We use the distribution of gravitationally-lensed image separations observed in the Cosmic Lens All-Sky Survey ( CLASS ) and the PMN-NVSS Extragalactic Lens Survey ( PANELS ) , which are ( nearly ) complete for the image separation range 0 ^ { \prime \prime } .3 \leq \Delta \theta \leq 6 ^ { \prime \prime } , to constrain a model velocity dispersion function ( VF ) of early-type galaxies .
Assuming a current concordance cosmological model and adopting a singular isothermal ellipsoid ( SIE ) model for galactic potentials , we consider constraining both a characteristic velocity dispersion ( parameter \sigma _ { * } ) and the shape of the function ( parameters \alpha and \beta ; Sheth et al .
2003 ) for 0.3 \lesssim z \lesssim 1 .
If all three parameters are allowed to vary , then none of the parameters can be tightly constrained by the lensing data because of the small size of the sample .
If we fix the shape of the function by either the SDSS local stellar VF or an inferred local stellar VF based on the SSRS2 galaxy sample , then the constrained values of \sigma _ { * } are nearly equal to the corresponding stellar values ; we have f _ { SIE / center } ( \equiv \sigma _ { * { SIE } } / \sigma _ { * { center } } ) = 0.90 \pm 0 % .18 { ( SDSS ) } or 1.04 \pm 0.19 { ( SSRS 2 ) } assuming non-evolution of the function between the present epoch and z \sim 0.6 .
Finally , using only the CLASS statistical sample ( Browne et al .
2003 ) and thus including an absolute multiple-imaging probability , we find that the SDSS stellar VF may have significantly underestimated the abundance of morphologically early-type galaxies .