The statistics of gravitational lenses in a clumpy Universe

Abstract

We evaluate the effect of small-scale inhomogeneities on large-scale observations within the statistics of gravitationally lensed quasars. To this end, we consider a cosmological model whose large-scale properties (dynamics, matter distribution) are the same as in Friedmann–Lemaître models, but whose matter distribution is locally inhomogeneous. We use the well-known Dyer–Roder distances to allow a simple analytical expression of the optical depth τ, and pay particular attention to the different roles played by the different notions of distance (filled beam angular diameter distance and Dyer–Roder distances) when calculating this quantity, following the 1986 prescription of Ehlers & Schneider for a coherent formalism. We find that the expected number of gravitationally lensed quasars is a decreasing function of the clumpiness parameter α.