A comprehensive analysis of the compact phase space for Hu-Sawicki $f(R)$ dark energy models including spatial curvature

TL;DR

This paper conducts a detailed dynamical systems analysis of the Hu-Sawicki $f(R)$ dark energy model, revealing novel phase space features, viable cosmological regions, and comparing its evolution to the standard $ ext{Lambda}$CDM model.

Contribution

It introduces a comprehensive phase space analysis for the Hu-Sawicki $f(R)$ model, including invariant submanifolds and fixed points, and compares its cosmological evolution with $ ext{Lambda}$CDM.

Findings

Identification of invariant submanifolds and fixed points in the phase space.

Determination of the physically viable region for the model.

Numerical comparison showing differences and similarities with $ ext{Lambda}$CDM evolution.

Abstract

We present a comprehensive dynamical systems analysis of homogeneous and isotropic Friedmann-La\^{i}matre-Robertson-Walker cosmologies in the Hu-Sawicki $f(R)$ dark energy model for the parameter choice $\{n,C_1\}=\{1,1\}$. For a generic $f(R)$ theory, we outline the procedures of compactification of the phase space, which in general is 4-dimensional. We also outline how, given an $f(R)$ model, one can determine the coordinate of the phase space point that corresponds to the present day universe and the equation of a surface in the phase space that represents the $\Lambda$CDM evolution history. Next, we apply these procedures to the Hu-Sawicki model under consideration. We identify some novel features of the phase space of the model such as the existence of invariant submanifolds and 2-dimensional sheets of fixed points. We determine the physically viable region of the phase space, the fixed point corresponding to possible matter dominated epochs and discuss the possibility of a non-singular bounce, re-collapse and cyclic evolution. We also provide a numerical analysis comparing the $\Lambda$CDM evolution and the Hu-Sawicki evolution.