Order statistics applied to the most massive and most distant galaxy
  clusters
  In this work we present for the first time an analytic framework for
calculating the individual and joint distributions of the n-th most massive or
n-th highest redshift galaxy cluster for a given survey characteristic allowing
to formulate LCDM exclusion criteria. We show that the cumulative distribution
functions steepen with increasing order, giving them a higher constraining
power with respect to the extreme value statistics. Additionally, we find that
the order statistics in mass (being dominated by clusters at lower redshifts)
is sensitive to the matter density and the normalisation of the matter
fluctuations, whereas the order statistics in redshift is particularly
sensitive to the geometric evolution of the Universe. For a fixed cosmology,
both order statistics are efficient probes of the functional shape of the mass
function at the high mass end. To allow a quick assessment of both order
statistics, we provide fits as a function of the survey area that allow
percentile estimation with an accuracy better than two per cent. Furthermore,
we discuss the joint distributions in the two-dimensional case for different
combinations of order.
  Having introduced the theory, we apply the order statistical analysis to the
SPT massive cluster sample and MCXC catalogue and find that the ten most
massive clusters in the sample are consistent with LCDM and the Tinker mass
function. In turn, by assuming the LCDM reference cosmology, order statistics
can also be utilised for consistency checks of the completeness of the observed
sample and of the modelling of the survey selection function. [abridged]
