Scaling properties of non-linear gravitational clustering

Hamilton et al. recently proposed the idea that the growth of density perturbations in an expanding universe is governed by a general scaling law, and showed agreement with existing numerical simulations. We examine the possible origin of this scaling behaviour in more detail. The underlying equations of motion are cast in a suggestive form, and motivate a conjecture that the scaled pair velocity, h(x, a) ≡ – [v/(ȧx)], depends on the expansion factor a and comoving coordinate x only through the density contrast $$\bar\xi(x,a)$$ (the two-point correlation averaged over a sphere of radius x). This leads naturally to the proposed scaling law – the true non-linear density contrast is a universal function of the density contrast $$\bar\xi_L(l,a)$$, computed in the linear theory and evaluated at a scale l which is derived to be $$l=x(1+\bar\xi)^{1/3}$$. Apart from basing the proposed scaling form on an explicit dynamical hypothesis, this gives a convenient solution for the scaling function in terms of the input pair velocity. Possibilities for further elaboration of this approach in interpreting simulations of nonlinear gravitational clustering are briefly discussed.