Observational Constraints and Dark-Matter Fraction

Primordial-black-hole constraints do not form one universal exclusion curve. Each bound applies to a specified mass distribution, spatial distribution, cosmic history, observable, calibration, selection function, and nuisance model. A defensible combination begins with those likelihoods and common parameters; it does not take the lowest published line at every mass. This page gives the recasting algebra and an evidence map without freezing a mutable numerical envelope into evergreen exposition.

Helpful background. Einstein Equations supplies compact-object gravity; Relativistic Matter and Compact Objects supplies environmental interactions; Black-Hole Dynamics and Collapse supplies merger and accretion histories; Relativistic Cosmology supplies cosmic transfer and observables; and Probability and Statistical Inference supplies likelihoods, priors, covariance, and selection effects.

Define the population before applying a bound

An abundance without an epoch is ambiguous. For PBHs that survive to the present, define the mass distribution

$$\begin{aligned} \phi_0(M_0) &\equiv\frac{d f_{{\rm PBH},0}}{d\ln M_0} =\frac{1}{\rho_{{\rm DM},0}} \frac{d\rho_{{\rm PBH},0}}{d\ln M_0},\\ f_{{\rm PBH},0} &\equiv\frac{\Omega_{{\rm PBH},0}}{\Omega_{{\rm DM},0}} =\int_{\mathcal S_0}d\ln M_0\;\phi_0(M_0). \end{aligned}$$

where $M_0$ is the present mass and $\phi_0(M_0)\ge0$. Only surviving PBHs in the domain $\mathcal S_0$ contribute to this fraction. Any stable-remnant prescription changes that domain and must be stated. A population that has completely evaporated without a remnant has no present-mass branch, so it cannot be assigned a present-day $f_{{\rm PBH},0}$.

Evaporation constraints instead begin with a formation-era abundance. Write

$$\psi_f(M_f)\equiv\frac{d\beta'}{d\ln M_f}, \qquad \beta(t_f)\equiv \frac{\Omega_{\rm PBH}(t_f)}{\Omega_{\rm tot}(t_f)}.$$

Here $M_f$ and $t_f$ are the formation mass and time. The symbol $\beta'$ denotes the convention-adjusted formation fraction used by the source; its relation to $\beta$ can depend on the assumed expansion and thermal history. That conversion must therefore be copied from the paper rather than treated as a universal formula. The integral over $\psi_f$ defines one formation fraction only for a common formation hypersurface and denominator. If different mass cohorts form at different times, retain a joint distribution in $(M_f,a_f)$ or transport them to one reference epoch before summing. Auffinger 2023, §1, printed pp. 2–3, and §4, p. 17 separates already-evaporated populations from surviving dark matter and uses a formation fraction for the more general comparison.

For a surviving branch, mass and spin evolution define a history-dependent map

$$M_0=\mathcal F(M_f,a_{*,f};\text{cosmic history}).$$

If this map is monotonic and PBHs neither merge nor split, first transport the physical number distribution through its conserved comoving measure:

$$a_0^3\frac{d n_0}{d\ln M_0} =a_f^3\frac{d n_f}{d\ln M_f} \left\lvert\frac{d\ln M_f}{d\ln M_0}\right\rvert, \qquad \phi_0(M_0)=\frac{M_0}{\rho_{{\rm DM},0}} \frac{d n_0}{d\ln M_0}.$$

Deterministic accretion can be included in the same monotonic mass map. With a spin distribution, several formation branches, mergers, or stochastic environment-dependent accretion, the change of variables instead becomes an integral evolution kernel. Complete evaporation terminates the branch before today; its observable products are computed from $\psi_f$ and the time-dependent emission history, not from $\phi_0$.

Suppose now that an observable is linear in independent PBH contributions. The numerator and the monochromatic upper bound must use the same abundance variable, mass coordinate, epoch, halo or background model, calibration, efficiency, and nuisance assumptions. A present-day survivor constraint can then be recast as

$$\int d\ln M_0\; \frac{\phi_0(M_0)}{f_{\max,0}(M_0)}\le1,$$

whereas a formation-abundance constraint, after translation to one stated $\beta'$ convention, can be recast as

$$\int d\ln M_f\; \frac{\psi_f(M_f)}{\beta'_{\max,f}(M_f)}\le1.$$

Neither formula licenses putting $\phi_0$ over a formation-era bound or $\psi_f$ over a present-day bound. Carr et al. 2017, §II, especially eqs. (7)–(12), and §§III–IV (Open PDF) derive the extended-mass recast for independent linearized constraints and illustrate its use. Merger observables are nonlinear in pairs of masses; accretion changes both the mass function and radiation history; a multi-bin CMB spectrum is not one scalar count; and clustering couples objects. Those cases require the original forward model. Green and Kavanagh 2021, § III.J, eqs. (19)–(20) review the construction and its failure modes.

