On the evolution of universes in quadratic theories of gravity

Abstract

We use a dynamical systems approach to investigate Bianchi type I and II universes in quadratic theories of gravity. Due to the complicated nature of the equations of motion we focus on the stability of exact solutions and find that there exists an isotropic FRW universe acting as a past attractor. This may indicate that there is an isotropisation mechanism at early times for these kind of theories. We also discuss the Kasner universes, elucidate the associated centre manifold structure, and show that there exists a set of non-zero measure which has the Kasner solutions as a past attractor. Regarding the late-time behaviour, the stability shows a dependence of the parameters of the theory. We give the conditions under which the de Sitter solution is stable and also show that for certain values of the parameters there is a possible late-time behaviour with phantom-like behaviour. New types of anisotropic inflationary behaviour are found which do not have counterparts in general relativity.