AVERAGING INHOMOGENEOUS NEWTONIAN COSMOLOGIES

Idealizing matter as a pressureless fluid and representing its motion by a peculiar-velocity field superimposed on a homogeneous and isotrop ic Hubble expansion, we apply (Lagrangian) spatial averaging on an arb itrary domain D to the (nonlinear) equations of Newtonian cosmology an d derive an exact, general equation for the evolution of the (domain d ependent) scale factor a(D)/(t). We consider the effect of inhomogenei ties on the average expansion and discuss under which circumstances th e standard description of the average motion in terms of Friedmann's e quation holds. We find that this effect vanishes for spatially compact models if one averages over the whole space. For spatially infinite i nhomogeneous models obeying the cosmological principle of large-scale isotropy and homogeneity, Friedmann models may provide an approximatio n to the average motion on the largest scales, whereas for hierarchica l (Charlier-type) models the general expansion equation shows how inho mogeneities might appreciably affect the expansion at all scales. An a veraged vorticity evolution law is also given. Since we employ spatial averaging, the problem of justifying ensemble averaging does not aris e. A generalization of the expansion law to general relativity is stra ightforward for the case of irrotational flows and will be discussed. The effect may have important consequences for a variety of problems i n large-scale structure modeling as well as for the interpretation of observations.