The clustering of local maxima in random noise

A mixture of analytic and numerical techniques is used to study the clustering properties of local maxima of random noise. Technical complexities restrict us to the case of 1D noise, but the results obtained should give a reasonably accurate picture of the behaviour of cosmologjcal density peaks in noise defined on a 3D domain. We give estimates of ξpk−pk(r), the two-point correlation function of local maxima, for both Gaussian and non-Gaussian noise and show that previous approximations are not accurate. Furthermore, we show that the strong dependence of ξpk−pk(r) on the shape of the underlying correlation function, ξ(r), ensures that no simple approximations to ξpk−pk(r) are obtainable for general ξ(r). We find that zero-crossings of ξpk−pk(r) do not, in general, coincide with those of ξ(r). This poses a problem for the CDM model, in that the cluster–cluster correlation function is clearly positive at distances where we expect it to be negative if clusters are identified with peaks of a Gaussian random field. Using a log-normal field to model the density distribution obtained after non-linear evolution from Gaussian initial conditions, we find that a moderate amount of non-linear evolution, as expected on cluster scales, does not have a drastic effect on the bias achieved. We also study the distribution of nearest-neighbour distances for local maxima and find that, for high maxima, this distribution is very flat, leading to a scaling of the mean nearest-neighbour distance with sample size, similar to that observed by Einasto & Einasto.