Analytical insight into dark matter subhalo boost factors for
  Sommerfeld-enhanced $s$- and $p$-wave $\gamma$-ray signals
  As searches for thermal and self-annihilating dark matter (DM) intensify, it
becomes crucial to include as many relevant physical processes and ingredients
as possible to refine signal predictions, in particular those which directly
relate to the intimate properties of DM. We investigate the combined impact of
DM subhalos and the (velocity-dependent) Sommerfeld enhancement of the
annihilation cross section, in both the $s$- and $p$-wave cases. Both features
are expected to play an important role in searches for thermal DM particle
candidates with masses around or beyond TeV, or in scenarios with a light dark
sector. We provide a detailed analytical description of the phenomena at play,
and show how they scale with the subhalo masses and the main Sommerfeld
parameters. We derive approximate analytical expressions for the overall boost
factors resulting from these combined effects, from which the intricate
phenomenology can be better understood, and which strongly increase gamma-ray
signal predictions for typical targets of different masses (from dwarf galaxies
to galaxy clusters). DM subhalos lead to an increase of the Sommerfeld effect
by several orders of magnitude (for both the $s$- and $p$-wave cases),
especially on resonances, which makes them critical to get sensible
predictions.
