Axion dark matter, solitons, and the cusp-core problem
  Self-gravitating bosonic fields can support stable and localised field
configurations. For real fields, these solutions oscillate in time and are
known as oscillatons. The density profile is static, and is soliton. Such
solitons should be ubiquitous in models of axion dark matter, with the soliton
characteristic mass and size depending on some inverse power of the axion mass.
Stable configurations of non-relativistic axions are studied numerically using
the Schr\"{o}dinger-Poisson system. This method, and the resulting soliton
density profiles, are reviewed. Using a scaling symmetry and the uncertainty
principle, the core size of the soliton can be related to the central density
and axion mass, $m_a$, in a universal way. Solitons have a constant central
density due to pressure-support, unlike the cuspy profile of cold dark matter
(CDM). One consequence of this fact is that solitons composed of ultra-light
axions (ULAs) may resolve the `cusp-core' problem of CDM. In DM halos,
thermodynamics will lead to a CDM-like Navarro-Frenk-White profile at large
radii, with a central soliton core at small radii. Using Monte-Carlo techniques
to explore the possible density profiles of this form, a fit to
stellar-kinematical data of dwarf spheroidal galaxies is performed. In order
for ULAs to resolve the cusp-core problem (without recourse to baryon feedback
or other astrophysical effects) the axion mass must satisfy $m_a<1.1\times
10^{-22}\text{ eV}$ at 95\% C.L. On the other hand, ULAs with $m_a\lesssim
1\times 10^{-22}\text{ eV}$ are in some tension with cosmological structure
formation. An axion solution to the cusp-core problem thus makes novel
predictions for future measurements of the epoch of reionisation. On the other
hand, this can be seen as evidence that structure formation could soon impose a
\emph{Catch 22} on axion/scalar field DM, similar to the case of warm DM.
