Constraints on the mass-concentration relation of cold dark matter haloes with 11 strong gravitational lenses

The mass-concentration relation of dark matter haloes reflects the assembly history of objects in hierarchical structure formation scenarios and depends on fundamental quantities in cosmology such as the slope of the primordial matter power spectrum. This relation is unconstrained by observations on sub-galactic scales. We derive the first measurement of the mass-concentration relation using the image positions and flux ratios from 11 quadruple-image strong gravitational lenses (quads) in the mass range $10{6}-10{10} {, mathrm{M}_{odot }}$, assuming cold dark matter. We model both subhaloes and line-of-sight haloes, finite-size background sources, and marginalize over nuisance parameters describing the lens macromodel. We also marginalize over the logarithmic slope and redshift evolution of the mass-concentration relation, using flat priors that encompass the range of theoretical uncertainty in the literature. At z = 0, we constrain the concentration of $10{8} , mathrm{M}-{odot }$ haloes $c=12-{-5}{+6}$ at $68 {{ rm per cent}}$ CI, and $c=12-{-9}{+15}$ at $95 {{ rm per cent}}$ CI. For a $10{7} {, mathrm{M}-{odot }}$ halo, we obtain $68 {{ rm per cent}}$ ($95 {{ rm per cent}}$) constraints $c=15-{-8}{+9}$ ($c=15-{-11}{+18}$), while for $10{9} , mathrm{M}-{odot }$ haloes $c=10-{-4}{+7}$ ($c=10-{-7}{+14}$). These results are consistent with the theoretical predictions from mass-concentration relations in the literature and establish strong lensing by galaxies as a powerful probe of halo concentrations on sub-galactic scales across cosmological distance.