  Self-interacting dark matter scenarios have recently attracted much
attention, as a possible means to alleviate the tension between N-body
simulations and observations of the dark matter distribution on galactic and
sub-galactic scales. The presence of internal structure for the dark matter ---
for example, a nearly-degenerate state in the spectrum that could decay, or be
collisionally excited or de-excited --- has also been proposed as a possible
means to address these discrepancies. Such internal structure can be a source
of interesting signatures in direct and indirect dark matter searches, for
example providing a novel explanation for the 3.5 keV line recently observed in
galaxies and galaxy clusters. We analyze a simple model of dark matter
self-scattering including a nearly-degenerate excited state, and develop an
accurate analytic approximation for the elastic and inelastic s-wave cross
sections, which is valid outside the perturbative regime provided the particle
velocity is sufficiently low (this condition is also required for the s-wave to
dominate over higher partial waves). We anticipate our results will be useful
in incorporating inelastic self-scattering into N-body simulations, in order to
study the quantitative impact of nearly-degenerate states in the dark matter
spectrum on galactic structure and dynamics, and in computing the indirect
signatures of multi-state dark matter.
