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 $\Omega_M$ and the normalized cosmological constant $\Omega_{\Lambda}$, 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 $z\sim 3$ can be moderately ($\mu \sim 1.3$) lensed by galaxies even in a $\Lambda$-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.