Strong Gravitational Lensing and Velocity Function as Tools to Probe   Cosmological Parameters: Current Constraints and Future Predictions
  Constraints on cosmological models from strong gravitational lensing statistics are investigated. We pay particular attention to the role of the velocity function in the calculation of the lensing probability. The velocity function derived from the observed galaxy luminosity function, which is used in most previous work, is unable to predict the large separation lensing events. In this paper, we also use the Press-Schechter theory to construct a velocity function theoretically. Model predictions are compared with the observed velocity function and the HST snapshot survey. Comparison with the latter observation shows that the predictions based on the theoretical velocity function are consistent with the observed large separation events in COBE normalized low-density models, especially with a non-vanishing cosmological constant. Adopting the COBE normalization, however, we could find no model which simultaneously satisfies both the observed velocity function and the HST snapshot survey. We systematically investigate various uncertainties in the gravitational lensing statistics including finite core radius, the distance formula, magnification bias, and dust obscuration. The results are very sensitive to these effects as well as theoretical models for the velocity function, implying that current limits on the cosmological parameters should be interpreted with caution. Predictions for future surveys are also presented. 
Takahiro T. Nakamura and Yasushi Suto (University of Tokyo)