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\begin{document}
\centerline{\normalsize\bf Mathematical Features of Several Cosmological
Models}
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\centerline{\footnotesize Luis P. Chimento and Alejandro S. Jakubi}
\baselineskip=13pt
\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\'on I, }
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\centerline{\footnotesize\it 1428 Buenos Aires, Argentina. }
\baselineskip=13pt\centerline{\footnotesize E-mail: chimento@df.uba.ar,
jakubi@df.uba.ar}
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\abstracts{Einstein equations for several matter
sources in homogeneous, isotropic metric are shown to reduce to a second order
nonlinear ordinary
differential equation. An analysis of
its solutions is made in an important case.
}
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\section{Introduction}
Exact solutions of the Einstein equations are difficult to obtain due to their
nonlinear nature. In this paper we show that the system of
equations for homogeneous, isotropic cosmological models with a
variety of matter sources reduce to particular cases of the ordinary
differential equation
\begin{equation} \label{1}
\ddot y+y^n\dot y+\beta y^{2n+1}+cy^n =0
\end{equation}
\noindent
where $\beta$, $c$ and $n$ are constants.
The problem of a causal viscous fluid with the bulk viscosity coefficient
$\zeta$ proportional to $\rho^{1/2}$ corresponds to $n=1$, $c=0$, $y\propto H$
in the truncated Extended Irreversible Thermodynamics theory,\cite{visco} and
$n=-1/r$, $c=0$, $y\propto H^{-r}$ in the full theory, where the relation
between temperature and energy density is assumed to be of the form $T\sim
\rho^r$.\cite{Maar} Also, the behavior near the singularity, when the
relaxation term is much more important than the viscous term in the transport
equation, corresponds to $n=1$, $c=0$, $y\propto H$ for generic power-law
relation $\zeta=\alpha\rho^m$.\cite{Zak} For a perfect fluid source, with an
equation of state $p=(\gamma-1)\rho$, and a cosmological constant $\Lambda$,
we recover Eq.(\ref{1}) with $n=1$, $c=-\gamma\Lambda$ provided we derive
twice the $00$ Einstein equation
\begin{equation} \label{2}
H^2=\frac{1}{3}\frac{\rho_0}{a^{3\gamma}}-\frac{k}{a^2}+\frac{\Lambda}{3}
\end{equation}
\noindent We also find Eq.(\ref{1}) with $n=1$, $c\propto\Lambda$, $y\propto
H$ for a fluid with equation of state $3p=-\rho-C/a^2$, $\rho=C \log a/a^2$,
$C$ a constant. For two scalar fields, one free and the other
self--interacting with a potential $V(\phi)=V_0\exp(-A\phi)$, the change
$y=a^{A^2/2}$ in the Einstein equation leads to Eq.(\ref{1}) with $n=-6/A^2$
and $c=0$, as can be seen in the paper "Two-scalar field cosmologies"
appearing in this volume. For a time decaying cosmological "constant",
$\dot\Lambda\sim -H^3\Lambda$ with $y\propto H$, the case with $n=1$, $c=0$
also arises.\cite{Reut}
Thus, it turns out to be of great interest to analyze Eq.(\ref{1}) from the
mathematical point of view. Its general solution will be studied elsewhere, and
we concentrate here on the families of real solutions of the case $n=1$, $c=0$.
\section{Analysis of the solutions for $n=1$, $c=0$}
Unless $\beta=1/9$, Eq.(\ref{1}), for $n=1$ and $c=0$ has only two point Lie
symmetries and it is not equivalent to a second order linear equation
under a point transformation.\cite{Lea} So, we consider the
nonlocal transformation
\begin{equation} \label{21}
z = y^2 ,\qquad \eta =\int y dt
\end{equation}
\noindent
which turns Eq.(\ref{1}) into the equation of a damped linear oscillator
\begin{equation} \label{22}
\frac{d^2z}{d\eta^2} + \frac{dz}{d\eta} + 2\beta z = 0
\end{equation}
\noindent
and we obtain the general solution of in a parametrized
form $(t(\eta),y(\eta))$. The real solutions of (\ref{22}) can be classified as
follows:
\noindent
a. $\beta<1/8$ (strong damping).
