Gravitational lensing statistics with extragalactic surveys. I. A lower limit on the cosmological constant

We reanalyse optical gravitational lens surveys from the literature in order to determine relative probabilities in the lambda_ {0}-Omega_ {0} plane, using a softened singular isothermal sphere lens model. In addition, we examine a portion of the lambda_ {0}-Omega_ {0} plane which includes all viable cosmological models; this is vital for comparison with other cosmological tests. The results are, within the errors, consistent with those of more specialised analyses, such as those concerning upper limits on lambda_ {0} in a flat universe. We note that gravitational lensing statistics can provide a quite robust lower limit on the cosmological constant as well, which could prove important in confirming current claims of a positive cosmological constant. At 95% confidence, our lower and upper limits on lambda_ {0}-Omega_ {0}, using lens statistics information alone, are respectively -3.17 and 0.3. For a flat universe, these correspond to lower and upper limits on lambda_ {0} of respectively -1.09 and 0.65.