Weak lensing calibrated Cluster Number Counts

Recent cosmological results of weak gravitational lensing calibrated cluster number counts selected by the South Pole Telescope and by eROSITA have demonstrated the potential of this cosmological probe. We review the cosmological and astrophysical aspects underlying this measurement. We highlight the multi-wavelength nature of galaxy clusters, which appear as overdensities in the galaxy field, extended X-ray emission, SZe signatures, and weak gravitational lensing signal (WL). We present how this wealth of data is observed and analyzed with a Bayesian Population model. Fitting this model allows us to constrain cosmology while simultaneously accounting for the major systematic effect of cluster cosmological studies: mass calibration. This latter point is addressed by WL, which we describe in more detail. The resulting cosmological results from the eROSITA sample can measure 4 later time parameters on their own. At the same time, the SPT analysis maps the redshift evolution of the growth of structure over an unprecedented redshift range. In conclusion, some preliminary implications from the comparison with other cosmological probes are drawn.