On the structural stability of a simple cosmological model in $R+\alpha R^{2}$ theory of gravity

Abstract

The theory of gravity with a quadratic contribution of scalar curvature is investigated using a dynamical systems approach. The simplest Friedmann--Robertson--Walker metric is employed to formulate the dynamics in both the Jordan frame and the conformally transformed Einstein frame. We show that, in both frames, there are stable de Sitter states where the expansion of the Hubble function naturally includes terms corresponding to an effective dark matter component. Using the invariant center manifold, we demonstrate that, in the Einstein frame, there exists a zero-measure set of initial conditions that lead from an unstable to a stable de Sitter state. Additionally, the initial de Sitter state is associated with a parallelly propagated singularity. We show that the formulations of the theory in the Jordan frame and the Einstein frame are physically nonequivalent.