# Mechanisms of Producing Primordial Black Holes and Their Evolution

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Production Mechanisms

#### 2.1. Initial Density Inhomogeneities

#### 2.2. First-Order Phase Transitions

#### 2.3. Second-Order Phase Transitions

#### 2.4. PBH Production in $f\left(R\right)$-Gravity

## 3. Mass Accretion Mechanisms

#### 3.1. Bondi Accretion

#### 3.2. Accretion Inside Neutron Star

#### 3.3. Eddington Limit

#### 3.4. Accretion in Schwarzschild Spacetime

#### 3.5. McVittie Solution

## 4. Discussion

- Initial density inhomogeneities:Advantages: a broad mass spectrum.Disadvantages: it leads to strong inhomogeneities and there is no possibility to form clusters of PBHs.
- First-order phase transitions:Advantages: it does not require any assumption beyond standard Big Bang physics.Disadvantages: The mass spectrum in this model is close to monochromatic, so it is impossible to explain the existence of black holes of various masses. It is also impossible to produce clusters of PBHs within this model.
- Second-order phase transitions:Advantages: it has a broad mass spectrum and it also could be coupled with external processes, e.g., baryogenesis.Disadvantages: a fine-tuning of the initial conditions is required.

- Bondi accretion:Advantages: a simple conventional model which is accurate enough.Disadvantages: it does not take into account possible relativistic effects and spacetime curvature.
- Accretion in Schwarzschild spacetime:Advantages: it takes into account spacetime curvature.Disadvantages: cosmic expansion is not considered in this case.
- McVittie solution:Advantages: it takes into account spacetime curvature with cosmic expansion.Disadvantages: it exhibits a superluminal motion of the fluid at a distance of $\overline{r}=Gm\left(t\right)/2$.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zel’dovich, Y.B.; Novikov, I.D. The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model. Astron. Zhurnal
**1967**, 10, 602. [Google Scholar] - Hawking, S. Gravitationally Collapsed Objects of Very Low Mass. Mon. Not. R. Astron. Soc.
**1971**, 152, 75–78. [Google Scholar] [CrossRef] - Carr, B.J.; Hawking, S.W. Black Holes in the Early Universe. Mon. Not. R. Astron. Soc.
**1974**, 168, 399–415. [Google Scholar] [CrossRef] - Hawking, S.W.; Moss, I.G.; Stewart, J.M. Bubble collisions in the very early universe. Phys. Rev. D
**1982**, 26, 2681–2693. [Google Scholar] [CrossRef] - Falomo, R.; Bettoni, D.; Karhunen, K.; Kotilainen, J.K.; Uslenghi, M. Low-redshift quasars in the Sloan Digital Sky Survey Stripe 82. The host galaxies. Mon. Not. R. Astron. Soc.
**2014**, 440, 476–493. [Google Scholar] [CrossRef] - Dokuchaev, V.; Eroshenko, Y.N.; Rubin, S. Origin of supermassive black holes. arXiv
**2007**, arXiv:0709.0070. [Google Scholar] - The LIGO Scientific Collaboration, T.V.C. Search for intermediate mass black hole binaries in the first and second observing runs of the Advanced LIGO and Virgo network. Phys. Rev. D
**2019**, 100, 18. [Google Scholar] [CrossRef] - Carr, B.; Kühnel, F. Primordial black holes as dark matter candidates. SciPost Phys. Lect. Notes
**2022**, 48, 53. [Google Scholar] [CrossRef] - Luca, V.D.; Franciolini, G.; Pani, P.; Riotto, A. Primordial black holes confront LIGO/Virgo data: Current situation. J. Cosmol. Astropart. Phys.
