Measuring the curvature of the universe with gravitational lensing

The image separations of the gravitational lens systems as a function of the source redshift depends on the curvature of the universe if lenses can be approximated by singular isothermal spheres. In a flat k = 0 cosmology, image separations should be uncorrelated with the source redshift. In an open k = -1 cosmology such image separations become smaller with increasing source redshift and vice versa for closed k = +1 cosmology. The observed lens systems do show negative correlation between the image separation versus the source redshift. However, the negative correlation seen in the data is much stronger than any correlation expected in open or even in empty universe. Therefore, the curvature effect alone cannot explain the observed negative correlation. So we explore other possibilities that may explain this strong correlations: steeper galaxy mass profile, lens evolution, like merging and infall, and cluster helping. None of them produce strong enough negative correlation seen in the data, leaving us with a puzzle. If a rather unlikely assumption that all lens systems with image separations lager than 3.0″ are "false" lenses [15], correlation becomes insignificant and nothing is troublesome. However, we show that under such image separation distribution roughly 300 lens systems suffice to test the curvature of Ω = 0.4 open universe with 95% confidence, which is well within the reach of future surveys.