Constraints on Dark Matter with a moderately large and velocity-dependent DM-nucleon cross-section

We derive constraints on a possible velocity-dependent DM-nucleon scattering cross section, for Dark Matter in the 10 MeV -- 100 GeV mass range, using the XQC, DAMIC, and CRESST 2017 Surface Run experiments. We report the limits on cross sections of the form $\sigma=\sigma_0\,v^n$, for a range of velocity dependencies with $n\in\{-4,-2,-1,0,1,2\}$. We point out the need to measure the efficiency with which nuclear recoil energy in the sub-keV range thermalizes, rather than being stored as Frenkel pairs in the semi-conductor lattice. The possibility of a significant inefficiency leaves open a considerable `hole' in the limits for mass in the $\sim$ 0.2 -- 2 GeV range, which XQC and CRESST can potentially fill when the thermalization efficiency is measured. We call attention to the asymmetry between a conventional lower limit cross section and the `upper-reach cross section' imposed by attenuation in an overburden -- an upper boundary being extremely sharp but quite insensitive to the statistics of the experiment. Considering the recent interest to use dark matter-baryon interaction with velocity dependence $n=-4$ to explain the EDGES 21 cm anomaly, we also derive the limits on milli-charged DM that scatters off protons and electrons under a Coulomb-like interaction. We find that much but not all of the region of interest for the EDGES anomaly can be excluded.