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 \Omega = 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 is 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 < \Omega < 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 : q _ { 0 } > -2.0 .