Fs. Labini et L. Pietronero, MULTIFRACTALITY AS A LINK BETWEEN LUMINOSITY AND SPACE DISTRIBUTION OF VISIBLE MATTER, The Astrophysical journal, 469(1), 1996, pp. 26-39
We discuss how luminosity and space distributions of galaxies are natu
rally linked in view of their multifractal properties. In particular,
we show that the mass (luminosity) function corresponding to a multifr
actal distribution in a given observed volume consists of a power law
followed by an exponential cutoff. This implies that the amplitude phi
of the Schechter function scales with the sample depth, as confirmed
by various observational data. This effect is analogous to the scalin
g of the space density due to the fractal, nature of the space distrib
ution. This has the important consequence that the luminosity function
can be defined properly 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 t
hey should be considered from a different perspective with respect to
the usual one, Such a concept allows us to unify the space and luminos
ity distributions as being shaped by a single cause: multifractality,
which should therefore claim a central stage in theoretical investigat
ions.