  The axion, motivated as a solution to the strong CP problem, is also a viable
dark matter candidate. We use N-body simulations to study the formation of
substructures from white-noise density fluctuations. The density profiles of
our relaxed axion minihalos can be described by the Navarro-Frenk-White
profile, and the minihalos' concentration number agrees well with a simple,
physically-motivated model. We develop a semi-analytic formula to fit the mass
function from our simulation, which agrees broadly at different redshifts and
only differs at factor of two level from classic halo mass functions. This
analytic mass function allows us to consider uncertainties in the
post-inflation axion scenario, as well as extrapolate our high-redshift
simulations results to the present.
  Our work estimates the present-day abundance of axion substructures, as is
necessary for predicting their effect on cosmological microlensing caustics and
pulsar timing. Our calculations suggest that if pulsar timing and microlensing
probes can reach recent sensitivity forecasts, they may be sensitive to the
post-inflation axion dark matter scenario, even when accounting for
uncertainties pertaining to axion strings. For pulsar timing, the most
significant caveat is whether axion minihalos are disrupted by stars, which our
estimates show is mildly important at the most relevant masses. Finally, as our
gravitational simulations are scale invariant, the results can be extended to
models where the dark matter is comprised of other axion-like particles and
even clusters of primordial black holes.
