\right)=0\;.$$ Eqs. (\[effectivefriedmann\]) correspond to the equations (\[averageraychaudhuri\]), (\[averagehamilton\]) and (\[integrability\]), respectively.

Given an equation of state of the form $p^{{\cal D}}_{\rm eff} = \beta (\varrho^{{\cal D}}_{\rm eff}, a_{\cal D})$ that relates the effective sources (\[equationofstate\]) with a possible explicit dependence on the volume scale factor, the effective Friedmann equations (\[effectivefriedmann\]) can be solved (one of the equations (\[effectivefriedmann\]) is redundant). Therefore, any question posed that is related to the evolution of scalar characteristics of inhomogeneous universe models may be ‘reduced’ to finding the [*cosmic state*]{} on a given spatial scale. Although formally similar to the situation in Friedmannian cosmology, here the equation of state depends on the details of the evolution of inhomogeneities. In general it describes non–equilibrium states.

We finally wish to emphasize that these equations are limited to [*regular*]{} solutions: as in the non–averaged case, the matter model ‘dust’ generically leads to shell–crossing singularities. In the averaged equations this fact would be mirrored in a break of the boundary of the averaging domain or a merging of two boundaries (Legendrian singularities), thus inducing a jump of the Euler–characteristic of the boundary. This would especially happen for small collapsing domains and is related to the fragmentation and merging 