What each observable actually constrains

Observable class	Forward-model ingredients	Dominant conditional boundary
Hawking products	Formation-time mass/spin distribution and abundance convention, greybody spectra, particle content, secondary cascades, cosmic propagation, deposition, and backgrounds	Semiclassical evaporation and the expansion history; a fully evaporated population has no present-day dark-matter fraction
Microlensing and compact-object lensing	Source catalog, cadence, magnification threshold, detection efficiency, halo velocities, finite-source and wave effects	Survey-specific selection and spatial model; clustering can invalidate an isolated-lens treatment
CMB and intergalactic accretion	Gas velocity and temperature, accretion geometry, luminosity, feedback, energy deposition, recombination response	Disk versus spherical flow, duty cycle, halo growth, radiative efficiency, and degeneracy with cosmological parameters
Stellar and dynamical systems	Initial stellar distribution, ages, compact-object encounters, relaxation, and survival probability	Object membership, system history, dark-matter profile, and environmental perturbations
Gravitational-wave mergers	Binary channels, mass/spin/redshift population, disruption, detector selection, waveform and catalog likelihood	Astrophysical mixture, clustering, rate normalization, selection, and pairwise mass dependence

Evaporation constraints require especially careful particle modeling. Auffinger 2023, §§ 6–11 tracks how BBN, CMB, radiation backgrounds, cosmic rays, and extended mass or spin distributions depend on primary and secondary spectra. Gravitational constraints apply to any compact object only within their stated spatial and population assumptions; indirect constraints on the primordial curvature spectrum apply only to PBHs formed from those perturbations.

Combine likelihoods, not plotted envelopes

For datasets $D_i$, population parameters $\boldsymbol\vartheta$, and shared nuisances $\boldsymbol\eta$, the correct joint statement is generative:

$$\mathcal L_{\rm joint}(\{D_i\}\mid\boldsymbol\vartheta) =\int d\boldsymbol\eta\;\pi(\boldsymbol\eta) \mathcal L(\{D_i\}\mid\boldsymbol\vartheta,\boldsymbol\eta).$$

A product $\prod_i\mathcal L_i$ is justified only after conditional independence and shared calibration have been addressed. Published upper curves may use different confidence constructions, priors, Galactic halos, background models, or monochromatic conventions. Taking their pointwise minimum silently changes model from mass to mass. Adding all curves as independent penalties can double-count a dataset or a correlated systematic.

A reproducible recast therefore records the exact data or catalog version, calibration and response, sample selections and completeness, likelihood statistic and interval convention, nuisance priors, covariance, mass and spatial distribution, and code or machine-readable product. The inspected source set supplies reviews and individual references, but not a single versioned public joint likelihood spanning all constraint classes. No combined numerical region is reproduced here.

Results and limitations through 2026-08-19

As of 2026-08-19, observations strongly restrict many specified PBH populations, but the allowed dark-matter fraction for a general extended and clustered population remains unresolved. “All dark matter” and “no dark matter” are both too broad without a mass range and model class.

Evidence or result	Current implication	Scope and limitation
Evaporation-product null searches	Bound the formation-era abundance of sufficiently light populations through photons, leptons, neutrinos, BBN, and CMB effects	Depend on the stated $\beta'$ convention, Hawking emission, mass/spin evolution, propagation, deposition, backgrounds, and expansion history; they are not present-day $f_{{\rm PBH},0}$ bounds for fully evaporated populations
Lensing surveys	Restrict isolated compact objects over survey-dependent mass intervals	Halo kinematics, cadence, efficiency, finite sources, wave optics, and clustering set the interpretation
Accretion, dynamical, and structure studies	Can restrict heavier populations through heating, radiation, or gravitational disturbance	Astrophysical calibration and environmental histories can dominate the uncertainty
Gravitational-wave catalogs and searches	Constrain selected merger mixtures and unusual mass populations	Merger channels, disruption, astrophysical backgrounds, priors, and detector selection remain degenerate
Extended-mass recasts	Properly accumulate a linear observable across mass	Invalid for nonlinear, evolving, multivariate, or correlated responses without the original likelihood
O4a PBH population analysis	The versioned primary analysis derives model-conditional bounds over $0.6$–$100\,M_\odot$, dominated by resolvable mergers rather than its stochastic-background branch	Monochromatic or lognormal PBH populations, binary-formation history, catalog selection, and the astrophysical-mixture treatment define the result; no durable public analysis environment was verified
LVK O1–O4a stochastic null and official products	No isotropic ground-band background is detected; released cross spectra and upper-limit products support selected early-Universe translations	This is an indirect PBH constraint only after a prescribed scalar spectrum, formation map, and thermal history; it is not a universal bound on $f_{\rm PBH}$

