Inner radius and energy conditions of dark matter halos surrounding Schwarzschild black holes
We study a class of analytic models for a dark matter halo surrounding a Schwarzschild black hole sitting at the center of a galaxy, with a variable inner radius $r_{\text{in}}$ at which the density profile of the dark matter halo vanishes. We examine in detail how the three energy conditions are satisfied in such models. In particular, independent of concrete profile, we find that the three energy conditions are satisfied when $r_{\text{in}}\ge5M/2$, where $M$ denotes the mass of the black hole. This indicates it is crucial to include inner radius when discussing dark matter distributions. All our solutions expressed explicitly in closed form are particularly valuable for the studies of the gravitational waveforms of extreme/intermediate mass ratio inspirals and the nature of dark matter in galaxies.