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Observational investigations of the abundance of massive precursors of localgalaxy clusters ("proto-clusters") allow us to test the growth of densityperturbations, to constrain cosmological parameters that control it, to testthe theory of non-linear collapse and how the galaxy formation takes place indense environments. The Planck collaboration has recently published a catalogueof >~ 2000 cold extra-galactic sub-millimeter sources, i.e. with coloursindicative of z >~ 2, almost all of which appear to be over-densities ofstar-forming galaxies. They are thus considered as proto-cluster candidates.Their number densities (or their flux densities) are far in excess ofexpectations from the standard scenario for the evolution of large-scalestructure. Simulations based on a physically motivated galaxy evolution modelshow that essentially all cold peaks brighter than S_{545GHz} = 500 mJy foundin Planck maps after having removed the Galactic dust emission can beinterpreted as positive Poisson fluctuations of the number of high-z dustyproto-clusters within the same Planck beam, rather then being individual clumpsof physically bound galaxies. This conclusion does not change if an empiricalfit to the luminosity function of dusty galaxies is used instead of thephysical model. The simulations accurately reproduce the statistic of thePlanck detections and yield distributions of sizes and ellipticities inqualitative agreement with observations. The redshift distribution of thebrightest proto-clusters contributing to the cold peaks has a broad maximum at1.5 <~ z <~ 3. Therefore follow-up of Planck proto-cluster candidates willprovide key information on the high-z evolution of large scale structure.

On the statistics of proto-cluster candidates detected in the Planck all-sky survey