Light Propagation in Inhomogeneous Universes. III. Distributions of Image Separations

Using an analytical model, we compute the distribution of image separations resulting from gravitational lensing of distant sources, for seven COBE-normalized cold dark matter models with various combinations of Ω₀ and λ₀. Our model assumes that multiple imaging results from strong lensing by individual galaxies. We model galaxies as nonsingular isothermal spheres, whose parameters are functions of the luminosity and morphological type, and take into account the finite angular size of the sources. Our model neglects the contribution of the background matter distribution to lensing and assumes that lensing is entirely caused by galaxies. To test the validity of this assumption, we performed a series of ray-tracing experiments to study the effect of the background matter on the distribution of image separations. Our results are the following: (1) The presence of the background matter tends to increase the image separations produced by lensing galaxies, making the distributions of image separations wider. However, this effect is rather small and independent of the cosmological model. (2) Simulations with galaxies and background matter often produce a secondary peak in the distribution of image separations at large separations. This peak does not appear when the background matter is excluded from the simulations. (3) The effect of the background matter on the magnification distribution is negligible in low-density universes (Ω₀ = 0.2) with small density contrast (σ₈ = 0.4), but becomes very important as Ω₀ and σ₈ increase, resulting in a significant widening of the distribution. (4) Multiple imaging is caused primarily by early-type galaxies (elliptical and S0), with a negligible contribution from spiral galaxies. (5) Our analytical model, which has only two free parameters, is in good agreement with the results of ray-tracing experiments, successfully reproducing the distributions of image separations, and also the multiple-imaging probability, for all cosmological models considered. (6) The analytical model predicts that the distributions of image separations are virtually indistinguishable for flat, cosmological-constant models with different values of Ω₀. (7) For models with no cosmological constant, the distributions of image separations do depend upon Ω₀, but this dependence is weak. We conclude that while the number of multiple-imaged sources can put strong constraints on the cosmological parameters, the distribution of image separations does not constrain the cosmological models in any significant way and mostly provides constraints on the structure of the galaxies responsible for lensing.