Gravitational Lens Statistics for Generalized Nfw Profiles: Parameter Degeneracy and Implications for Self-interacting Cold Dark Matter

Strong lensing is a powerful probe of the distribution of matter in the cores of clusters. Recent data suggests that the cold dark matter model predicts cores that are denser than those observed in galaxies, groups and clusters. One possible resolution of the discrepancy is that the dark matter has strong interactions (SIDM), which leads to lower central densities. A generalized form of the Navarro, Frenk and White profile (Zhao profile) may be used to describe these halos. In this paper we examine lensing statistics for this class of model. The optical depth to multiple imaging is a very sensitive function of the profile parameters in the range of interest for SIDM halos around clusters of galaxies. Less concentrated profiles, which result from larger self-interaction cross-sections, produce significantly fewer lensed pairs. Furthermore, profiles that result in a small optical depth exhibit reduced typical splittings, but produce multiple images that are more highly magnified. We find that lensing statistics based on these profiles obtained from fits out to the virial radius are dependent on the minimization scheme adopted, and may be seriously in error. However, profile fits weighted towards the core region have parameter degeneracies that are approximately equivalent to those for strong lensing cross-sections. Lensing statistics provide a powerful test for SIDM. More realistic and observationally oriented calculations remain to be done, however larger self-interaction cross-sections may well be ruled out by the very existence of strong lenses on galaxy cluster scales. In future statistical studies, it will be important to properly take account of the scatter among halo profiles since the optical depth to multiple imaging is dominated by the more concentrated members of a cluster population.