Gravitational lensing of extended high-redshift sources by dark matter haloes

High-redshift galaxies and quasi-stellar objects (QSOs) are most likely to be strongly lensed by intervening haloes between the source and the observer. In addition, a large fraction of lensed sources is expected to be seen in the submillimetre region, as a result of the enhanced magnification bias on the steep intrinsic number counts. We extend in three directions Blain's earlier study of this effect. First, we use a modification of the Press-Schechter mass function and detailed lens models to compute the magnification probability distribution. We compare the magnification cross-sections of populations of singular isothermal spheres and Navarro, Frenk & White (NFW) haloes and find that they are very similar, in contrast to the image-splitting statistics which were recently investigated in other studies. The distinction between the two types of density profile is therefore, irrelevant for our purposes. Secondly, we discuss quantitatively the maximum magnification, mu(max), that can be achieved for extended sources (galaxies) with realistic luminosity profiles, taking into account the possible ellipticity, of the lensing potential. We find that mu(max) plausibly falls into the range 10-30 for sources of 1-10 h(-1) kpc effective radius at redshifts within 1-4. Thirdly, we apply our model for the lensing magnification to a class of sources following the luminosity, evolution typical for a unified scheme of QSO formation. As a result of the peculiar steepness of their intrinsic number counts, we find that the lensed source counts at a fiducial wave length of 850 mum can exceed the unlensed counts by several orders of magnitude at flux densities greater than or similar to100 mJy, even with a conservative choice of the maximum magnification.