$J$-factors for Velocity-dependent Dark Matter

If dark matter annihilates with a velocity-dependent cross section within a subhalo, then the magnitude and angular distribution of the resulting photon signal will change. These effects are encoded in the $J$-factor. In this work we compute the $J$-factor for a variety of choices for the cross section velocity-dependence, and for a variety of choices for the dark matter profile, including generalized Navarro-Frenk-White (NFW), Einasto, Burkert and Moore. We find that, for cuspy profiles, the angular distribution of a future signal can potentially be used to determine the velocity-dependence of the annihilation cross section, and that these results are robust to changes in the form of the profile. For a cored profile, determining the velocity-dependence of the cross section is more difficult, but potentially still possible. Interestingly, we find that for a density profile with an inner slope power law steeper than 4/3, Sommerfeld-enhanced annihilation in the Coulomb limit leads to a divergence at the center, requiring a more detailed treatment of departure from the Coulomb limit.