Dynamical system approach to FRW models in higher-order gravity theories

TL;DR

This paper analyzes the late-time behavior of positively curved FRW cosmological models with a scalar field in higher-order gravity theories, revealing conditions under which the universe avoids recollapse.

Contribution

It introduces a dynamical system approach to higher-order gravity models, applying advanced mathematical methods to analyze equilibrium solutions and their stability.

Findings

De Sitter space is a stable equilibrium in the model.

Positively curved FRW models can avoid recollapse under certain conditions.

The analysis uses center manifold and normal form theories for stability assessment.

Abstract

We study the late time evolution of positively curved FRW models with a scalar field which arises in the conformal frame of the $R+\alpha R^{2}$ theory. The resulted three-dimensional dynamical system has two equilibrium solutions corresponding to a de Sitter space and an ever expanding closed universe. We analyze the structure of the first equilibrium with the methods of the center manifold theory and, for the second equilibrium we apply the normal form theory to obtain a simplified system, which we analyze with special phase plane methods. It is shown that an initially expanding closed FRW spacetime avoids recollapse.