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 Q 0.6 -relation obtained in linear perturbation 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 Q 0.6 -relation depend very much on the special geometrical situation