2006MNRAS.373.1369C

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,vir. Gravitational lensing probability and separation distribution on the other hand are sensitive to the velocity dispersion (or circular velocity) at about the effective radius. We adopt the singular isothermal ellipsoid (SIE) lens model to estimate the velocity dispersion at the effective radius. The velocity dispersion from strong lensing based on the SIE, σ_SIEis then closely related to the observational central stellar velocity dispersion, σ_cent; we have empirically σ_SIE~ σ_cent. We use radio lenses from the Cosmic Lens All-Sky Survey and the PMN-NVSS Extragalactic Lens Survey to study how the velocity dispersions, σ_SIE, are related to v_c,vir; if the galaxy were a singular isothermal sphere up to the virial radius,. When we include both the lensing probability and separation distribution as our lensing constraints, we find for 200km/s ≲ σ_SIE≲ 260km/s; at σ_SIE= 200km/s, the ratio is about 1.65^+0.57_–0.37[68 per cent confidence limit (CL)] but decreases to 0.65^+0.15_–0.12(68 per cent CL) for σ_SIE= 260km/s. These results are consistent with those of Seljak obtained from galaxy-galaxy weak lensing for galaxies of around L_*. However, our results clearly suggest that the ratio must vary significantly as σ_SIEis varied and are marginally discrepant with the Seljak results at σ_SIE= 260km/s. The scaling σ_SIE ∝ v^0.22±0.05_c,viris broadly consistent with those from galaxy occupation statistics studies and the most recent galaxy-galaxy weak lensing study. We discuss briefly the implications of our results for galaxy formations and structures. These constraints can be significantly strengthened when larger lens samples become available and the accuracy of semi-analytical model predictions improves.