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 α 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 α analysis is built over mock spectra extracted from hydrodynamical simulations. The resulting constraints exhibit a tension (∼3.3σ) between the low σ8 values preferred by the low-redshift cluster data, σ8=0.74+0.03-0.04, and the higher one preferred by the high-redshift Lyman α data, σ8=0.91+0.03-0.03. We present a detailed analysis to understand the origin of this tension and to establish whether it arises from systematic uncertainties related to the assumptions underlying the analyses of cluster counts and/or Lyman α forest. We found this tension to be robust with respect to the choice of modelling of the IGM, even when including possible systematics from unaccounted sub-Damped Lyman α (DLA) and Lyman-limit systems (LLS) in the Lyman α data. We conclude that to solve this tension would require a large bias on the cluster mass estimate, or large unaccounted errors on the Lyman α mean fluxes. 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 ΛCDM paradigm.