**2020**, 2020, 044. [Google Scholar] [CrossRef] - Carr, B.; Kühnel, F. Primordial Black Holes as Dark Matter: Recent Developments. Annu. Rev. Nucl. Part. Sci.
**2020**, 70, 355–394. [Google Scholar] [CrossRef] - Carr, B.; Kohri, K.; Sendouda, Y.; Yokoyama, J. Constraints on primordial black holes. Rep. Prog. Phys.
**2021**, 84, 116902. [Google Scholar] [CrossRef] - Luca, V.D.; Desjacques, V.; Franciolini, G.; Riotto, A. The clustering evolution of primordial black holes. J. Cosmol. Astropart. Phys.
**2020**, 2020, 028. [Google Scholar] [CrossRef] - Belotsky, K.M.; Dokuchaev, V.I.; Eroshenko, Y.N.; Esipova, E.A.; Khlopov, M.Y.; Khromykh, L.A.; Kirillov, A.A.; Nikulin, V.V.; Rubin, S.G.; Svadkovsky, I.V. Clusters of Primordial Black Holes. Eur. Phys. J. C
**2019**, 79, 246–266. [Google Scholar] [CrossRef] - Berezin, V.; Dokuchaev, V.; Eroshenko, Y.; Smirnov, A. Formation and Clustering of Primordial Black Holes in Brans-Dicke Theory. Universe
**2020**, 6, 158. [Google Scholar] [CrossRef] - García-Bellido, J.; Clesse, S. Constraints from microlensing experiments on clustered primordial black holes. Phys. Dark Univ.
**2018**, 19, 144–148. [Google Scholar] [CrossRef] - Calcino, J.; García-Bellido, J.; Davis, T.M. Updating the MACHO fraction of the Milky Way dark halowith improved mass models. Mon. Not. R. Astron. Soc.
**2018**, 479, 2889–2905. [Google Scholar] [CrossRef] - Konoplich, R.V.; Rubin, S.G.; Sakharov, A.S.; Khlopov, M.Y. Formation of black holes in first-order phase transitions as a cosmological test of symmetry-breaking mechanisms. Phys. Atom. Nucl.
**1999**, 62, 1593–1600. [Google Scholar] - Hawking, S. Black holes from cosmic strings. Phys. Lett. B
**1989**, 231, 237–239. [Google Scholar] [CrossRef] - García-Bellido, J.; Linde, A.; Wands, D. Density perturbations and black hole formation in hybrid inflation. Phys. Rev. D
**1996**, 54, 6040–6058. [Google Scholar] [CrossRef] - Rubin, S.G.; Khlopov, M.Y.; Sakharov, A.S. Primordial black holes from nonequilibrium second order phase transition. Grav. Cosmol.
**2000**, 6, 51–58. [Google Scholar] - Rubin, S.G.; Sakharov, A.S.; Khlopov, M.Y. The formation of primary galactic nuclei during phase transitions in the early universe. J. Exp. Theor. Phys.
**2001**, 92, 921–929. [Google Scholar] [CrossRef] - Khlopov, M.Y. Primordial black holes. Res. Astron. Astrophys.
**2010**, 10, 495–528. [Google Scholar] [CrossRef] - Barack, L.; Cardoso, V.; Nissanke, S.; Sotiriou, T.P.; Askar, A.; Belczynski, C.; Bertone, G.; Bon, E.; Blas, D.; Brito, R.; et al. Black holes, gravitational waves and fundamental physics: A roadmap. Class. Quantum Gravity
**2019**, 36, 143001. [Google Scholar] [CrossRef] - Nikulin, V.V.; Krasnov , M.A.; Rubin , S.G. Compact extra dimensions as the source of primordial black holes. Front. Astron. Space Sci.
