Primordial Black Holes

Primordial black holes connect rare early-universe fluctuations to nonlinear collapse, semiclassical evaporation, population dynamics, and observational inference. The chapter’s durable problem is to carry one explicitly defined initial ensemble through that whole chain without silently changing its mass function, spatial statistics, or background history. No abundance or dark-matter statement is meaningful until those choices and the relevant evidence date are declared.

Helpful background. Einstein Equations and relativistic matter supply gravity and fluid dynamics; black-hole dynamics and collapse introduces trapped surfaces and critical behavior. The cosmology overview locates the background and perturbation chapters, while probability and inference provides rare-event and likelihood reasoning. Start with the linked formation page below for the specific compaction and threshold definitions.

One population, several physical maps

The teaching thread begins with a stochastic curvature field. A window and slicing map it to a profile-dependent mass excess; relativistic dynamics decides whether that profile collapses; critical behavior maps its distance above threshold to a black-hole mass. Cosmic expansion then converts the formation fraction into a later matter fraction. Hawking emission, accretion, clustering, disruption, and mergers can change the mass, spin, position, or number distribution before any catalog is observed.

Four distinctions prevent most conceptual errors:

• the horizon mass is not automatically the final black-hole mass;
• a power spectrum does not determine a rare non-Gaussian tail;
• a formation fraction is not the present dark-matter fraction; and
• a monochromatic upper limit is not automatically a limit on an extended, clustered, evolving population.

The formation and statistics chain is synthesized in Escrivà, Kühnel, and Tada 2025, Chapters II–III (Open PDF). The observational appraisal in Green and Kavanagh 2021, §§III.A–III.J (Open PDF) shows why every bound retains its spatial, mass-function, and astrophysical assumptions.

The six problems in order

1. Formation from Enhanced Density Perturbations defines compaction, profile shape, smoothing, collapse threshold, and the exponentially sensitive rare-tail abundance.
2. Horizon Mass, Critical Collapse, and Mass Functions derives $M_H=(2GH)^{-1}$, applies critical scaling, and distinguishes distributions per mass from distributions per logarithmic mass.
3. Non-Gaussianity and Alternative Formation Channels compares heavy tails, early matter domination, phase transitions, defects, walls, bubbles, and condensate fragmentation through common physical outputs.
4. Hawking Evaporation and Light Primordial Black Holes derives temperature and lifetime scaling and propagates greybody-weighted particle emission into cosmological energy injection.
5. Accretion, Clustering, and Merger Histories separates early binaries, late captures, disruption, and accretion, with a converged dimensionless benchmark that does not impersonate a merger-rate forecast.
6. Observational Constraints and Dark-Matter Fraction recasts linear monochromatic limits correctly and explains why heterogeneous likelihoods cannot be combined by splicing plotted envelopes.

The order matters. Reading a current constraint first can hide the formation and evolutionary assumptions already embedded in it; deriving a formation fraction without the later pages can make an early-time number look like a present observable.

Two routes through the chapter

The formation route is for readers asking what primordial physics can make black holes. Read the first two pages in full, then the alternative-channel comparison. Continue to the gravitational-wave-background chapter when a curvature peak, phase transition, or defect predicts a correlated stochastic background.

The phenomenology route is for readers interpreting a population or bound. Begin with the mass-function definitions on the second page, then read evaporation, population histories, and constraints. Return to the formation and alternative-channel pages whenever a constraint is translated into a statement about a primordial spectrum or particle model.

A black-hole specialist can instead use the chapter as a bridge: the collapse volume develops general critical phenomena, quantum-field and semiclassical volumes derive Hawking radiation and backreaction, and this chapter applies those results to radiation-era initial data and cosmological populations.

Checks that recur on every route

Stage	Required check	What failure looks like
Perturbation to collapse	Match slicing, window, profile, $w$, and threshold variable	A quoted universal $\delta_c$ is combined with an unrelated variance
Collapse to mass	Verify dimensions, critical-law regime, and distribution normalization	$M_H$ is called the PBH mass or a $dM$ density is integrated over $d\ln M$
Formation to later population	Track number, mass, and radiated energy through the stated history	Accretion or mergers change masses without updating number density or selections
Population to observable	Bind calibration, efficiency, nuisance model, covariance, and evidence date	Incompatible upper curves are minimized or multiplied as if they were one likelihood

These checks also expose model boundaries. A converged quadrature cannot validate an astrophysical kernel; a null induced-wave search does not constrain PBHs made by a channel without that wave source; and an evaporation bound assumes the relevant semiclassical and particle-physics model.

Review the chapter

A model produces PBHs from a narrow curvature feature during radiation domination. It quotes a present mass fraction using a monochromatic mass $M=M_H$, then applies the pointwise minimum of several published limits. Identify at least four missing steps.

Solution

First, specify the curvature statistics beyond the power spectrum, the smoothing window, profile prescription, slice, and collapse threshold. Second, apply critical scaling and integrate the resulting extended mass function rather than setting every mass equal to $M_H$. Third, redshift the formation fraction through the declared thermal history and evolve evaporation, accretion, mergers, and clustering where relevant. Fourth, recast each observable onto that same evolved distribution using its calibration, selection, likelihood, and nuisance assumptions. Only conditionally independent likelihoods with shared covariance handled may be combined; the pointwise minimum of curves is not such a combination.

Chapter boundary

This chapter develops formation, abundance, evolution, signatures, and conditional constraints. It does not provide an undated exclusion plot, a complete inflationary-model catalog, a general proof of critical scaling, a quantum derivation of Hawking radiation, or an instrument-specific likelihood pipeline. Those deeper or mutable treatments remain with their canonical volumes and research products.

References

• Escrivà, Albert, Florian Kühnel, and Yuichiro Tada. 2025. “Primordial Black Holes.” arXiv:2211.05767v5. Open PDF.
• Green, Anne M., and Bradley J. Kavanagh. 2021. “Primordial Black Holes as a Dark Matter Candidate.” Journal of Physics G 48: 043001. DOI.