There is a well-known prediction that density fluctuations in the first moments of the Big Bang produced primordial black holes. Black holes from stellar collapse have their masses constrained by stellar dynamics, so stellar black holes are unlikely to be less massive than the T.O.V. limit for stable neutron-degenerate matter, and are also unlikely to be more massive than the most massive stars. Primordial black holes, however, date from the era when the density of the universe was approximately uniform, so the stellar-scale mass constraints didn't apply. Primordial black holes could be much more massive than stars, possibly including what we would now call "supermassive"; primordial black holes have been offered as candidates for "intermediate-mass" black holes, which seem unlikely to have formed from a reasonable number of stellar-mass mergers; primordial black holes might be less massive than the T.O.V. limit, perhaps even small enough that they are net emitters of Hawking radiation rather than net absorbers of the cosmic microwave background. The relative sizes of these populations are the subject of some computational literature.

Similarly, while white dwarfs and neutron stars have minimum masses which are set by the dynamics of stellar collapse, neutron-degenerate matter might be stable in compact objects as small as $0.1\text{–}0.2\,M_\odot$. I assume there is some similar lower limit for electron-degenerate matter, but I haven't tried very hard to look for it.

My question: how large is the phase space for "primordial" fluctuations which created electron- or neutron-degenerate objects, whose masses might be too small to have been produced in a stellar collapse? I suppose the very early universe must have been neutron- and electron-degenerate everywhere, so these primordial degenerate objects would have formed from density fluctuations as the universe expanded through the phase changes to non-degeneracy in the neutron and electron fields.

I have at least three guesses:

  • The phase space is zero, because the equation of state for degenerate matter has some temperature dependence which doesn't apply to the collapse of a black hole, and the universe was too hot or too cold when the density was right for primordially-degenerate objects to form.

  • The phase space is small, but the sub-stellar objects in the dense medium of the early universe would have been overwhelmingly likely to accrete matter from their surroundings and exceed their maximum mass limit. This accretion argument is also an argument against low-mass primordial black holes, so a comparison would be interesting.

  • The transition from a degenerate to a non-degenerate density for neutrons and/or electrons happened while there was still a substantial amount of not-yet-annihilated antimatter, so any temporarily degenerate object would have been disrupted by ongoing annihilation.

I'm assuming that all of these guesses are wrong in interesting ways.

  • $\begingroup$ Arising from. $\endgroup$
    – rob
    Jul 28, 2022 at 19:40
  • 2
    $\begingroup$ Here is a recent (Sep. 22) paper: Primordial neutron star; a new candidate of dark matter The author theorizes formation of BCS-like neutrino–neutron pair condensate pre-BBN, which would serve as seed for primordial neutron stars. This seems far fetched to me and since it is just appeared on arXiv the paper has not been properly peer-reviewed, but still it is interesting. $\endgroup$
    – A.V.S.
    Sep 29, 2022 at 7:07

1 Answer 1


The challenge to the formation of any kind of structure other than black holes is that up until matter-radiation equality at a time of about about 52000 years, the radiation bath dominated the energy density of the universe. Radiation has such high pressure that it rapidly equalizes any density variations. Since radiation dominates, it drives the gravitational potential. For example, this is the fate of fluctuations in the gravitational potential after they enter the horizon.

gravitational potential decays after entering the horizon

Basically what is happening here is that at times before "horizon entry" on the plot, the potential fluctuation (arising from a density fluctuation at the same scale) is too large to be causally connected, so it cannot evolve or undergo any dynamics. It sits at a constant value set by the initial conditions (e.g. inflation). As the horizon grows, the fluctuation gradually becomes dynamical, as indicated by "horizon entry". When this happens, the potential fluctuation quickly decays to zero. With zero fluctuations in the gravitational potential, nothing can gravitationally cluster.

For example, the above argument makes it difficult for dark matter clusters to form during the radiation epoch. But for baryonic compact objects, the situation is even worse, because the baryonic matter that would form your compact objects is tightly coupled to the radiation by electromagnetic forces. The radiation itself will not cluster due to its pressure, so neither will the baryonic matter.

Primordial black holes get around this problem because roughly speaking, they form from primordial fluctuations that are so extreme that at horizon entry, the gravitational attraction overpowers the radiation pressure. They form out of radiation, not matter!


The plot shows $\Phi(k,\tau)/\Phi(k,0)$ as a function of $k\tau$, where $\Phi(k,\tau)$ is the Newtonian gauge metric perturbation as a function of conformal time $\tau$ and wavenumber $k$. Specifically I'm showing an analytic approximation (e.g. Dodelson & Schmidt equation 8.46) that assumes the radiation is a perfect fluid (no anisotropic stress) and the perturbation is much smaller than 1.

Also, this answer applies to the formation of compact objects by direct gravitational condensation, which is (usually) the production mechanism for primordial black holes. Exotic "seed" masses may still be able to grow into primordial neutron stars and the like.

  • $\begingroup$ Thanks for this great answer. Can you add a source for your figure? $\endgroup$
    – rob
    Sep 28, 2022 at 3:03
  • $\begingroup$ I made the figure, but I have added some further notes about it. $\endgroup$
    – Sten
    Sep 28, 2022 at 7:01
  • $\begingroup$ What about exotic stuff like Q-balls/strangelets/fifth force etc.? These could lead to formation of structures via nongravitational means in earlier epochs. Could such clumps serve as seeds for ordinary neutron stars? $\endgroup$
    – A.V.S.
    Sep 29, 2022 at 6:44
  • $\begingroup$ I wouldn't rule it out. My answer only applies to gravitational condensation of compact objects, which is how primordial black holes would form. Nongravitational condensation of an exotic seed (including the neutrino-neutron condensation discussed in the article you linked in another comment) requires totally different calculations. But once such an exotic seed forms, I guess it should be able to accrete baryons as long as the temperature is well below the gravitational potential energy of a baryon on its surface. $\endgroup$
    – Sten
    Sep 29, 2022 at 10:51

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