Statistics of gravitational lenses in the clumpy Universe
  We evaluate the effect of small scale inhomogeneities on large scale observations within the statistics of gravitationally lensed quasars. At this aim, 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, and pay particular attention on the different role played by the different notions of distances (filled beam angular diameter distance and Dyer-Roder distances) when calculating this quantity, following the prescription from Ehlers & Schneider for a coherent formalism. We find that the expected number of gravitationally lensed quasars is a decreasing function of the clumpiness parameter. 
G. Covone (1), M. Sereno (2) and R. de Ritis ((1) Laboratoire   d'Astrophysique de Marseille, (2) Universita' degli Studi di...