$R^2$ Gravity Effects on the Kinetic Axion Phase Space

In this work we consider the effect of an $R^2$ term on the kinetic misalignment axion theory. By using the slow-roll assumptions during inflation and the field equations, we construct an autonomous dynamical system for the kinetic axion, including the effects of the $R^2$ term and we solve numerically the dynamical system. As we demonstrate, the pure kinetic axion attractor is transposed to the right in the field phase space, and it is no longer $(\phi,\dot{\phi})=(\langle \phi \rangle,0)$, but it is $(\phi,\dot{\phi})=(\langle \phi '\rangle,0)$, with $\langle \phi '\rangle\neq 0$ some non-zero value of the scalar field with $\langle \phi '\rangle> \langle \phi \rangle$. This feature indicates that the kinetic axion mechanism is enhanced, and the axion oscillations are further delayed, compared with the pure kinetic axion case. The phenomenological implications on the duration of the inflationary era, on the commencing of the reheating era and the reheating temperature, are also discussed.