Resonant Scattering between Dark Matter and Baryons: Revised Direct
  Detection and CMB Limits
  Traditional dark matter models, eg. WIMPs, assume dark matter is weakly
coupled to the standard model so that elastic scattering between dark matter
and baryons can be described perturbatively by Born approximation. Most direct
detection experiments are analyzed according to that assumption. We show that
when the fundamental DM-baryon interaction is attractive, dark matter-nucleus
scattering is non-perturbative in much of the relevant parameter range. The
cross section exhibits rich resonant behavior with a highly non-trivial
dependence on atomic mass; furthermore, the extended rather than point-like
nature of nuclei significantly impacts the cross sections. The repulsive case
also requires full numerical calculation. These non-perturbative effects change
existing constraints.
  Near a resonance value of the parameters, the cross section has non-trivial
velocity dependence rather than the usual $\sigma \sim v^0$; we take the
velocity dependence into account. (However doing so has little impact on
current constraints.)
  We report the corrected exclusion regions superseding previous limits from
XQC, CRESST Surface Run, CMB power spectrum and extensions with Lyman-$\alpha$
and Milky Way satellites, and Milky Way gas clouds. Some limits become weaker
than previous bounds in the literature, while others become stronger. Gaps
which open by correct treatment of some particular constraint can sometimes be
closed using a different constraint. We also discuss the dependence on mediator
mass and give approximate expressions for the velocity dependence near a
resonance. Sexaquark ($uuddss$) DM with mass around 2 GeV, which exchanges QCD
mesons with baryons, remains unconstrained for most of the parameter space of
interest.
  A statement in the literature that a DM-nucleus cross section larger than
$10^{-25}\,{\rm cm}^2$ implies dark matter is composite, is corrected.