\begin{equation} \label{23}
z(\eta) = C \exp (\lambda_{\pm} \eta)
\end{equation}
\begin{equation} \label{24}
z(\eta) = 2C \exp (- \eta/2) \cosh \left(\delta \eta/2 + \phi\right)
\end{equation}
\begin{equation} \label{25}
z(\eta) = 2C \exp (- \eta/2) \sinh \left(\delta \eta/2 + \phi\right)
\end{equation}
\noindent
b. $\beta=1/8$ (critical damping).
\begin{equation} \label{26}
z(\eta) = C \exp (- \eta/2)
\end{equation}
\begin{equation} \label{27}
z(\eta) = C (\eta + \phi) \exp (- \eta/2)
\end{equation}
\noindent
c. $\beta>1/8$ (weak damping)
\begin{equation} \label{28}
z(\eta) = 2C \exp (- \eta/2) \sin (\delta \eta/2 + \phi)
\end{equation}
\noindent
where $\lambda_{\pm}$ are the roots of the characteristic polynomial,
$\delta=|1-8\beta|^{1/2}$, and $C$, $\phi$ are arbitrary integration
constants.
Through the transformation (\ref{21}), both Lie point symmetries of
(\ref{1}) have a simple equivalent: $t(\eta)$ is defined up to an arbitrary
integration constant $t_0$ , and this reflects the invariance of (\ref{1})
under $t\to t+t_0$ . Also, the invariance $z\to z/A^2$ $(A\neq 0)$, of
(\ref{22}) is
equivalent to the symmetry transformation $t\to |A|t$, $y\to y/|A|$. Besides,
the permutation between the two branches of $z$ leads to the
discrete symmetry transformation $y\to -y$, $t\to -t$.
Whenever $z(\eta)$ has a zero, extremum or inflexion point at $\eta_1$ ,
$y(t)$
has a zero, extremum or inflexion point at $t_1 =t(\eta_1 )$. Besides, it
can be seen that $\dot y$ is finite at any zero point, and so $y(t)$ is odd in a
neighborhood of all finite zero points (see below).
There are two groups of solutions $z(\eta)$:
\noindent
i) Those that never vanish, i.e., (\ref{23}),(\ref{24}), and (\ref{26}), so
that we may choose $z(\eta)>0$ for all $\eta$. For these solutions $y(t)$ is
nonvanishing, and it is obtained from any of the two branches of
$\sqrt{z}$ (depending on sign $y$).
\noindent
ii) Those that have (at least) one zero point, i.e., (\ref{25}),
(\ref{27}), and (\ref{28}). The requirement that $z(\eta)>0$ cannot be
satisfied on both sides of the zero point by the same solution (with a given value
of $C$). Therefore, $z(\eta)$ gives rise to two solutions y(t), one for each
sign of $C$. Since $y(t)$ is odd (see bellow), these solutions are obtained by
joining at the zero point both branches of $\sqrt{z}$.
From (\ref{21}), we see that $\eta(t)$ is even for odd $y(t)$ and has extrema
at the zero points of $y(t)$. Then, for the group (i), $\eta(t)$ is monotonic, that
is, there is a one to one mapping between the real axis $\eta$ and some
interval of the axis $t$. However , for the group (ii), the pair of branches
at each side of the zero point correspond to different mappings between
$\eta$ and $t$. Besides, each singularity of $y(t)$ (where $\eta(t)$ diverges
logarithmically) marks a boundary for the mapping $\eta\to t$. For solutions
(\ref{24}), (\ref{25}), (\ref{26}), and (\ref{28}), $t(\eta)$ can be
expressed in
terms of a hypergeometric function. Only for $\beta=1/9, 0$ or $-1$,
$t(\eta)$ can be inverted in closed form.
