Continuous Limit of Multiple Gravitational Lens Effect and Average Magnification Factor

Abstract

We show that the gravitational magnification factor averaged over all configurations of lenses in a locally inhomogeneous universe satisfy a second order differential equation with redshift $z$ by taking the continuous limit of multi-plane gravitational lens equation (the number $N$ of lenses $\to \infty$) and that the gravitationally magnified Dyer-Roeder distance in a clumpy universe becomes to that of the Friedmann-Lema\^ıtre universe for arbitrary values of the density parameter ${\Omega }_0$ and of a mass fraction $\bar{\alpha }$ (smoothness parameter).

13 pages, 2 figures