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 \times 10 ^ { -33 } ~ { } \mathrm { cm } ^ { 2 } for n = -2 and \sigma _ { 0 } > 1.7 \times 10 ^ { -41 } ~ { } \mathrm { 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 \lesssim 0.4 \% of dark matter is tightly coupled to baryons .
We discuss the implications of our results for present and future cosmological observations .