The Milky Way satellite velocity function is a sharp probe of
  small-scale structure problems
  Twenty years ago, the mismatch between the observed number of Milky Way
satellite galaxies and the predicted number of cold dark matter subhalos was
dubbed the "missing satellites problem". Although mostly framed since in terms
of satellite counts in luminosity space, the missing satellites problem was
originally posed in velocity space. The stellar velocity dispersion function
encodes information about the density profile of satellites as well as their
abundance. We compare the completeness-corrected MW satellite velocity function
down to its ultrafaint dwarfs (L > 340 L$_\odot$) against well-motivated,
semi-empirical predictions based on galaxy-halo scaling relations. For our most
conservative completeness correction, we find good agreement with a simple CDM
model in which massive, classical satellites (M$_{\rm vir} \gtrsim
10^9~$M$_\odot$) have baryon-driven cores, while low-mass, ultrafaint
satellites (M$_{\rm vir} \lesssim 10^9~$M$_\odot$) inhabit cuspy halos that are
not strongly tidally stripped. This bifurication is required to explain a
non-power-law feature in the velocity function at $\sigma_{\rm los}^* \approx
10$ km/s. Intriguingly, this feature could point to a flattening of the
stellar-mass--halo-mass relation. Tidal destruction of satellites by the Milky
Way's disk must be minimal, or the corrected velocity function exceeds any
plausible prediction -- a "too many satellites" problem. We rule out
non-core-collapsing self-interacting dark matter models with a constant cross
section $\gtrsim$ 0.3 cm$^2$/g. Constraints on warm dark matter are stronger
than those based on the luminosity function on account of the velocity
function's additional sensitivity to the central densities of subhalos.
Reducing uncertainties on stellar kinematics and the amount of tidal stripping
experienced by the faintest dwarfs is key to determining the severity of the
too many satellites problem.
