Averaging generalized scalar-field cosmologies III: Kantowski-Sachs and closed Friedmann-Lemaitre-Robertson-Walker models

Scalar-field cosmologies with a generalized harmonic potential and matter with energy density rho(m), pressure p(m), and barotropic equation of state (EoS) p(m) = (gamma - 1).m, gamma is an element of [0, 2] in Kantowski-Sachs (KS) and closed Friedmann-Lemaitre-Robertson-Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function D. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global latetime attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for 0 <= gamma <= 2, and flat FLRW matter-dominated universe if 0 <= gamma <= 2/3. For closed FLRW metric late-time attractors of full and averaged systems are a flat matterdominated FLRW universe for 0 <= gamma <= 2/3 as in KS and Einstein-de Sitter solution for 0 <= gamma < 1. Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein-Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally for the whole D-range for KS and closed FLRW (if 0 <= gamma < 1) too. However, for gamma >= 1 closed FLRW solutions of the full system depart from the solutions of the averaged system as D is large. Our results are supported by numerical simulations.

Automated summary

Scalar-field cosmologies with a generalized harmonic potential and matter are investigated in different metrics, with a focus on global results. Different attractors are found for Kantowski-Sachs and closed Friedmann-Lemaitre-Robertson-Walker metrics, and the importance of time-averaged system in determining future asymptotics is highlighted.