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

H. Dehnen·V. R. Gavrilov·V. N. Melnikov
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Abstract

We study integrability by quadrature of a spatially flat Friedmann model containing both a minimally coupled scalar field $ϕ$ with an exponential potential $V(ϕ)\sim\exp[-\sqrt{6}σκϕ]$, $κ=\sqrt{8πG_N}$, of arbitrary sign and a perfect fluid with barotropic equation of state $p=(1-h)ρ$. 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 $σ$ 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}. $σ$ is arbitrary, $h=0$; {\bf B}. $σ=1-h/2$, $0<h<2$; {\bf C1}. $σ=1-h/4$, $0<h\leq 2$; {\bf C2}. $σ=|1-h|$, $0<h\leq 2$, $h\neq 1,4/3$. We discuss the properties of the exact solutions near the initial singularity and at the final stage of evolution.

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H. Dehnen, V. R. Gavrilov, V. N. Melnikov. 2002-12-26. General solutions for flat Friedmann universe filled by perfect fluid and scalar field with exponential potential. https://arxiv.org/abs/gr-qc/0212107

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