Measuring the Deviation from the Linear and Deterministic Bias through Cosmic Gravitational Lensing Effects

Since gravitational lensing effects directly probe inhomogeneities of darkmatter, lensing-galaxy cross-correlations can provide us important informationon the relation between dark matter and galaxy distributions, i.e., the bias.In this paper, we propose a method to measure the stochasticity/nonlinearity ofthe galaxy bias through correlation studies of the cosmic shear and galaxynumber fluctuations. Specifically, we employ the aperture mass statistics$M_{ap}$ to describe the cosmic shear. We divide the foreground galaxy redshift$z_f^2/< N_g^2(z_f)>$ foreach redshift bin. Then the ratio of the summation of $^2/$ over the bins to $$ gives a measure of thenonlinear/stochastic bias. Here $N_g(z_f)$ is the projected surface numberdensity fluctuation of foreground galaxies at redshift $z_f$, and $M_{ap}$ isthe aperture mass from the cosmic-shear analysis. We estimate that for amoderately deep weak-lensing survey with $z_s=1$, source galaxy surface numberdensity $n_b=30 \hbox {gal}/\hbox {arcmin}^2$ and a survey area of $25 \hbox{deg}^2$, the effective $r$-parameter that represents the deviation from thelinear and deterministic bias is detectable in the angular range of 1'-10' if$|r-1|\gsim 10%$. For shallow, wide surveys such as the Sloan Digital SkySurvey with $z_s=0.5$, $n_b=5 \hbox {gal}/\hbox {arcmin}^2$, and a survey areaof $10^4 \hbox {deg}^2$, a 10% detection of $r$ is possible over the angularrange $1'-100'$.Comment: ApJ in pres