Closed cosmologies with a perfect fluid and a scalar field

TL;DR

This paper analyzes the long-term behavior of closed, homogeneous cosmological models with a perfect fluid and scalar field, revealing conditions for recollapse or eternal expansion, using dynamical systems methods.

Contribution

It provides a detailed dynamical systems analysis of closed FRW and Kantowski-Sachs models with scalar fields, including new insights into their asymptotic behaviors.

Findings

Models either recollapse or expand forever towards inflation.

Kantowski-Sachs models show intermediate dynamical behaviors.

Conditions for inflationary solutions are identified.

Abstract

Closed, spatially homogeneous cosmological models with a perfect fluid and a scalar field with exponential potential are investigated, using dynamical systems methods. First, we consider the closed Friedmann-Robertson-Walker models, discussing the global dynamics in detail. Next, we investigate Kantowski-Sachs models, for which the future and past attractors are determined. The global asymptotic behaviour of both the Friedmann-Robertson-Walker and the Kantowski-Sachs models is that they either expand from an initial singularity, reach a maximum expansion and thereafter recollapse to a final singularity (for all values of the potential parameter kappa), or else they expand forever towards a flat power-law inflationary solution (when kappa^2<2). As an illustration of the intermediate dynamical behaviour of the Kantowski-Sachs models, we examine the cases of no barotropic fluid, and of a massless scalar field in detail. We also briefly discuss Bianchi type IX models.