LAUSR

Milky Way (MW) satellites exhibit a diverse range of internal kinematics, reflecting in turn a diverse set of subhalo density profiles. These profiles include large cores and dense cusps, which any successful dark matter model must explain simultaneously. A plausible driver of such diversity is self-interactions between dark matter particles (SIDM) if the cross section passes the threshold for the gravothermal collapse phase at the characteristic velocities of the MW satellites. In this case, some of the satellites are expected to be hosted by subhalos that are still in the classical SIDM core phase, while those in the collapse phase would have cuspy inner profiles, with a SIDM-driven intermediate mass black hole (IMBH) in the centre as a consequence of the runaway collapse. We develop an analytical framework that takes into account the cosmological assembly of halos and is calibrated to previous simulations; we then predict the timescales and mass scales ($M_{\rm BH}$) for the formation of IMBHs in velocity-dependent SIDM (vdSIDM) models as a function of the present-day halo mass, $M_0$. Finally, we estimate the region in the parameter space of the effective cross section and $M_0$ for a subclass of vdSIDM models that result in a diverse MW satellite population, as well as their corresponding fraction of SIDM-collapsed halos and those halos' inferred IMBH masses. We predict the latter to be in the range $0.1-1000~ {\rm M_\odot}$ with a $M_{\rm BH}-M_0$ relation that has a similar slope, but lower normalization, than the extrapolated empirical relation of super-massive black holes found in massive galaxies.