New dynamical system approach to Palatini $f(R)$ theories and its application to exponential gravity

TL;DR

This paper introduces a new dynamical system approach for Palatini $f(R)$ theories, including exponential gravity, enabling analysis of cosmological solutions and their stability without invertibility constraints.

Contribution

It proposes an alternative variable choice for Palatini $f(R)$ models, allowing a complete phase space analysis of exponential gravity, which was not possible with previous methods.

Findings

Identifies attractor points with specific effective equations of state.

Shows the phase space structure for exponential gravity models.

Validates analytical results with numerical solutions.

Abstract

The approach of dynamical systems is a useful tool to investigate the cosmological history that follows from modified theories of gravity. It provides qualitative information on the typical background solutions in a parametrized family of models, through the computation of the fixed points and their characters (attractor, repeller or saddle), allowing, for instance, the knowledge of which regions on the parameter space of the models generate the desired radiation, matter and dark energy dominated eras. However, the traditional proposal for building dynamical systems for an $f(R)$ theory in the Palatini formalism assumes the invertibility of a function that depends on the specific Lagrangian functional form, which is not true, for example, for the particular theory of exponential gravity ($f(R)=R-\alpha R_*(1-e^{-R/R_*})$). In this work, we propose an alternative choice of variables to treat $f(R)$ models in their Palatini formulation, which do include exponential gravity. We derive some general results that can be applied to a given model of interest and present a complete description of the phase space for exponential gravity. We show that Palatini exponential gravity theories have a final attractor critical point with an effective equation of state parameter $w_{\text{eff}} = -1$ (for $\alpha>1$), $w_{\text{eff}} = -2/3$ (for $\alpha=1$) and $w_{\text{eff}} = 0$ (for $\alpha<1$). Finally, our analytical results are compared with numerical solutions of the field equations.