The statistics of gravitational lenses in a clumpy Universe

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-Lemaitre 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 tau, 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 alpha.