Statistics of Weak Gravitational Lensing in Cold Dark Matter Models; Magnification Bias on Quasar Luminosity Functions

We compute statistical properties of weak gravitational lensing by large-scale structure in three Cold Dark Matter models. We use a P$^3$M $N$-body code to simulate the formation and evolution of large-scale structure in the universe. We perform $1.1\times10^7$ ray-tracing experiments for each model using the multiple lens-plane algorithm. From the results of these experiments, we calculate the probability distribution functions (PDF) of the convergences, shears, and magnifications, and their root-mean-square (rms) values. We find that the rms values of the convergence and shear agree with the predictions of a nonlinear analytical model. We also find that the PDFs of the magnifications $\mu$ have a peak at values slightly smaller than $\mu=1$, and are strongly skewed toward large magnifications. In particular, for the high-density model, a power-law tail appears in the magnification distribution at large magnifications for sources at redshifts $z_s>2$. The rms values of the magnifications essentially agree with the nonlinear analytical predictions for sources at low redshift, but exceed these predictions for high redshift sources, once the power-law tail appears. We study the effect of magnification bias on the luminosity functions of high-redshift quasars, using the calculated PDFs of the magnifications. We show that the magnification bias is moderate in the absence of the power-law tail in the magnification distribution, but depends strongly on the value of the density parameter. In presence of the power-law tail, the bias becomes considerable, especially at the bright end of the luminosity functions.