We show that the relic abundance and expected mass range of the QCD axion, a
hypothetical particle that can potentially constitute the cosmic dark matter
(DM), are greatly modified if the axion field resulting from the evaporation of
primordial black holes (PBHs) begins to oscillate just before the onset of Big
Bang Nucleosynthesis (BBN). We predominantly explore the PBHs in the mass range
$(10^6 – 5\times 10^8)\,$g. We investigate the relation between the relic
abundance of DM axion and the primordial population of black holes. We
numerically solve the set of Boltzmann equations that governs the cosmological
evolution during the radiation bath and the PBH-dominated epoch, providing the
bulk energy content of the early Universe. We further solve the equation of
motion of the axion field in addition to obtaining its present abundance. If
the QCD axion is ever discovered, it will give us insights into the early
Universe and probe into the physics of the PBH-dominated era. Light QCD axions,
alongside non-relativistic particles, are generated from PBHs evaporation
through Hawking radiation and could make up a fraction of dark radiation (DR).
We estimate the bounds on the model from DR axions produced via said PBH
evaporation and thermal decoupling, and we account for isocurvature bounds
during the period of inflation where the Peccei-Quinn symmetry is broken. We,
additionally, take the results obtained and put them against the available CMB
data and state our observations. We briefly study the forecasts from
gravitational wave searches. We comment on the consequences of PBH accretion
and on the uncertainties it may further add to particle physics modeling.
