The Delta theta-z$_{s}$ Relation as a Cosmological Test

Recently, it was noted by Park \& Gott (1997) that there is a statistically significant, strong, negative correlation between the image separation $\Delta\theta$ and source redshift $z_{\mathrm{s}}$ for gravitational lenses. This is somewhat puzzling if one believes in a flat ($k=0$) universe, since in this case the typical image separation is expected to be independent of the source redshift, while one expects a negative correlation in a $k=-1$ universe and a positive one in a $k=+1$ universe. Park \& Gott explored several effects which could cause the observed correlation, but no combination of these can explain the observations with a realistic scenario. Here, I explore this test further in three ways. First, I show that in an inhomogeneous universe a negative correlation is expected regardless of the value of $k$. Second, I test whether the $\Delta\theta$-$z_{\mathrm{s}}$ relation can be used as a test to determine $\lambda_{0}$ and $\Omega_{0}$, rather than just the sign of $k$. Third, I compare the results of the test from the Park \& Gott sample to those using other samples of gravitational lenses, which can illuminate (unknown) selection effects and probe the usefulness of the $\Delta\theta$-$z_{\mathrm{s}}$ relation as a cosmological test.