Constraints on the velocity profiles of galaxies from strong lensing statistics and semi-analytical model of galaxy formation

Semi-analytical models of galaxy formation can be used to predict the evolution of the number density of early-type galaxies as a function of the circular velocity at the virial radius, v_c. Gravitational lensing probability and separation distribution on the other hand are sensitive to the central velocity dispersions, \sigma. We use radio lenses from the Cosmic Lens All-Sky Survey and the PMN-NVSS Extragalactic Lens Survey to study how the central velocity dispersions, \sigma, are related to v_c; for a singular isothermal sphere, v_c = \sqrt{2} \sigma. When we include both the lensing probability and separation distribution as our lensing constraints, we find \sigma/(200 km/s) = [(1.17_{-0.26}^{+0.40}) v_c/ (200 km/s)]^{0.22^{+0.05}_{-0.04}} for 200 km/s < \sigma < 260 km/s; at \sigma=200 km/s, the ratio \sqrt{2}\sigma/ v_c is about 1.65^{+0.57}_{-0.37} (68% CL) but decreases to 0.65_{-0.12}^{+0.15} (68% CL) for \sigma=260 km/s. These results are consistent with those of Seljak (2002) obtained from galaxy-galaxy weak lensing for galaxies of around $L_*$. However, our results clearly suggest that the ratio must vary significantly as \sigma is varied and are marginally discrepant with the Seljak results at \sigma=260 km/s. The scaling \sigma ~ v_c^{0.22+/-0.05} is broadly consistent with those from galaxy occupation statistics studies and the most recent galaxy-galaxy weak lensing study. These constraints can be significantly strengthened when larger lens samples become available and the accuracy of semi-analytical model predictions improves.