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Numerical solution of the exact background collisional Boltzmann equation for dark matter-baryon scattering
Suroor Seher Gandhi, Yacine Ali-Haïmoud
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Linear cosmological observables can be used to probe elastic scattering of dark matter (DM) with baryons. Availability of high-precision data requires a critical reassessment of any assumptions that may impact the accuracy of constraints. The standard formalism for constraining DM-baryon scattering pre-recombination is based on assuming a Maxwell-Boltzmann (MB) velocity distribution for DM. This assumption is not always justified and does not allow for probing DM self-interactions in addition to its interactions with baryons. Lifting the MB assumption requires solving the full collisional Boltzmann equation (CBE), which is highly non-trivial. Earlier work proposed a more tractable Fokker-Planck (FP) approximation to the CBE, but its accuracy remained unknown. In this work, we numerically solve the exact CBE for the first time, in a homogeneous expanding background. We consider DM-baryon scattering cross-sections that are positive power-laws of relative velocity. We derive analytical expressions for the collision operator in the case of isotropic differential scattering cross-sections. We then solve the background CBE numerically and use our solution for the DM velocity distribution to compute the DM-baryon heat-exchange rate, which we compare against those obtained with the MB assumption and FP approximation. Over a broad range of DM-to-baryon mass ratios, we find that the FP approximation leads to a maximum error of 17%, significantly better than the up to 160% error introduced by the MB assumption. While our results strictly apply only to the background evolution, the accuracy of the FP approximation is likely to carry over to perturbations. This motivates its implementation into cosmological Boltzmann codes, where it can supersede the much less accurate MB assumption, and allow for a more general exploration of DM interactions with baryons and with itself.
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