  In this paper we study the density profiles of self-interacting dark matter
(SIDM) haloes spanning the full observable mass range, from dwarf galaxies to
galaxy clusters. Using realistic simulations that model the baryonic physics
relevant for galaxy formation, we compare the density profiles of haloes
simulated with either SIDM or cold and collisionless dark matter (CDM) to those
inferred from observations of stellar velocity dispersion, gas rotation curves,
weak and strong gravitational lensing, and/or X-ray maps. We make our
comparison in terms of the maximal surface density of haloes, circumventing the
need for semi-analytic or parametric models for dark matter density profiles.
We find that the maximal surface density as a function of halo mass is well
reproduced by CDM simulations that include baryons, while for SIDM with a
velocity-independent cross-section of 1 cm$^2/$g, the simulated galaxy clusters
have mean maximal surface densities that are below those of observed systems by
an amount greater than the standard deviation of the observed maximal surface
density at fixed mass. For less massive systems both CDM and SIDM agree with
the observation equally well.
