Constraining self-interacting dark matter with scaling laws of observed halo surface densities

The observed surface densities of dark matter halos are known to follow a simple scaling law, ranging from dwarf galaxies to galaxy clusters, with a weak dependence on their virial mass. Here we point out that this can not only be used to provide a method to determine the standard relation between halo mass and concentration, but also to use large samples of objects in order to place constraints on dark matter self-interactions that can be more robust than constraints derived from individual objects. We demonstrate our method by considering a sample of about 50 objects distributed across the whole halo mass range, and by modelling the effect of self-interactions in a way similar to what has been previously done in the literature. Using additional input from simulations then results in a constraint on the self-interaction cross section per unit dark matter mass of about $\sigma/m_\chi\lesssim 0.3$ cm$^2$/g. We expect that these constraints can be significantly improved in the future, and made more robust, by i) an improved modelling of the effect of self-interactions, both theoretical and by comparison with simulations, ii) taking into account a larger sample of objects and iii) by reducing the currently still relatively large uncertainties that we conservatively assign to the surface densities of individual objects. The latter can be achieved in particular by using kinematic observations to directly constrain the average halo mass inside a given radius, rather than fitting the data to a pre-selected profile and then reconstruct the mass. For a velocity-independent cross-section, our current result is formally already somewhat smaller than the range $0.5-5$ cm$^2$/g that has been invoked to explain potential inconsistencies between small-scale observations and expectations in the standard collisionless cold dark matter paradigm.