Implications of the lens redshift distribution of strong lensing
  systems: cosmological parameters and the global properties of early-type
  galaxies
  In this paper, we assemble a well-defined sample of early-type gravitational
lenses extracted from a large collection of 158 systems, and use the redshift
distribution of galactic-scale lenses to test the standard cosmological model
($\Lambda$CDM) and the modified gravity theory (DGP). Two additional
sub-samples are also included to account for possible selection effect
introduced by the detectability of lens galaxies. Our results show that
independent measurement of the matter density parameter ($\Omega_m$) could be
expected from such strong lensing statistics. Based on future measurements of
strong lensing systems from the forthcoming LSST survey, one can expect
$\Omega_m$ to be estimated at the precision of $\Delta\Omega_m\sim 0.006$,
which provides a better constraint on $\Omega_m$ than \textit{Planck} 2015
results. Moreover, use the lens redshift test is also used to constrain the
characteristic velocity dispersion of the lensing galaxies, which is well
consistent with that derived from the optical spectroscopic observations. A
parameter $f_E$ is adopted to quantify the relation between the lensing-based
velocity dispersion and the corresponding stellar value. Finally, the
accumulation of detectable galactic lenses from future LSST survey would lead
to more stringent fits of $\Delta f_E\sim10^{-3}$, which encourages us to test
the global properties of early-type galaxies at much higher accuracy.
