Statistical properties of gravitational lenses with a nonzero cosmological constant

We investigate statistical properties of gravitational lensing in the presence of a cosmological constant, with emphasis given to the study of uncertainties in cosmological factors arising from different possible physical and statistical formulations of the problem. We find that a substantial uncertainty associated with the distance formulation for the prediction of the lensing optical-depth makes the discrimination between low- and high-density universe models difficult for high-redshift lens-quasar systems. We find, however, that the cosmological constant, if it dominates over the mass density, increases the optical depth greatly, and its effect is much larger than the uncertainty arising from details of the problem's formulation. Therefore, the lensing frequency can provide a simple and very useful test for the cosmological constant. For a low-redshift system (z_s ≲ 2) the formulation uncertainties are rather moderate, though the difference among different cosmology models is not large. The optical-depth redshift distribution is also very sensitive to cosmological models, but it is less sensitive to statistical formulations. A realistic prediction is also made for the lensing frequency taking account of various selection effects for some particular samples. We also study the gravitational lens effect with a cosmological constant on the quasar-galaxy correlation and on fluctuations in the cosmic microwave background radiation. This paper thus attempts to present a systematic and reasonably complete discussion of statistical problems in gravitational lensing for A ≠ 0 cosmological models.