Geometric inflation and dark energy with axion F(R) gravity

We present a model of F(R) gravity in the presence of a string theory motivated misalignment axion like particle materialized in terms of a canonical scalar field minimally coupled with gravity, and we study the cosmological phenomenology of the model, emphasizing mainly on the late-time era. The main result of the paper is that inflation and the dark energy era may be realized in a geometric way by an F(R) gravity, while the axion is the dark matter constituent of the Universe. The F(R) gravity model consists of an R-2 term, which as we show dominates the evolution during the early time, thus producing a viable inflationary phenomenology, and a power law term similar to R-delta with delta << 1 and positive, which eventually controls the latetime era. The axion field remains frozen during the inflationary era, which is an effect known for misalignment axions, but as the Universe expands, the axion starts to oscillate, and its energy density scales eventually as we show, as rho(a) similar to a(-3). After appropriately rewriting the gravitational equations in terms of the redshift z, we study in detail the late-time phenomenology of the model, and we compare the results with the Lambda CDM model and the latest Planck 2018 data. As we show, the model for small redshifts 0 < Z < 5 is phenomenologically similar to the Lambda CDM model, however at large redshifts and deeply in the matter domination era, the results are different from those of the Lambda CDM model due to the dark energy oscillations. For the late-time study we investigate the behavior of several well-known statefmder quantities, like the deceleration parameter, the jerk and Om(z), and we demonstrate that the statefinders which contain lower derivatives of the Hubble rate have similar behavior for both the Lambda CDM and the axion F(R) gravity model. We conclude that the axion F(R) gravity model can unify in a geometric way the inflationary epoch with the dark energy era, and with the axion being the main dark matter constituent.