Chirality of Gravitational Waves in Chern-Simons $f(R)$ Gravity Cosmology

In this paper we shall consider an axionic Chern-Simons corrected $f(R)$ gravity theoretical framework, and we shall study the chirality of the generated primordial gravitational waves. Particularly, we shall consider two main axion models, the canonical misalignment axion model and the kinetic axion model, both of which provide an interesting particle phenomenology, in the presence of $R^2$ terms in the inflationary Lagrangian. The axion does not affect significantly the background evolution during the inflationary era, which is solely controlled by $R^2$ gravity. However, the due to the presence of the Chern-Simons term, the tensor perturbations are directly affected, and our aim is to reveal the extent of effects of the Chern-Simons term on the gravitational waves modes, for both the axion models. We aim to produce analytical descriptions of the primordial tensor modes behavior, and thus we solve analytically the evolution equations of the tensor modes, for a nearly de Sitter primordial evolution controlled by the $R^2$ gravity. We focus the analytical study on superhorizon and subhorizon modes. For the misalignment model, we were able to produce analytic solutions for both the subhorizon and superhorizon modes, in which case we found the behavior of the circular polarization function. Our results indicate that the produced tensor spectrum is strongly chiral. For the kinetic axion model though, analytic solutions are obtained only for the superhorizon modes. In order to have a grasp of the behavior of the chirality of the tensor modes, we studied the chirality of the superhorizon modes, however a more complete study is needed, which is impossible to do analytically though.