We analyse strongly lensed images in 8 galaxy clusters to measure their dark matter density profiles in the radial region between 10 kpc and 150 kpc, and use this to constrain the self-interaction cross-section of dark matter (DM) particles. We infer the mass profiles of the central DM haloes, bright central galaxies, key member galaxies, and DM subhalos for the member galaxies for all 8 clusters using the QLenscode. The inferred DM halo surface densities are fit to a self-interacting dark matter (SIDM) model, which allows us to constrain the self-interaction cross-section over mass σ/m. When our full method is applied to mock data generated from two clusters in the Illustris-TNG simulation, we find results consistent with no dark matter self-interactions as expected. For the eight observed clusters with average relative velocities of $1458_{-81}^{+80}$ km/s, we infer $\sigma /m = 0.082_{-0.021}^{+0.027} \rm cm^2/g$ and $\sigma /m < 0.13~ \rm cm^2/g$ at the 95 per cent confidence level.

We test the ability of strong lensing data to constrain the size of a central core in the dark matter halos of galaxy clusters, using Abell 611 as a prototype. Using simulated data, we show that modeling a cluster halo with ellipticity in the gravitational potential can bias the inferred mass and concentration, which may bias the inferred central density when weak lensing or X-ray data are added. We also the highlight the possibility for spurious constraints on the core size if the radial density profile is different from the assumed model. These systematics can be ameliorated if central images are present in the data. Applying our methodology to Abell 611 and imposing a reasonable prior on the stellar mass-to-light ratio restricts the core size to be less than about 4 kpc, with a minimum reduced χ 2 of 0.28 for 0. 2 positional errors. Such small cores imply a constraint on the dark matter self-interaction cross section of the order of 0.1 cm 2 /g at relative velocities of about 1500 km/s.