Kinetic Axion f(R) Gravity Phase Space

In this work, we studied the phase space of f(R) gravity in the presence of a misalignment axion, including parity violating Chern-Simons terms. We construct the autonomous dynamical system by using appropriate dimensionless variables and find the cosmological attractors of the phase space, which are basically the fixed points of the autonomous dynamical system. We focus on the R2 model and the misalignment axion potential near the minimum. We demonstrate that the Chern-Simons terms have no effect on the phase space. We found four distinct, possibly unstable fixed points with physical significance. Specifically, we found two identical de Sitter fixed points, one radiation domination fixed point, and one dark matter dominated fixed point. Thus, in the presence of a kinetic misalignment axion, the vacuum f(R) gravity contains all of the cosmological fixed points that can characterize all of the known evolution eras of our universe.