On the probability of magnification by cosmologically distributed gravitational lenses

We present an analytical method for calculating the statistical properties of source magnification caused by gravitational lenses randomly distributed throughout the universe. Our method is based on the assumption that the total magnification accumulated in a random direction is multiplicative along successive redshift intervals. By considering two lenses at different redshifts, we show that such an assumption is a statistically adequate approximation. The general formulae derived here are applied to point-mass lenses with both point and extended sources. We find that our analytical results of the magnification probability for point sources are accurate to within 10% in comparison with the available numerical simulations to moderate redshifts of z < ~2. In terms of the flux conservation, Our results are accurate to within 18% at a redshift of z = 6 with respect to the Dyer-Roeder model of a clumpy universe. We conclude that our formulae are adequate for statistical studies of magnification by random gravitational lenses on cosmological scales. Several new results are presented here for a three-dimensional case of random point-mass gravitational lensing. First, the lensing optical depth would be nonlinear with respect to the cosmological density of the lenses if the Dyer-Roeder model of a clumpy universe is assumed. Second, for point sources, the shape of the differential probability distribution at large magnifications may become progressively shallower as the lensing optical depth increases. Third, the mean magnification by point-mass lenses is independent of the source size. Fourth, for extended sources, the slope of the differential probability distribution at large magnifications is a continuous function of the source size. Finally, for astrophysical objects of finite size, the magnification effect of gravitational lensing would vanish if the mass of individual lenses approaches zero, as expected for a smooth Friedmann universe.