On the statistics of proto-cluster candidates detected in the Planck
  all-sky survey
  Observational investigations of the abundance of massive precursors of local
galaxy clusters ("proto-clusters") allow us to test the growth of density
perturbations, to constrain cosmological parameters that control it, to test
the theory of non-linear collapse and how the galaxy formation takes place in
dense environments. The Planck collaboration has recently published a catalogue
of >~ 2000 cold extra-galactic sub-millimeter sources, i.e. with colours
indicative of z >~ 2, almost all of which appear to be over-densities of
star-forming galaxies. They are thus considered as proto-cluster candidates.
Their number densities (or their flux densities) are far in excess of
expectations from the standard scenario for the evolution of large-scale
structure. Simulations based on a physically motivated galaxy evolution model
show that essentially all cold peaks brighter than S_{545GHz} = 500 mJy found
in Planck maps after having removed the Galactic dust emission can be
interpreted as positive Poisson fluctuations of the number of high-z dusty
proto-clusters within the same Planck beam, rather then being individual clumps
of physically bound galaxies. This conclusion does not change if an empirical
fit to the luminosity function of dusty galaxies is used instead of the
physical model. The simulations accurately reproduce the statistic of the
Planck detections and yield distributions of sizes and ellipticities in
qualitative agreement with observations. The redshift distribution of the
brightest proto-clusters contributing to the cold peaks has a broad maximum at
1.5 <~ z <~ 3. Therefore follow-up of Planck proto-cluster candidates will
provide key information on the high-z evolution of large scale structure.
