On dynamical systems approaches and methods in $f(R)$ cosmology

TL;DR

This paper applies global dynamical systems techniques to $f(R)$ cosmology models, providing comprehensive solution space analyses in both Jordan and Einstein frames, highlighting differences and methodological advantages.

Contribution

It introduces globally covering state space variables and demonstrates their use in analyzing $f(R)$ models, including the $R + eta R^2$ case, with new regular dynamical systems formulations.

Findings

Complete solution space pictures in both frames.

Not all Jordan frame solutions are in the Einstein frame.

Comparison of dynamical systems methods in $f(R)$ cosmology.

Abstract

We discuss dynamical systems approaches and methods applied to flat Robertson-Walker models in $f(R)$-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, $f(R) = R + \alpha R^2$, $\alpha > 0$, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving, e.g., blow ups of nilpotent fixed points, center manifold analysis, averaging, and use of monotone functions. As a result of applying dynamical systems methods to globally state space adapted dynamical systems formulations, we obtain pictures of the entire solution spaces in both the Jordan and the Einstein frames. This shows, e.g., that due to the domain of the conformal transformation between the Jordan and Einstein frames, not all the solutions in the Jordan frame are completely contained in the Einstein frame. We also make comparisons with previous dynamical systems approaches to $f(R)$ cosmology and discuss their advantages and disadvantages.