Closing up the cluster tension

The exquisite measurements of the cosmic microwave background (CMB) fluctuations by Planck allows us to tightly constrain the amplitude of matter fluctuations at redshift $\sim 1100$ in the $\Lambda$-cold dark matter ($\Lambda$CDM) model. This amplitude can be extrapolated to the present epoch, yielding constraints on the value of the $\sigma_8$ parameter. On the other hand the abundance of Sunyaev-Zeldovich (SZ) clusters detected by Planck, with masses inferred by using hydrostatic equilibrium estimates, leads to a significantly lower value of the same parameter. This discrepancy is often dubbed the "$\sigma_8$ tension" in the literature and is sometimes regarded as a possible sign of new physics. Here, we examine a direct determination of $\sigma_8$ at the present epoch in $\Lambda$CDM, and thereby the cluster mass calibrations using cosmological data at low redshift, namely the measurements of $f\sigma_8$ from the analysis of the completed Sloan Digital Sky Survey (SDSS): we combine redshift-space distortions measurements with Planck CMB constraints, X-ray, and SZ cluster counts within the $\Lambda$CDM framework, but leaving the present day amplitude of matter fluctuations as an independent parameter (i.e. no extrapolation is made from high-redshift CMB constraints). The calibration of X-ray and SZ masses are therefore left as free parameters throughout the whole analysis. Our study yields tight constraints on the aforementioned calibrations, with values entirely consistent with results obtained from the full combination of CMB and cluster data only. Such agreement suggests an absence of tension in the $\Lambda$CDM model between CMB-based estimates of $\sigma_8$ and constraints from low-redshift on $f\sigma_8$ but indicates a tension with the standard calibration of clusters masses.