How can primordial black holes be formed?

By Arjen Siegers

Summary

It is shown that there is a good theoretical basis on which one can assume that black holes did form in the very early universe. Several articles have been used to find theoretical and observational evidence for the existence of primordial black holes. The oldest theory for the creation of PBHs is the collapse of a region in the early niverse to within its own Schwarzschild radius. PBHs can also be formed during a period of inflation, during which the universe experiences a phase transition which may cause fluctuations which can collapse to form back holes. Observational evidence for the existence of primordial black holes is to be found in the background radiation. The setup of this article is as follows: Part 1. will concern itself with the historical perspective of Primordial black holes(PBHs), part 2 with the creation of PBHs, part 3 with the role of PBHs in a few cosmological theories and part 4 will present the conclusions.

1. Historical perspective

In 1971, Stephen Hawking published an article in which he showed that in the very early Universe, fluctuations in the density of energy in region could cause those regions to collapse o within their Schwarzschild radii and thus form black holes. Later, this theory was expanded by Hawking and others. In some of their theories PBHs were said to have caused the formation of at least some of the galaxies and clusters which we see today, while in another theory PBHs heated the early universe by means of accretion. These theories will be dealt with at a later stage, they are only mentioned here to give a preview of this research.

2. The creation of Primordial Black Holes(PBH's)
Several theories for the creation of PBHs hve been formulated. In Hawking (1971) a theory is mentioned involving density fluctuations in a region of th early universe, which cause that part of the universe to collapse. This happens when a fluctuation causes the gravitational energy

within the region to considerably exceed the kinetic energy

of the expansion of this region. Where µ is the energy density of the region and R is the radius of the region. Units are such that G=c=1.(Carr&Hawking, 1974).

In a k=0 Friedmann universe, the sum of these energies is zero. Thus:

If the density in a region is somewhat higher than average, that is, if it exceeds the Jeans length, or the rate of expansion somewhat lower, the gravitational forces may be able to end the expansion by overcoming the kinetic energy of the expansion and the pressure forces, which means that the gravitational energy should be larger than the internal energy. If this is the case, the region collapses and forms a black hole.
This is an entirely classical process, so the black hole will never be smaller than the Planck lentght (Ghc-3)1/2~10-33cm. An equation for the age of PBHs is:

(Canuto,1978) This equation gives a critical mass (the mass which a pbh must at least have in order not to have been annihilated) Mc=1015g. In which g stands for gramme.

Another theory for the formation of PBHs is that they were formed in a period of inflation. The inflatio theory states that the universe underwent a very short period of very rapid expansion.
During such an inflationary epoch, the energy density of the universe would probably decrease dramatically, which could result in a cosmological phase transition.A cosmological phase transition is a process comparable to the freezing of water: energy is distracted from a substance, and several specifications of the substance, such as mass density change. If this occured, it would cause some irregularities. These have been referred to by MacGibbon and Carr (1991) as bubbles of broken symmetry. If a phase transition is to produce primordial black holes, then the production rate of these bubbles needs to be finely tuned. If it is too high, the entire universe will undergo a phase transition, and if it is too low, they will never collide. (A phase transition does not necessarily occur during a period of inflation.)
Canuto(1978) uses a phase transition as an alternative to a cold model for the early universe (see part 3). He shows that, assuming a Gaussian distribution for the density fluctuations, the probability that any particular region will ever evolve into a black hole is:
where and is the amplitude of the fluctuations on the horizon scale. Here, a very soft equation of state will increase P, while a stiffening of the equation will produce the opposite result. This relation can unfortunately not be used near a phase transition, since the density is not constant in such a situation. Canuto prefers to use the correlation function, a function which describes the correlation between regiopns as a function of the regions

, where , b=a positive constant. When not at a critical point, and v(r) has a very short range and correlations exist only among nearby regions which act independently. When near a phase transition however, the compressibility and a->0. Thus v(r) acquires a longer range. Fewer regions are thus independent of one another and therefor a larger fraction of the universe can be in black holes.
If the equation of state,

softens, that is: . This could also happen if the mass of the universe was channeled into grains which would have been massive enough to be non relativistic. This is shown by Khlopov and Polnarev in their 1985 paper. If this is the case PBH formation depends on the spherical symmetry of th bounded (collapsing) regions. PBH formation would result in a fairly small mass spectrum.

3. PBHs in cosmology
The existence of PBHs has been used by several scientists(Carr, Canuto, Hawking) in a variety of cosmological theories.

