The Velocity-Dependent $J$-factor of the Milky Way Halo: Does What Happens in the Galactic Bulge Stay in the Galactic Bulge?

Abstract: We consider the angular distribution of the photon signal which could arise from velocity-dependent dark matter annihilation within the Galactic bulge. We find that, for the case of Sommerfeld-enhanced annihilation, dark matter annihilation within the bulge is dominated by slow speed particles which never leave the bulge, allowing one to find a simple analytic relationship between the dark matter profile within the Galactic bulge and the angular distribution. On the other hand, for the case $p$- or $d$-wave annihilation, we find that the small fraction of high-speed particles which can leave the bulge provide a significant, often dominant, contribution to dark matter annihilation within the bulge. For these scenarios, fully understanding dark matter annihilation deep within the Galactic bulge, and the angular distribution of the resulting photon signal, requires an understanding of the dark matter profile well outside the bulge. We consider the Galactic Center excess in light of these results, and find that an explanation of this excess in terms of $p$-wave annihilation would require the dark matter profile within the bulge to have a much steeper profile than usually considered, but with uncertainties related to the behavior of the profile outside the bulge.