THE EVOLUTION OF CORRELATION-FUNCTIONS AND POWER SPECTRA IN GRAVITATIONAL CLUSTERING

Hamilton et al. proposed a simple formula relating the non-linear auto correlation function of the mass distribution to the primordial spectr um of density fluctuations for gravitational clustering in an Ohm = 1 universe. We present high-resolution N-body simulations which show thi s formula to work well for scale-free spectra P(k) proportional to k(n ) when n similar to 0, but not when n < -1. We show that a modified ve rsion of the formula can work well provided that its form is allowed t o depend on n. Our modified formula is easy to apply and is a good fit to our N-body simulations which have 0 less than or equal to n less t han or equal to -2. It can also be applied to non-power-law initial sp ectra such as that of the cold dark matter model by using the local sp ectral index at the current non-linear scale as the effective value of n at any given redshift. We give analytic expressions both for the no n-linear correlation function and for the non-linear power spectrum.