Polynomial $f(R)$ Palatini cosmology -- dynamical system approach

TL;DR

This paper applies dynamical system methods to analyze the cosmological evolution in polynomial $f(R)$ gravity within the Palatini formulation, revealing different phase space structures in Einstein and Jordan frames and addressing singularities.

Contribution

It introduces a novel approach using piece-wise smooth dynamical systems to study Palatini $f(R)$ cosmology, highlighting the importance of non-smooth Newtonian-type systems.

Findings

Dynamical system reduces to a 2D Newtonian type system.

Trajectories can be sewn for $C^0$-extendibility of the metric.

Different phase space topologies in Einstein and Jordan frames.

Abstract

We investigate cosmological dynamics based on $f(R)$ gravity in the Palatini formulation. In this study we use the dynamical system methods. We show that the evolution of the Friedmann equation reduces to the form of the piece-wise smooth dynamical system. This system is is reduced to a 2D dynamical system of the Newtonian type. We demonstrate how the trajectories can be sewn to guarantee $C^0$ extendibility of the metric similarly as `Milne-like' FLRW spacetimes are $C^0$-extendible. We point out that importance of dynamical system of Newtonian type with non-smooth right-hand sides in the context of Palatini cosmology. In this framework we can investigate singularities which appear in the past and future of the cosmic evolution. We consider cosmological systems in both Einstein and Jordan frames. We show that at each frame the topological structures of phase space are different.