THE RELATION BETWEEN PECULIAR VELOCITY AND DENSITY PARAMETER FOR A CLASS OF SOLUTIONS IN NEWTONIAN COSMOLOGY

We derive a relation between the peculiar velocity and the density parameter for a class of inhomogeneous three-dimensional solutions for the motion of dust in the framework of Newtonian cosmology. The domain of validity of these solutions is confined to the early non-linear regime of the formation of large-scale structure. We find that, despite of the solution's fully non-linear property, the OMEGA-0.6-relation obtained in linear pertubation theory holds. This is also true for the approximate ansatz of pancake theory, (the 'Zel'dovich approximation'), which is formally contained in a subclass. In this sense, this paper forms a generalization of a recent paper on the one-dimensional Zel'dovich solution and shows again that the non-linear corrections of the OMEGA-0.6-relation depend very much on the special geometrical situation.