  We use high-resolution $N$-body simulations to investigate the optical depth
of giant arcs with length-to-width ratio larger than 7.5 and 10 in the
`standard' $\LCDM$ model with $\sigma_8=0.9$ and $\Omega_{\rm m,0}=0.3$ and a
model based on three-year Wilkinson Microwave Anisotropy Probe (WMAP) data. We
find that, in dark-matter only simulations, the lensing probability in the
three-year WMAP model (with $\sigma_8=0.74$ and $\Omega_{\rm m,0}=0.238)$
decreases by a factor of $\sim 6$ compared with that in the `standard' $\LCDM$
model. The effects of baryonic cooling, star formation and feedbacks are
uncertain, but we argue that baryons will only increase the the lensing
cross-section by a moderate factor, $\sim 2$. We conclude that the low central
value of $\sigma_8$ and $\Omega_{\rm m,0}$ preferred by the WMAP three-year
data may be too low to be compatible with observations if conventional
assumptions of the background source population are correct.
