deletions | additions       diff --git a/section_Perturbations_The_linear_perturbations__.tex b/section_Perturbations_The_linear_perturbations__.tex  index 2c4025f..7137d2c 100644  --- a/section_Perturbations_The_linear_perturbations__.tex  +++ b/section_Perturbations_The_linear_perturbations__.tex 
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\end{equation}  where a prime denotes differentiation with respect to the scale factor.  This equation reduces to the standard $\Lambda CDM$ equation for the scale factor if we assume a vanishing $\delta G$, but to follow the true evolution of the perturbations in this model we will assume $\delta G \not= 0$.  In this case the perturbations for $\rho_\Lambda$ and $\rho_m$ are related to $\delta G$ due to the constraint imposed by the Bianchi identity \ref{bianchiconstraint}: (\ref{bianchiconstraint}):  \begin{equation}  \label{perturbationrelations}  \frac{\delta \rho_\Lambda}{\rho_\Lambda} = - \frac{\delta G}{G}, ~~~   \frac{\delta \rho_m}{\rho_m} = - \frac{(\delta G(a))'}{G'(a)}  \end{equation}  we can now use these relations to substitute the $\delta G$ terms in \ref{scalefactorfirst} (\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}  \begin{split}