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We present an analysis aimed at combining cosmological constraints fromnumber counts of galaxy clusters identified through the Sunyaev-Zeldovicheffect, obtained with the South Pole Telescope (SPT), and from Lyman-$\alpha$spectra obtained with the MIKE/HIRES and X-shooter spectrographs. The SPTcluster analysis relies on mass calibration based on weak lensing measurements,while the Lyman-$\alpha$ analysis is built over a suite of hydrodynamicalsimulations for the extraction of mock spectra. The resulting constraintsexhibit a tension ($\sim 3.3\sigma$) between the low $\sigma_8$ valuespreferred 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 tounderstand the origin of this tension and, in particular, to establish whetherit arises from systematic uncertainties related to the assumptions underlyingthe analyses of cluster counts and/or Lyman-$\alpha$ forest. We found thistension to be robust with respect to the choice of modeling of the IGM, evenwhen including possible systematics from unaccounted sub-Damped Lyman-$\alpha$(DLA) and Lyman-limit systems (LLS) in the Lyman-$\alpha$ data. We concludethat to solve this tension from the SPT side would require a large bias on thecluster mass estimate, or from the Lyman-$\alpha$ side large unaccounted errorson the Lyman-$\alpha$ mean fluxes, respectively. Our results have importantimplications for future analyses based on cluster number counts from futurelarge photometric surveys (e.g. Euclid and LSST) and on larger samples ofhigh-redshift quasar spectra (e.g. DESI and WEAVE surveys). If confirmed at themuch higher statistical significance reachable by such surveys, this tensioncould represent a significant challenge for the standard $\Lambda$CDM paradigm.

Weighing Cosmic Structures with Clusters of Galaxies and the Intergalactic Medium