Gravitational lensing statistics in a flat universe

The probability distribution of lens image separations is calculated for the "standard" gravitational lensing statistics model in an arbitrary, flat Robertson-Walker universe, where lensing galaxies are singular isothermal spheres that follow the Schechter luminosity function. In a flat universe, the probability distribution is independent of the source distribution in space and in brightness. The distribution is compared with observed multiple-image lens cases through Monte-Carlo simulations and the Kolmogorov-Smirnov test. The predicted distribution depends sensitively on the shape of the angular selection bias used, which 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. Although the "standard" model predicts a distribution which is different from the observed one, the statistical significance of the discrepancy is not large enough to invalidate the "standard" model. Only the radio data reject the model at the 95% confidence level. However, this is based on four/three lens cases. Therefore, we cannot say that the observational data reject the "standard" model with statistical confidence. However, if we take the velocity dispersion of dark matter without 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 survey data reject the "standard" model with ≳90% confidence.