Extensive analyses of statistical strong gravitational lensing are performed based on the final Cosmic Lens All-Sky Survey ( CLASS ) well-defined statistical sample of flat-spectrum radio sources and current estimates of galaxy luminosity functions per morphological type .
The analyses are done under the assumption that galactic lenses are well-approximated by singular isothermal ellipsoids and early-type galaxies evolved passively since redshift z \sim 1 .
Two goals of the analyses are : ( 1 ) to constrain cosmological parameters independently of other techniques ( e.g .
Type Ia supernovae magnitude-redshift relation , cosmic microwave background anisotropies , galaxy matter power spectra ) ; and ( 2 ) to constrain the characteristic line-of-sight velocity dispersion and the mean projected mass ellipticity for the early-type galaxy population .
Depending on how the late-type galaxy population is treated ( i.e. , whether its characteristic velocity dispersion is constrained or not ) , we find for a flat universe with a classical cosmological constant that the matter fraction of the present critical density \Omega _ { m } = 0.31 ^ { +0.27 } _ { -0.14 } ( 68 % ) for the unconstrained case or 0.40 ^ { +0.28 } _ { -0.16 } ( 68 % ) for the constrained case , with an additional systematic uncertainty of \approx 0.11 arising from the present uncertainty in the distribution of CLASS sources in redshift and flux density .
For a flat universe with a constant equation of state for dark energy w = p _ { x } ( pressure ) / \rho _ { x } ( energy density ) and the prior constraint w \geq - 1 , we find that -1 \leq w < -0.55 ^ { +0.18 } _ { -0.11 } ( 68 % ) for the unconstrained case or -1 \leq w < -0.41 ^ { +0.28 } _ { -0.16 } ( 68 % ) for the constrained case , where w = -1 corresponds to a classical cosmological constant .
The determined value of the early-type characteristic velocity dispersion ( \sigma _ { * } ^ { ( e ) } ) depends on the faint-end slope of the early-type luminosity function ( \alpha ^ { ( e ) } ) and the intrinsic shape distribution of galaxies ; for equal frequencies of oblates and prolates , we find that \sigma _ { * } ^ { ( e ) } ( 0.3 \la z \la 1 ) = 198 ^ { +22 } _ { -18 } km s ^ { -1 } ( 68 % ) for a ‘ steep ’ \alpha ^ { ( e ) } = -1 or \sigma _ { * } ^ { ( e ) } ( 0.3 \la z \la 1 ) = 181 ^ { +18 } _ { -15 } km s ^ { -1 } ( 68 % ) for a ‘ shallow ’ \alpha ^ { ( e ) } = -0.54 .
Finally , from the relative frequencies of doubly-imaged sources and quadruply-imaged sources , we find that a mean projected mass ellipticity of early-type galaxies \bar { \epsilon } _ { mass } = 0.42 with a 68 % lower limit of 0.28 assuming equal frequencies of oblates and prolates .