The strongest gravitational lenses: IV. The order statistics of the
  largest Einstein radii with cluster mergers
  Based on techniques developed in the previous papers of this series, we
investigate the impact of galaxy-cluster mergers on the order statistics of the
largest Einstein radii. We show that the inclusion of mergers significantly
shifts the extreme value distribution of the largest Einstein radius to higher
values, typically increasing the expected value by ${\sim}10\%$. A comparison
with current data reveals that the largest observed Einstein radius agrees
excellently well with the theoretical predictions of the $\Lambda$CDM model at
redshifts $z > 0.5$. At redshifts $z < 0.5$, our results are somewhat more
controversial. Although cluster mergers also increase the expected values of
the order statistics of the $n$ largest Einstein radii by ${\sim}10\%$, the
theoretically expected values are notably lower (${\sim}3\sigma$ deviation for
$n = 12$) than the largest Einstein radii of a selected sample of SDSS clusters
in the redshift range $0.1 \leq z \leq 0.55$. The uncertainties of the observed
Einstein radii are still large, however, and thus the measurements need to be
carefully revised in future works. Therefore, given the premature state of
current observational data, overall, there is still no reliable statistical
evidence for observed Einstein radii to exceed the theoretical expectations of
the standard cosmological model.