**2022**, 9, 8. [Google Scholar] [CrossRef] - Bronnikov, K.A.; Rubin, S.G. Self-stabilization of extra dimensions. Phys. Rev. D
**2006**, 73, 124019. [Google Scholar] [CrossRef] - Capozziello, S.; De Laurentis, M. The Dark Matter problem from f(R) gravity viewpoint. Ann. Phys.
**2012**, 524, 545–578. [Google Scholar] [CrossRef] - Bronnikov, K.A.; Konoplich, R.V.; Rubin, S.G. The diversity of universes created by pure gravity. Class. Quantum Gravity
**2007**, 24, 1261–1277. [Google Scholar] [CrossRef] - Bronnikov, K.; Budaev, R.; Grobov, A.; Dmitriev, A.; Rubin, S.G. Inhomogeneous compact extra dimensions. J. Cosmol. Astropart. Phys.
**2017**, 2017, 001. [Google Scholar] [CrossRef] - Bronnikov, K.A.; Popov, A.A.; Rubin, S.G. Inhomogeneous compact extra dimensions and de Sitter cosmology. Eur. Phys. J. C
**2020**, 80, 10. [Google Scholar] [CrossRef] - De Felice, A.; Tsujikawa, S. f(R) Theories. Living Rev. Relativ.
**2010**, 13, 161. [Google Scholar] [CrossRef] [PubMed] - Capozziello, S.; De Laurentis, M. Extended Theories of Gravity. Phys. Rep.
**2011**, 509, 167–321. [Google Scholar] [CrossRef] - Frank, J.; King, A.; Raine, D. Accretion Power in Astrophysics; Cambridge University Press: Cambridge, MA, USA, 2002. [Google Scholar]
- Bondi, H. On spherically symmetrical accretion. Mon. Not. R. Astron. Soc.
**1952**, 112, 195. [Google Scholar] [CrossRef] - Babichev, E.; Dokuchaev, V.; Eroshenko, Y. Black Hole Mass Decreasing due to Phantom Energy Accretion. Phys. Rev. Lett.
**2004**, 93, 4. [Google Scholar] [CrossRef] - Faraoni, V.; Jacques, A. Cosmological expansion and local physics. Phys. Rev. D
**2007**, 76, 16. [Google Scholar] [CrossRef] - McVittie, G.C. The Mass-Particle in an Expanding Universe. Mon. Not. R. Astron. Soc.
**1933**, 93, 325–339. [Google Scholar] [CrossRef] - Lora-Clavijo, F.; Guzmá n, F.; Cruz-Osorio, A. PBH mass growth through radial accretion during the radiation dominated era. J. Cosmol. Astropart. Phys.
**2013**, 2013, 015. [Google Scholar] [CrossRef] - Shakura, N.I.; Sunyaev, R.A. Black holes in binary systems. Observational appearance. Astron. Astrophys.
**1973**, 24, 337–355. [Google Scholar] - Page, D.N.; Thorne, K.S. Disk-Accretion onto a Black Hole. Time-Averaged Structure of Accretion Disk. Astrophys. J.
**1974**, 191, 499–506. [Google Scholar] [CrossRef] - Beloborodov, A.M. Accretion disk models. ASP Conf. Ser.
**1999**, 161, 295. [Google Scholar] - Khlopov, M.Y.; Konoplich, R.V.; Rubin, S.G.; Sakharov, A.S. First order phase transitions as a source of black holes in the early universe. Grav. Cosmol.
**1999**, 2, S1. [Google Scholar] - Baker, M.J.; Breitbach, M.; Kopp, J.; Mittnacht, L. Primordial Black Holes from First-Order Cosmological Phase Transitions. arXiv
**2021**, arXiv:2105.07481. [Google Scholar] - Liu, J.; Bian, L.; Cai, R.G.; Guo, Z.K.; Wang, S.J. Primordial black hole production during first-order phase transitions. Phys. Rev. D
**2022**, 105, L021303. [Google Scholar] [CrossRef] - Jedamzik, K.; Niemeyer, J.C. Primordial black hole formation during first-order phase transitions. Phys. Rev. D
**1999**, 59, 124014. [Google Scholar] [CrossRef] - Bai, Y.; Korwar, M. Cosmological constraints on first-order phase transitions. Phys. Rev. D
