Universal scaling relations in scale-free structure formation

A large number of astronomical phenomena exhibit remarkably similar scaling relations. The most well-known of these is the mass distribution dN/dM ∝ M−2 which (to first order) describes stars, protostellar cores, clumps, giant molecular clouds, star clusters, and even dark matter haloes. 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 dA/dln Σ ∝ Σ−1. In the case where structure formation is controlled by gravity the two-point correlation becomes ξ2D ∝ R−1. Furthermore, structures formed by such processes (e.g. young star clusters, DM haloes) tend to a ρ ∝ 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.