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\subsection{LogR-$\gamma$ relation}
Using the simplified relations we can then compute the logR statistics at each redshift (let's start from one ..the one already computed by George z=3.211). This should correspond to the orange circles computed in the preliminary tests.
In this plot each dot represents a different simulation model. The $\gamma$ values of the contaminant foreground Ly-$\alpha$ at z=2.553 ($\gamma_{\alpha f}$) are almost the same as the Ly-$\beta$ at z=3.211 ($\gamma_{beta}$) for each model. Similarly, $T0_{\alpha f}\sim T0_{beta}$ because the simulations predict a small evolution in T0 and $\gamma$ as a function of redshift. The density fields ($\rho$) corresponding to each model, for both the redshifts of the foreground Ly-$\alpha$ and the Ly-$\beta$ forests, are the ones used to compute the LogR-$\gamma$ relation presented in the paper (that we can call "original relation" from now on) without any further modification.
For example the value of logR corresponding to the D15 model is computed using the density field of this simulation for both the redshifts z=2.553 and z=3.211 and imposing a power law with T0$\sim18200$ and $\gamma\sim 1.5$.
Comparing the ``original relation" with this simplified relation, obtained by imposing the power law, we can try to understand how much possible ``non-power law effects" can affect the results at different redshifts. We expect the biggest difference to appear at the lowest redshifts (z$\lesssim$ 2.55) where the ly-$\beta$ forest starts to probe higher overdensities ($\Delta>10$). The T($\Delta$) values corresponding to these high overdensities may in fact not follow the underlying T-$\rho$ relation. We can study as a function of redshift how much the original relation differs from the one obtained imposing a pure power-law .
