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 cor es of clusters of galaxies. Recent studies suggest that the cold dark matte r model predicts cores that are denser than those observed in galaxies, gro ups, and clusters. One possible resolution of the discrepancy is that the d ark matter has strong interactions (SIDM), which leads to lower central den sities. A generalized form of the Navarro, Frenk, & White profile (Zhao pro file) can be used to describe these halos. In this paper we examine gravita tional lensing statistics for this class of model. The optical depth to mul tiple imaging is a very sensitive function of the profile parameters in the range of interest for SIDM halos around clusters of galaxies. Less concent rated profiles, which result from larger self-interaction cross sections, c an produce many fewer lensed pairs. Furthermore, profiles that result in a small optical depth exhibit reduced typical splittings, but produce multipl e images that are more highly magnified. However, the resulting increased m agnification bias does not alter our conclusions. We find that lensing stat istics based on profile parameters obtained from fits out to the virial rad ius are dependent on the minimization scheme adopted, and may be seriously in error. However, profile fits weighted toward the core region have parame ter degeneracies that are approximately equivalent to those for strong-lens ing cross sections. Lensing statistics provide a powerful test for SIDM. Mo re realistic and observationally oriented calculations remain to be done; h owever, larger self-interaction cross sections may well be ruled out by the very existence of strong lenses on galaxy cluster scales. The inclusion of centrally dominant cluster galaxies should boost the cross section to mult iple imaging. However, our preliminary calculations suggest that the additi onal multiple imaging rate is small with respect to the differences in mult iple imaging rate for different halo profiles. In future statistical studie s, it will be important to properly account for the scatter among halo prof iles, since the optical depth to multiple imaging is dominated by the most concentrated members of a cluster population.