Phase space analysis of $f(R)$ cosmologies: revisiting attractor-to-GR

Abstract

Phase space analysis of $f(R)$-Gravity, a simple generalization of Einstein's General theory of Relativity has been of interest especially in the context of Cosmology. Beyond its success in providing some viable descriptions of both early and late time cosmology, $f(R)$ gravity is considered to be an ideal set up for investigating some generic features of modified gravitational dynamics because of its mathematical simplicity, freedom from higher-derivative ghost instabilities and also having a natural dual description as scalar-tensor gravity in the Einstein frame representation. In this work, we have considered a class of $f(R)$-Gravity theories with $f(R)=R+αR^n$ for both positive and negative integral values of $n$ and studied some general aspects of their cosmological phase space dynamics in the Einstein frame. We have used standard techniques from dynamical systems to obtain finite fixed points and assess their stability non-perturbatively by the Lyapunov method, whereas fixed points at infinity have been addressed using Poincaré projection. We have introduced techniques from Hybrid dynamical systems to the models with odd $n$, which do not admit standard methods of analysis in the Einstein frame representation due to peculiarities of the conformal transformation involved. Based on both analytical and numerical results we show the claimed convergence of all such $f(R)$ theories to GR (plus a dark energy like component for inverse power models) in a universe with a future-singularity free expansion history, assuming an isotropic and homogeneous background - via the dynamical freeze-out of the extra scalar degree of freedom to stable fixed points. At the end we discuss some observational constraints on the parameters of such theories and discuss their consistency with respect to our work.