Numerical solution of the exact background collisional Boltzmann
  equation for dark matter-baryon scattering
  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. All existing cosmological constraints of DM-baryon scattering
assume that DM has a Maxwell-Boltzmann (MB) velocity distribution in order to
compute heat- and momentum-exchange rates. 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.
