form of the effective sources in Eq. (\[equationofstate\]) shows that, for vanishing averaged scalar curvature, backreaction obeys a stiff equation of state as suggested by the fluid analogy with a free scalar field [@madsen], [@buchert:grgfluid]. Moreover, if we also model the averaged curvature by an effective scalar field potential, we find that the integrability condition (\[integrability\]) provides the evolution equation for the scalar field in this potential, and it is identical to the Klein–Gordon equation, as we explain now.

Setting–up the correspondence
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We now propose to model the effective sources arising from geometrical degrees of freedom due to backreaction (i.e., roughly the ‘vacuum’ part of the effective equations) by a scalar field $\Phi_{{\cal D}}$ evolving in an effective potential $U_{{\cal D}}:= U (\Phi_{{\cal D}})$[^2], both domain–dependent, as follows (recall that we have no matter pressure source here): $$\label{morphon:sources}
\varrho^{{\cal D}}_{\rm eff} =: \langle\varrho\rangle_{\cal D} +
\varrho^{{\cal D}}_{\Phi}\;\;\;;\;\;\;
p^{{\cal D}}_{\rm eff} =: p^{{\cal D}}_{\Phi}\;\;,$$ with [@madsen], [@mukhanov] $$\label{morphon:field}
\varrho^{{\cal D}}_{\Phi}=\epsilon \frac{1}{2}{{\dot\Phi}_{{\cal D}}}^2 + U_{{\cal D}}\;\;\;;\;\;\;p^{{\cal D}}_{\Phi} =
\epsilon \frac{1}{2}{{\dot\Phi}_{{\cal D}}}^2 - U_{{\cal D}}\;\;,$$ where $\epsilon=+1$ for a standard scalar field (with positive kinetic energy), and $\epsilon=-1$ for a phantom scalar field (with negative kinetic energy). Thus, in view of Eq. 