  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,
$h(a,x)\equiv -[v/(\dot{a}x)]$ 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
$<(\delta\rho/\rho)^{2}_{x}>^{1/2}=\sigma(a,x)$ is a universal function of the
density contrast $\sigma_L(a,l)$, computed in the linear theory and evaluated
at a scale $l$ where $l=x(1+\sigma^2)^{1/3}$. 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.