Due to the symmetries of (\ref{1}), if $y(t)$ is a solution, $A y(A\Delta
t)$ is
also a solution, where $\Delta t=t-t_0$ . In particular, if $y(\Delta t)$ satisfies
$y=0$ and $\dot y\neq 0$ at $t=t_0$ , $-y(-\Delta t)$ is also a solution with the same
initial data. However, as (\ref{1}) satisfies a Lipschitz condition,
given these initial data the solution is unique. Thus, we conclude
that $y(\Delta t)$ is odd. Further, it is easy to see that $y(t)$ is analytic
at $t_0$ , so that there is an interval where its Taylor series
converges. As $y(t)$ must be odd also about any further zero point within
the interval of convergence of the series, it comes out that there
are only two possibilities for an interval with a zero point:
\noindent
a) The interval contains only one zero point.
\noindent
b) There are infinitely many equispaced zero points; that is, $y(t)$ is an
oscillatory periodic function (the radius of convergence is
infinite).
Solutions which exhibit behavior (a) occur only for $\beta<1/8$, while the solutions for
$\beta>1/8$ have behavior (b). So, for $\beta<1/8$,
$y(t)$
has either one or no zero point in the interval where it is bounded.
The solutions (\ref{23}) lead to the two one-parameter families
of solutions for $\beta<1/8$:
\begin{equation} \label{31}
y_{\pm} (t) = \alpha /\Delta t ,\quad \alpha_{\pm} = -2/\lambda_{\pm},\quad
\lambda_+\neq 0;\qquad
y_+ = K ,\quad \lambda_+ = 0
\end{equation}
\noindent
and we wish to investigate small departures from them. Let us consider first the case when
$\exp[(\lambda_- -\lambda_+ )\eta-2\phi]\ll 1$. As $\lambda_+ >\lambda_-$ ,
this occurs for any $\phi$ if $\eta$ is big
enough. Then to first order we get the approximated solution
\begin{equation} \label{33}
y(t) = \frac{\alpha}{\Delta t} \left( 1 + \gamma {\Delta t}^r \right),\quad
\lambda_+\neq 0;
\qquad
y(t) = K \left(1 + \gamma \exp(-K t) \right ) ,\quad \lambda_+ = 0
\end{equation}
\noindent
where $\gamma\propto\exp(-2\phi)$, $r=4-\alpha$ is the
Kowalevski exponent and $\alpha=\alpha_+$ in this case.\cite{Yosh} In the opposite
case, that is for $\eta\to -\infty$, we get also (\ref{33}a), but now
$\alpha=\alpha_-$ and $\gamma\propto \exp(2\phi)$. Whenever
$y(t)$ has a singularity, $z(\eta)$ diverges, and this occurs for
$\eta\to\infty$ as well as for $\eta\to -\infty$ if $\beta<0$. Hence, using
(\ref{21}), we find that any singularity is located at a finite time, and
in a neighborhood of it, $y(t)$ has the asymptotic form (\ref{33}a) with
$r>0$. When $\beta>0$, $z(\eta)\to 0$ and $|t(\eta)|\to\infty$ for
$\eta\to\infty$. Then, $y(t)$ vanishes at infinity with the asymptotic
behavior (\ref{33}a), where $\alpha=\alpha_+$ and $r<0$.
The two-parameter families of solutions arise from (\ref{24}), (\ref{25}),
(\ref{27}), and (\ref{28}). We classify them in two groups: those which have
a singularity at a finite time, and those which are regular for all
time; and we give the main features of their behavior.
\noindent
a. Singular solutions.
\noindent
$0\le\beta<1/8, C>0$.
\noindent
a1. They have a singularity, where the leading behavior is (\ref{31}--),
decrease monotonically and vanish at infinity with leading behavior
(\ref{31}+) unless $\beta=0$, when they have a nonvanishing limit.
