptotically stable solutions
such that both densities keep a constant ratio, or one of them dominates for
large times.
Most of our solutions cannot be obtained by means of the slow-roll
approximation.~\cite{Lin}$^{,}$ \cite{Kolb} Calculations of the primordial spectrum of perturbations, based
on recent measurements of the cosmological background radiation show that
tensorial perturbations may have played an important role in the formation of
cosmic structures. However, the amount of gravitational perturbations
predicted by means of the slow-roll approximation is very small.~\cite{Lit}
Thus exact solutions which lay outside the slow-roll regime may lead to an
improved understanding of the evolution of the early universe.
In a future paper we will extend our procedure to more general models with
dissipative fluids.
\section{References}
\begin{thebibliography}{9}
%\leftmargin 2.5em
\bibitem{Lin}
A.~D.~Linde ,
{\it Particle Physics and Inflationary Cosmology\/} (Harwood, Chur, 1990).
\bibitem{Kolb}
E.~W.~Kolb and M.~S.~Turner,
{\it The Early Universe \/} (Addison-Wesley, New York, 1990).
\bibitem{Rat}
B.~Ratra and P.~J.~E.~Peebles,
{\it Phys. Rev. \/} {\bf 37} (1988) 3406.
\bibitem{Bar93}
J.~D.~Barrow ,
{\it Class. Quantum Grav.\/} {\bf 10} (1993) 279.
\bibitem{Kra}
N.~N.~ Krasovskii,
{\it Stability of Motion \/} (Stanford University Press, Stanford, 1963).
\bibitem{Luc}
F.~Lucchin and S.~Matarrese,
{\it Phys. Rev.\/} D {\bf 32} (1985) 1316.
\bibitem{Ell}
G.~F.~R.~Ellis and M.~S.~Madsen ,
{\it Class. Quantum Grav.\/} {\bf 8} (1991) 667.
\bibitem{Lit}
A.~R.~Liddle and D.~H.~Lyth,
{\it Phys. Rep. \/} {\bf 231} (1993) 1.
\end{thebibliography}
\end{document}
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