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The Einstein radius plays a central role in lens studies as it characterizes the strength of gravitational lensing. In particular, the distribution of Einstein radii near the upper cut-off should probe the probability distribution of the largest mass concentrations in the universe. Adopting a triaxial halo model, we compute expected distributions of large Einstein radii. To assess the cosmic variance, we generate a number of Monte Carlo realizations of all-sky catalogues of massive clusters. We find that the expected largest Einstein radius in the universe is sensitive to parameters characterizing the cosmological model, especially σ_{8}: for a source redshift of unity, they are 42^{+9}_{−7}, 35^{+8}_{−6} and 54^{+12}_{−7} arcsec (errors denote 1σ cosmic variance), assuming best-fitting cosmological parameters of the *Wilkinson Microwave Anisotropy Probe* five-year (*WMAP*5), three-year (*WMAP*3) and one-year (*WMAP*1) data, respectively. These values are broadly consistent with current observations given their incompleteness. The mass of the largest lens cluster can be as small as ∼ 10^{15} M_{⊙}. For the same source redshift, we expect in all sky ∼35 (*WMAP*5), ∼15 (*WMAP*3) and ∼150 (*WMAP*1) clusters that have Einstein radii larger than 20 arcsec. For a larger source redshift of 7, the largest Einstein radii grow approximately twice as large. Whilst the values of the largest Einstein radii are almost unaffected by the level of the primordial non-Gaussianity currently of interest, the measurement of the abundance of moderately large lens clusters should probe non-Gaussianity competitively with cosmic microwave background experiments, but only if other cosmological parameters are well measured. These semi-analytic predictions are based on a rather simple representation of clusters, and hence calibrating them with *N*-body simulations will help to improve the accuracy. We also find that these ‘superlens’ clusters constitute a highly biased population. For instance, a substantial fraction of these superlens clusters have major axes preferentially aligned with the line-of-sight. As a consequence, the projected mass distributions of the clusters are rounder by an ellipticity of ∼0.2 and have ∼ 40–60 per cent larger concentrations compared with typical clusters with similar redshifts and masses. We argue that the large concentration measured in A1689 is consistent with our model prediction at the 1.2σ level. A combined analysis of several clusters will be needed to see whether or not the observed concentrations conflict with predictions of the flat Λ-dominated cold dark matter model.

*Keywords: *
gravitational lensing;
galaxies: clusters: general;
cosmology: theory;
dark matter

*Journal Article.*
*10125 words.*
*Illustrated.*

*Subjects: *
Astronomy and Astrophysics

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