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 $\mathrm{\Lambda }$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 $1\text{arcsec}$. We find that the optical depth scales as the source redshift approximately as ${\tau }_{1^{\mathrm{\prime }\mathrm{\prime }}}=2.25\times {10}^{-6}\mathrm{/}[1+(\text{zs}\mathrm{/}3.14{)}^{-3.42}]$ ($0.6<\text{zs}<7$). The amplitude is about 50% higher for an effective source diameter of $0.5\text{arcsec}$. The optimal lens redshift for giant arcs with the length-to-width ratio ($L\mathrm{/}W$) larger than 10 increases from 0.3 for $\text{zs}=1$, to 0.5 for $\text{zs}=2$, and to 0.7-0.8 for $\text{zs}>3$. 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 $L\mathrm{/}W$ ratio. Finite source size and ellipticity have modest effects on the optical depth. We also found that the number of highly magnified (with magnification $\mathrm{\mid }\mu \mathrm{\mid }>10$) and ``undistorted’’ images (with $L\mathrm{/}W<3$) is comparable to the number of giant arcs with $\mathrm{\mid }\mu \mathrm{\mid }>10$ and $L\mathrm{/}W>10$. 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.