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
non-minimal coupling to the axion scalar field. The axion field is described by
the misalignment model, in which the primordial $U(1)$ Peccei-Quinn symmetry is
broken during inflation and the $f(R)$ gravity is described by the $R^2$ model,
and in addition, the non-minimal coupling has the form $\sim
h(\phi)R^{\gamma}$, with $0<\gamma<0.75$. By appropriately constraining the
non-minimal 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 non-minimal 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 $H\sim
\Lambda^{1/2}$, where $\Lambda$ 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
pre-inflationary primordial stiff era, in which the energy density of the axion
scales as $\rho_a\sim a^{-6}$.
