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 $\mathrm{d}
N/\mathrm{d} M\propto M^{-2}$ 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.
