D}}^{6}{\cal Q}_{{\cal D}}\right)+
a_{{\cal D}}^{-2}\partial_{t}\left(a_{{\cal D}}^{2}{\left\langle {{\cal R}}\right\rangle_{{\cal D}}}\right)\,=\,0\;\;,$$ a first scaling solution of that equation is obviously provided by ([@buchert:grgdust] Appendix B): $$\label{ndiffpsolution}
{\cal Q}_{{\cal D}}={\cal Q}_{{{{{\cal D}}_{\rm \bf i}}} }\; a_{{\cal D}}^{-6}\;\;\;\;;\;\;\;\;
{\left\langle {{\cal R}}\right\rangle_{{\cal D}}}={\cal R}_{{{{{\cal D}}_{\rm \bf i}}}}\; a_{{\cal D}}^{-2}\;\;.$$

![\[fig:r-n\] Scaling law $n$ for the backreaction as a function of the parameter $r$.](F1.eps){width="10cm"}

Moreover, this is clearly the only solution with $n\neq p$. In the case $n=p$, we define a new ‘backreaction parameter’ $r^{{\cal D}}$ (that can be chosen differently for a chosen domain of averaging[^5]) such that ${\cal Q}_{{{{{\cal D}}_{\rm \bf i}}}}\propto
{\cal R}_{{{{{\cal D}}_{\rm \bf i}}}}$; the solution reads: $$\label{backreactionparameter}
{\cal Q}_{{\cal D}}\;=\;r \;{\left\langle {{\cal R}}\right\rangle_{{\cal D}}}\;=\;r \;{\cal R}_{{{{{\cal D}}_{\rm \bf i}}}} \,a_{{\cal D}}^{n}\;\;,$$ where ([*cf.*]{} Fig. \[fig:r-n\]) $$\label{neqpsolution}
n\;=\;-2 \frac{(1+3r)}{(1+r)}\;\;,$$ with $r\neq-1$. The case $r=-1$ is not represented as a solution in this class; this line of states degenerates to a point corresponding to a model with Einstein–de Sitter kinematics, i.e. it has vanishing backreaction and vanishing averaged curvature. Note here, that the vanishing of ${\cal Q}_{{\cal D}}$, if required on all domains, is necessary, but also sufficient for the reduction of the averaged equations to Friedmann–Lemaître cosmologies. This limiting case also 