curvature.

We shall now exemplify this correspondence for a new class of scaling solutions that contains this and other known subcases.

A family of scaling solutions of the backreaction problem
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In the following, we shall study a class of solutions that prescribes backreaction and averaged curvature functionals in the form of scaling laws of the volume scale factor $a_{{{\cal D}}}$. If not explicitly stated otherwise, we restrict attention to the case $\Lambda = 0$ throughout, and treat the cosmological constant as a particular morphon.

Exact scaling solutions
-----------------------

In this subsection we shall present a systematic classification of scaling behaviors for the cosmological models introduced previously. The averaged dust matter density $\langle\varrho\rangle_{{\cal D}}$ evolves, for a restmass preserving domain ${{\cal D}}$, as $\langle\varrho\rangle_{{\cal D}}=\langle\varrho\rangle_{{{{{\cal D}}_{\rm \bf i}}}}\; a_{{\cal D}}^{-3}$. Let us suppose that the backreaction term and the averaged curvature also obey scaling laws, that is: $$\label{prescription}
{\cal Q}_{{\cal D}}={\cal Q}_{{{{{\cal D}}_{\rm \bf i}}}}\; a_{{\cal D}}^{n}\;\;\;\;;\;\;\;\;
{\left\langle {{\cal R}}\right\rangle_{{\cal D}}}={\cal R}_{{{{{\cal D}}_{\rm \bf i}}}}\; a_{{{\cal D}}}^{p}\;\;,$$ where ${\cal Q}_{{{{{\cal D}}_{\rm \bf i}}}}$ and ${\cal R}_{{{{{\cal D}}_{\rm \bf i}}}}$ denote the initial values of ${\cal Q}_{{\cal D}}$ and ${\left\langle {{\cal R}}\right\rangle_{{\cal D}}}$, respectively.

Rewriting the integrability condition (\[integrability\]), $$a_{{\cal D}}^{-6}\partial_{t}\left(a_{{\cal 