*Primordial black holes in a cold early universe

Carr(1977) suggests that the early universe must have been cold (photonless), because the only hot models for the early universe tend to produce them in either too large or numbers to match observations or too few to be interesting. A cold model would, according to Carr, imply the existence of PBHs, because they would have heated the universe by accretion.
There are major differences between PBH formation in a cold and a hot universe, the most obvious being the fact that in a hot universe regions collapse to black holes when they exceed their Jeans length when they stop expanding. In most cases (Carr treats several scenarios for various values for initial conditions of the universe) in a cold early universe however, regions lighter than 105 solar massescollapse until the temperatures are high enough to start nuclear reactions. Then they evolve into main-sequence stars. The heavier regions tend to collapse to black holes.
The formation of primordial stars is an essential part of the cold early universe model, because in a cold early universe there can be no cosmological nucleosynthesis, and thus the observed 25-30% Helium abundance should be produced in primordial stars.
Another feature of this model is that many PBHs will only form when the primordial stars have gone trough their evolution, which will take approximately 106 years. Carr also links the 3K background radiation to the PBHs formed after 106. Canuto(1978) opposes this theory, by showing that through the exchange of f0-mesons the equation of state can be softened and a phase transition can exist.

*Primordial black holes and the cosmic background radiation

As mentioned in the previous paragraph, PBHs may have contributed to the cosmic background radiation. MacGibbon and Carr(1991) mention several processes trough which PBH's can contribute to the cosmic background radiation. This radiation is the result of evaporation of the black holes, a process discovered by Hawking. An evaporating black hole 'sucks' one half of a particle-antiparticle pair out of the vacuum, and the other half is emitted as radiation. Generally, black holes tend to radiate a spectrum comparable to that of a blackbody. Among the particles emitted are photons, neutrinos, electrons and gravitons. But as the temperature decreases (that is, if the mass decreases) heavier particles, such as hadrons may also be emitted.

MacGibbon and Carr(1991) have found that these emmisions may contribute significantly to the extragalctic photon and interstellar cosmic-ray electron, positron spectra around 0.1-1 GeV. Especially the gamma-ray spectrum between 50 and 170 MeV fits well to the PBH emission.
These results are of great importance, since the origin of cosmic rays around 100 MeV is still unknown. It might be possible that PBHs are responsible for a large part of the gamma background. This radiation is mainly due to the evaporation of a PBH. The gamma burst which marks the end of its existence is highly unlikely to be detected, since it lasts for a very short time(10-7s).

* PBHs can be MACHOs.

A MACHO is a massive compact halo object. These have been observed through gravitational microlensing.(Yokoyama,1997)
Yokoyama has produced a method capable of creating PBHs which would account for these objects. This method could also be used to show that PBHs could exist in active galactic nuclei. However, Wright(1995) states that PBHs in the glactic halo would produce a large anisotropic flux that woul be easily observed. He also gives an estimated ratio for PBH annihilation in the galoctic halo of .
Wright suspects the PBH's may be tracers for the Cold Dark Matter(CDM) in the halo. He thinks PBHs would be part of a large scale structure formation of CDM, and that the gamma rays from these PBHs are the first nongravitational evidence for CDM.

*PBHs as an explanation for the irregularity of the current universe.

There are a number of scientists (at least in the 70s), especially Carr and Hawking(eg. Carr&Hawking 1974, Carr 1975), who think that since PBHs are the ultimate irregularities in the early universe, they may have contributed to the irregularity of the current universe. When we look at the universe, we see a very irregular distribution of mass. It is mostly packed into relatively high concentrations such as stars, galaxies and clusters. If PBHs have contributed to the irregular distribution of matter, then it might very well be possible that they have collected concentrations of matter around them. In other words, PBHs might form the core of possibly most galaxies. This theory falls outside the scope of my research, and the reader is referred to the articles of my fellow students who did some research in this area.

4. Conclusions

Theoretically speaking, PBHs can exist. They have played a significant role in several cosmological theories and are thought to have contributed to the cosmic background radiation.

Acknowledgments

I would like to thank my fellow students who have provided an ever present platform for useful discussion and sometimes not so useful debate.

References
*Hawking, Gravitational collapsed objects of very low mass,
Mon. Not. R. astr. Soc.(1971), 152, 75-78

* Hawking& Carr, Black Holes in the early universe,
Mon. Not. R. astr. Soc.(1974) 168, 399-415

* Carr, The primordial black hole mass spectrum,
The astrophysical journal, 201:1-19,1975 October 1

* Carr, black hole and galaxy formation in the early universe,
Mon. Not. R. astr. Soc. (1977) 181, 293-309

* Canuto, On the origin of Hawking mini black-holes and the cold early universe
Mon. Not. R. astr. Soc. (1978)184, 721-725

* MacGibbon& Carr, Cosmic rays from primordial black holes
The astrophysical journal, 371:447:469, 1991 April 20

* Khopov& Malomed& Zeldovich, Gravitational instability and the formation of primordial black holes
Mon. not. R. astr. Soc. (1985) 215 575-589

*Wright, On the density of primordial black holes in the galactic halo
The astrophysical journal, 459:487-490, 1996 march 10