Is the Number of Giant Arcs in LCDM Consistent With Observations?

We use high-resolution N-body simulations to study the galaxy-cluster cross-sections and the abundance of giant arcs in the $Lambda$CDM model. Clusters are selected from the simulations using the friends-of-friends method, and their cross-sections for forming giant arcs are analyzed. The background sources are assumed to follow a uniform ellipticity distribution from 0 to 0.5 and to have an area identical to a circular source with diameter $1arcsec$. We find that the optical depth scales as the source redshift approximately as $tau_{1} = 2.25 times 10^{-6}/[1+(zs/3.14)^{-3.42}]$ ($0.6<zs<7$). The amplitude is about 50% higher for an effective source diameter of $0.5arcsec$. The optimal lens redshift for giant arcs with the length-to-width ratio ($L/W$) larger than 10 increases from 0.3 for $zs=1$, to 0.5 for $zs=2$, and to 0.7-0.8 for $zs>3$. The optical depth is sensitive to the source redshift, in qualitative agreement with Wambsganss et al. (2004). However, our overall optical depth appears to be only $sim$ 10% to 70% of those from previous studies. The differences can be mostly explained by different power spectrum normalizations ($sigma_8$) used and different ways of determining the $L/W$ ratio. Finite source size and ellipticity have modest effects on the optical depth. We also found that the number of highly magnified (with magnification $|mu|>10$) and ``undistorted images (with $L/W<3$) is comparable to the number of giant arcs with $|mu|>10$ and $L/W>10$. We conclude that our predicted rate of giant arcs may be lower than the observed rate, although the precise `discrepancy is still unclear due to uncertainties both in theory and observations.