$f(R)$ Gravity Inflation with String-Corrected Axion Dark Matter

It is quite well known for some time that string inspired axionic terms of the form $\nu (\phi)\tilde{R}R$, known also as Chern-Simons terms, do not affect the scalar perturbations and the background evolution for a flat Friedman-Robertson-Walker Universe. In this paper we study and quantify the implications of the presence of the above term in the context of vacuum $f(R)$. Particularly, we assume that axionic dark matter is present during inflation, and we examine in a quantitative way the effects of axionic Chern-Simons terms on the tensor perturbations. The axion field is quantified in terms of a canonical scalar field, with broken Peccei-Quinn symmetry. The model perfectly describing axions as potential dark matter candidates is based on the so-called misalignment mechanism, in which case the axion is frozen near its non-zero vacuum expectation value during early times in which $H\gg m_a$. In effect, the inflationary era is mainly controlled by the $f(R)$ gravity and the Chern-Simons term. As we demonstrate, the Chern-Simons term may achieve to make a non-viable $f(R)$ gravity theory to be phenomenologically viable, due to the fact that the tensor-to-scalar ratio is significantly reduced, and the same applies to the spectral index of the tensor perturbations $n_T$. Also by studying the Starobinsky model in the presence of the Chern-Simons term, we demonstrate that it is possible to further reduce the amount of primordial gravitational radiation. The issues of having parity violating gravitational waves, also the graceful exit from inflation due to axion oscillations and finally the unification of dark energy-inflation and axion dark matter in the same $f(R)$ gravity-axion dark matter model, are also briefly discussed.