equations) and the simplest matter model ‘irrotational dust’ in order to exemplify the correspondence; third, we realize the correspondence for a particular class of scaling solutions with single–power laws. In this framework we show that the correspondence holds with a standard minimally coupled scalar field that can play the role of a quintessence field. If we include the full coordinate degrees of freedom as well as an inhomogeneous pressure source in a perfect fluid energy momentum tensor (or even imperfect fluid sources), more sophisticated scalar field cosmologies would arise. We comment on this possibility at the end of the paper.

The proposed correspondence can also be inverted, e.g. we may start with a known model of quintessence and try to recover the corresponding solution of the averaged Einstein equations. We shall not exemplify this inversion in the present paper, we just note that free parameters in a given quintessence model can be determined through this correspondence.

We proceed as follows. Section 2 recalls the basic equations and relations needed for the present work. Section 3 sets up the correspondence between kinematical backreaction and scalar field models. Section 4 investigates a family of scaling solutions to the backreaction problem. Section 5 