2-Component Cosmological Models with Perfect Fluid and Scalar Field: Exact Solutions

Abstract

We study integrability by quadrature of a spatially flat Friedmann model containing both a perfect fluid with barotropic equation of state p = (1− h)ρ and a minimally coupled scalar field φ with either a single exponential potential V(φ) ~ exp[−√6σκφ], $K = \sqrt {8\pi {G_N}}$, of arbitrary sign or a simplest multiple exponential potential V(φ) = W ₀ − V ₀ sinh (√6σκφ), where the parameters W ₀ and V ₀ are arbitrary. 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. For the single exponential potential we present 4 classes of general solutions with the parameters obeying the following relations: A. σ is arbitrary, h = 0; B. σ = 1− h/2, 0 < h < 2; C1. σ =1−h/4, 0 < h ≤ 2; C2. σ = |1−h|, 0 < h ≤ 2, h ≠ 1, 4/3. The properties of the exact solutions near the initial singularity and at the final stage of evolution are analyzed. For the multiple exponential potential the model is integrated with h = 1 and σ = 1/2 and all exact solutions describe the recollapsing universe. We single out the exact solution describing the evolution within the time approximately equal to 2H ⁻¹₀ with the present-day values of the acceleration parameter q ₀ = 0.5 and the density parameter Ω _{ρ 0} = 0.3.