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The observed surface densities of dark matter halos are known to follow asimple scaling law, ranging from dwarf galaxies to galaxy clusters, with a weakdependence on their virial mass. Here we point out that this can not only beused to provide a method to determine the standard relation between halo massand concentration, but also to use large samples of objects in order to placeconstraints on dark matter self-interactions that can be more robust thanconstraints derived from individual objects. We demonstrate our method byconsidering a sample of about 50 objects distributed across the whole halo massrange, and by modelling the effect of self-interactions in a way similar towhat has been previously done in the literature. Using additional input fromsimulations then results in a constraint on the self-interaction cross sectionper unit dark matter mass of about $\sigma/m_\chi\lesssim 0.3$ cm$^2$/g. Weexpect that these constraints can be significantly improved in the future, andmade more robust, by i) an improved modelling of the effect ofself-interactions, both theoretical and by comparison with simulations, ii)taking into account a larger sample of objects and iii) by reducing thecurrently still relatively large uncertainties that we conservatively assign tothe surface densities of individual objects. The latter can be achieved inparticular by using kinematic observations to directly constrain the averagehalo mass inside a given radius, rather than fitting the data to a pre-selectedprofile and then reconstruct the mass. For a velocity-independentcross-section, our current result is formally already somewhat smaller than therange $0.5-5$ cm$^2$/g that has been invoked to explain potentialinconsistencies between small-scale observations and expectations in thestandard collisionless cold dark matter paradigm.

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