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

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We reanalyse optical gravitational lens surveys from the literature in order to determine relative probabilities in the λ₀-Ω₀ plane, using a softened singular isothermal sphere lens model. In addition, we examine a portion of the λ₀-Ω₀ 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 λ₀ 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 λ₀-Ω₀, using lens statistics information alone, are respectively -3.17 and 0.3. For a flat universe, these correspond to lower and upper limits on λ₀ of respectively -1.09 and 0.65.