> I'm curious what math led to that funny exponent in eq.(5): ρ(r) ~ r^{-9/4}. The actual exponent and its workings-out almost certainly comes from Gott 1975,

> I'm curious what math led to that funny exponent in eq.(5): ρ(r) ~ r^{-9/4}. The actual exponent and its workings-out almost certainly comes from Gott 1975, which I do not remember ever having read <https://adsabs.harvard.edu/full/1975ApJ...201..296G https://adsabs.harvard.edu/full/1975ApJ...201..296G>, so I cannot do it any sort of justice at this time, but see my comment on Gunn 1977 below. Theres an early history of pre-NFW power-law dark matter density profiles swept up in ref [34] of the Planet 9 preprint, which is cited just before the equation that piqued your curiosity. [34] is <https://ui.adsabs.harvard.edu/abs/1985ApJS...58...39B/abstract https://ui.adsabs.harvard.edu/abs/1985ApJS...58...39B/abstra...> which turns out to be part of a 1984 Princeton doctoral thesis, and I am unfamiliar with it although I recognize its author Edmund Bertschinger as well-known from his later 1980s-2000s work in cosmological perturbation theory and cosmological simulations. [34] also lists Gunn & Gott 1972, Gunn 1977 and Gott 1975 in the references section (bottom of 1st to top of 2nd column). Surprisingly the Planet 9 preprint lists none of these papers. However, in a way that could be serendipitous: you might enjoy the "funny exponent"s in many the equations of [34], since some involve powers of -8/9, 8/3, -5/2, among others. I wrote but abandoned an attempt to tease out the detail using the following two seminal papers. Gott & Gunn 1972, "On the Infall of Matter into Clusters of Galaxies and Some Effects on their Evolution" <https://ui.adsabs.harvard.edu/abs/1972ApJ...176....1G/abstra... https://ui.adsabs.harvard.edu/abs/1972ApJ...176....1G/abstra...> (PDF via top right box), contemplating a Friedmann universe with a spherical homogeneous overdensity that collapses into a galaxy cluster. Gunn 1977 <https://ui.adsabs.harvard.edu/abs/1977ApJ...218..592G/abstract https://ui.adsabs.harvard.edu/abs/1977ApJ...218..592G/abstra...> contemplates a spherical but inhomogeneous perturbation, and arrives at a power law of density \propto r^{-9/4}. It relies on Gott 1975 for that, however. This paper is the basis for a "universality" result which indicates that the shape of an overdensity's boundary is essentially preserved over time, even as the matter within the overdensity collapses into various structures. That universality was convenient at the time to avoid having to treat spirals specially, and is in effect claimed as applicable to a PBH equipped with a DM microhalo in the preprint linked at the top. The "funny exponent" appears in the text just after eqns (8)-(9b) in Gunn 1977. All of the NASA/ADS abstracts linked in my comment have PDFs available in the top right box on the page. From the abandoned first attempt (which grew in length and messiness) I'll save two things: the "curious" equation's r_eq and \rho_eq are fixed at the transition from radiation-domination to matter-domination at z ~ 3200 (47 thousand years after the big bang), after which radiation pressure is insufficient to keep perturbations from growing. The transition is also when scale factor a \propto t^{2/3}, and energy density \rho \propto a^{-3(1+w)} where w is close to 0; self-gravitation in the overdensity causes its \rho to drop more slowly, and furthermore draws in matter from the surrounding ~average density shell.