  We use semi-analytic modelling of the galaxy-cluster population and its
strong lensing efficiency to explore how the expected abundance of large
gravitational arcs on the sky depends on $\sigma_8$. Our models take all
effects into account that have been shown to affect strong cluster lensing
substantially, in particular cluster asymmetry, substructure, merging, and
variations in the central density concentrations. We show that the optical
depth for long and thin arcs increases by approximately one order of magnitude
when $\sigma_8$ increases from 0.7 to 0.9, owing to a constructive combination
of several effects. Models with high $\sigma_8$ are also several orders of
magnitude more efficient in producing arcs at intermediate and high redshifts.
Finally, we use realistic source number counts to quantitatively predict the
total number of arcs brighter than several magnitude limits in the R and I
bands. We confirm that, while $\sigma_8\sim0.9$ may come close to the known
abundance of arcs, even $\sigma_8\sim0.8$ falls short by almost an order of
magnitude in reproducing known counts. We conclude that, should
$\sigma_8\sim0.8$ be confirmed, we would fail to understand the strong-lensing
efficiency of the galaxy cluster population, and in particular the abundance of
arcs in high-redshift clusters. We argue that early-dark energy or non-Gaussian
density fluctuations may indicate one way out of this problem.
