Boltzmann-Fokker-Planck formalism for dark-matter--baryon scattering
  Linear-cosmology observables, such as the Cosmic Microwave Background (CMB),
or the large-scale distribution of matter, have long been used as clean probes
of dark matter (DM) interactions with baryons. It is standard to model the DM
as an ideal fluid with a thermal Maxwell-Boltzmann (MB) velocity distribution,
in order to compute the heat and momentum-exchange rates relevant to these
probes. This approximation only applies in the limit where DM self-interactions
are frequent enough to efficiently redistribute DM velocities. It does not
accurately describe weakly self-interacting particles, whose velocity
distribution unavoidably departs from MB once they decouple from baryons. This
article lays out a new formalism required to accurately model DM-baryon
scattering, even when DM self-interactions are negligible. The ideal fluid
equations are replaced by the collisional Boltzmann equation for the DM
phase-space distribution. The collision operator is approximated by a
Fokker-Planck operator, constructed to recover the exact heat and momentum
exchange rates, and allowing for an efficient numerical implementation.
Numerical solutions to the background evolution are presented, which show that
the MB approximation can over-estimate the heat-exchange rate by factors of ~
2-3, especially for light DM particles. A Boltzmann-Fokker-Planck hierarchy for
perturbations is derived. This new formalism allows to explore a wider range of
DM models, and will be especially relevant for upcoming ultra-high-sensitivity
CMB probes.
