Constraints on the velocity profiles of galaxies from strong lensing statistics and semi-analytical modelling 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,V-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, sigma(SIE) is then closely related to the observational central stellar velocity dispersion, sigma(cent); we have empirically sigma(SIE) approximate to sigma(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, sigma(SIE), are related to V-c,V-vir; if the galaxy were a singular isothermal sphere up to the virial radius, V-c,V-vir = root 2 sigma(SIE). When we include both the lensing probability and separation distribution as our lensing constraints, we find sigma(SIE)/(200 km s(-1)) = [(1.17(-0.26)(+0.40)) v(c,vir)/(200 km s(-1))](0.22)(+0.05)(-0.04) for 200 km s(-1) less than or similar to sigma(SIE) less than or similar to 260 km s(-1); at sigma(SIE) = 200km s(-1)' the ratio root 2 sigma(SIE)/V-c,V-vir is about 1.65(-0.37)(+0.57) [68 per cent confidence limit (CL)] but decreases to 0.65(-0.12)(+0-15) (68 per cent CL) for Or SIE = 260 km s(-1). 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 sigma(SIE) is varied and are marginally discrepant with the Seljak results at sigma(SIE) = 260 km s(-1). The scaling sigma(SIE) proportional to V-c,vir(0.22 +/- 0.05) is 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.