> Renormalization

Huh? Explain. How does that fit?

It's just a power-law distribution for the density of a halo that forms early around the "Planet 9" primordial black hole.

The density ~ r^{{3/2}^2} is a result from 1970s studies of structure formation shortly after Vera Rubin's work indicated that dark matter halos surrounded spiral galaxies. Rubin & Ford 1970 <https://ui.adsabs.harvard.edu/abs/1970ApJ...159..379R/abstra...> emissions regions of M31, begat Gott & Gunn 1972 <https://ui.adsabs.harvard.edu/abs/1972ApJ...176....1G/abstra...> infall of matter onto galaxy clusters begat Gott 1975 <https://adsabs.harvard.edu/full/1975ApJ...201..296G> structure formation and elliptical galaxies begat Gunn 1977 <https://ui.adsabs.harvard.edu/abs/1977ApJ...218..592G/abstra...> formation of massive galactic halos.

These are foundational to the paper at the top's ref [35], which is quoted just before the "funny exponent".

Roughly, let's start with an expanding Friedmann universe, where we treat galaxy clusters as motes of dust, and smear that out into one or more fluids representing nonrelativistic matter and radiation and other relativistic matter, with an identical energy-density at every point in space in an equatorial slicing, with each slice succeeding a spatially-smaller slice and preceeding a spatially-larger slice. At small scales, radiation pressure stabilizes any overdensities or underdensities in the nonrelativistic and relativistic fluids. Eventually expansion causes a transition from radiation-dominance to matter-dominance, allowing overdensities and underdensities to grow.

The papers by Gott and Gunn above consider the evolution of a spherical perturbation, an overdensity, starting at that transition, studying whether shortly after the transition from radiation-dominance, halos can support the evolution of generic galaxies and galaxy clusters. They can; and the "Planet 9" paper at the very top (albeit by way of its ref [34]) takes that further and applies this halo logic to primordial black holes based on that "universality" result (in Gunn 1977, penultimate paragraph).

Essentially what happens is that the nonrelativistic matter in an overdensity self-gravitates and so sticks around as an overdensity from one slice to the next bigger slice. Moreover, neither cold dark matter nor a primordial black hole radiates that early in the universe, so the overdensity can't revert to average through dissipation. Furthermore, the overdensity's gravitation draws in matter from the "shell" outside it (i.e., the rest of the universe); that's the "infall". If the infall is dissipationless, it sticks around in shells well outside the centre of the overdense perturbation.

All of this combines so that the density of an overdense perturbation drops much more slowly than the density in the rest of the universe. The latter drops like cosmic_time^{3/2} while the perturbation's density drops like cosmic_time^{9/4} (in these 1970s papers; in the age of fast computers one would use a profile like NFW <https://en.wikipedia.org/wiki/Navarro%E2%80%93Frenk%E2%80%93...>, and in this "Planet 9 is a PBH" context one might want to think about "bottom-up" hierarchical mergers of microhalos at all length scales: in that case, is "Planet 9" more likely to be a microhalo without a small black hole?).

In the case of the "Planet 9" PBH the (very small) black hole and its (relatively dense) halo of dark matter are essentially noninteracting except for gravitation, because the horizon is so small and cold dark matter basically feels nothing but gravitation. Gravitation (by way of the equivalence principle) can impart some fairly gentle accelerations that can alter the halo, as discussed in the paper linked at the very top.

Practically all of the results above were found using a linear approximation to General Relativity, checked by N-body numerical simulations (they got up to N=32k or so by the time non-power-law halo distributions were found to fit observations better in the 1990s). For the "Planet 9" PBH it is easy to imagine nonlinearities from interactions with early supernovae, however, if the "capture mechanism" is in-place-formation or an early entrainment to the local standard of rest. Even thinking about using a full nonlinear solution, I don't understand how renormalization might be used and or how it would generate the "funny exponent".