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. The numerical
results from our semi-analytical method are in good agreement with those from
fully relativistic N-body simulations. 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.
