  Assuming axions are potential dark matter (DM) candidate that make up all of
the DM abundance, we discuss production of axions via (i) standard misalignment
mechanism during the period of reheating and (ii) graviton-mediated 2-to-2
scattering of the inflaton and bath particles, where the inflaton $\phi$
oscillates in a monomial potential $V(\phi)\propto\phi^k$ with a general
equation of state. Considering reheating takes place purely gravitationally,
mediated by massless gravitons, we explore the viable region of the parameter
space that agrees with the observed DM relic abundance, satisfying bounds from
big bang nucleosynthesis (BBN) and cosmic microwave background radiation (CMB).
We also discuss complementarity between dedicated axion search experiments and
futuristic gravitational wave search facilities in probing the viable parameter
space.
