On dynamical systems approaches and 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.

36 pages, 7 figures. v2: references added, matches published version

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• Finite-time Cosmological Singularities and the Possible Fate of the Universe
• Dynamical systems approach and generic properties of $f(T)$ cosmology
• Scalar-Fluid interacting dark energy: cosmological dynamics beyond the exponential potential
• Effects of Spatial Curvature on the $f(R)$ Gravity Phase Space: no Inflationary Attractor?
• The Classical and Loop Quantum Cosmology Phase Space of Interacting Dark Energy and Superfluid Dark Matter
• New class of generalized coupling theories
• Dynamical equivalence of $f(R)$ gravity in Jordan and Einstein frames
• Cosmological dynamics of magnetic Bianchi I in viable $f(R)$ models of gravity
• Similarity solutions for the Wheeler-DeWitt equation in $f\left(R\right)$-cosmology
• Dynamics of classes of barotropic fluids in spatially curved FRW spacetimes
• Vacuum self similar anisotropic cosmologies in $F(R)-$gravity