Multifractality as a Link Between Luminosity and Space Distribution of Visible Matter

Abstract

We discuss how luminosity and space distribution of galaxies are naturally linked in view of their multifractal properties. In particular we show that the mass (luminosity) function corresponding to a multifractal distribution in a given observed volume, consists of a power law followed by an exponential cut-off. This implies that the amplitude ${\varphi }^{\ast }$ of the Schechter function scales with the sample depth, as confirmed by various observational data. This effect is analogous to the scaling of the space density due to the fractal nature of the space distribution. This has the important consequence that the luminosity function can be properly defined only in a volume limited sample. Also the so-called "luminosity segregation" and the concept of bias correspond to a natural consequence of multifractality. This implies however that they should be considered from a different perspective with respect to the usual one. Such a concept allows us to unify the space and luminosity distributions as being shaped by a single cause:multifractality which should therefore claim a central stage in theoretical investigations.

38 pages, Latex, 9 figures available under request at [email protected]. To be published in the Astrophysical Journal, 1996, Vol.468 September 20

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