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 1 \arcsec .
We find that the optical depth scales as the source redshift approximately as \tau _ { 1 ^ { \prime \prime } } = 2.25 \times 10 ^ { -6 } / [ 1 + ( z _ { s } / 3.14 ) ^ { -3.42 } ] ( 0.6 < z _ { s } < 7 ) .
The amplitude is about 50 % higher for an effective source diameter of 0.5 \arcsec .
The optimal lens redshift for giant arcs with the length-to-width ratio ( L / W ) larger than 10 increases from 0.3 for z _ { s } = 1 , to 0.5 for z _ { s } = 2 , and to 0.7-0.8 for z _ { s } > 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 .