**2022**, 105, 095015. [Google Scholar] [CrossRef] - Dolgov, A.; Silk, J. Baryon isocurvature fluctuations at small scales and baryonic dark matter. Phys. Rev. D
**1993**, 47, 4244–4255. [Google Scholar] [CrossRef] - Affleck, I.; Dine, M. A new mechanism for baryogenesis. Nucl. Phys. B
**1985**, 249, 361–380. [Google Scholar] [CrossRef] - Dolgov, A.; Postnov, K. Why the mean mass of primordial black hole distribution is close to 10M
_{⊙}. J. Cosmol. Astropart. Phys.**2020**, 2020, 63. [Google Scholar] [CrossRef] - Dolgov, A.D. Massive and supermassive black holes in the contemporary and early Universe and problems in cosmology and astrophysics. Phys.-Uspekhi
**2018**, 61, 115–132. [Google Scholar] [CrossRef] - Kirillov, A.A.; Rubin, S.G. On Mass Spectra of Primordial Black Holes. Front. Astron. Space Sci.
**2021**, 8, 777661. [Google Scholar] [CrossRef] - Matacz, A. Inflation and the fine tuning problem. Phys. Rev. D
**1997**, 56, 1836–1840. [Google Scholar] [CrossRef] - Garriga, J.; Vilenkin, A.; Zhang, J. Black holes and the multiverse. J. Cosmol. Astropart. Phys.
**2016**, 2016, 064. [Google Scholar] [CrossRef] - Deng, H.; Garriga, J.; Vilenkin, A. Primordial black hole and wormhole formation by domain walls. J. Cosmol. Astropart. Phys.
**2017**, 2017, 050. [Google Scholar] [CrossRef] - Khlopov, M.Y.; Rubin, S.G. Cosmological Pattern of Microphysics in the Inflationary Universe; Fundamental Theories of Physics; Springer: Dordrecht, The Netherlands, 2004. [Google Scholar] [CrossRef]
- Lyakhova, Y.; Popov, A.A.; Rubin, S.G. Classical evolution of subspaces. Eur. Phys. J. C
**2018**, 78, 1–13. [Google Scholar] [CrossRef] - Fabris, J.C.; Popov, A.A.; Rubin, S.G. Multidimensional gravity with higher derivatives and inflation. Phys. Lett. B
**2020**, 806, 135458. [Google Scholar] [CrossRef] - Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. Planck 2018 results-X. Constraints on inflation. A&A
**2020**, 641, A10. [Google Scholar] [CrossRef] - Hoyle, F.; Lyttleton, R.A. The effect of interstellar matter on climatic variation. Math. Proc. Camb. Philos. Soc.
**1939**, 35, 405–415. [Google Scholar] [CrossRef] - Ricotti, M. Bondi Accretion in the Early Universe. Astrophys. J.
**2007**, 662, 53. [Google Scholar] [CrossRef] - Fukue, J. Bondi Accretion onto a Luminous Object. Publ. Astron. Soc. Jpn.
**2001**, 53, 687–692. [Google Scholar] [CrossRef] - Korol, V.; Ciotti, L.; Pellegrini, S. Bondi accretion in early-type galaxies. Mon. Not. R. Astron. Soc.
**2016**, 460, 1188–1200. [Google Scholar] [CrossRef] - Richards, C.B.; Baumgarte, T.W.; Shapiro, S.L. Accretion onto a small black hole at the center of a neutron star. Phys. Rev. D
**2021**, 103, 104009. [Google Scholar] [CrossRef] - Jiang, Y.F.; Stone, J.M.; Davis, S.W. Super-Eddington Accretion Disks around Supermassive Black Holes. Astrophys. J.
**2019**, 880, 67. [Google Scholar] [CrossRef] - Regan, J.A.; Downes, T.P.; Volonteri, M.; Beckmann, R.; Lupi, A.; Trebitsch, M.; Dubois, Y. Super-Eddington accretion and feedback from the first massive seed black holes. Mon. Not. R. Astron. Soc.
