The statistics of gravitational lenses : the distributions of image angular separations and lens redshifts.

The authors compute analytically the probability dτ that the light from a distant quasar at redshift z_Q will be lensed by objects in redshift range dz_L and find that for nearby objects τ goes as z_Q² for point lenses and z_Q³ for galaxy lenses (modeled as singular isothermal spheres). At fixed z_Q the most probable lens redshift is typically small, z_L ≈ Min(z_Q/2, 0.8), so the lensing objects (if they are galaxies) should be detectable. A variety of numerical simulations are made to determine the expected properties of double optical or radio quasars. These simulations take into account realistic selection biases which affect the expected properties of observed lens systems. The results are insensitive to the assumed cosmology but highly dependent on the central velocity dispersions of bright elliptical galaxies and to the maximum surface densities in great clusters. The inferred luminosity of an imaged quasar is usually much greater than its true luminosity, with the average flux amplification a factor of 4 in an unbiased sample.