Scaling properties of non-linear gravitational clustering

The growth of density perturbations in an expanding universe in thenon-linear regime is investigated. The underlying equations of motion are castin a suggestive form, and motivate a conjecture that the scaled pair velocity,$h(a,x)\equiv -[v/(\dot{a}x)]$ depends on the expansion factor $a$ and comovingcoordinate $x$ only through the density contrast $\sigma(a,x)$. This leads tothe result that the true, non-linear, density contrast$<(\delta\rho/\rho)^{2}_{x}>^{1/2}=\sigma(a,x)$ is a universal function of thedensity contrast $\sigma_L(a,l)$, computed in the linear theory and evaluatedat a scale $l$ where $l=x(1+\sigma^2)^{1/3}$. This universality is supported byexisting numerical simulations with scale-invariant initial conditions havingdifferent power laws. We discuss a physically motivated ansatz$h(a,x)=h[\sigma^2(a,x)]$ and use it to compute the non-linear density contrastat any given scale analytically. This provides a promising method for analysingthe non-linear evolution of density perturbations in the universe and forinterpreting numerical simulations.Comment: 14 pages 2 figures available on request, TeX, IUCAA-12/9