The Effective J-Factor of the Galactic Center for Velocity-Dependent Dark Matter Annihilation

We present the effective $J$-factors for the Milky Way for scenarios in which dark matter annihilation is p-wave or d-wave suppressed. We find that the velocity suppression of dark matter annihilation can have a sizable effect on the morphology of a potential dark matter annihilation signal in the Galactic Center. The gamma-ray flux from the innermost region of the Galactic Center is in particular suppressed. We find that for dark matter density profiles with steep inner slopes, the morphology of the Inner Galaxy gamma-ray emission in p-wave models can be made similar to the morphology in standard s-wave models. This similarity may suggest that model discrimination between s-wave and p-wave is challenging, for example, when fitting the Galactic Center excess. However, we show that it is difficult to simultaneously match s- and p-wave morphologies at both large and small angular scales. The $J$-factors we calculate may be implemented with astrophysical foreground models to self-consistently determine the morphology of the excess with velocity-suppressed dark matter annihilation.