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In this paper we shall consider an axionic Chern-Simons corrected $f(R)$gravity theoretical framework, and we shall study the chirality of thegenerated primordial gravitational waves. Particularly, we shall consider twomain axion models, the canonical misalignment axion model and the kinetic axionmodel, both of which provide an interesting particle phenomenology, in thepresence of $R^2$ terms in the inflationary Lagrangian. The axion does notaffect significantly the background evolution during the inflationary era,which is solely controlled by $R^2$ gravity. However, the due to the presenceof the Chern-Simons term, the tensor perturbations are directly affected, andour aim is to reveal the extent of effects of the Chern-Simons term on thegravitational waves modes, for both the axion models. We aim to produceanalytical descriptions of the primordial tensor modes behavior, and thus wesolve analytically the evolution equations of the tensor modes, for a nearly deSitter primordial evolution controlled by the $R^2$ gravity. We focus theanalytical study on superhorizon and subhorizon modes. For the misalignmentmodel, we were able to produce analytic solutions for both the subhorizon andsuperhorizon modes, in which case we found the behavior of the circularpolarization function. Our results indicate that the produced tensor spectrumis strongly chiral. For the kinetic axion model though, analytic solutions areobtained only for the superhorizon modes. In order to have a grasp of thebehavior of the chirality of the tensor modes, we studied the chirality of thesuperhorizon modes, however a more complete study is needed, which isimpossible to do analytically though.

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