Upcoming cosmological surveys such as Euclid, LSST, and CMB-S4 will have both the sensitivity and the area to push cluster detection to z > 2. The resulting cluster catalogs will contain of the order of 100,000 cluster detections, which is two orders of magnitudes more than the number of clusters detected by the Planck satellite. As the largest gravitationally bound systems in the universe, galaxy clusters provide a low-redshift cosmological probe that is complementary to BAO, SN Ia, and CMB. Thus, it will be essential to use these objects to alleviate inherent degeneracies between cosmological parameters estimated with each individual probe, to minimise the impact of systematic effects, and to unveil potential new tensions with the standard cosmological model that are hitherto not significant. This will only be feasible if all sources of systematic uncertainties associated with cluster cosmological constraints are characterized in details. In particular, the mass-observable scaling relation, that links the observable considered in a given survey to the total mass of galaxy clusters, as well as the halo mass function, that traces the abundance of halos as a function of mass and redshift, are both key ingredients driving the precision and accuracy of the final cosmological contours. High-angular resolution millimetre and X-ray follow-up studies of clusters detected in cosmological surveys enable investigating the Intra-Cluster Medium (ICM) properties at high redshift and greatly improve our understanding of cluster formation. Such studies are fundamental to precisely calibrate the mass-observable scaling relation and the sub-grid models considered in the numerical simulations that infer the halo mass function.
I will present results obtained in the context of the NIKA2 galaxy cluster program aiming at calibrating the mass-observable scaling relation at high redshift for future millimetre surveys such as CMB-S4. I will then highlight the importance of observational contraints from cluster analyses at high redshift in order to calibrate ICM models in numerical simulations. To this end, I will describe recent results from two on-going programs realised with the South Pole Telescope and the Chandra satellite.