Calibrating the self-interacting dark matter gravothermal catastrophe with N -body simulations

Self-interacting dark matter (SIDM) theories predict that dark matter halos experience core-collapse in late-stage evolution, a process where the halo's inner region rapidly increases in density and decreases in size. This process can be modeled by treating the dark matter as a gravothermal fluid, and solving the fluid equations to predict the density profile evolution. This model is incomplete without calibration to N-body simulations, through a constant factor $β$ included in the thermal conductivity for the long-mean-free-path limit. The value of $β$ employed in the gravothermal fluid formalism has varied between studies, with no clear universal value in the literature. In this work, we use the N-body code Arepo to conduct a series of isolated core-collapse simulations across a range of scattering cross-sections, halo concentrations, and halo masses to calibrate the heat transfer parameter $β$. We find that $β$ is independent of cross-section, halo concentration, and halo mass for velocity independent elastic scattering cross-sections. We present a model for an effective $β$ as a function of a dimensionless cross-section, to describe halo evolution in the long mean free path limit, and show that it accurately captures halo evolution as long as the cross section is not too large. This effective model facilitates comparisons between simulations and the gravothermal model, and enables fast predictions of the dark matter density profile at any given time without running N-body simulations.