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

Scale-invariant intergalactic dynamics governed by a statistically homogeneous cascade process generically yields multifractal luminosity distributions with highly inhomogeneous realizations (the standard nonfractal and fractal models are special limiting cases). The main obstacles for extending scaling 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 with the more usual Hubble law governing the expansion of the universe) and show that the theory is compatible with both the observed apparent and absolute luminosities. The results of multifractal analysis on two galaxy catalogues (depth 150 h−1 Mpc each) show that the observed form of the dimension function follows if only matter in sufficiently dense (and sparse) concentrations is luminous (with critical dimension Dc≈1.85), i.e., mass and luminosity are tightly correlated only above a critical mass density singularity threshold (γc≈0.4). Since this critical 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” δ≈0.75. This implies that the ratio of luminous to dark matter is Λ′δ where Λ′ is the ratio of the outer and inner cascade scales; taking Λ′ in the range 10–100 we find that 85–97% of the matter is dark (Λ′≈10 is the value most compatible with the microwave background and standard cosmologies and with the data used here, Λ′≈100 is apparently compatible with some galaxy catalogues). The model also includes a multifractal phase transition associated with very bright self-organized critical galaxies whose luminosity we find to be algebraic with critical exponent ≈4 (not exponential as is often assumed). A basic problem with the scaling models proposed to date is that there is no satisfactory way of reconciling the high heterogeneity of luminous matter (fractal dimension 1.85) with the apparently low heterogeneity of the mass as inferred from the cosmic background or the small peculiar velocities. Our model concretely shows that the fractal dimension of the regions making the dominant contribution to the mean density may be as large as D1≈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 Λ′=10), as required by the observations.