6 ms·

This is a very "dense" paper. So: if dark matter is something that annihilates itself, that's a two-particle interaction, so its macroscopic rate scales as dens

This is a very "dense" paper. So: if dark matter is something that annihilates itself, that's a two-particle interaction, so its macroscopic rate scales as density squared ρ². If there's a black hole in the solar system, the dark matter halo immediately surrounding it would be compressed far denser than around a normal object, so it'd have a much brighter annihilation signal. Nothing's known about dark matter annihilation. They speculate, if its cross section is large enough, this annihilation could be observable by gamma-ray telescopes like Fermi or the upcoming CTA. And that would distinguish a black hole from something else. That's all I could understand. It'd be a discovery of a solar system black hole, and new dark matter physics, two-for-one. I'm curious what math led to that funny exponent in eq.(5): ρ(r) ~ r^{-9/4}.

kurthr 3y ago
Just a thought, if a black hole that close were discovered there would be another space race to get there. It just sounds so cool to the politicians (who've seen the movies) that any scientist would have no problem convincing them to loosen the purse strings.
- deleted 3y ago
  [deleted]
- cgriswald 3y ago
  For reference, if it exists, it’s closest approach to the sun is more than six times farther than Pluto’s furthest approach and it took New Horizons nearly a decade of travel time to reach Pluto. A probe would be a decades long project. Sending humans would be effectively impossible.
  - pierat 3y ago
    > Sending humans would be effectively impossible. I can think of PLENTY of billionaires to send :)
    - eru 3y ago
      Why? I assume you object to billionaires because they use a disproportionate share of the world's resources? Spending even more resources on them doesn't seem like it would make that better?
  - MaxikCZ 3y ago
    Given such a scientifically interesting object so close to us, I can't imagine any other reality than massive world-scale spending to not just "get there first", but first to perform other low-hanging publicity stuns (think first space walk, first moon landing). The fact you _can't_ even land there is a massive accessibility boost. The thought is so exciting. The understanding of physics would be destined to leapfrog, distinctively changing humanity, and nature. Can anyone imagine something concrete?
    - ClumsyPilot 3y ago
      First man to die by black hole?
      - yellowapple 3y ago
        "First human to experience spaghettification firsthand" would be quite the honor, I suppose.
        jncfhnb 3y ago
        Some say he’s still experiencing it…
    - dpe82 3y ago
      Related; it's instructive to learn that a significant motivation behind the Apollo program was its national defense implications: https://youtu.be/xZFnTBSRKcg?t=137 https://youtu.be/xZFnTBSRKcg?t=137
    - jncfhnb 3y ago
      The majority of people would say it was dumb and wasteful much like they did for the moon landing and they would be pretty clearly correct.
      - eru 3y ago
        The moon landings were widely popular at the time, weren't they? Even the Soviets didn't say they were dumb and wasteful. (I tend to be a bit milder, and just say that the moon landings should be thought of as spending on entertainment, not on science. Manned space flight in general is very cost-ineffective, if you care about scientific bang for your buck.)
  - kzrdude 3y ago
    Do we even have a guess where it is right now? The orbit is huge, so the search area is ginormous, even if we would know approximately where in its orbit it would currently be.
  - skykooler 3y ago
