The strongest gravitational lenses Ill. The order statistics of the largest Einstein radii

Context. The Einstein radius of a gravitational lens is a key characteristic. It encodes information about decisive quantities such as halo mass, concentration, triaxiality, and orientation with respect to the observer. Therefore, the largest Einstein radii can potentially be utilised to test the predictions of the ACDM model. Aims. Hitherto, studies have focussed on the single largest observed Einstein radius. We extend those studies by employing order statistics to formulate exclusion criteria based on the n largest Einstein radii and apply these criteria to the strong lensing analysis of 12 MACS clusters at z > 0.5. Methods. We obtain the order statistics of Einstein radii by a Monte Carlo approach, based on the semi-analytic modelling of the halo population on the past lightcone. After sampling the order statistics, we fit a general extreme value distribution to the first-order distribution, which allows us to derive analytic relations for the order statistics of the Einstein radii. Results. We find that the Einstein radii of the 12 MACS clusters are not in conflict with the ACDM expectations. Our exclusion criteria indicate that, in order to exhibit tension with the concordance model, one would need to observe approximately twenty Einstein radii with theta(eff) greater than or similar to 30 '', ten with theta(eff) greater than or similar to 35 '', five with theta(eff) greater than or similar to 42 '', or one with theta(eff) greater than or similar to 74 '' in the redshift range 0.5 <= z <= 1.0 on the full sky (assuming a source redshift of z(s) = 2). Furthermore, we find that, with increasing order, the haloes with the largest Einstein radii are on average less aligned along the line-of-sight and less triaxial. In general, the ctunulative distribution functions steepen for higher orders, giving them better constraining power. Conclusions. A framework that allows the individual and joint order distributions of the n-largest Einstein radii to be derived is presented. From a statistical point of view, we do not see any evidence of an Einstein ring problem even for the largest Einstein radii of the studied MACS sample. This conclusion is consolidated by the large uncertainties that enter the lens modelling and to which the largest Einstein radii are particularly sensitive.