Scaling properties of the gravitational clustering in the nonlinear regime

The growth of density perturbations in an expanding universe in the non-linear regime is investigated. The underlying equations of motion are cast in a suggestive form, and motivate a conjecture that the scaled pair velocity, $h(a,x)\equiv -[v\mathrm{/}(\dot{a}x)]$ depends on the expansion factor $a$ and comoving coordinate $x$ only through the density contrast $\sigma (a,x)$. This leads to the result that the true, non-linear, density contrast $<(\delta \rho \mathrm{/}\rho {)}_x^2{>}^{1\mathrm{/}2}=\sigma (a,x)$ is a universal function of the density contrast ${\sigma }_L(a,l)$, computed in the linear theory and evaluated at a scale $l$ where $l=x(1+{\sigma }^2{)}^{1\mathrm{/}3}$. This universality is supported by existing numerical simulations with scale-invariant initial conditions having different 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 contrast at any given scale analytically. This provides a promising method for analysing the non-linear evolution of density perturbations in the universe and for interpreting numerical simulations.