Clustering of Faint Galaxies: $\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 $\w$ 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.