*Summary:*

*According to Zel’dovich and Novikov, a primordial black hole (PBH) would not grow much at all if it were much smaller than the cosmological horizon at formation but it could grow at the same rate as the universe if its initial size were comparable to it.**But this idea was disproved by Carr and Hawking, as there were no self-similar solutions at that time. But, they also claimed that it is possible, if we connect black hole interior to an exact Friedmann exterior via a sound-wave. If we will have such a solution, it would represent a black hole growing at the same**rate as the universe.**Like these, many physically realistic self-similar solutions have been proposed, in which a primordial black hole is attached to an exact or asymptotically Friedmann model for an equation of state of the form p = (γ − 1)ρc².**One of these solutions suggested by Carr and colleagues in this review, in which black hole might grow as fast as the cosmological horizon in a quintessence-dominated universe in some circumstances, supporting the proposal that accretion onto primordial black holes may have played a role in the production of the supermassive black holes in galactic nuclei.*

Over the last 40 years there has been much interest in how fast a black hole formed in the early universe, when the density is usually radiation-dominated, would grow. As first pointed out by Zel’dovich and Novikov, a simple Bondi-type accretion analysis suggests that a primordial black hole (PBH) would not grow much at all if it were much smaller than the cosmological horizon at formation but that it could grow at the same rate as the universe if its initial size were comparable to it. (The term “cosmological horizon” should here be interpreted as the Hubble horizon if the PBHs form after an inflationary period but the particle horizon otherwise.) One might expect the latter situation to apply, since a PBH must be bigger than the Jeans length at formation, so this suggests that any PBH might grow to the horizon mass at the end of the radiation era, which is around 10^{17} M. Since there is no evidence for such enormous black holes, for a while it was assumed that no PBHs ever formed.

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PBH might in principle continue to grow as fast as the cosmological horizon until the end of the radiation-dominated era, when its mass would be of order 10^{17}M.”— wrote Zel’dovich and Novikov, in their paper

However, the validity of the Zel’dovich-Novikov calculation is questionable when the black hole size is comparable to the horizon size because it neglects the expansion of the universe and is not fully relativistic. Indeed, the conclusion that a PBH could grow at the same rate as the universe in the radiation-dominated era was disproved by Carr and Hawking. They demonstrated this by proving that there is no self-similar solution which contains a black hole attached to an exact flat Friedmann background via a sonic point (i.e. in which the black hole forms by purely causal processes). The Zel’dovich Novikov prediction is therefore definitely misleading in this case. Since the PBH must soon fall well below the horizon size, when their argument should be valid, this suggests that PBHs would not grow much at all.

This gave the subject of PBHs a new lease of life and motivated Hawking to consider the quantum effects associated with black holes (since only PBHs could be small enough for these to be significant). Ultimately, this led to his discovery of black hole radiation, so it is ironic that a consideration of PBH accretion led to the conclusion that they evaporate!

“For the radiation-dominated universe considered by Carr and Hawking, the sound speed is V = 1/ √ 3. Since the Friedmann solution itself has such a point, one might envisage a self-similar solution in which one attaches a black hole interior to an exact Friedmann exterior via a sound-wave. If such a solution existed, it would represent a black hole growing at the same rate as the universe.“— told Harada, second author of the study

Friends, Carr and Hawking though proved at that time that there is no self-similar solution. But, they also claimed/showed that there are self-similar solutions which are asymptotically – rather than exactly – Friedmann at large distances from the black hole. However, these correspond to special acausal initial conditions, in which matter is effectively thrown into the black hole at every distance; they do not contain a sonic point because they are supersonic everywhere. Indeed, such solutions exist in the “dust” case, when the cosmological fluid is pressureless. They even found solutions in which the whole universe is in some sense inside a black hole; these are now termed “universal” black holes.

Similar to these, many physically realistic self-similar solutions have been proposed now, in which a primordial black hole is attached to an exact or asymptotically Friedmann model for an equation of state of the form p = (γ − 1)ρc², where p is the pressure, ρ is the mass density and γ is a constant (4/3 in the radiation case). Carr and colleagues, briefly reviewed these solutions in their paper.

One of them we have already seen above, in the positive pressure case (1 < γ < 2), there is no such solution when the black hole is attached to an exact Friedmann background via a sonic point. However, it has been claimed that there is a one-parameter family of asymptotically Friedmann black hole solutions providing the ratio of the black hole size to the cosmological horizon size is in a narrow range above some critical value. There are also “universal” black holes in which the black hole has an apparent horizon but no event horizon. It turns out that both these types of solution are only asymptotically quasi-Friedmann, because they contain a solid angle deficit at large distances, but they are not necessarily excluded observationally. Such solutions may also exist in the 2/3 ≤ γ < γ < 2/3), corresponding to a dark-energy-dominated universe, there is a one-parameter family of black hole solutions which are properly asymptotically Friedmann (in the sense that there is no angle deficit) and such solutions may arise naturally in the inflationary scenario. The ratio of the black hole size to the cosmological horizon size must now be below some critical value, so the range is more extended than in the positive pressure case and one needs less fine-tuning. If one tries to make a black hole which is larger than this, one finds a self-similar solution which connects two asymptotic regions, one being exactly Friedmann and the other asymptotically quasi-Friedmann. This might be regarded as a cosmological wormhole solution providing one defines a wormhole throat quasi-locally in terms of a non-vanishing minimal area on a spacelike hypersurface.

The possibility of self-similar black holes in phantom fluids (γ < 0), where the black hole shrinks as the big rip singularity is approached, or tachyonic fluids (γ > 2) remains unclear.

Carr and colleagues also considered the possibility of self-similar black hole solutions in a universe dominated by a scalar field. If the field is massless, the situation resembles the stiff fluid case, so any black hole solution is again contrived, although there may still be universal black hole solutions. The situation is less clear if the scalar field is rolling down a potential and therefore massive, as in the quintessence scenario.

The situation is more complicated – but also more interesting – if there is a scalar potential (i.e. if the scalar. field is no longer massless). In particular, this applies in the quintessence scenario, in which the scalar field rolls down a flat potential. In this context, Bean and Magueijo have used a variant of the Zel’dovich Novikov argument to claim that PBHs could grow up to ∼ 100M through the accretion of quintessence before nucleosynthesis and this could be large enough to provide the seeds for the supermassive black holes found in galactic nuclei.— told Harada, second author of the study

Although no explicit asymptotically Friedmann black hole solutions of this kind are known, they are not excluded and comparison with the 0 < γ < 2/3 perfect fluid case suggests that they should exist if the black hole is not too large. This implies that a black hole might grow as fast as the cosmological horizon in a quintessence-dominated universe in some circumstances, supporting the proposal that accretion onto primordial black holes may have played a role in the production of the supermassive black holes in galactic nuclei.

**Reference***: B J Carr, Tomohiro Harada and Hideki Maeda, “Can a primordial black hole or wormhole grow as fast as the universe?”, Class. Quantum Grav. ***27***(18) 183101, 2011. **https://iopscience.iop.org/article/10.1088/0264-9381/27/18/183101*

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