Since gravitational lensing effects directly probe inhomogeneities of dark matter , lensing-galaxy cross-correlations can provide us important information on the relation between dark matter and galaxy distributions , i.e. , the bias .
In this paper , we propose a method to measure the stochasticity/nonlinearity of the galaxy bias through correlation studies of the cosmic shear and galaxy number fluctuations .
Specifically , we employ the aperture mass statistics M _ { ap } to describe the cosmic shear .
We divide the foreground galaxy redshift z _ { f } < z _ { s } into several bins , where z _ { s } is the redshift of the source galaxies , and calculate the quantity < M _ { ap } N _ { g } ( z _ { f } ) > ^ { 2 } / < N _ { g } ^ { 2 } ( z _ { f } ) > for each redshift bin .
Then the ratio of the summation of < M _ { ap } N _ { g } ( z _ { f } ) > ^ { 2 } / < N _ { g } ^ { 2 } ( z _ { f } ) > over the bins to < M _ { ap } ^ { 2 } > gives a measure of the nonlinear/stochastic bias .
Here N _ { g } ( z _ { f } ) is the projected surface number density fluctuation of foreground galaxies at redshift z _ { f } , and M _ { ap } is the aperture mass from the cosmic-shear analysis .
We estimate that for a moderately deep weak-lensing survey with z _ { s } = 1 , source galaxy surface number density n _ { b } = 30 \hbox { gal } / \hbox { arcmin } ^ { 2 } and a survey area of 25 \hbox { deg } ^ { 2 } , the effective r -parameter that represents the deviation from the linear and deterministic bias is detectable in the angular range of 1 ^ { \prime } - 10 ^ { \prime } if |r - 1 | \lower 4.0 pt \hbox { $ { \buildrel \displaystyle > \over { \sim } } $ } 10 \% .
For shallow , wide surveys such as the Sloan Digital Sky Survey with z _ { s } = 0.5 , n _ { b } = 5 \hbox { gal } / \hbox { arcmin } ^ { 2 } , and a survey area of 10 ^ { 4 } \hbox { deg } ^ { 2 } , a 10 \% detection of r is possible over the angular range 1 ^ { \prime } -100 ^ { \prime } .