Dynamics of quadratic f(R) cosmology with a perfect fluid: Jordan and Einstein frames

TL;DR

This paper analyzes the global dynamics of quadratic f(R) gravity in cosmology, comparing Jordan and Einstein frames through dynamical systems, fixed points, and asymptotic solutions, revealing the structure of cosmological evolutions.

Contribution

It introduces a comprehensive dynamical systems framework for quadratic f(R) gravity in both frames, providing a global description and identifying conditions for solutions to map between frames.

Findings

Global flow structure characterized in both frames

Identification of fixed points and their stability

Conditions for solutions to be conformally related

Abstract

We investigate the global dynamics of the field equations of (pure) quadratic theories of gravity which generalise Einstein's theory in spatially flat homogeneous and isotropic cosmological models with a perfect fluid. We introduce global and regular 3-dimensional dynamical systems' formulations, on both the Jordan frame and the conformally related Einstein frame. The analysis in the Jordan frame explores the monotonicity properties of the interior flow which, together with the characterisation of the orbit structure on the 2-dimensional invariant boundaries and the desingularisation of non-hyperbolic fixed points, provides a global description of the flow and its limit sets. In the Einstein frame, the analysis uses the skew-product structure of the Einstein state space and the characterisation of the orbit structure on the 2-dimensional invariant boundaries. Furthermore, by obtaining asymptotic expansions we identify the solutions that are global conformally mapped from the Jordan frame to the Einstein frame and those that are not.