PREPRINT
4E35183E-5B8C-4385-B9E0-246AA7D080BD

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

Kelly MacDevette, Peter Dunsby, Saikat Chakraborty

Submitted on 31 August 2022

## Abstract

We present a comprehensive dynamical systems analysis of homogeneous and isotropic Friedmann-La\^{i}matre-Robertson-Walker cosmologies in the Hu-Sawicki $f\left(R\right)$ dark energy model for the parameter choice $\left\{n,{C}_{1}\right\}=\left\{1,1\right\}$. For a generic $f\left(R\right)$ theory, we outline the procedures of compactification of the phase space, which in general is 4-dimensional. We also outline how, given an $f\left(R\right)$ 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 $\mathrm{\Lambda }$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 $\mathrm{\Lambda }$CDM evolution and the Hu-Sawicki evolution.

## Preprint

Comment: 23 pages, 9 figures

Subjects: General Relativity and Quantum Cosmology; Astrophysics - Cosmology and Nongalactic Astrophysics