Light Propagation in Inhomogeneous Universes

The propagation of light bundles in a clumpy universe is studied by combining a Monte Carlo method used by Refsdal with recent analytical results. A simple numerical procedure allows us to study a large number of light bundles. We derive an integral equation for the angular diameter distance measured through the "empty cones" of a clumpy universe, which is shown to be equivalent to the Dyer and Roeder differential equation. The probability distribution for the linear component of the cumulative shear along light rays is derived; it is shown that the cumulative shear can be dominated by nonlinear components, especially for light rays in empty cones. The numerical simulations are used to find the fraction of empty cones in a clumpy universe. The high- amplification tail of the amplification probability distribution is compared to analytic results; these linear investigations are shown to underestimate the high-amplification probability and, hence, the importance of the amplification bias in source counts. The distribution of the ellipticity of images of infinitesimal circular sources is derived, and it is shown that this can be dominated by the nonlinear contributions to the cumulative shear. Comparing the results of the Monte Carlo simulation with those of the ray-shooting method (Schneider and Weiss), we can estimate the contribution of the diffuse flux to the amplification of sources behind empty cones. Finally, we briefly discuss the various methods to study nonlinear statistical lens problems.