  We present and compare several cosmological constraints on the cross section
for elastic scattering between dark matter (DM) and baryons, for cross sections
with a range of power-law dependences on the DM-baryon relative velocity $v$,
especially focusing on the case of $\sigma \propto v^{-4}$. We study
constraints spanning a wide range of epochs in cosmological history, from
pre-recombination distortions to the blackbody spectrum and anisotropies of the
cosmic microwave background (CMB), to modifications to the intergalactic medium
temperature and the resulting 21cm signal, and discuss the allowed signals in
the latter channels given the constraints from the former. We improve previous
constraints on DM-baryon scattering from the CMB anisotropies, demonstrate via
principal component analysis that the effect on the CMB can be written as a
simple function of DM mass, and map out the redshifts dominating this signal.
We show that given high-redshift constraints on DM-baryon scattering, a
$v^{-4}$ scaling of the cross section for light DM would be sufficient to
explain the deep 21cm absorption trough recently claimed by the EDGES
experiment, if 100% of the DM scatters with baryons. For millicharged DM models
proposed to explain the observation, where only a small fraction of the DM
interacts, we estimate that a PIXIE-like future experiment measuring CMB
spectral distortion could test the relevant parameter space.
