intrinsic curvature of the hypersurfaces), an averaged pressure gradient (or an averaged acceleration divergence), and frame fluctuation terms, i.e. coordinate effects (like the averaged variance of the lapse function in the ADM setting). In this framework and for vanishing shift, but arbitrary lapse and arbitrary 3–metric, the general equations were given in [@buchert:grgfluid]. However, we further restrict the analysis to a universe model filled with an irrotational fluid of dust matter [@buchert:grgdust] in order to provide the most transparent framework for the purpose of setting up the correspondence with a scalar field cosmology.

Averaged ADM equations for constant lapse and vanishing shift
-------------------------------------------------------------

We employ a foliation of spacetime into flow–orthogonal hypersurfaces with the 3–metric $g_{ij}$ in the line–element $ds^2 = -dt^2 + g_{ij}\,dX^i dX^j$. We define the following averager, restricting attention to scalar functions $\Psi (t,X^i )$: $$\label{eq:average-GR}
{\left\langle \Psi (t, X^i) \right\rangle_{{\cal D}}}:=
\frac{1}{V_{{\cal D}}}\int_{{\cal D}}\Psi (t, X^i)\;J d^3 X  \;\;;\;\;V_{{\cal D}}= \int_{{\cal D}}J d^3 X \;\;,$$ with $J:=\sqrt{\det(g_{ij})}$; $g_{ij}$ is an arbitrary metric of the spatial hypersurfaces, and $X^i$ are coordinates that are constant along flow lines, which are here spacetime geodesics. Defining a [*volume scale factor*]{} by the volume $V_{{\cal D}}$ of a simply–connected domain ${{\cal 