and ${\cal Q}_{{{\cal D}}}$, respectively. In order to compare this pair of parameters with the ‘Friedmannian curvature parameter’ that is employed to interprete observational data, we can alternatively introduce the pair $$\label{omeganewton}
\Omega_{k}^{{{\cal D}}} := - \frac{k_{{{{{\cal D}}_{\rm \bf i}}}}}{a_{{\cal D}}^2 H_{{{\cal D}}}^2 }\;\;;\;\;
\Omega_{{\cal Q}N}^{{{\cal D}}} := \frac{1}{3 a_{{\cal D}}^2 H_{{\cal D}}^2}
\int_{{{t_{\rm \bf i}}}}^t \rmd t'\ {\cal Q}_{{\cal D}}\frac{\rmd }{\rmd t'} a^2_{{\cal D}}(t')\;\;,$$ being related to the previous parameters by $\;\Omega_{k}^{{{\cal D}}} +\Omega_{{\cal Q}N}^{{{\cal D}}}\;=\; 
\Omega_{\cal R}^{{{\cal D}}} + \Omega_{\cal Q}^{{{\cal D}}}$.

These parameters arise from inserting the first integral of Eq. (\[integrability\]), $$\label{integrabilityintegral}
\frac{k_{{{{{\cal D}}_{\rm \bf i}}}}}{a_{{\cal D}}^2} - \frac{1}{3 a_{{\cal D}}^2} \int_{t_i}^t \,dt' \;
{\cal Q}_{{\cal D}}\; \frac{d}{dt'} a^2_{{\cal D}}(t')
= \frac{1}{6}\left(\,{\left\langle {{\cal R}}\right\rangle_{{\cal D}}} + {\cal Q}_{{\cal D}}\,\right) \;\;,$$ into (\[averagehamilton\]): $$\label{averagefriedmann}
\frac{\dot{a}_{{\cal D}}^2 + k_{{{{{\cal D}}_{\rm \bf i}}}}}{a_{{\cal D}}^2 } - \frac{8\pi G {\left\langle \varrho \right\rangle_{{\cal D}}}}{3}
- \frac{\Lambda}{3} = \frac{1}{3 a_{{\cal D}}^2} \int_{{{t_{\rm \bf i}}}}^t \rmd t'\ {\cal Q}_{{\cal D}}\frac{\rmd }{\rmd t'} a^2_{{\cal D}}(t')\;\;.$$ This equation is formally equivalent to its Newtonian counterpart [@buchertehlers], [@bks]. It shows that, by eliminating the averaged curvature, the whole history of the averaged kinematical fluctuations acts as a source of a generalized Friedmann equation.

Like the volume 