  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.
