exploits the proposed correspondence. Here we explicitly reconstruct the potential for the scaling solutions, explore the solution space of inhomogeneous cosmologies with the help of the scaling solutions, and discuss some particular cases, which have been advanced before in the literature, and which are now interpreted with the help of a morphon field. We also give a concrete model and confront it with observational constraints. Section 6 contains a summary and an outlook on possible generalizations.

The backreaction context
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The averaging problem in relativistic cosmology involves a variety of approaches and problems. Generally, research work in this field deals with averaging inhomogeneities in matter and/or geometry. Mostly, spatial averaging is envisaged, but also averaging on the light–cone is accessible by some averaging techniques (compare discussions and references in [@ellis], [@ellisbuchert], [@carfora:RG; @klingon; @buchertcarfora:PRL].)

In this work we focus on the comparatively simple approach of averaging the scalar parts of Einstein’s equations on a given foliation of spacetime. The time–evolution of integral properties of the cosmological model on compact spatial domains can be extracted from Einstein’s equation without any perturbative assumptions by Riemannian volume integration. The simplest example is the time–evolution of the volume. Thus, we do not aim at 