Regular Primordial Black Holes

• Regular primordial black holes are non-singular early Universe objects that use modified metrics (e.g., Bardeen or Hayward) to avoid curvature singularities.
• Their formation involves collapse of over-dense regions, hybrid inflationary resonances, and phase-transition mechanisms, which affect their thermodynamic evolution.
• RPBHs offer insights into dark matter, gravitational-wave signals, and cosmic structure, though their observable signatures are tightly constrained by gamma-ray and CMB data.

Regular primordial black holes (RPBHs) are a theoretical class of primordial black holes (PBHs) that are free of interior curvature singularities, typically as a result of incorporating new geometrical structures or quantum gravity effects into their spacetime models. RPBHs are hypothesized to form in the early Universe and have been explored extensively in the context of cosmology, dark matter, cosmic structure formation, gravitational-wave astronomy, and fundamental questions concerning the resolution of classical singularities. Both their formation channels and astrophysical implications are sensitive to the specific regularization mechanism adopted—Bardeen-type cores, Hayward geometries, loop quantum gravity scales, or more exotic features.

1. Concept and Mathematical Structure of Regular Primordial Black Holes

Regular primordial black holes are characterized by central regions that avoid curvature singularities present in the classic Schwarzschild or Kerr solutions. This regularity is achieved by introducing an extra length parameter (denoted ℓ, g, or similar), which modifies the metric to provide a non-singular (often de Sitter or Minkowski) core. Three archetypal regular, time-radial–symmetric metrics frequently discussed are:

Metric	Lapse Function f(r) (with mass M, regularization scale ℓ)	Core Type
Bardeen	$f_B(r) = 1 - \frac{2Mr^2}{(r^2+\ell^2)^{3/2}}$	de Sitter
Hayward	$f_H(r) = 1 - \frac{2Mr^2}{r^3+2M\ell^2}$	de Sitter
CGSV/SV	$f_{CGSV}(r) = 1 - \frac{2M}{r} e^{-\ell/r}$	Minkowski

In all cases, the regularization scale ℓ corresponds to a minimum length introduced by quantum gravity, Planck-scale physics, or non-linear electromagnetic interactions. For ℓ → 0, these metrics reduce to the familiar Schwarzschild metric. Near r = 0, $f(r) \propto r^2$ for the Bardeen and Hayward cases, mimicking de Sitter space and regularizing the central curvature.

The surface temperature of the black hole is given, as in standard general relativity, by the surface gravity:

$T = \frac{f^{\prime}(r_H)}{4\pi}$

where $r_H$ is the horizon radius. However, in regular models, the temperature reaches a maximum for some intermediate mass, then decreases, and vanishes in the extremal (degenerate horizon) limit. This distinguishes RPBHs thermodynamically from Schwarzschild PBHs, for which the temperature diverges as the mass tends to zero.

2. Formation Mechanisms in the Early Universe

RPBHs can form through several processes, all generically linked to high-energy events in the primordial universe:

• Collapse of Large Density Perturbations: In radiation-dominated epochs, sufficiently rare, high-σ primordial overdensities—either Gaussian or induced by inflationary features, topological defects, or baryogenesis inhomogeneities—collapse under gravity. The critical overdensity for collapse is model-dependent and can be lowered during phase transitions (e.g., at the QCD transition where the equation of state softens) (Gonin et al., 8 May 2025, Amendola et al., 2017, Escrivà et al., 2022).
• Hybrid Inflation with Parametric Resonance: Post-inflation parametric resonance between inflaton fields and “waterfall” fields can exponentially amplify certain Fourier modes, leading to arbitrarily large inhomogeneities that can collapse to PBHs across an extended mass range—from sub-solar to supermassive (Frampton, 2015).
• Phase-Transition-Induced Collapse: First-order phase transitions (QCD, cosmic string network collapse, bubble collisions) provide mechanisms for the formation of PBHs with mass spectra peaked at special scales corresponding to the horizon mass at the transition (Gonin et al., 8 May 2025, Escrivà et al., 2022).

