A Critique of Scaling Behavior in Nonlinear Structure Formation Scenarios

Moments of the BBGKY equations for spatial correlation functions of cosmological density perturbations are used to obtain a differential equation for the evolution of the dimensionless function, h = -( v / x ), where v is the mean relative pair velocity. The BBGKY equations are closed using a hierarchical scaling Ansatz for the three-point correlation function. Scale-invariant solutions derived earlier by Davis and Peebles are then used in the nonlinear regime, along with the generalized stable-clustering hypothesis ( h → const), to obtain an expression for the asymptotic value of h , in terms of the power-law index of clustering, γ, and the tangential and radial velocity dispersions. The Davis-Peebles solution is found to require that tangential dispersions are larger than radial ones, in the nonlinear regime; this can be understood on physical grounds. Finally, stability analysis of the solution demonstrates that the allowed asymptotic values of h , consistent with the stable-clustering hypothesis, lie in the range 0 ≤ h ≤ 1/2. Thus, if the Davis-Peebles scale-invariant solution (and the hierarchical model for the three-point function) is correct, the standard stable-clustering picture ( h → 1 as → ∞ ) is not allowed in the nonlinear regime of structure formation.