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 $Ω_{0}$ and of a mass fraction $\barα$ (smoothness parameter).