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 $\sim h(\varphi )R^{\gamma }$, 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 $m_a\gg H$, the axion field oscillates. By assuming a slowly varying evolution of the axion field, the axion energy density scales as ${\rho }_a\sim a^{-3}$, 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 ${\rho }_a\sim a^{-3}$, however, the Hubble rate of the expansion and thus the dynamical evolution of the Universe is controlled by terms containing the higher derivatives of $\sim R^{\gamma }$, 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 $H\sim {\mathrm{\Lambda }}^{1/2}$, where $\mathrm{\Lambda }$ 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 ${\rho }_a\sim a^{-6}$.

• Received 11 April 2019

DOI:https://doi.org/10.1103/PhysRevD.99.104070