Effective Models for Statistical Studies of Galaxy-Scale Gravitational
  Lensing
  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.
