Hints against the cold and collisionless nature of dark matter from the galaxy velocity function

The observed number of dwarf galaxies as a function of rotation velocity is significantly smaller than predicted by the $\Lambda$CDM model. This discrepancy cannot be simply solved by assuming strong baryonic processes, since they would violate the observed relation between maximum circular velocity ($v_{\rm max}$) and baryon mass of galaxies. A speculative but tantalising possibility is that the mismatch between observation and theory points towards the existence of non-cold or non-collisionless dark matter (DM). In this paper, we investigate the effects of warm, mixed (i.e warm plus cold), and self-interacting DM scenarios on the abundance of dwarf galaxies and the relation between observed HI line-width and maximum circular velocity. Both effects have the potential to alleviate the apparent mismatch between the observed and theoretical abundance of galaxies as a function of $v_{\rm max}$. For the case of warm and mixed DM, we show that the discrepancy disappears, even for luke-warm models that evade stringent bounds from the Lyman-$\alpha$ forest. Self-interacting DM scenarios can also provide a solution as long as they lead to extended ($\gtrsim 1.5$ kpc) dark matter cores in the density profiles of dwarf galaxies. Only models with velocity-dependent cross sections can yield such cores without violating other observational constraints at larger scales.