Magnification cross-sections of gravitational lensing by galaxies in general FLRW cosmologies

For a wide variety of cosmological models characterized by the cosmic mass density ΩM and the normalized cosmological constant ΩΛ, we derive an analytic expression for the estimate of magnification cross-sections by an ensemble of isothermal spheres as models of galactic mass distributions. This provides a simple approach to demonstrate how the lensing probability by galaxies depends on the cosmological parameters. An immediate consequence is that, while a non-zero cosmological constant indeed leads to a significant increase of the lensing probability as it has been shown in the literature, only a small fraction of sky to z ∼ 3 can be moderately (μ ∼ 1.3) lensed by galaxies even in aΛ-dominated flat universe. Therefore, whether or not there is a nonzero cosmological constant, it is unlikely that the overall quasar counts have been seriously contaminated by the presence of galactic lenses.