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Linear cosmological observables can be used to probe elastic scattering ofdark matter (DM) with baryons. Availability of high-precision data requires acritical reassessment of any assumptions that may impact the accuracy ofconstraints. The standard formalism for constraining DM-baryon scatteringpre-recombination is based on assuming a Maxwell-Boltzmann (MB) velocitydistribution for DM. This assumption is not always justified and does not allowfor probing DM self-interactions in addition to its interactions with baryons.Lifting the MB assumption requires solving the full collisional Boltzmannequation (CBE), which is highly non-trivial. Earlier work proposed a moretractable Fokker-Planck (FP) approximation to the CBE, but its accuracyremained unknown. In this work, we numerically solve the exact CBE for thefirst time, in a homogeneous expanding background. We consider DM-baryonscattering cross-sections that are positive power-laws of relative velocity. Wederive analytical expressions for the collision operator in the case ofisotropic differential scattering cross-sections. We then solve the backgroundCBE numerically and use our solution for the DM velocity distribution tocompute the DM-baryon heat-exchange rate, which we compare against thoseobtained with the MB assumption and FP approximation. Over a broad range ofDM-to-baryon mass ratios, we find that the FP approximation leads to a maximumerror of 17%, significantly better than the up to 160% error introduced by theMB assumption. While our results strictly apply only to the backgroundevolution, the accuracy of the FP approximation is likely to carry over toperturbations. This motivates its implementation into cosmological Boltzmanncodes, where it can supersede the much less accurate MB assumption, and allowfor a more general exploration of DM interactions with baryons and with itself.

Numerical solution of the exact background collisional Boltzmann equation for dark matter-baryon scattering