A New Method for Determining Cumulative Gravitational Lensing Effects in Inhomogeneous Universes

We present a new approach to calculating the statistical distributions for magnification, shear, and rotation of images of cosmological sources due to gravitational lensing. In this approach one specifies an underlying Robertson-Walker cosmological model together with relevant information on the clumping of matter on scales much smaller than the Hubble radius. The geodesic deviation equation is then integrated backwards in time until the desired redshift is reached, using a Monte Carlo procedure wherein each photon beam in effect ``creates its own universe'' as it propagates. The approach is somewhat similar to that used in ``Swiss cheese'' models, but the ``cheese'' has been completely eliminated, the matter distribution in the ``voids'' need not be spherically symmetric, and the total mass in each void need equal the corresponding Robertson-Walker mass only on average. We show that when the matter in the universe consists of point masses, the statistical distributions of the lensing images are essentially independent of both the mass spectrum and clustering properties of the point masses, provided that the clustering is spherical. We apply our results (i) to argue that the positive correlation recently found between quasar luminosity and the number of absorption line systems is not likely to be due to lensing, and (ii) to determine the amount of ``noise'' and possible bias produced by lensing in measurements of $q_0$ using distant supernovae.