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 of the (velocity-dependent) Sommerfeld enhancement of the annihilation cross section. 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 that can be used to estimate the overall boost factors resulting from these combined effects, from which the intricate phenomenology can be better understood. DM subhalos lead to an increase of the Sommerfeld effect by several orders of magnitude (for both $s$- and $p$-wave annihilation processes), especially on resonances, which makes them critical to get sensible gamma-ray signal predictions for typical targets of different masses (from dwarf galaxies to galaxy clusters).