Self-interacting dark matter halos and the gravothermal catastrophe

We study the evolution of an isolated spherical halo of self-interacting dark matter (SIDM) in the gravothermal fluid formalism. We show that the thermal relaxation time t(r) of an SIDM halo with the central density and velocity dispersion of a typical dwarf galaxy is significantly shorter than its age. We find a self-similar solution for the evolution of an SIDM halo in the limit where the mean free path between collisions, lambda, is longer than the gravitational scale height H everywhere. Typical halos formed in this long mean free path regime relax to a quasi-stationary gravothermal density profile characterized by a nearly homogeneous core and a power-law halo where rho proportional to r(-2.19). We solve the more general time-dependent problem and show that the contracting core evolves to sufficiently high density that lambda inevitably becomes smaller than H in the innermost region. The core undergoes secular collapse to a singular state ( the gravothermal catastrophe) in a time t(coll) approximate to 290t(r), which is longer than the Hubble time for a typical dark matter dominated galaxy core at the present epoch. Our model calculations are consistent with previous more detailed N-body simulations for SIDM, providing a simple physical interpretation of their results and extending them to higher spatial resolution and longer evolution times. At late times, mass loss from the contracting dense inner core to the ambient halo is significantly moderated, so that the final mass of the inner core may be appreciable when it becomes relativistic and radially unstable to dynamical collapse to a black hole.