The multifractal behaviour of hierarchical density distributions

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The observed power-law shape of the two-point correlation function for the distribution of cosmic structures, ξ(r) ∝ r–γ as well as the hierarchical behaviour displayed by higher order correlation functions, argues for a self-similar fractal structure of the non-linear galaxy clustering. In this paper we investigate the multifractal structure implied by a list of non-Gaussian, hierarchical distributions already introduced in the literature. For each distribution we work out the multifractal spectrum of generalized dimensions, Dq, implied by the small-scale non-linear clustering. We find that positive-order dimensions, which describe the scaling properties of the distribution inside the overdense regions, turn out to be Dq = 3 – γ, independent of the details of the hierarchical model. In contrast, negative-order dimensions, which are related to the scaling inside the underdense regions, depend on the shape of the distribution. In particular, the negative binomial and the hierarchical Poissonian models exhibit monofractal behaviour, while the generalized thermodynamical and the BBGKY models have rather different scaling in the overdense and in the underdense regions, with a resulting non-trivial multifractal behaviour. This remarkable result emphasizes how important it is to characterize properly the statistics inside the devoid regions of the galaxy distribution in order to discriminate between different theoretical models. We also compare the Dq spectrum of the above models with the results of the multifractal analysis of N-body simulations realized by Valdarnini, Borgani & Provenzale. The rather complex behaviour displayed by the simulations for negativeorder dimensions rules out most of the hierarchical models, although all of them seem to be equally efficient at accounting for the flat shape of the Dq curve at positive q. At moderate values of |q|, which characterize the geometry rather than the clustering of the point distribution, the ‘lognormal’ model introduced by Jones, Coles & Martínez is quite good at reproducing the output of non-linear clustering. Although the multifractal behaviour of the generalized thermodynamical model could be quite promising as an explanation for the output of the N-body simulations, the lack of convergence of the observed dimension spectrum to the D– ∞ value prevents us from drawing any definite conclusion. This further indicates the importance of properly characterizing the statistics inside the most underdense regions.