Strong lensing signatures of self-interacting dark matter in low-mass haloes

Core formation and runaway core collapse in models with self-interacting dark matter (SIDM) significantly alter the central density profiles of collapsed haloes. Using a forward modelling inference framework with simulated data-sets, we demonstrate that flux ratios in quadruple image strong gravitational lenses can detect the unique structural properties of SIDM haloes, and statistically constrain the amplitude and velocity dependence of the interaction cross-section in haloes with masses between 10(6) and 10(10) M-circle dot. Measurements on these scales probe self-interactions at velocities below 30 km s(-1), a relatively unexplored regime of parameter space, complimenting constraints at higher velocities from galaxies and clusters. We cast constraints on the amplitude and velocity dependence of the interaction cross-section in terms of sigma(20), the cross-section amplitude at 20 km s(-1). With 50 lenses, a sample size available in the near future, and flux ratios measured from spatially compact mid-IR emission around the background quasar, we forecast sigma(20) < 11-23 cm(2)g(-1) at 95 per cent CI, depending on the amplitude of the subhalo mass function, and assuming cold dark matter (CDM). Alternatively, if sigma(20) = 19.2 cm(2)g(-1) we can rule out CDM with a likelihood ratio of 20:1, assuming an amplitude of the subhalo mass function that results from doubly efficient tidal disruption in the Milky Way relative to massive elliptical galaxies. These results demonstrate that strong lensing of compact, unresolved sources can constrain SIDM structure on sub-galactic scales across cosmological distances, and the evolution of SIDM density profiles over several Gyr of cosmic time.