Critical assessment of CMB limits on dark matter-baryon scattering: New treatment of the relative bulk velocity

We perform an improved cosmic microwave background (CMB) analysis to search for dark matter- proton scattering with a momentum-transfer cross section of the form sigma(0)v(n) for n = -2 and n = -4. In particular, we present a new and robust prescription for incorporating the relative bulk velocity between the dark matter and baryon fluids into the standard linear Boltzmann calculation. Using an iterative procedure, we self-consistently include the effects of the bulk velocities in a cosmology in which dark matter interacts with baryons. With this prescription, we derive CMB bounds on the cross section, excluding sigma(0) > 2.3 x 10(-33) cm(2) for n = -2 and sigma(0) > 1.7 x 10(-41) cm(2) for n = -4 at 95% confidence, for dark matter masses below 10 MeV. Furthermore, we investigate how these constraints change when only a subcomponent of dark matter is interacting. We show that Planck limits vanish if less than or similar to 0.4% of dark matter is tightly coupled to baryons. We discuss the implications of our results for present and future cosmological observations.