\noindent
$\beta<0, C>0$.
\noindent
a2. They have two singularities, one with leading
behavior (\ref{31}--) and the other with leading behavior
(\ref{31}+). There is a minimum (maximum) between them.
\noindent
a3. They have a zero point between two singularities, where the leading
behavior is (\ref{31}+). They increase monotonically and have
three inflexion points.
\noindent
$\beta\le 1/8, C<0$.
\noindent
a4. They have a zero point between two singularities, where the
leading behavior is (\ref{31}--) for $\beta<1/8$ and
\begin{equation} \label{39}
y(t)\sim \frac{4}{\Delta t} \left[1+\frac{1}{2\ln |\Delta t|}+
\frac{A-\ln\ln(1/|\Delta t|)}{\left(2\ln|\Delta t|\right)^2}\right]
\end{equation}
for $\beta=1/8$. They decrease monotonically.
\noindent
b. Regular solutions.
\noindent
$0<\beta\le 1/8, C>0$.
\noindent b1. They have a zero point between two extrema. They have three
inflexion points and they vanish at infinity, with leading behavior
(\ref{31}+) for $\beta<1/8$ and like a4., with the replacement
$\ln(1/|\Delta t|)\to \ln|\Delta t |$, for $\beta=1/8$.
\noindent
$\beta=0, C>0$.
\noindent
b2. They have a zero point at $t_0$ and increase monotonically with a
nonvanishing limit at infinity.
\noindent
$\beta>1/8$.
\noindent
b3.
They are oscillatory periodic, and its period and amplitude have a
relation of the form $AT^2=f(\delta)$.
The period diverges as $\beta\to 1/8^+$ and has the limit $T\to
2\pi^{3/2}/(\sqrt{C}|\Gamma(3/4)|^2)$, $\beta\to\infty$.
These results agree with the phase space analysis,\cite{Min} and confirm
the numerical simulations.\cite{Lea2}
\section{ Conclusions}
We have obtained the general solution of Eq.(\ref{1}) for $n=1$ and $c=0$ in
a parametrized
form by means of Eq.(\ref{21})
The solutions have moving singularities and
depending on whether these points are real or not, two
groups of real solutions arise: the singular and the regular ones.\cite{Ince}
Both one-parameter families of solutions for $\beta <1/8$ are
singular, unless $\beta =0,$ when one of them turns into a constant. They
coalesce for $\beta =1/8,$ and there is no real one-parameter family of
solutions for $\beta >1/8.$ These one-parameter solutions give the
leading behavior of the solutions about a
singularity. Only for $\beta=1/9,0,-1$ two-parameter solutions are functions
on the complex plane and real solutions can be expressed in closed form.
In general, the problem of the construction of explicit solutions
of a given integrable nonlinear differential equation remains
open. One direction along which one can attempt to proceed is
linearization, i.e. the reduction of the equation to a linear
ordinary differential equation, which is, by definition,
integrable.
Only for $\beta =1/9$ Eq.(\ref{1}) possesses eight symmetries and
is linearizable by a point transformation. On the other
hand, the transformation (\ref{21}) linearizes it for any
value of $\beta$. Thus, although it has only two Lie point symmetries,
it possesses eight nonlocal symmetries. We think that it is of
utmost importance to study this kind of linearizing
transformations, which have received up to now little attention.
\section{References}
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%\leftmargin 2.5em
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R.~Maartens,
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M.~Zakari and D.~Jou,
{\it Phys. Lett. A \/} {\bf 175} (1993) 395.
\bibitem{Reut}
M.~ Reuter and C.~Wetterich
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F.~M.~Mahomed and P.~G.~L. Leach,
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H.~Yoshida,
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N.~Minorsky,
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P.~G.~L.~Leach, M.~R.~Feix and S.~Bouquet,
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\end{thebibliography}
\end{document}