deletions | additions       diff --git a/section_Perturbations_The_linear_perturbations__.tex b/section_Perturbations_The_linear_perturbations__.tex  index 5f31bc0..d974813 100644  --- a/section_Perturbations_The_linear_perturbations__.tex  +++ b/section_Perturbations_The_linear_perturbations__.tex 
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we can now use these relations to substitute the $\delta G$ terms in \ref{scalefactorfirst} after another differentiation to obtain a third order differential equation for the growth factor that will depend only on the cosmological quantities already introduced:  \begin{equation}  \label{scalefactorsecond}  D'''(a)+\frac{1}{2}\left( \begin{split}  &D'''(a)+\frac{1}{2}\left(  16-9\tilde{\Omega}(a) \right) \frac{D''(a)}{a} + \\ \frac{3}{2} &\frac{3}{2}  \left( 8 - 11 \tilde{\Omega}_m(a) + 3 \tilde{\Omega}_m^2(a) - a \tilde{\Omega}_m'(a) \right) \frac{D'(a)}{a^2} = 0 \end{split}  \end{equation}  This equation lends itself to a numerical solution whose behavior is shown in [FIGURE] where the growth factor is shown as a function of redshift alongside with the $\Lambda CDM$ solution.   It is possible to see how the model predicts and enhancement of the growth for the $\nu < 0$ solution, this is due to the strengthening of the gravitational coupling at high redshift that allows to overcome the "repulsion" associated with the expansion of cause by the vacuum energy density.