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An Explanation about the Primordial Black Holes Genesis


Cosmic Strings [1] can emerge from symmetry-breaking [2] phase transitions in the early Universe (further information in here). These one-dimensional structures have been proposed as catalysts for large-scale cosmic structures, such as Galaxy(ies) (Figure 1) and galaxy clusters. Theoretical investigations often require cosmic strings to have substantial linear mass densities to effectively influence matter. A key parameter, [math]\small{G\mu/c^2}[/math], where [math]\small{G}[/math] is the gravitational constant (for further information upgrade here), is typically expected to be of the order of 10-6.


However, a critical aspect of cosmic string dynamics involves the formation of black holes (further information in here, section 3) from cosmic-string loops [3] As a loop contracts, it may cross its own gravitational horizon, leading to the creation of a black hole. Hawking Radiation from these black holes could then emit energy, primarily in the form of gamma rays, with implications for the observable gamma-ray background [4] (Figure 2). However, it's possible to set an upper limit on [math]\normalsize{\mu}[/math], considering the potential creation and evaporation of black holes from cosmic strings and their impact on gamma-ray observations.


Black Hole Formation from Cosmic-String Loops


A cosmic-string loop possesses energy [math]\small{E}[/math], and its gravitational radius (also known as the Schwarzschild Radius, further information in here, section 1) is given by:


[math]\LARGE{R_g = 2GE = 2G\mu l,}[/math]


[math]\Large{R_g = 2GE = 2G\mu l,}[/math]

Equation 1.   Schwarzschild Radius form for a string loop


where [math]\small{l}[/math] is the Length of the String Loop, and [math]\normalsize{\mu}[/math] is the Linear Mass Density of the cosmic string. If the loop contracts to within this gravitational radius, it can form a black hole. This black hole could then evaporate due to the Hawking radiation mechanism, emitting energy primarily in the form of gamma rays. The density of primordial black holes created by cosmic-string loops would then be a significant factor in understanding the gamma-ray background.


The condition for black hole formation is therefore met when the loop's radius becomes less than or equal to the Schwarzschild radius. This condition is a basic criterion and represents a lower bound for black hole formation. The interesting point about this criterion is that it depends on the one-dimensional nature of the string. The ratio of the Schwarzschild radius to the loop length is given by:


[math]\LARGE{f = \frac{R_g}{l} = 2G\mu}[/math]


[math]\Large{f = \frac{R_g}{l} = 2G\mu}[/math]

Equation 2.   Relation between the Schwarzschild Radius and the loop length


which does not depend on the fundamental length, [math]\small{l}[/math] of the loop.


craiyon_180439_milky_way_galaxy (1)

Figure 1.   The Milky Way in a Dark Night Sky


Parameterization and Probability of Black Hole Formation


The formation probability of black holes from cosmic-string loops depends on several factors, including the loop's energy and its shape. The probability, [math]\normalsize{p}[/math], is related to the ratio [math]\small{R_g/l}[/math], which is proportional to [math]\small{2G\mu}[/math]. This relationship can be expressed as:


[math]\LARGE{p = \kappa (G\mu)^{2 + q}}[/math]


[math]\Large{p = \kappa (G\mu)^{2 + q}}[/math]

Equation 3.   The equation for Black Holes Formation Probability


in which [math]\normalsize{\kappa}[/math] and [math]\normalsize{q}[/math] are coefficients to be determined. The parameter [math]\normalsize{q}[/math] represents the uncertainty in the parametrization of the loop's shape and evolution. This uncertainty is critical in estimating the upper limit for [math]\small{G\mu/c^2}[/math], as it influences the probability of black-hole formation from cosmic-string loops.


Impact on Gamma-Ray Background


When black holes evaporate [5] through Hawking radiation, they emit energy across the electromagnetic spectrum (further information in here, section 2), with a significant contribution to gamma rays. The density of these black holes, [math]\normalsize{n_{BH}}[/math], is dependent on the cosmic-string loop density, [math]\normalsize{n_{loop}}[/math], and the probability of black-hole formation:


[math]\LARGE{n_{BH} = p \cdot n_{loop}}[/math]


[math]\Large{n_{BH} = p \cdot n_{loop}}[/math]

Equation 4.   Black holes density - loop density expression


As the Universe expands [6], the energy density of black holes, [math]\normalsize{\rho_{BH}}[/math] decreases at a slower rate than the energy density of radiation (Equation 5), implying that even a low initial density of primordial black holes could lead to a measurable gamma-ray background. The observation of this background, along with gamma-ray bursts, allows for setting upper limits on [math]\small{G\mu/c^2}[/math].



