Towards an improved model of self-interacting dark matter haloes

In this work, we discuss the relation between the strength of the self-interaction of dark matter particles and the predicted properties of the inner density distributions of dark matter haloes. We present the results of $N$-body simulations for 28 haloes performed with the same initial conditions for cold dark matter and for self-interacting dark matter, for a range of cross-sections. We provide a simple phenomenological description of these results and compare with the semi-analytical model typically used in the literature. Using these results, we then predict how the inner dark matter surface density and core radius should depend on the self-interaction cross-section for observed haloes.