Averaging inhomogeneous Newtonian cosmologies

Idealizing matter as a pressureless fluid and representing its motion by a peculiar--velocity field superimposed on a homogeneous and isotropic Hubble expansion, we apply (Lagrangian) spatial averaging on an arbitrary domain $\cal D$ to the (nonlinear) equations of Newtonian cosmology and derive an exact, general equation for the evolution of the (domain dependent) scale factor $a_{\cal D}(t)$. We consider the effect of inhomogeneities on the average expansion and discuss under which circumstances the standard description of the average motion in terms of Friedmann's equation holds. We find that this effect vanishes for spatially compact models if one averages over the whole space. For spatially infinite inhomogeneous models obeying the cosmological principle of large--scale isotropy and homogeneity, Friedmann models may provide an approximation to the average motion on the largest scales, whereas for hierarchical (Charlier--type) models the general expansion equation shows how inhomogeneities might appreciably affect the expansion at all scales. An averaged vorticity evolution law is also given. Since we employ spatial averaging, the problem of justifying ensemble averaging does not arise. A generalization of the expansion law to general relativity is straightforward for the case of irrotational flows and will be discussed. The effect may have important consequences for a variety of problems in large--scale structure modeling as well as for the interpretation of observations.