  Axion-like particles (ALPs) are leading candidates to explain the dark matter
in the universe. Their production via the misalignment mechanism has been
extensively studied for cosine potentials characteristic of
pseudo-Nambu-Goldstone bosons. In this work we investigate ALPs with
non-periodic potentials, which allow for large misalignment of the field from
the minimum. As a result, the ALP can match the relic density of dark matter in
a large part of the parameter space. Such potentials give rise to
self-interactions which can trigger an exponential growth of fluctuations in
the ALP field via parametric resonance, leading to the fragmentation of the
field. We study these effects with both Floquet analysis and lattice
simulations. Using the Press-Schechter formalism, we predict the halo mass
function and halo spectrum arising from ALP dark matter. These halos can be
dense enough to produce observable gravitational effects such as astrometric
lensing, diffraction of gravitational wave signals from black hole mergers,
photometric microlensing of highly magnified stars, perturbations of stars in
the galactic disk or stellar streams. These effects would provide a probe of
dark matter even if it does not couple to the Standard Model. They would not be
observable for halos predicted for standard cold dark matter and for ALP dark
matter in the standard misalignment mechanism. We determine the relevant
regions of parameter space in the (ALP mass, decay constant)-plane and compare
predictions in different axion fragmentation models.
