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.
