Curvature of the universe and observed gravitational lens image separations versus redshift

In a flat, k = 0, cosmology with galaxies that approximate singular isothermal spheres, gravitational lens image separations should be uncorrelated with source redshift. But in an open, k = -1, cosmology, such gravitational lens image separations become smaller with increasing source redshift. The observed separations do become smaller with increasing source redshift, but the effect is even stronger than that expected in an Ω= 0 cosmology. The observations are thus not compatible with the "standard" gravitational lensing statistics model in a flat universe. We try various open and flat cosmologies, galaxy mass profiles, galaxy merging and evolution models, and lensing aided by clusters to explain the correlation. We find the data are not compatible with any of these possibilities within the 95% confidence limit, leaving us with a puzzle. If we regard the observed result as a statistical fluke, it is worth noting that we are about twice as likely to observe it in an open universe (with 0 < Ω < 0.4) as we are to observe it in a flat one. Finally, the existence of an observed multiple-image lens system with a source at z = 4.5 places a lower limit on the deceleration parameter: q₀ > -2.0.