We investigate the transfer of power between different scales and coupling of modes during non-linear 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 .
Non-Linear 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 . 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 n < n _ { c } , power in this range grows faster than linear rate , while if n > n _ { 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 .
We demonstrate that non-linear small scales do not effect the growth of perturbations at larger scales .