Clustering of Faint Galaxies: $\text{w}$, Induced by Weak Gravitational Lensing

Weak gravitational lensing by large scale structure affects the number counts of faint galaxies through the magnification bias'' and thus affects the measurement of the angular two-point correlation function $\w $. At faint magnitudes the clustering amplitude will decrease differently with limiting magnitude than expected from Limber's equation. The amplitude will hit a minimum and then rise with limiting magnitude. This behavior occurs because $\w$ due to clustering decreases with distance, while the magnification bias’’ due to weak lensing increases with distance. The apparent magnitude $m_{min}$ at which the magnification bias starts to dominate the observed clustering is model and color dependent. It is given by $\omega(m=m_{min},\theta=5^\prime) \approx (1\ -\ 2)\times 10^{-3}(5s-2)^2 \Omega_0^2 \sigma_8^2$,where$s$ is the logarithmic slope of the number counts. Already published measurements of $\text{w}$ at $R=25$ may be strongly influenced by the magnification bias''. An experiment using the ratio of blue and red number counts across the sky can be designed such that the effects of the true’’ clustering is minimized. The magnification bias is a measurement of the clustering of the mass. This weak lensing experiment does not require measuring shapes and position angles of galaxies. I derive a revised Limber’s Equation including the effects of magnification bias.