split into kinematical invariants through the decomposition $\Theta_{ij} = \frac{1}{3}g_{ij}\theta + \sigma_{ij}$, with the rate of expansion $\theta =\Theta^i_{\;i}$, the shear tensor $\sigma_{ij}$, and the rate of shear $\sigma^2 :=1/2 \sigma_{ij}\sigma^{ij}$; note that vorticity is absent in the present foliation; we adopt the summation convention.

The time–derivative of the averaged Hamiltonian constraint (\[averagehamilton\]) agrees with the averaged Raychaudhuri equation (\[averageraychaudhuri\]) by virtue of the following [*integrability condition*]{}: $$\label{integrability}
\partial_t {\cal Q}_{{\cal D}}+ 6 H_{\cal D} {\cal Q}_{{\cal D}}+  
\partial_t {\left\langle {{\cal R}}\right\rangle_{{\cal D}}}+ 2 H_{\cal D} {\left\langle {{\cal R}}\right\rangle_{{\cal D}}} = 0 \;\;,$$ where we have introduced a [*volume Hubble functional*]{} $H_{\cal D}:= 
{\dot a}_{{\cal D}}/ a_{{\cal D}}$. The above equations can formally be recast into standard Friedmann equations for an effective perfect fluid energy momentum tensor with new effective sources [@buchert:grgfluid][^1]: $$\label{equationofstate}
\fl
\varrho^{{{\cal D}}}_{\rm eff} = {\left\langle \varrho \right\rangle_{{\cal D}}}-\frac{1}{16\pi G}{\cal Q}_{{\cal D}}- 
\frac{1}{16\pi G}{\left\langle {{\cal R}}\right\rangle_{{\cal D}}}
\;\;\;;\;\;\;{p}^{{{\cal D}}}_{\rm eff} =  -\frac{1}{16\pi G}{\cal Q}_{{\cal D}}+ \frac{1}{48\pi
G}{\left\langle {{\cal R}}\right\rangle_{{\cal D}}}\;\;.$$ $$\label{effectivefriedmann}
\fl
3\frac{{\ddot a}_{{\cal D}}}{a_{{\cal D}}} = \Lambda - 4\pi G (\varrho^{{{\cal D}}}_{\rm eff}
+3{p}^{{{\cal D}}}_{\rm eff})\;\;;\;\;
3H_{{\cal D}}^2 =\Lambda + 8\pi G \varrho^{{{\cal D}}}_{\rm eff}\;\;;\;\;
{\dot\varrho}^{{{\cal D}}}_{\rm eff} + 
3H_{{\cal D}}\left(\varrho^{{{\cal D}}}_{\rm eff}
+{p}^{{{\cal D}}}_{\rm eff} 