The velocity-dependent J-factor of the Milky Way halo: does what happens in the galactic bulge stay in the galactic bulge?

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