GRAVITATIONAL LENSING STATISTICS IN A FLAT UNIVERSE

the probability distribution of lens image separations is calculated f or the ''standard'' gravitational lensing statistics model in an arbit rary, flat Robertson-Walker universe, where lensing galaxies are singu lar isothermal spheres that follow the Schecter luminosity function. I n a flat universe, the probability distribution is independent of the source distribution in space and in brightness. The distribution is co mpared with observed multiple-image lens cases through Monte-Carlo sim ulations and the Kolmogorov-Smirnov test. The predicted distribution d epends sensitively on the shape of the angular selection bias used, wh ich varies for different observations. The test result also depends on which lens systems are included in the samples. We mainly include the lens systems where a single galaxy is responsible for lensing. Althou gh the ''standard'' model predicts a distribution which is different f rom the observed one, the statistical significance of the discrepancy is not large enough to invalidate the ''standard'' model. Only the rad io data reject the model at the 95% confidence level. However, this is based on four/three lens cases. Therefore, we cannot say that the obs ervational data reject the ''standard'' model with statistical confide nce. However, if we take the velocity dispersion of dark matter withou t the conversion factor (3/2)(1/2) from that of luminous matter [1,2], the discrepancy is quite severe, and even the ground-based optical su rvey data reject the ''standard'' model with greater than or similar t o 90% confidence.