Weak annihilation cusp inside the dark matter spike about a black hole

We reinvestigate the effect of annihilations on the distribution of collisionless dark matter (DM) in a spherical density spike around a massive black hole. We first construct a very simple, pedagogic, analytic model for an isotropic phase space distribution function that accounts for annihilation and reproduces the "weak cusp" found by Vasiliev for DM deep within the spike and away from its boundaries. The DM density in the cusp varies as r^-1/2 for s-wave annihilation, where r is the distance from the central black hole, and is not a flat "plateau" profile. We then extend this model by incorporating a loss cone that accounts for the capture of DM particles by the hole. The loss cone is implemented by a boundary condition that removes capture orbits, resulting in an anisotropic distribution function. Finally, we evolve an initial spike distribution function by integrating the Boltzmann equation to show how the weak cusp grows and its density decreases with time. We treat two cases, one for s-wave and the other for p-wave DM annihilation, adopting parameters characteristic of the Milky Way nuclear core and typical WIMP models for DM. The cusp density profile for p-wave annihilation is weaker, varying like ~r^-0.34, but is still not a flat plateau.