Universal Scaling Relations in Scale-Free Structure Formation

TL;DR

This paper demonstrates that scale-free structure formation naturally produces universal power-law relations in astronomical systems, explaining observed similarities across diverse phenomena.

Contribution

It provides a unified theoretical framework showing that scale-free processes lead to common power-law distributions in mass, column density, and density profiles in astrophysics.

Findings

Mass function proportional to M^{-2} across systems

Column density distribution with a  \,A/\d \ln\Sigma  \Sigma^{-1} tail

Structures tend to a  \rho/\d R^{-3} density profile

Abstract

A large number of astronomical phenomena exhibit remarkably similar scaling relations. The most well-known of these is the mass distribution $\mathrm{d} N/\mathrm{d} M\propto M^{-2}$ which (to first order) describes stars, protostellar cores, clumps, giant molecular clouds, star clusters and even dark matter halos. In this paper we propose that this ubiquity is not a coincidence and that it is the generic result of scale-free structure formation where the different scales are uncorrelated. We show that all such systems produce a mass function proportional to $M^{-2}$ and a column density distribution with a power law tail of $\mathrm{d} A/\mathrm{d} \ln\Sigma\propto\Sigma^{-1}$. In the case where structure formation is controlled by gravity the two-point correlation becomes $\xi_{2D}\propto R^{-1}$. Furthermore, structures formed by such processes (e.g. young star clusters, DM halos) tend to a $\rho\propto R^{-3}$ density profile. We compare these predictions with observations, analytical fragmentation cascade models, semi-analytical models of gravito-turbulent fragmentation and detailed "full physics" hydrodynamical simulations. We find that these power-laws are good first order descriptions in all cases.