  Axions and axion-like particles are compelling candidates for the missing
dark matter of the universe. As they undergo gravitational collapse, they can
form compact objects such as axion stars or even black holes. In this paper, we
study the formation and distribution of such objects. First, we simulate the
formation of compact axion stars using numerical relativity with aspherical
initial conditions that could represent the final stages of axion dark matter
structure formation. We show that the final states of such collapse closely
follow the known relationship of initial mass and axion decay constant $f_a$.
Second, we demonstrate with a toy model how this information can be used to
scan a model density field to predict the number densities and masses of such
compact objects. In addition to being detectable by the LIGO/VIRGO
gravitational wave interferometer network for axion mass of $10^{-9} < m_a <
10^{-11}$ eV, we show using peak statistics that for $f_a < 0.2M_{pl}$, there
exists a "mass gap" between the masses of axion stars and black holes formed
from collapse.
