Scaling properties of nonlinear gravitational clustering

Abstract

Hamilton et al. recently proposed the idea that the growth of density perturbations in an expanding universe is govemed 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, α)≡ -[v/(αx)], depends on the expansion factor α and comoving coordinate x only through the density contrast ξ-(x, α) (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 ξ-L(l,a), computed in the linear theory and evaluated at a scale lwhich is derived to be l =x(1 +ξ-)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 non-linear gravitational clustering are briefly discussed.