    Sending humans would be difficult (technically we have the technology already [see Project Orion], it's just illegal), but it's certainly possible to get a probe out there far faster than New Horizons. Travel speed is mainly determined by the delta-v of the upper stage. New Horizons used a Star-48B solid rocket motor to get to its escape trajectory. However, if the probe were smaller, or the launch vehicle bigger, it could have multiple upper stages, significantly increasing the ultimate speed. It might also be feasible to use an ion engine stage with solar panels and a route that dips down below Venus' orbit to get even more speed without increasing the probe mass significantly. In other words, with a large enough budget, we could easily get a probe to those distances in under a decade with no new technology required.
    - seventytwo 3y ago
      “… with a large enough budget, we could easily…”
      - skykooler 3y ago
        If there were a new space race, budgets would probably increase.
        aurareturn 3y ago
        It doesn't have to be political. Discovering a blackhole reachable by humans would be one of the most exciting discoveries possible. It might even unite nations to work together to get there as soon as possible.
      - aurareturn 3y ago
        If there's really a black hole there, I'd personally donate at least 10% of my year's salary to the program if it means we make it there within my lifetime. Studying it might allow us to finally unify GR and QM.
      - ada1981 3y ago
        Currently sums up my last romantic relationship…
- zakki 3y ago
  If there is a race to be the first to reach this “black hole” and the racers are careless, is it going to make “the black hole“ alive and getting bigger?
- robofanatic 3y ago
  If it is a black hole then won’t the nearest planets get sucked into it?
  - deleted 3y ago
    [deleted]
  - dingosity 3y ago
    Only if the other planet's orbits take them near enough to the black hole to get sucked in. The main difference between a primordial black hole at about the Earth's mass and a planet at about the Earth's mass is the presumed density of dark matter around it. And I'm guessing there's more hawking radiation coming from the black hole than the planet. But small black holes with the same mass as the Earth have the same sucking power as the Earth since it's the mass that does the sucking.
  - dreamcompiler 3y ago
    If our sun magically turned into a black hole of the same mass tomorrow, nothing would change on earth or in our solar system, except it would get very dark. Nothing would get "sucked in". We'd all freeze to death in a few days, but the earth would go right on orbiting as if nothing happened. If our moon turned into a black hole of its same mass tomorrow there would be even less of an effect. We'd notice we could no longer see the moon, but we'd still have tides just like before.
- gofixurcode 3y ago
  I think it’s hilarious that there could be a space race to get to a golf ball sized invisible object. I see the importance, but at the same time you must admit from a layman’s perspective it’s kinda comical.
peteradio 3y ago
> I'm curious what math led to that funny exponent in eq.(5): ρ(r) ~ r^{-9/4}. Renormalization.
- raattgift 3y ago
  > 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/abstract 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/abstract 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 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/abstract 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%93White_profile 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".
dmix 3y ago
> or the upcoming CTA I'm guessing this refers to https://en.wikipedia.org/wiki/Cherenkov_Telescope_Array https://en.wikipedia.org/wiki/Cherenkov_Telescope_Array ?
emmelaich 3y ago
The -9/4 immediately reminded me of Kepler's ratio. K = T^2/r^3.
deleted 3y ago
[deleted]
fithisux 3y ago
Can you suggest a book for these kind of calculations? (or lecture notes , preferably free)
raattgift 3y ago
> 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.

