Do the lensing cross-sections of faint galaxies cover the whole sky ?

Abstract

Very deep galaxy surveys have revealed a considerably large population of faint galaxies, which leads to the speculation that all distant objects are moderately magnified by the gravitational lensing effects of galaxies (Fried 1997). In this letter, we present a simple estimate of the lensing amplitudes by all galaxies up to redshift $z=2$ in terms of galaxy merging and answer the question whether the sky is fully covered by the lensing cross-sections of galaxies. It is shown that, as a result of the combination of the increase of galaxy number density and the decrease of galaxy velocity dispersion with lookback time, less than $\sim1/10$ of the sky to $z=2$ can be moderately affected by galaxies acting as lenses with magnification $\mu>1.1$. This conclusion is independent of the galaxy limiting magnitude. In other words, no matter how high the surface number density of faint galaxies becomes, it is unlikely that their lensing cross-sections of $\mu>1.1$ can cover the whole sky.