the Universe with the help of averaged matter sources only. We call the scalar field a [*morphon*]{} only if it arises from the proposed mean field description; we can of course still consider other scalar fields as sources of Einstein’s equations. We expect, and we shall demonstrate this for quintessence fields, that the [*morphon*]{} is capable of acting like any other scalar field model, e.g., assigned to an inflaton, a curvaton, a dilaton, etc., depending on our capability to exploit the proposed correspondence, i.e. the possibility to find solutions of the averaged Einstein equations and to reconstruct the scalar field potential for this solution.

We here provide a first step that should help to appreciate the various possibilities, but this investigation does not claim to be exhaustive. In this line we are going to choose a minimal parameterization of the scalar field cosmology in the following sense: first, we do not include the cosmological constant and the constant curvature term in the ‘Friedmannian part’ of the averaged equations, since both arise as subcases of special solutions for the backreaction terms; second, we choose the simplest foliation (constant lapse and vanishing shift in the ADM (Arnowitt Deser Misner) formulation of Einstein’s 