Is the Number of Giant Arcs in ΛCDM Consistent with Observations?

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Abstract
We use high-resolution N-body simulations to study the galaxy-cluster cross sections and the abundance of giant arcs in the ΛCDM 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''. We find that the optical depth scales as the source redshift approximately as τ = 2.25 × 10-6/[1 + (zs/3.14)-3.42] (0.6 < zs < 7). The amplitude is about 50% higher for an effective source diameter of 05. The optimal lens redshift for giant arcs with the length-to-width (L/W) ratio 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 ~10%-70% of those from previous studies. The differences can be mostly explained by the different power spectrum normalizations (σ8) 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 |μ| > 10) and "undistorted" images (with L/W < 3) is comparable to the number of giant arcs with |μ| > 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 in both theory and observations.