Power transfer in nonlinear gravitational clustering and asymptotic universality

We study the non linear gravitational clustering of collisionless particles in an expanding background using an integro-differential equation for the gravitational potential. In particular, we address the question of how the nonlinear mode-mode coupling transfers power from one scale to another in the Fourier space if the initial power spectrum is sharply peaked at a given scale. We show that the dynamical equation allows self similar evolution for the gravitational potential $\phi_{\bf k}(t)$ in Fourier space of the form $\phi_{\bf k}(t) = F(t)D({\bf k}) $ where the function $F(t)$ satisfies a second order non-linear differential equation. We provide a complete analysis of the relevant solutions of this equation thereby determining the asymptotic time evolution of the gravitational potential and density contrast. The analysis shows that both $F(t) $ and $D({\bf k})$ have well defined asymptotic forms indicating that the power transfer leads to a universal power spectrum at late times. The analytic results are compared with numerical simulations showing good agreement.