  Elucidating the particle physics nature of dark matter (DM) is one of the
great challenges in modern science. The current lack of any direct DM
detections in the laboratory heightens the need for astrophysical constraints,
extending the search to DM models beyond the popular weakly interacting massive
particle (WIMP) scenario. We here apply the classical Rees-Ostriker-Silk
cooling criterion for galaxy formation to models with ultralight axion DM, also
known as fuzzy dark matter (FDM). The resulting constraints provide a heuristic
framework for upcoming observations, and our approximate analysis motivates the
need for future, self-consistent simulations of FDM structure formation. We use
observational constraints for the DM hosts of ultra faint dwarf (UFD) galaxies
in the Local Group, together with the redshift constraints for the onset of
primordial star formation from the recent EDGES 21-cm cosmology measurement, to
illustrate this approach. We find that the existing constraints are
straightforward to reconcile with standard $\Lambda$CDM, but disfavour FDM
axion masses below $\sim 10^{-21}\,{\rm eV}/c^2$. The future potential for
harnessing astrophysical probes of DM particle physics is compelling.
