On Fractal Modeling in Astrophysics: The Effect of Lacunarity on the Convergence of Algorithms for Scaling Exponents

Fractals and multifractals are used to model hierarchical, inhomogeneous structures in several areas of astrophysics, notably the distribution of matter at various scales in the universe. Current analysis techniques used to assert fractality or multifractality and extract scaling exponents from astrophysical data, however, have significant limitations and caveats. It is pointed out that some of the difficulties regarding the convergence rates of algorithms used to determine scaling exponents for a fractal or multifractal, are intrinsically related to its texture, in particular, to its lacunarity. A novel approach to characterize prefactors of cover functions, in particular, lacunarity, based on the formalism of regular variation (in the sense of Karamata) is proposed. This approach allows deriving bounds on convergence rates for scaling exponent algorithms and may provide more precise characterizations for fractal-like objects of interest for astrophysics. An application of regular variation based fractal modeling to apollonian packing, which can be useful in cosmic voids morphology modeling, is also suggested.