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 $sigma_8sim0.9$ may come close to the known abundance of arcs, even $sigma_8sim0.8$ falls short by almost an order of magnitude in reproducing known counts. We conclude that, should $sigma_8sim0.8$ 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.
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