The amplitude of the initial density fluctuation spectrum from lensing statistics

A complete catalogue of galaxy cluster models can be used to constrain cosmological parameters like ${\sigma }_8$, the amplitude of the initial density fluctuation spectrum. We focus here on two strong lensing statistics for clusters of galaxies: the fraction of rich Abell clusters that are critical to lensing, and the abundance of arcs in such clusters, especially the X-ray luminous ones. The specific aim of this paper is to constrain ${\sigma }_8$ for the standard Cold Dark Matter scenario within an Einstein-de Sitter universe.