Untying the growth index to relieve the $\sigma_8$ discomfort

The matter fluctuation parameter $\sigma_8$ is, by model construction, degenerate with the growth index $\gamma$. Here, we study the effect on the cosmological parameter constraints by treating each independently from one another, considering $\sigma_8$ as a free and non-derived parameter along with a free $\gamma$. We then try to constrain all parameters using three probes that span from deep to local redshifts, namely the CMB spectrum, the growth measurements from redshift space distortions and the galaxy cluster counts. We also aim to assess the impact of this relaxation on the $\sigma_8$ tension. We also propose a more sophisticated correction, along with the classical one, that takes into account the impact of cosmology on the growth measurements by adjusting the growth to keep the observed power spectrum invariant with the background evolution. We found that untying the two parameters does not shift the maximum likelihood of either $\sigma_8$ or $\gamma$, but rather enables larger bounds with respect to when $\sigma_8$ is a derived parameter. More precisely, we obtain $\sigma_8 = 0.809\pm 0.043 $ and $\gamma = 0.613\pm 0.046$ in agreement with Planck's constraint for the former and compatible with $\Lambda$CDM for the latter but with bounds wide enough to accommodate both values subject to the tensions. On the other hand, considering a tiered correction yields $\sigma_8 = 0.734\pm 0.013$ close to the inferred local values albeit with a growth index of $\gamma = 0.636\pm 0.022$, while allowing for massive neutrinos yielded $\sigma_8 = 0.756\pm 0.024$, still preferring low values but with looser constraints on $\gamma$ and a slight preference for $\Sigma m_\nu \sim 0.19$. We conclude that untying $\sigma_8$ and $\gamma$ helps in relieving the discomfort on the former and that careful analysis should be followed when using data products treated in a model-dependent way.(abridged)