HETEROTOPIC CLUSTERING

The observed galaxy clustering appears to display a rich scaling structure that can be characterized as a multifractal. We present two mathematical models of the non-linear clustering regime based on these interwoven fractal point sets, and show how the multifractal description provides some important insights into what is going on in non-linear clustering. This approach contrasts with earlier hierarchical models for non-linear galaxy clustering in which each level of the hierarchy is a rescaled version of its ancestor level; such hierarchies yield simple fractal distributions characterized by a single scaling exponent, The possibility of using a multifractal model to characterize the scaling properties of the clustering has important consequences. We show that being a multifractal is related to the galaxy counts-in-cells following a skewed statistical distribution such as the Lognormal. The model predicts the scaling properties of the mean and variance of the cell-count distribution with cell size. Heterotopic point distributions are a manifestation of point sets that are Lognormally distributed; in these models the voids are associated with a kind of spatial intermittency ('heterotopicity'). A multifractal model also provides insights into the way the non-linear clustering process works. We present a non-linear fragmentation model that is manifestly hierarchical in nature and that reproduces the complex multifractal scaling behaviour observed in galaxy clustering in the non-linear regime. Unlike the simple scaling hierarchies which describe non-linear clustering in terms of a single scaling parameter, this model has four scaling parameters that describe the statistical redistribution of material at each level of the clustering hierarchy.