2.3. Number Count of Quasar Lensing

This is a promising new version of the classical number density test. Strong sensitivity to arises when is positive and comparable to _m. In this case, the universe should have gone through a phase of slower expansion in the recent cosmological past, which should be observed as an accumulation of objects at a specific redshift of order unity. In particular, it should be reflected in the observed rate of lensing of high-redshift quasars by foreground galaxies [7]. The probability of lensing of a source at redshift z_s by a population of isothermal spheres of constant comoving density as a function of the cosmological parameters is [1]:

(3)

where d_a(z₁, z₂) is the angular diameter distance from z₁ to z₂. The contours of constant lensing probability in the _m - plane for z_s ~ 2 [1] happen to almost coincide with the lines _m - = const. The limits from lensing are thus similar in nature to the limits from SNIa.

Pro: This test shares all the advantages of direct geometrical measures, e.g., being independent of dark matter, fluctuations, GI and galaxy biasing. The high redshifts involved bring about a unique sensitivity to , compared to the negligible effect that has on the structure observed at z << 1.

Con: The constraint is weakened if the lensed images are obscured by dust in the early-type galaxies that are responsible for the lensing, especially if E galaxies at z ~ 1 have nuch more dust than low-redshift ones [8]. There is no clear evidence for a strong effect. A similar uncertainty arises if these galaxies had rapid evolution between z ~ 1 and the present; current evidence suggests a weak evolution. The method was criticized for it's sensitivity to the velocity dispersion assumed for the typical lenses and thus to the galaxy luminosity function [9], but the requirement that the distribution of lenses should simultaneously produce the observed distribution of image separations largely invalidates this criticism [10].

Current Results: From the failure to detect the accumulation of lenses, the current limit for a flat model is < 0.66 (or _m > 0.36) at 95% confidence [10] (Fig. 1). If = 0, this test provides only a weaker bound, _m > 0.2 at 90% confidence. Several new lenses are found each year, promising slow but continuous improvement.