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\subsection{Log(R0 evolution for varying $\gamma$ scenarios}
Previous results show a strong sensitivity of the values of $\log\langle R_{\kappa}\rangle$ to the variation of the parameter $\gamma$ in the foreground contaminating Ly-$\alpha$ forest (see section 4 in this document).
Using the correlation between $\Delta\gamma(foreground)$ and $\Delta\log\langle R_{\kappa}\rangle$ we can attempt a prediction for the expected log(R) evolution in different scenarios of possible $\gamma$ evolution with redshift.
We show how a possible considerable departure from a little evolution in $\gamma$ would manifest itself in a detectable footprint in the evolution of $\log\langle R_{\kappa}\rangle$ with redshift that our observations do not show.
