Critical Index and Fixed Point in the Transfer of Power in Nonlinear Gravitational Clustering

We investigate the transfer of power between different scales and coupling of modes during nonlinear evolution of gravitational clustering in an expanding universe. We start with a power spectrum of density fluctuations that is exponentially damped outside a narrow range of scales and use numerical simulations to study evolution of this power spectrum. Nonlinear effects generate power at other scales with most power flowing from larger to smaller scales. The ``cascade'' of power leads to equipartition of energy at smaller scales, implying a power spectrum with index $n\approx -1$. We find that such a spectrum is produced in the range $1 < \delta < 200$ for density contrast $\delta$. {\it This result continues to hold even when small scale power is added to the initial power spectrum}. Semi-analytic models for gravitational clustering suggest a tendency for the effective index to move towards a critical index $n_c\approx -1$ in this range. For $nn_c$, it grows at a slower rate -- thereby changing the index closer to $n_c$. At scales larger than the narrow range of scales with initial power, a $k^4$ tail is produced which evolves to $k^{-3/8}$ later on.