Gravitational lensing: effects of cosmology and of lens and source profiles

We present detailed calculations of the magnification distribution, including both weak and strong lensing, using very recent solutions of the Dyer-Roeder (1973) equation for light propagation in a inhomogeneous universe with a cosmological constant and up-to-date models for the evolving cosmological distribution of dark matter halos. We address in particular the effect on the magnification distribution of a non-zero cosmological constant, of different density profiles of lenses, and of finite sizes of lensed sources, three important issues not yet fully settled. We show that, if dark matter fluctuations are normalized to the local cluster abundance, in the presence of a cosmological constant the optical depth for lensing decreases compared to the case of an Einstein-de Sitter universe, because halos in the relevant mass range are less abundant over a large redshift interval. We also discuss the differences in the magnification probability distributions produced by Navarro, Frenk & White (NFW) and by Singular Isothermal Sphere (SIS) density profiles of lenses. We find that the NFW lens is more efficient for moderate magnifications (2 < ∼ A < ∼ 4), and less efficient for larger magnifications. Moreover, we discuss quantitatively the maximum magnification, Amax, that can be achieved in the case of extended sources (galaxies) with realistic luminosity profiles, taking into account the possible ellipticity of the lens potential. We find that plausible values of Amax are in the range 10–30. Finally, we apply our results to a class of sources following the luminosity evolution typical for a unified scheme of QSO formation. We find that the lensed source counts at 850μm can be larger than the unlensed ones by several orders of magnitude at flux densities >∼ 100mJy.