Weighing Cosmic Structures with Clusters of Galaxies and the Intergalactic Medium

We present an analysis aimed at combining cosmological constraints from number counts of galaxy clusters identified through the Sunyaev-Zeldovich effect, obtained with the South Pole Telescope (SPT), and from Lyman-$\alpha$ spectra obtained with the MIKE/HIRES and X-shooter spectrographs. The SPT cluster analysis relies on mass calibration based on weak lensing measurements, while the Lyman-$\alpha$ analysis is built over a suite of hydrodynamical simulations for the extraction of mock spectra. The resulting constraints exhibit a tension ($\sim 3.3\sigma$) between the low $\sigma_8$ values preferred by the low-redshift cluster data, $\sigma_8=0.74 ^{+0.03}_{-0.04}$, and the higher one preferred by the high-redshift Lyman-$\alpha$ data, $\sigma_8=0.91 ^{+0.03}_{-0.03}$. We present a detailed analysis in order to understand the origin of this tension and, in particular, to establish whether it arises from systematic uncertainties related to the assumptions underlying the analyses of cluster counts and/or Lyman-$\alpha$ forest. We found this tension to be robust with respect to the choice of modeling of the IGM, even when including possible systematics from unaccounted sub-Damped Lyman-$\alpha$ (DLA) and Lyman-limit systems (LLS) in the Lyman-$\alpha$ data. We conclude that to solve this tension from the SPT side would require a large bias on the cluster mass estimate, or from the Lyman-$\alpha$ side large unaccounted errors on the Lyman-$\alpha$ mean fluxes, respectively. Our results have important implications for future analyses based on cluster number counts from future large photometric surveys (e.g. Euclid and LSST) and on larger samples of high-redshift quasar spectra (e.g. DESI and WEAVE surveys). If confirmed at the much higher statistical significance reachable by such surveys, this tension could represent a significant challenge for the standard $\Lambda$CDM paradigm.