kurthr 3y ago

Just a thought, if a black hole that close were discovered there would be another space race to get there. It just sounds so cool to the politicians (who've seen the movies) that any scientist would have no problem convincing them to loosen the purse strings.

deleted 3y ago
[deleted]
cgriswald 3y ago
For reference, if it exists, it’s closest approach to the sun is more than six times farther than Pluto’s furthest approach and it took New Horizons nearly a decade of travel time to reach Pluto. A probe would be a decades long project. Sending humans would be effectively impossible.
- pierat 3y ago
  > Sending humans would be effectively impossible. I can think of PLENTY of billionaires to send :)
  - eru 3y ago
    Why? I assume you object to billionaires because they use a disproportionate share of the world's resources? Spending even more resources on them doesn't seem like it would make that better?
- MaxikCZ 3y ago
  Given such a scientifically interesting object so close to us, I can't imagine any other reality than massive world-scale spending to not just "get there first", but first to perform other low-hanging publicity stuns (think first space walk, first moon landing). The fact you _can't_ even land there is a massive accessibility boost. The thought is so exciting. The understanding of physics would be destined to leapfrog, distinctively changing humanity, and nature. Can anyone imagine something concrete?
  - ClumsyPilot 3y ago
    First man to die by black hole?
    - yellowapple 3y ago
      "First human to experience spaghettification firsthand" would be quite the honor, I suppose.
      - jncfhnb 3y ago
        Some say he’s still experiencing it…
  - dpe82 3y ago
    Related; it's instructive to learn that a significant motivation behind the Apollo program was its national defense implications: https://youtu.be/xZFnTBSRKcg?t=137 https://youtu.be/xZFnTBSRKcg?t=137
  - jncfhnb 3y ago
    The majority of people would say it was dumb and wasteful much like they did for the moon landing and they would be pretty clearly correct.
    - eru 3y ago
      The moon landings were widely popular at the time, weren't they? Even the Soviets didn't say they were dumb and wasteful. (I tend to be a bit milder, and just say that the moon landings should be thought of as spending on entertainment, not on science. Manned space flight in general is very cost-ineffective, if you care about scientific bang for your buck.)
- kzrdude 3y ago
  Do we even have a guess where it is right now? The orbit is huge, so the search area is ginormous, even if we would know approximately where in its orbit it would currently be.
- skykooler 3y ago
  Sending humans would be difficult (technically we have the technology already [see Project Orion], it's just illegal), but it's certainly possible to get a probe out there far faster than New Horizons. Travel speed is mainly determined by the delta-v of the upper stage. New Horizons used a Star-48B solid rocket motor to get to its escape trajectory. However, if the probe were smaller, or the launch vehicle bigger, it could have multiple upper stages, significantly increasing the ultimate speed. It might also be feasible to use an ion engine stage with solar panels and a route that dips down below Venus' orbit to get even more speed without increasing the probe mass significantly. In other words, with a large enough budget, we could easily get a probe to those distances in under a decade with no new technology required.
  - seventytwo 3y ago
    “… with a large enough budget, we could easily…”
    - skykooler 3y ago
      If there were a new space race, budgets would probably increase.
      - aurareturn 3y ago
        It doesn't have to be political. Discovering a blackhole reachable by humans would be one of the most exciting discoveries possible. It might even unite nations to work together to get there as soon as possible.
    - aurareturn 3y ago
      If there's really a black hole there, I'd personally donate at least 10% of my year's salary to the program if it means we make it there within my lifetime. Studying it might allow us to finally unify GR and QM.
    - ada1981 3y ago
      Currently sums up my last romantic relationship…
zakki 3y ago
If there is a race to be the first to reach this “black hole” and the racers are careless, is it going to make “the black hole“ alive and getting bigger?
robofanatic 3y ago
If it is a black hole then won’t the nearest planets get sucked into it?
- deleted 3y ago
  [deleted]
- dingosity 3y ago
  Only if the other planet's orbits take them near enough to the black hole to get sucked in. The main difference between a primordial black hole at about the Earth's mass and a planet at about the Earth's mass is the presumed density of dark matter around it. And I'm guessing there's more hawking radiation coming from the black hole than the planet. But small black holes with the same mass as the Earth have the same sucking power as the Earth since it's the mass that does the sucking.
- dreamcompiler 3y ago
  If our sun magically turned into a black hole of the same mass tomorrow, nothing would change on earth or in our solar system, except it would get very dark. Nothing would get "sucked in". We'd all freeze to death in a few days, but the earth would go right on orbiting as if nothing happened. If our moon turned into a black hole of its same mass tomorrow there would be even less of an effect. We'd notice we could no longer see the moon, but we'd still have tides just like before.
gofixurcode 3y ago
I think it’s hilarious that there could be a space race to get to a golf ball sized invisible object. I see the importance, but at the same time you must admit from a layman’s perspective it’s kinda comical.

peteradio 3y ago

> I'm curious what math led to that funny exponent in eq.(5): ρ(r) ~ r^{-9/4}. Renormalization.

raattgift 3y ago
> 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/abstract 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/abstract 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 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/abstract 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%93White_profile 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".

dmix 3y ago

> or the upcoming CTA I'm guessing this refers to https://en.wikipedia.org/wiki/Cherenkov_Telescope_Array https://en.wikipedia.org/wiki/Cherenkov_Telescope_Array ?

emmelaich 3y ago

The -9/4 immediately reminded me of Kepler's ratio. K = T^2/r^3.

deleted 3y ago

[deleted]

fithisux 3y ago

Can you suggest a book for these kind of calculations? (or lecture notes , preferably free)

raattgift 3y ago

> 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.