(\[equationofstate\]), we obtain the following correspondence: $$\label{correspondence1}
-\frac{1}{8\pi G}{\cal Q}_{{\cal D}}\;=\; \epsilon {\dot\Phi}^2_{{\cal D}}- U_{{\cal D}}\;\;\;;\;\;\;
-\frac{1}{8\pi G}{\left\langle {{\cal R}}\right\rangle_{{\cal D}}}= 3 U_{{\cal D}}\;\;.$$

We see that the averaged scalar curvature directly represents the potential, whereas the kinematical backreaction term represents ‘kinetic energy density’ directly, if the averaged scalar curvature vanishes. This representation of ${\cal Q}_{{\cal D}}$ is physically sensible, since it expresses a balance between ‘kinetic energy’ $E_{\rm kin}^{{\cal D}}:= 1/2 {\dot\Phi}^2_{{\cal D}}V_{{\cal D}}$ and ‘potential energy’ $E_{\rm pot}^{{\cal D}}:= -U_{{\cal D}}V_{{\cal D}}$. For ${\cal Q}_{{\cal D}}=0$ we obtain the ‘virial condition’[^3] $2\epsilon E_{\rm kin}^{{\cal D}}+ E_{\rm pot}^{{\cal D}}=0$, and so kinematical backreaction is identified as causing deviations from ‘equilibrium’ (defined through this balance[^4]). Note that Friedmann–Lemaître cosmologies correspond to the vanishing of ${\cal Q}_{{\cal D}}$ (established ‘virial equilibrium’ or vanishing relative information entropy according to [@hosoya:infoentropy]) [*on all scales*]{}. The scale–dependent formulation allows to identify states in ‘virial equilibrium’ on particular spatial scales. Below we learn that this ‘virial balance’ can be stable [*or*]{} unstable.

Inserting (\[correspondence1\]) into the integrability condition (\[integrability\]) then implies that $\Phi_{{\cal D}}$, for ${\dot\Phi}_{{\cal D}}\ne 0$, obeys the Klein–Gordon equation: $$\label{kleingordon}
{\ddot\Phi}_{{\cal D}}+ 3 H_{\cal D}{\dot\Phi}_{{\cal D}}+ 
\epsilon\frac{\partial}{\partial \Phi_{{\cal D}}}U(\Phi_{{\cal D}})\;=\;0\;\;.$$ With this 