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

TL;DR

This paper analyzes the stability of cosmological solutions in an $R + eta R^2$ gravity model using dynamical systems, revealing stable de Sitter states, initial condition sensitivities, and frame inequivalence.

Contribution

It provides a detailed dynamical systems analysis of $R + eta R^2$ gravity, highlighting differences between Jordan and Einstein frames and identifying conditions for stable de Sitter solutions.

Findings

Existence of stable de Sitter states in both frames.

Initial conditions can lead from unstable to stable states.

Formulations in Jordan and Einstein frames are physically nonequivalent.

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.