General solutions for flat Friedmann universe filled by perfect fluid and scalar field with exponential potential

Abstract

We study integrability by quadrature of a spatially flat Friedmann model containing both a minimally coupled scalar field $\phi$ with an exponential potential $V(\phi)\sim\exp[-\sqrt{6}\sigma\kappa\phi]$, $\kappa=\sqrt{8\pi G_N}$, of arbitrary sign and a perfect fluid with barotropic equation of state $p=(1-h)\rho$. From the mathematical view point the model is pseudo-Euclidean Toda-like system with 2 degrees of freedom. We apply the methods developed in our previous papers, based on the Minkowsky-like geometry for 2 characteristic vectors depending on the parameters $\sigma$ and $h$. In general case the problem is reduced to integrability of a second order ordinary differential equation known as the generalized Emden-Fowler equation, which was investigated by discrete-group methods. We present 4 classes of general solutions for the parameters obeying the following relations: {\bf A}. $\sigma$ is arbitrary, $h=0$; {\bf B}. $\sigma=1-h/2$, $0<h<2$; {\bf C1}. $\sigma=1-h/4$, $0<h\leq 2$; {\bf C2}. $\sigma=|1-h|$, $0<h\leq 2$, $h\neq 1,4/3$. We discuss the properties of the exact solutions near the initial singularity and at the final stage of evolution.