Scaleinvariant matter distribution in the universe. II  Bifractal behaviour
Abstract
We calculate the multifractal dimensions of the galaxy and matter distribution in the universe, from the unique assumption that the Nbody correlation functions behave as r^(N1)γ^ with the same index γ. We discuss the evidence from observations, theory and numerical simulations for this hypothesis. It leads to a correlation length independent of the size of the sample and it is already amply justified by its ability to predict the galaxy counts in cells, the largescale void distribution, as well as the cluster counts and correlations. We find a typical scale in the universe, of the order of a few times the correlation length, determined by the scale at which the present universe turns nonlinear, below which galaxy and matter distribution is scalefree. Although simple, the induced powerlaw behaviour leads to a rich structure for galaxy and matter clustering properties. Nontrivial fractal behaviours are found between bounds which can be evaluated. The galaxy distribution is seen to be bifractal between l_c_ ~ 0.1 and l_v_ ~ 10h^1^ Mpc. The lower bound l_c_ is the average distance between galaxies in a cluster and appears because galaxies are discrete objects. The upper bound l_v_ interpreted as a typical void size, represents the scale at which a transition occurs for a randomly placed box, from being probably empty to being probably occupied. The occurrence of a bifractal behaviour is due to the coexistence at the same scales of clusters and of voids with a sizeable probability. The two fractal dimensions are 3  γ for the dense, clustered regions, and (3  γ), for the underdense, but still populated regions. The index π is related to the large scale behaviour of the void probability as well as to the parameter [alpha} = π  2 introduced by Schechter to fit the galaxy luminosity function. A simple description of galaxy clustering in the universe comes out: most galaxies are organized in selfsimilar clusters with the fractal dimension 3  γ; these structures are surrounded by loosely populated or void regions which occupy most of the space. The latter have the dimension (3  γ)π, which is Hausdorff's dimension, smaller than the correlation dimension. Although described by correlation functions with the same powerlaw, the matter distribution is quite different, since no voids are expected in this case. It also displays a bifractal behaviour, but with different dimensions: 3  γ in the dense, and 3 in the underdense regions. The latter behaviour may possibly hold for galaxies in case the very faint, overwhelmingly numerous ones are in the number counts. The main results are reviewed in Sect. 7 which is selfcontained.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 December 1989
 Bibcode:
 1989A&A...226..373B
 Keywords:

 Cosmology;
 Fractals;
 Galactic Clusters;
 Interstellar Matter;
 Universe;
 Bbgky Hierarchy;
 Computational Astrophysics;
 Probability Distribution Functions;
 Spatial Distribution;
 Astrophysics