Characterising superclusters with the galaxy cluster distribution
  Superclusters are the largest, observed matter density structures in the
Universe. Recently Chon et al.(2013) presented the first supercluster catalogue
constructed with a well-defined selection function based on the X-ray
flux-limited cluster survey, REFLEX II. For the construction of the sample we
proposed a concept to find the large objects with a minimum overdensity such
that most of their mass will collapse in the future. The main goal of the paper
is to provide support for our concept using simulations that we can, on the
basis of our observational sample of X-ray clusters, construct a supercluster
sample defined by a certain minimum overdensity, and to test how superclusters
trace the underlying dark matter distribution. Our results confirm that an
overdensity in the number of clusters is tightly correlated with an overdensity
of the dark matter distribution. This enables us to define superclusters such
that most of the mass will collapse in the future and to get first-order mass
estimates of superclusters on the basis of the properties of the member
clusters. We also show that in this context the ratio of the cluster number
density and dark matter mass density is consistent with the theoretically
expected cluster bias. Our previous work provided evidence that superclusters
are a special environment for density structures of the dark matter to grow
differently from the field as characterised by the X-ray luminosity function.
Here we confirm for the first time that this originates from a top-heavy mass
function at high statistical significance provided by a Kolmogorov-Smirnov
test. We also find in close agreement with observations that the superclusters
occupy only a small volume of few percent while they contain more than half of
the clusters in the present day Universe.
