Primordial Black Holes

Primordial black holes (PBHs) are hypothetical black holes formed by nonstellar processes in the early Universe. General relativity permits such collapse, but no population-level PBH detection is established. The central object is therefore not a monochromatic mass point or an exclusion plot: it is a formation model, an extended mass function, spatial and velocity statistics, and an abundance that must be propagated through every observable with compatible astrophysical assumptions.

Early-Universe black-hole population

Property	Working statement
Object class	Astrophysical class
Physical status	Hypothetical cosmological population
Mass range	Formation-model dependent and potentially many orders of magnitude
Proposed channels	Collapse of enhanced density perturbations and other early-Universe nonstellar mechanisms
Candidate signatures	Hawking emission, lensing, dynamical effects, accretion, gravitational waves, and compact-object demographics
Current evidence	Constraints and candidate interpretations are model dependent; no population detection is established
Evidence cutoff	20 August 2026

The word primordial refers to origin, not mass. A PBH could overlap the stellar-remnant interval, and mass alone would not distinguish the populations. Any affirmative claim must supply an observable that is difficult for astrophysical black holes or other matter to reproduce, together with a consistent rate and selection model.

Formation and mass functions

In the standard radiation-era picture, an overdense region can collapse after horizon entry if its smoothed perturbation exceeds a profile-dependent threshold. Near threshold, critical collapse broadens the relation between horizon mass and black-hole mass. The abundance depends exponentially on the tail of the perturbation distribution, so window function, transfer function, non-Gaussianity, perturbation profile, equation of state, and nonlinear mapping all matter; Escrivà, Kühnel, and Tada 2022, §§3–4 reviews these formation ingredients. The complete calculation is treated in Horizon Mass, Critical Collapse, and Mass Functions.

A useful abundance convention is a present-day differential fraction per logarithmic mass,

$$\psi(M)=\frac{1}{\rho_{\rm DM}}\frac{d\rho_{\rm PBH}}{d\ln M}, \qquad f_{\rm PBH}=\int \psi(M)\,d\ln M.$$

Authors also normalize distributions per unit mass or to the PBH fraction itself. Those conventions differ by factors of $M$ and $f_{\rm PBH}$; they must be translated before curves are compared. A “monochromatic” population is an idealized narrow limit, not a generic prediction. The dependence of dark-matter conclusions on mass-function and astrophysical assumptions is surveyed by Green and Kavanagh 2021, §§ 2–3.

Two controlled limits are informative. In the rare, unclustered limit, individual events may be modeled with a smooth halo phase-space distribution. In a strongly clustered limit, binary formation, lensing statistics, accretion environments, and local number density all change, but clustering must arise from a consistent formation history rather than being inserted independently into each constraint. Similarly, the monochromatic limit can validate a response kernel, while a broad mass function tests whether the published constraint is linear and local in abundance.

Hawking emission and surviving masses

Semiclassical Hawking temperature scales as $T_H\propto M^{-1}$, and the evaporation time grows approximately as $M^3$ when the particle content is fixed. Sufficiently low-mass PBHs would therefore have evaporated by the present epoch; higher masses can still contribute photons, charged particles, neutrinos, or heating during their evolution. Greybody factors, particle species, secondary emission, cosmological propagation, foregrounds, and detector response enter the prediction, as reviewed by Auffinger 2023, §§ 3.1–3.4 and 6–10. The field-theoretic derivation belongs with Hawking Radiation from Gravitational Collapse.

Evaporation constraints are not interchangeable with lensing or accretion limits. Each probes a different functional of $\psi(M)$ and a different environment. The relevant question is whether the same mass function and abundance survive all applicable likelihoods.

Constraints as forward models

Microlensing searches count or exclude events after cadence, blending, finite-source, wave-optics, parallax, halo density, and velocity distributions are specified. Dynamical bounds depend on the survival and initial conditions of stars, clusters, or wide systems. Accretion constraints require relative velocities, gas thermodynamics, feedback, duty cycles, and radiative efficiency. Gravitational-wave constraints require binary-formation physics, merger delay, detector selection, and separation from astrophysical channels.

For a sufficiently weak PBH contribution, an experiment can sometimes be represented by a response kernel $K_i(M)$,

$$\mu_i[\psi]=\int K_i(M)\,\psi(M)\,d\ln M,$$

but many observables are nonlinear in abundance or depend on clustering and feedback. One cannot generally take the pointwise minimum of published monochromatic upper-limit curves and call the remaining region allowed for an extended distribution. Compatible nuisance parameters and covariances must be carried through a joint or deliberately conservative synthesis. That workflow is developed in Combining Primordial-Black-Hole Constraints. Carr et al. 2017, § 3, eqs. (8)–(12) gives the integral prescription for translating monochromatic response limits to extended mass functions and shows why curve intersection is insufficient.

Candidate evidence and discriminants

A compact-object merger with an unusual mass is not by itself primordial. Plausible discriminants include a statistically supported substellar-mass black-hole population, redshift evolution incompatible with stellar formation, correlations tied to a predicted primordial spectrum, or a multimessenger signature with no viable astrophysical counterpart. Franciolini et al. 2024, §§5–7 reviews how such gravitational-wave signatures depend on primordial binary formation and competing astrophysical populations. Each requires a detection pipeline, background model, trials accounting, population selection, and a prediction for other channels.

The most permissive interpretation consistent with some constraints is not automatically a detection, and the strongest single exclusion is not automatically universal. Competing analyses often differ in halo models, accretion prescriptions, source sizes, wave effects, statistical thresholds, or whether an extended distribution was folded correctly. The appropriate output is a dated allowed posterior under declared assumptions.

Limitations and future tests

At the evidence cutoff, PBHs remain a viable model in portions of parameter space, with the allowed fraction and intervals dependent on formation and astrophysical assumptions. They are not established as all or any fixed fraction of dark matter. This assessment should change after a materially improved survey likelihood, a corrected constraint, a robust positive population signal, or a formation calculation that alters the mapping between primordial perturbations and $\psi(M)$.

References

• Auffinger, Jérémy. “Primordial Black Hole Constraints with Hawking Radiation—A Review.” Progress in Particle and Nuclear Physics 131 (2023): 104040. DOI.
• Carr, Bernard, Martti Raidal, Tommi Tenkanen, Ville Vaskonen, and Hardi Veermäe. “Primordial Black Hole Constraints for Extended Mass Functions.” Physical Review D 96 (2017): 023514. DOI.
• Escrivà, Albert, Florian Kühnel, and Yuichiro Tada. “Primordial Black Holes.” In Handbook of Gravitational Wave Astronomy. Springer, 2022; updated arXiv version 2025. arXiv.
• Franciolini, Gabriele, et al. “Primordial Black Holes and Their Gravitational-Wave Signatures.” Living Reviews in Relativity 27 (2024): 1. DOI.
• Green, Anne M., and Bradley J. Kavanagh. “Primordial Black Holes as a Dark Matter Candidate.” Journal of Physics G 48 (2021): 043001. DOI.