The Evolution of Correlation Functions and Power Spectra in Gravitational Clustering

Hamilton et al. (1991) proposed a simple formula relating the nonlinear autocorrelation function of the mass distribution to the primordial spectrum of density fluctuations for gravitational clustering in an $\Omega=1$ universe. High resolution N-body simulations show this formula to work well for scale-free spectra $P(k) \propto k^n$ when the spectral index $n\sim 0$, but not when $n \lsim -1$. We show that a modified version of the formula can work well provided its form depends on $n$. This dependence can be derived from a simple physical model for collapse from Gaussian initial conditions. Our modified formula is easy to apply and is an excellent fit to N-body simulations with $0 \leq n \leq -2$. It can also be applied to non-power law initial spectra such as that of the standard Cold Dark Matter model by using the local spectral index at the current nonlinear scale as the effective value of $n$ at any given redshift. We give analytic expressions both for the nonlinear correlation function and for the nonlinear power spectrum.