Andrés-Carcasona et al. 2026v2, §§2–3, §4, and conclusion (Open PDF) supply the O4a population analysis. Their astrophysical-population-agnostic treatments modestly relax the solar-mass bounds, while a joint fit that freely moves an astrophysical low-mass cutoff can spuriously favor a PBH component; controlling that choice removes the significant preference. The result is a primary, model-conditional constraint, not evidence that any catalog event is primordial.

The independent ground-band null comes from LIGO–Virgo–KAGRA 2025, §I, printed pp. 1–3, §VII, pp. 15–17, and §X, pp. 22–23 (Open PDF). Its official LIGO-G2502630-v1 product supplies O1–O4a cross spectra, posterior samples, upper-limit arrays, power-law-integrated curves, and scripts. It does not supply a public joint likelihood spanning the other PBH constraint classes, and no separately versioned model-likelihood environment was verified for the early-Universe translation.

Escrivà, Kühnel, and Tada 2025, Chapter VI–Chapter VIII survey current constraints and signatures while retaining mass-function, clustering, and formation dependence. The competing interpretations concern conservative versus aggressive foreground subtraction, spherical versus disk accretion, smooth versus clustered populations, early versus late merger channels, and monochromatic versus extended mass functions. No one regime axis resolves all of them.

The interpretation could change after a public likelihood or corrected data release supersedes a major constraint, or after a robust joint analysis exposes shared covariance and consistently recasts the same extended, evolving, and clustered population across multiple observable classes.

Diagnose an extended-mass recast

A surviving two-bin population contributes $f_{0,1}=0.04$ where $f_{\max,0,1}=0.10$ and $f_{0,2}=0.20$ where $f_{\max,0,2}=0.50$. Assume one linear observable, present-mass bins, and the same nuisance model. Is it allowed by the recast, and why is checking each bin separately insufficient?

Solution

The accumulated response is

$$\frac{f_{0,1}}{f_{\max,0,1}}+ \frac{f_{0,2}}{f_{\max,0,2}} =\frac{0.04}{0.10}+ \frac{0.20}{0.50}=0.8.$$

It satisfies the bound. Testing $f_{0,i}\lt f_{\max,0,i}$ separately would miss accumulation: changing the second contribution to $0.35$ would leave both bins individually below their maxima but give $0.4+0.7=1.1$, which is excluded under the linear assumptions. For a merger or evolving-accretion observable, even this sum would be invalid and the full forward likelihood would be required. The same arithmetic could be used for formation-era bins only after replacing both numerator and denominator by one consistently defined $\beta'$ convention.

Scope boundary

This page organizes conditional constraints; it does not publish a static exclusion plot, certify a current allowed window, or combine incompatible confidence statements. A survivor fraction and a formation fraction are never combined until a stated cosmic-history model translates both to one epoch and mass variable. Formation-model priors, mutable parameter summaries, and instrument-specific likelihood pipelines remain separate from this durable method.

References

• Andrés-Carcasona, M., A. J. Iovino, E. Vallejo-Pagès, V. Vaskonen, H. Veermäe, M. Martínez, and Ll. M. Mir. 2026. “Constraints on Primordial Black Holes from the First Part of LIGO–Virgo–KAGRA Fourth Observing Run.” arXiv:2605.15749v2. Open PDF.
• Auffinger, Jérémy. 2023. “Primordial Black Hole Constraints with Hawking Radiation—A Review.” Progress in Particle and Nuclear Physics 131: 104040. DOI.
• Carr, Bernard, Martti Raidal, Tommi Tenkanen, Ville Vaskonen, and Hardi Veermäe. 2017. “Primordial Black Hole Constraints for Extended Mass Functions.” Physical Review D 96: 023514. DOI. Open PDF.
• 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.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. 2025. “Cosmological and High Energy Physics Implications from Gravitational-Wave Background Searches in LIGO–Virgo–KAGRA’s O1–O4a Runs.” arXiv:2510.26848v2. Open PDF. Official O1–O4a products.

Dated Research assessments. For open questions and recent results, follow the links below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Combining Primordial-Black-Hole Constraints
• Can Primordial Black Holes Be All the Dark Matter?