D}}$ in a t–hypersurface, normalized by the volume of the initial domain ${{{{\cal D}}_{\rm \bf i}}}$, $$\label{scalefactor}
a_{{\cal D}}:= \left(\frac{V_{{\cal D}}}{V_{{{{{\cal D}}_{\rm \bf i}}}}}\right)^{1/3}\;\;,$$ the following exact equations can be derived [@buchert:grgdust] (an overdot denotes partial time–derivative). First, by averaging [*Raychaudhuri’s equation*]{} we obtain: $$\label{averageraychaudhuri}
3\frac{{\ddot a}_{{\cal D}}}{a_{{\cal D}}} + 4\pi G \langle\varrho\rangle_{{\cal D}}- \Lambda
= {\cal Q}_{{\cal D}}\;\;\;;\;\;
 \langle\varrho\rangle_{{\cal D}}= \frac{M_{{\cal D}}}{V_{{{{{\cal D}}_{\rm \bf i}}}}a_{{\cal D}}^3}\;\;.$$ The first integral of the above equation is directly given by averaging the [*Hamiltonian constraint*]{}: $$\label{averagehamilton}
\left( \frac{{\dot a}_{{\cal D}}}{a_{{\cal D}}}\right)^2 - \frac{8\pi G}{3}\langle\varrho\rangle_{{\cal D}}- \frac{\Lambda}{3}=- \frac{{\left\langle {{\cal R}}\right\rangle_{{\cal D}}} +{\cal Q}_{{\cal D}}}{6} \;\;,$$ where the total restmass $M_{{\cal D}}$, the averaged spatial 3–Ricci scalar ${\left\langle {{\cal R}}\right\rangle_{{\cal D}}}$ and the [*kinematical backreaction term*]{} ${\cal Q}_{{\cal D}}$ are domain–dependent and, except the mass, time–dependent functions. The backreaction source term is given by $$\label{eq:Q-GR} 
{\cal Q}_{{\cal D}}: = 2 {\left\langle {{II}}\right\rangle_{{\cal D}}} - \frac{2}{3}{\left\langle {{I}}\right\rangle_{{\cal D}}}^2 =
\frac{2}{3}{\left\langle \left(\theta - {\left\langle \theta \right\rangle_{{\cal D}}}\right)^2  \right\rangle_{{\cal D}}} - 
2{\left\langle \sigma^2 \right\rangle_{{\cal D}}}\;\; ;$$ here, ${{I}}= \Theta^i_{\;i}$ and ${{II}}= \frac{1}{2}[\,(\Theta^i_{\;i})^2 - 
\Theta^i_{\;j}\Theta^j_{\;i}\,]$ denote the principal scalar invariants of the expansion tensor, defined as minus the extrinsic curvature tensor $K_{ij}:=-\Theta_{ij}$. In the second equality above it was 