[math]\LARGE{\frac{\rho_{BH}}{\rho_{matter}} = \frac{(G\mu)^{2 + q}\cdot \nu \kappa \cdot \mu^3 + M^{-2}}{\rho_{matter}|_{t \ = \ t_{BH}}}}[/math]


[math]\Large{\frac{\rho_{BH}}{\rho_{matter}} = \frac{(G\mu)^{2 + q}\cdot \nu \kappa \cdot \mu^3 + M^{-2}}{\rho_{matter}|_{t \ = \ t_{BH}}}}[/math]

Equation 5.   Ratio between black hole energy density and the matter density


[math]\small{M}[/math] is the Black Hole Mass, [math]\small{t_{BH}}[/math] is the Black Hole Formation Time and [math]\normalsize{\rho_{matter}}[/math] is the Matter Density.


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Figure 2.   A pictorical Represention of the Gamma-Ray Background


Constraints on Cosmic-String Linear Mass Density


Combining theoretical predictions with observational constraints from gamma-ray background and gamma-ray bursts, we can derive an upper bound for the linear mass density of cosmic strings (Equation 6). This upper bound depends on the value of [math]\normalsize{q}[/math], which represents the uncertainty in the parameterization of the loop's shape and evolution. If [math]\normalsize{q \approx 0}[/math], then the upper bound for [math]\small{G\mu/c^2}[/math] aligns with theoretical expectations of the order of 10-6. However, variations in [math]\normalsize{q}[/math] could significantly alter this bound.



[math]\LARGE{G\mu \leq [(G\mu)_{max, \ q \ = \ 0}]^{(\frac{7}{7 + 2q})}}[/math]


[math]\Large{G\mu \leq [(G\mu)_{max, \ q \ = \ 0}]^{(\frac{7}{7 + 2q})}}[/math]

Equation 6.   The upper bound equation: the maximal value for [math]\normalsize{G\mu}[/math]


as [math]\normalsize{[(G\mu)_{max, \ q \ = \ 0}]}[/math] represents the Maximum Value for [math]\normalsize{G\mu}[/math] corresponding to [math]\normalsize{q = 0}[/math].


A "String" Cosmological Structure


The study of cosmic strings and their potential to form black holes has significant implications for cosmic structure formation and observational cosmology. By considering the gamma-ray background and gamma-ray bursts, one can derive an upper limit for the linear mass density of cosmic strings. The uncertainty in the parameter [math]\normalsize{q}[/math], representing the choice of parameterization and the dynamics of loop formation, is a critical factor in refining this upper limit. Further research into this parameterization and the evolution of cosmic strings is necessary to provide more accurate constraints on [math]\normalsize{G\mu/c^2}[/math].




  1. ResearchForLife7 (revisited from Canadian Science Publishing). "Solutions of the Schrödinger equation for pseudo-Coulomb potential plus a new improved ring-shaped potential in the cosmic string space–time" https://httpsresearchforlife7.com/wp-content/uploads/2024/12/Solutions_of_the_Schrodinger_equation_for_pseudo_Coulomb_potential_plus_a_new_improved_ring_shaped_potential_in_the_cosmic_string_space_time.pdf

  2. Physical Review Letter. "Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking" https://journals.aps.org/prd/abstract/10.1103/PhysRevD.74.063527

  3. Physical Review D. "Scaling of cosmic string loops" https://journals.aps.org/prd/pdf/10.1103/PhysRevD.74.063527

  4. IOPscience. "Measurements of the Galactic X-Ray/Gamma-Ray Background Radiation: Contribution of Discrete Sources" https://iopscience.iop.org/article/10.1086/308755/pdf

  5. Physical Review D. "Information loss in black holes" https://journals.aps.org/prd/pdf/10.1103/PhysRevD.72.084013

  6. arXiv. "Is the Expansion of the Universe Accelarating? All Signs Point to yes" https://arxiv.org/pdf/1610.08972


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