Dynamical Instability of Collapsed Dark Matter Halos

A self-interacting dark matter halo can experience gravothermal collapse, resulting in a central core with an ultrahigh density. It can further contract and collapse into a black hole, a mechanism proposed to explain the origin of supermassive black holes. We study dynamical instability of the core in general relativity. We use a truncated Maxwell-Boltzmann distribution to model the dark matter distribution and solve the Tolman-Oppenheimer-Volkoff equation. For given model parameters, we obtain a series of equilibrium configurations and examine their dynamical instability based on considerations of total energy, binding energy, fractional binding energy, and adiabatic index. Our numerical results indicate that the core can collapse into a black hole when the fractional binding energy reaches $0.035$ with a central gravitational redshift of $0.5$. We further show for the instability to occur in the classical regime, the boundary temperature of the core should be at least $10\%$ of the mass of dark matter particles; for a $10^9~{\rm M_\odot}$ seed black hole, the particle mass needs to be larger than a few keV. These results can be used to constrain different collapse models, in particular, those with dissipative dark matter interactions.