The statistics of multiple gravitational lensing and the application of a single lensing approximation

The statistics of multiple gravitational lensing in a locally inhomogeneous universe is taken into account, based on the expansion of the mean amplification into the summation of various lenses and the pointlike models for both background sources and lensing objects. By comparing the deviation of the effects of various lenses from mean amplification, the redshift ranges over which different lenses dominate are investigated. In particular, the accuracy of a single lensing approximation, which has been widely used in previous statistical lensing, is demonstrated. As the application of a single lensing approximation, the lowest correction to the Mattig's relation due to the inhomogeneities in universe and the lowest effect on the number counts of background sources due to amplification bias are reconsidered. It is emphasized the fact that the effect of statistical lensing, even in a single lensing approximation, is of significance for modern observational cosmology.