Arbitrating the $S_8$ discrepancy with growth rate measurements from Redshift-Space Distortions

Within the $\Lambda$CDM model, measurements from recent Cosmic Microwave Background (CMB) and weak lensing (WL) surveys have uncovered a $\sim 3\sigma$ disagreement in the inferred value of the parameter $S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}$, quantifying the amplitude of late-time matter fluctuations. Before questioning whether the $S_8$ discrepancy calls for new physics, it is important to assess the view of measurements other than CMB and WL ones on the discrepancy. Here, we examine the role of measurements of the growth rate $f(z)$ in arbitrating the $S_8$ discrepancy, considering measurements of $f\sigma_8(z)$ from Redshift-Space Distortions (RSD). Our baseline analysis combines RSD measurements with geometrical measurements from Baryon Acoustic Oscillations (BAO) and Type Ia Supernovae (SNeIa), given the key role of the latter in constraining $\Omega_m$. From this combination and within the $\Lambda$CDM model we find $S_8 = 0.762^{+0.030}_{-0.025}$, and quantify the agreement between RSD+BAO+SNeIa and \textit{Planck} to be at the $2.2\sigma$ level: the mild disagreement is therefore compatible with a statistical fluctuation. We discuss combinations of RSD measurements with other datasets, including the $E_G$ statistic. This combination increases the discrepancy with \textit{Planck}, but we deem it significantly less robust. Our earlier results are stable against an extension where we allow the dark energy equation of state $w$ to vary. We conclude that, from the point of view of combined growth rate and geometrical measurements, there are hints, but no strong evidence yet, for the \textit{Planck} $\Lambda$CDM cosmology over-predicting the amplitude of matter fluctuations at redshifts $z \lesssim 1$. From this perspective, it might therefore still be premature to claim the need for new physics from the $S_8$ discrepancy.