Unification of inflation with dark energy in f (R) gravity and axion dark matter

In this work we introduce an effective model of f (R) gravity containing a nonminimal coupling to the axion scalar field. The axion field is described by the misalignment model, in which the primordial U (1) Peccei-Quin symmetry is broken during inflation and the f (R) gravity is described by the R-2 model, and in addition, the nonminimal coupling has the form similar to h(phi)/R-gamma, with 0 < gamma < 0.75. By appropriately constraining the nonminimal coupling at early times, the axion field remains frozen in its primordial vacuum expectation value, and the R-2 gravity dominates the inflationary era. As the Universe expands, when H equals the axion mass m a and for cosmic times for which m(a) >> H, the axion field oscillates. By assuming a slowly varying evolution of the axion field, the axion energy density scales as rho(a) similar to a(-3), where a is the scale factor, regardless of the background Hubble rate, thus behaving as cold dark matter. At late times, the axion still evolves as rho(a) similar to a(-3), however, the Hubble rate of the expansion and thus the dynamical evolution of the Universe is controlled by terms containing the higher derivatives of similar to R-gamma, which are related to the nonminimal coupling, and as we demonstrate, the resulting solution of the Friedman equation at late times is an approximate de Sitter evolution. The late-time de Sitter Hubble rate scales as H similar to Lambda(1/2), where A is an integration constant of the theory, which has its allowed values very close to the current value of the cosmological constant. Finally, the theory has a prediction for the existence of a preinflationary primordial stiff era, in which the energy density of the axion scales as rho(a) similar to a(-6).