The clustering of peaks in a random Gaussian field

We consider the true peak–peak correlation function of a random Gaussian field in both one and three dimensions, as well as the validity of existing approximations to this. In particular, we show that in one dimension the peak–peak correlation function has zeros which do not, in general, coincide with the zeros of the underlying autocorrelation function: a Monte-Carlo simulation of the three-dimensional peak–peak correlation function shows the same behaviour. We show that approximations to ξpk−pk using the thresholded density field are in general inaccurate, but that ξpk−pk for a linear slice through a given density field appears to be an excellent approximation to the full three-dimensional result. Using the techniques described, we conclude that the canonical Ω = 1 CDM model has too low a correlation length when compared to the observed cluster–cluster correlation function: the predicted ξpk−pk for cluster-scale filtering goes negative at 25 h−1 Mpc. * For |h = 12|⁠, an open (Ω = 0. 2) CDM model fits reasonably well and a flat baryonic model is marginally too strongly clustered. However, these conclusions depend to a disturbing extent on the smoothing function used to filter the power spectrum: use of filters other than the conventional Gaussian choice can ameliorate or even eliminate the CDM large-scale clustering problem.