When regularizing modifications are imposed, these mechanisms do not alter the basic collapse threshold, but they do affect the subsequent thermodynamic evolution and evaporation rates.

3. Evaporation, Hawking Radiation, and Regularization

RPBHs radiate via Hawking evaporation, but the regularized interior/metric introduces qualitative modifications:

• Temperature Regulation: The Hawking temperature is bounded; it reaches a maximum, then decreases with decreasing mass, and vanishes for extremal RPBHs ($T\rightarrow 0$ as $M \rightarrow M_*$, where $M_*$ is the extremal mass) (Pacheco, 2018, Calzà et al., 4 Sep 2024, Khodadi, 24 Sep 2025). In contrast, Schwarzschild PBHs reach arbitrarily high temperatures as $M \rightarrow 0$.
• Evaporation Endpoint and Remnants: Some models (notably Hayward and Bardeen) admit extremal states with nonzero horizon and zero temperature, raising the prospect of stable remnants potentially as dark matter (Pacheco, 2018, Calzà et al., 4 Sep 2024). However, analysis shows the evaporation rate near the extremal limit behaves as $(M-M_*)^2$; thus, the time to reach remnant masses diverges logarithmically or worse. No true remnant is formed in finite cosmic time—evaporation asymptotes but does not halt (Khodadi, 24 Sep 2025).
• Photon/Fermion Emission Rate: The reduction in Hawking temperature leads to exponentially suppressed photon and particle emission in the low-mass regime. This feature relaxes observational bounds on the allowed abundance of RPBHs, especially in the asteroid-mass window, and expands the range in which RPBHs could constitute all of the dark matter (Calzà et al., 4 Sep 2024).

4. Astrophysical and Cosmological Implications

Dark Matter and Structure Formation

RPBHs have been proposed as dark matter candidates in various models (Pacheco, 2018, Calzà et al., 4 Sep 2024, Frampton, 2015, Escrivà et al., 2022, Dolgov, 2017). Their suitability as dark matter is subject to tight observational bounds:

• Gamma-Ray and CMB Constraints: The slow, but non-zero, cumulative radiation from cosmological populations of near-extremal RPBHs can overproduce extragalactic gamma rays and CMB spectral distortions. The integrated emission from these remnants rapidly exceeds the permitted backgrounds—ruling out low-mass RPBH populations as dominant dark matter (Khodadi, 24 Sep 2025).
• Mass Windows: The relaxation of evaporation-induced bounds for RPBHs means that the lower edge of the allowed asteroid-mass window for PBHs as dark matter candidates is shifted downward by an order of magnitude or more—potentially allowing a broader range of masses to survive as dark matter (Calzà et al., 4 Sep 2024). However, this is subject to the cumulative emission constraint above.
• Seeding of Supermassive Black Holes: High-mass RPBHs (≥10⁶ M_☉) formed at early times can act as seeds for quasars and galaxies, explaining supermassive black holes at high redshift and peculiar stars with unusual metal enrichment (Dubrovich et al., 2012, Dolgov, 2017, Dolgov, 2019, Frampton, 2015).

Gravitational-Wave and Astrophysical Signatures

Direct detection of events associated with RPBHs remains an open but active area:

• Gravitational-Wave Events: Binary mergers of RPBHs—especially in mass ranges not accessible to stellar evolution—are a target for current and future gravitational wave observatories. The predicted low spins and unusual mass ranges provide discriminants between RPBH mergers and conventional binaries (Escrivà et al., 2022, Mirbabayi et al., 2019).
• Gamma-Ray and Particle Showers: RPBH evaporation signatures (if any), such as diffuse gamma-ray backgrounds or rare bursts, serve as ancillary probes (Calzà et al., 4 Sep 2024, MacGibbon et al., 2015).
• Microlensing and Dynamical Effects: Stellar-mass and sublunar RPBHs may yield dynamical or lensing signatures, albeit exact predictions depend on their spatial and velocity distributions (Escrivà et al., 2022, Flores et al., 2023).

