  If dark matter has strong self-interactions, future astrophysical and
cosmological observations, together with a clearer understanding of baryonic
feedback effects, might be used to extract the velocity dependence of the dark
matter scattering rate. To interpret such data, we should understand what
predictions for this quantity are made by various models of the underlying
particle nature of dark matter. In this paper, we systematically compute this
function for fermionic dark matter with light bosonic mediators of vector,
scalar, axial vector, and pseudoscalar type. We do this by matching to the
nonrelativistic effective theory of self-interacting dark matter and then
computing the spin-averaged viscosity cross section nonperturbatively by
solving the Schrodinger equation, thus accounting for any possible Sommerfeld
enhancement of the low-velocity cross section. In the pseudoscalar case, this
requires a coupled-channel analysis of different angular momentum modes. We
find, contrary to some earlier analyses, that nonrelativistic effects only
provide a significant enhancement for the cases of light scalar and vector
mediators. Scattering from light pseudoscalar and axial vector mediators is
well described by tree-level quantum field theory.
