Cosmological dynamics inf(R)gravity

Abstract

In this paper, we study the cosmological viability conditions, the phase-space dynamics, and the cosmological evolution of $f(R)$ gravity. In contrast to most previous works in the literature, which analyzed the background dynamics of $f(R)$ gravity by means of a dynamical system, we proceed by focusing on the equivalent scalar field description of the theory, which we believe is a more intuitive way of treating the problem. In order to study how the physical solutions evolve in $f(R)$ cosmology, we explore the cosmological dynamics of a range of $f(R)$ models, including models that yield a large hierarchy of scales and are singularity free. We present generic features of the phase-space dynamics in $f(R)$ cosmology. We study the global structure of the phase space in $f(R)$ gravity by compactifying the infinite phase space into a finite space via the Poincar\'e transformation. On the expansion branch of the phase space, the constraint surface has a repeller and a de Sitter attractor, while on the contraction branch, the constraint surface has an attractor and a de Sitter repeller. Generally, the phase currents originate from the repeller and terminate at the corresponding attractor in each space. The trajectories between the repeller and the attractor in the presence of matter density are different from those in the vacuum case. The phase analysis techniques developed in this paper are very general, and can be applied to other similar dynamical systems.