Connection to Fundamental Physics

RPBHs are intimately linked to quantum gravity, singularity resolution, and even proposals involving exotic matter:

• Singularity Avoidance: RPBHs act as concrete, phenomenological models where curvature divergence is avoided, offering a testbed for quantum-corrected gravity and a laboratory for the interplay between the quantum and the classical (Calzà et al., 4 Sep 2024, Pacheco, 2018).
• Planck-Scale Remnants: Motivated by loop quantum gravity, string-inspired uncertainty principles, or NUT charges (gravitomagnetic monopoles), RPBHs with Planck-scale masses and radii are a frequent outcome (Pacheco, 2018, Chakraborty et al., 2022). The stability and phenomenology of such remnants remain subject to further quantum-gravitational analysis.
• Remnant Viability for Dark Matter: Theoretical analysis now indicates that the infinite evaporation time and cumulative low-level radiation from RPBH populations strongly disfavor the stable remnant dark matter scenario (Khodadi, 24 Sep 2025).

5. Observational Constraints and Future Prospects

The allowed abundance and mass spectrum of RPBHs are constrained by a combination of:

• Gamma-Ray Backgrounds and CMB Data: Any cumulative emission from slowly evaporating RPBH populations is limited by Fermi-LAT and Planck CMB observations (Calzà et al., 4 Sep 2024, Khodadi, 24 Sep 2025).
• Gravitational-Wave Observations: Current LIGO/Virgo/KAGRA data constrain merger rates for specific PBH mass windows; deviations could indicate exotic PBH populations, though modeling for regular interiors is at an early stage (Escrivà et al., 2022).
• Microlensing Searches: OGLE, EROS, MACHO, and future surveys bound the fraction of compact, non-luminous objects over a range of masses, with RPBH-specific characteristics lying within the same limits as Schwarzschild PBHs (Escrivà et al., 2022).

An expanded parameter space for RPBHs (especially in the asteroid-mass window) has recently been identified owing to the suppressed Hawking temperature and rate of evaporation. Yet, the cumulative effect of extremely slow, residual evaporation for even near-extremal RPBHs is sufficient, over the cosmic inventory, to violate background radiation limits unless the RPBH abundance is highly subdominant (Khodadi, 24 Sep 2025).

6. Theoretical and Model-Building Issues

• Evaporation Timescale and Remnant Issue: In all analytic regular PBH models studied under adiabatic and quasi-static assumptions, the evaporation time to the extremal, zero-temperature state is infinite. Accordingly, RPBHs persist as slowly leaking near-remnants, never actually reaching a true stable state within any finite cosmic timestep (Khodadi, 24 Sep 2025).
• Compatibility with Singular Schwarzschild/Kerr PBHs: For large ℓ, the regularized metrics deviate significantly from Schwarzschild; for ℓ → 0, the models converge to the classical singular solution. Physical guidance for selecting ℓ is often provided by quantum gravity or observational considerations (Calzà et al., 4 Sep 2024, Pacheco, 2018).
• Role of Nontrivial Interior Structure: If gravitational remnants include substantial NUT charge (gravitomagnetic monopole), as discussed in (Chakraborty et al., 2022), additional channels for suppression or modification of Hawking radiation arise, with effects sensitive to both quantum and global spacetime structure.

In summary, regular primordial black holes are a central and evolving theme in primordial black hole research. They provide a physically motivated alternative to singular black holes, avoid classical curvature divergences, and present rich phenomenology. Their distinctive thermodynamics—bounded evaporation, maximal temperature, possible extremality—opens new parameter space for black hole dark matter but simultaneously incurs deep observational and theoretical constraints. Contemporary developments indicate that, although RPBHs remain a crucial probe of singularity resolution and quantum gravitational effects, their direct viability as dark matter is tightly constrained by the integrated emission from a cosmological population and the infinities inherent in their evaporation timescales. The full cosmological implications of RPBHs thus continue to be at the forefront of both theoretical and observational investigation.