Gravitational clustering in a D-dimensional Universe

We consider the problem of gravitational clustering in a D-dimensional expanding Universe and derive scaling relations connecting the exact mean two-point correlation function with the linear mean correlation function, in the quasi-linear and non-linear regimes, using the standard paradigms of scale-invariant radial collapse and stable clustering. We show that the existence of scaling laws is a generic feature of gravitational clustering in an expanding background, in all dimensions except D=2 and comment on the special nature of the 2-dimensional case. The D-dimensional scaling laws derived here reduce, in the 3-dimensional case, to scaling relations obtained earlier from N-body simulations. Finally, we consider the case of clustering of 2-dimensional particles in a 2-D expanding background, governed by a force -GM/R, and show that the correlation function does not grow (to first order) until much after the recollapse of any shell.