Effective Models for Statistical Studies of Galaxy-Scale Gravitational Lensing

Abstract: We have worked out simple analytical formulae that accurately approximate the relationship between the position of the source with respect to the lens center and the amplification of the images, hence the lens cross section, for realistic lens profiles. We find that, for essentially the full range of parameters either observationally determined or yielded by numerical simulations, the combination of dark matter and star distribution can be very well described, for lens radii relevant to strong lensing, by a simple power-law whose slope is very weakly dependent on the parameters characterizing the global matter surface density profile and close to isothermal in agreement with direct estimates for individual lens galaxies. Our simple treatment allows an easy insight into the role of the different ingredients that determine the lens cross section and the distribution of gravitational amplifications. They also ease the reconstruction of the lens mass distribution from the observed images and, vice-versa, allow a fast application of ray-tracing techniques to model the effect of lensing on a variety of source structures. The maximum amplification depends primarily on the source size. Amplifications larger than ~20 are indicative of compact source sizes at high-z, in agreement with expectations if galaxies formed most of their stars during the dissipative collapse of cold gas. Our formalism has allowed us to reproduce the counts of strongly lensed galaxies found in the H-ATLAS SDP field. While our analysis is focussed on spherical lenses, we also discuss the effect of ellipticity and the case of late-type lenses (showing why they are much less common, even though late-type galaxies are more numerous). Furthermore we discuss the effect of a cluster halo surrounding the early-type lens and of a supermassive black hole at its center.