The evolution of correlation functions and power spectra in gravitational clustering

Hamilton et al. proposed a simple formula relating the non-linear autocorrelation function of the mass distribution to the primordial spectrum of density fluctuations for gravitational clustering in an Ω=1 universe. We present high-resolution N-body simulations which show this formula to work well for scale-free spectra P(k) ∝ kn when n ~ 0, but not when n < – 1. We show that a modified version of the formula can work well provided that its form is allowed to depend on n. Our modified formula is easy to apply and is is a good fit to N-body simulations which have 0 ≤ n ≤ − 2. It can also be applied to non-power law initial spectra such as that of the cold dark matter model by using the local spectral index at the current non-linear scale as the effective value of n at any given redshift. We give analytic expressions both for the non-linear correlation function and for the non-linear power spectrum.