The surprising accuracy of isothermal Jeans modelling of self-interacting dark matter density profiles

Recent claims of observational evidence for self-interacting dark matter (SIDM) have relied on a semi-analytic method for predicting the density profiles of galaxies and galaxy clusters containing SIDM. We present a thorough description of this method, known as isothermal Jeans modelling, and then test it with a large ensemble of haloes taken from cosmological simulations. Our simulations were run with cold and collisionless dark matter (CDM) as well as two different SIDM models, all with dark matter only variants as well as versions including baryons and relevant galaxy formation physics. Using a mix of different box sizes and resolutions, we study haloes with masses ranging from 3e10 to 3e15 Msun. Overall, we find that the isothermal Jeans model provides as accurate a description of simulated SIDM density profiles as the Navarro-Frenk-White profile does of CDM halos. We can use the model predictions, compared with the simulated density profiles, to determine the input DM-DM scattering cross-sections used to run the simulations. This works especially well for large cross-sections, while with CDM our results tend to favour non-zero (albeit fairly small) cross-sections, driven by a bias against small cross-sections inherent to our adopted method of sampling the model parameter space. The model works across the whole halo mass range we study, although including baryons leads to DM profiles of intermediate-mass (10^12 - 10^13 Msun) haloes that do not depend strongly on the SIDM cross-section. The tightest constraints will therefore come from lower and higher mass haloes: dwarf galaxies and galaxy clusters.