We obtain the first cosmological constraints on interactions between dark matter and protons within the formalism of nonrelativistic effective field theory developed for direct detection. For each interaction operator in the effective theory, parametrized by different powers of the relative velocity of the incoming particles, we use the Planck 2015 cosmic microwave background (CMB) temperature, polarization, and lensing anisotropy to set upper limits on the scattering cross section for all dark matter masses above 15 keV. We find that for interactions associated with a stronger dependence on velocity, dark matter and baryons stay thermally coupled for longer, but the interaction strengths are suppressed at the low temperatures relevant for Planck observations and are thus less constrained. At the same time, cross sections with stronger velocity dependencies are more constrained in the limit of small dark matter mass. In all cases, the effect of dark matter-proton scattering is most prominent on small scales in the CMB power spectra and in the matter power spectrum, and we thus expect substantial improvement over the current limits with data from ground-based CMB experiments and galaxy surveys.
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