The velocity dispersion function of early-type galaxies and its redshift
  evolution: the newest results from lens redshift test
  The redshift distribution of galactic-scale lensing systems provides a
laboratory to probe the velocity dispersion function (VDF) of early-type
galaxies (ETGs) and measure the evolution of early-type galaxies at redshift z
~ 1. Through the statistical analysis of the currently largest sample of
early-type galaxy gravitational lenses, we conclude that the VDF inferred
solely from strong lensing systems is well consistent with the measurements of
SDSS DR5 data in the local universe. In particular, our results strongly
indicate a decline in the number density of lenses by a factor of two and a 20%
increase in the characteristic velocity dispersion for the early-type galaxy
population at z ~ 1. Such VDF evolution is in perfect agreement with the
$\Lambda$CDM paradigm (i.e., the hierarchical build-up of mass structures over
cosmic time) and different from "stellar mass-downsizing" evolutions obtained
by many galaxy surveys. Meanwhile, we also quantitatively discuss the evolution
of the VDF shape in a more complex evolution model, which reveals its strong
correlation with that of the number density and velocity dispersion of
early-type galaxies. Finally, we evaluate if future missions such as LSST can
be sensitive enough to place the most stringent constraints on the redshift
evolution of early-type galaxies, based on the redshift distribution of
available gravitational lenses.
