Multifractal absolute galactic luminosity distributions and the multifractal Hubble 3/2 law

Scale-invariant intergalactic dynamics governed by a statistically homogene ous cascade process generically yields multifractal luminosity distribution s with highly inhomogeneous realizations (the standard nonfractal and fract al models are special limiting cases). The main obstacles for extending sca ling analyses to the spatial distribution of galactic absolute luminosities are the large "Malmquist" catalogue biases which - for multifractal galaxy distributions - we here show how to remove. We also derive the theoretical relation between absolute and apparent luminosity multifractal catalogues (the multifractal extension of the "Hubble 3/2" law; not to be confused wit h the more usual Hubble law governing the expansion of the universe) and sh ow that the theory is compatible with both the observed apparent and absolu te luminosities. The results of multifractal analysis on two galaxy catalog ues (depth 150 h(-1) Mpc each) show that the observed form of the dimension function follows if only matter in sufficiently dense (and sparse) concent rations is luminous (with critical dimension D-c approximate to 1.85), i.e. , mass and luminosity are tightly correlated only above a critical mass den sity singularity threshold (gamma (c) approximate to 0.4). Since this criti cal singularity is considerably larger than that which determines the mean mass, the clusters responsible for the mean mass are dark and we obtain a " dark mass exponent" delta approximate to 0.75. This implies that the ratio of luminous to dark matter is Lambda'(delta) where Lambda' is the ratio of the outer and inner cascade scales; taking Lambda' in the range 10-100 we f ind that 85-97% of the matter is dark (Lambda' approximate to 10 is the Val ue most compatible with the microwave background and standard cosmologies a nd with the data used here, Lambda' approximate to 100 is apparently compat ible with some galaxy catalogues). The model also includes a multifractal p hase transition associated with very bright self-organized critical galaxie s whose luminosity we find to be algebraic with critical exponent approxima te to 4 (not exponential as is often assumed). A basic problem with the sca ling models proposed to date is that there is no satisfactory way of reconc iling the high heterogeneity of luminous matter (fractal dimension less tha n or equal to1.85) with the apparently low heterogeneity of the mass as inf erred from the cosmic background or the small peculiar velocities. Our mode l concretely shows that the fractal dimension of the regions making the dom inant contribution to the mean density may be as large as D-1 approximate t o 2.97 which is very close to the space filling value 3. We show that this may give deviations from the Hubble law as small as 3-7% (for Lambda' = 10) , as required by the observations. (C) 2000 Elsevier Science B.V. All right s reserved.