Qualitative analysis of Kantowski-Sachs metric in a generic class of $f(R)$ models

Abstract

In this paper we investigate, from the dynamical systems perspective, the evolution of a Kantowski-Sachs metric in a generic class of $f(R)$ models. We present conditions (i. e., differentiability conditions, existence of minima, monotony intervals, etc.) for a free input function related to the $f(R)$, that guarantee the asymptotic stability of well-motivated physical solutions, specially, self-accelerated solutions, allowing to describe both inflationary- and late-time acceleration stages of the cosmic evolution. We discuss which $f(R)$ theories allows for a cosmic evolution with an acceptable matter era, in correspondence to the modern cosmological paradigm. We find a very rich behavior, and amongst others the universe can result in isotropized solutions with observables in agreement with observations, such as de Sitter, quintessence-like, or phantom solutions. Additionally, we find that a cosmological bounce and turnaround are realized in a part of the parameter-space as a consequence of the metric choice.