The Einstein radius plays a central role in lens studies as it characterises the strength of gravitational lensing .
In particular , the distribution of Einstein radii near the upper cutoff 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 realisations of all-sky catalogues of massive clusters .
We find that the expected largest Einstein radius in the universe is sensitive to parameters characterising the cosmological model , especially \sigma _ { 8 } : for a source redshift of unity , they are 42 { } ^ { +9 } _ { -7 } , 35 { } ^ { +8 } _ { -6 } , and 54 { } ^ { +12 } _ { -7 } arcseconds ( errors denote 1 \sigma cosmic variance ) , assuming best-fit cosmological parameters of the Wilkinson Microwave Anisotropy Probe five-year ( WMAP5 ) , three-year ( WMAP3 ) and one-year ( WMAP1 ) 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 \sim 10 ^ { 15 } M _ { \odot } .
For the same source redshift , we expect in all-sky \sim 35 ( WMAP5 ) , \sim 15 ( WMAP3 ) , and \sim 150 ( WMAP1 ) clusters that have Einstein radii larger than 20 ^ { \prime \prime } .
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 \sim 0.2 and have \sim 40 \% - 60 \% 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 1.2 \sigma level .
A combined analysis of several clusters will be needed to see whether or not the observed concentrations conflict with predictions of the flat \Lambda -dominated cold dark matter model .