**2019**, 486, 3892–3906. [Google Scholar] [CrossRef] - Willott, C.J.; Albert, L.; Arzoumanian, D.; Bergeron, J.; Crampton, D.; Delorme, P.; Hutchings, J.B.; Omont, A.; Reylé, C.; Schade, D. Eddington-Limited Accretion and the Black Hole Mass Function at Redshift 6. Astron. J.
**2010**, 140, 546. [Google Scholar] [CrossRef] - Wang, J.M.; Hu, C.; Yang, F.; Zhang, E.P.; Wu, M. The Effect of Radiative Efficiency on the Growth of the Black Hole Mass. Chin. Astron. Astrophys.
**2007**, 31, 109–116. [Google Scholar] [CrossRef] - Heinzeller, D.; Duschl, W.J. On the Eddington limit in accretion discs. Mon. Not. R. Astron. Soc.
**2006**, 374, 1146–1154. [Google Scholar] [CrossRef] - Abolmasov, P.; Chashkina, A. On the Eddington limit for relativistic accretion discs. Mon. Not. R. Astron. Soc.
**2015**, 454, 3432–3444. [Google Scholar] [CrossRef] - Babichev, E.O. The Accretion of Dark Energy onto a Black Hole. J. Exp. Theor. Phys.
**2005**, 100, 528. [Google Scholar] [CrossRef] - Babichev, E.O.; Dokuchaev, V.I.; Eroshenko, Y.N. Black Hole in a Radiation-Dominated Universe. Astron. Lett.
**2018**, 44, 491–499. [Google Scholar] [CrossRef] - Cheng-Yi, S. Dark Energy Accretion onto a Black Hole in an Expanding Universe. Commun. Theor. Phys.
**2009**, 52, 441–444. [Google Scholar] [CrossRef] - Gao, C.J.; Zhang, S.N. Reissner–Nordström metric in the Friedman–Robertson–Walker universe. Phys. Lett. B
**2004**, 595, 28–35. [Google Scholar] [CrossRef] - Hawking, S. Gravitational radiation in an expanding universe. J. Math. Phys.
**1968**, 9, 598–604. [Google Scholar] [CrossRef] - Hayward, S.A. Quasilocal gravitational energy. Phys. Rev. D
**1994**, 49, 831–839. [Google Scholar] [CrossRef] [PubMed] - Martin-Moruno, P.; Madrid, J.A.J.; Gonzalez-Diaz, P.F. Will black holes eventually engulf the universe? Phys. Lett. B
**2006**, 640, 117–120. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of a scalar field potential in which phase transitions of the first kind are possible.

**Figure 3.**Schematic representation of the problem statement. $\zeta $ is the impact parameter and ${v}_{\infty}$ is the velocity of the test particle at a large distance from the accreting object.

**Figure 5.**Relation between the initial mass of a PBH (${M}_{0}$) at time t, which is M, derived from Formula (46). The time t on the horizontal axis is the cosmic time. The dotted line shows the moment when the reheating ends (${10}^{-16}$ s, chosen for demonstration); the equation of state parameter at the reheating stage was chosen as $\omega =0$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Krasnov, M.A.; Nikulin, V.V.
Mechanisms of Producing Primordial Black Holes and Their Evolution. *Particles* **2023**, *6*, 580-594.
https://doi.org/10.3390/particles6020033

**AMA Style**

Krasnov MA, Nikulin VV.
Mechanisms of Producing Primordial Black Holes and Their Evolution. *Particles*. 2023; 6(2):580-594.
https://doi.org/10.3390/particles6020033

**Chicago/Turabian Style**

Krasnov, Maxim A., and Valery V. Nikulin.
2023. "Mechanisms of Producing Primordial Black Holes and Their Evolution" *Particles* 6, no. 2: 580-594.
https://doi.org/10.3390/particles6020033