Combining Primordial-Black-Hole Constraints
Scrollable formula. Use Left and Right arrow keys to reveal hidden content.
Scrollable table. Use Left and Right arrow keys to reveal additional columns.
Scrollable figure. Use Left and Right arrow keys to inspect hidden content.
A primordial-black-hole (PBH) constraint plot usually places curves from evaporation, lensing, dynamics, accretion, structure, and gravitational waves on one mass axis. That visual comparison is useful, but it is not automatically a joint statistical result. The curves may use different confidence conventions, nuisance parameters, population definitions, and data, and some responses are nonlinear in abundance. Through 2026-08-20, a conditional method for extended mass functions and valuable curated artifacts are available; a public likelihood-level synthesis with demonstrated covariance and domain control is not. The conditional method is supported, while a universal envelope is not.
Helpful background. Primordial-Black-Hole Observational Constraints explains why each observational mechanism responds to PBHs. This record develops the stricter rules for combining those heterogeneous responses into a reproducible population statement.
A joint limit is a defined statistical object
Section titled “A joint limit is a defined statistical object”The finite question is whether, for a declared PBH population, one can construct a versioned joint constraint whose component likelihoods are valid in that population’s domain and whose nuisance parameters, covariance, confidence semantics, and nonlinear response are explicit. Summing digitized inverse upper limits is a derived approximation; its conditions must be shown rather than assumed.
The central axes are raw likelihood versus published limit curve; monochromatic response versus extended distribution; linear versus nonlinear abundance response; independent versus overlapping data; conditional versus marginalized nuisance parameters; frequentist confidence versus Bayesian credibility; formation versus present abundance; and mass, spin, clustering, accretion, and merger evolution.
| Assessment at 2026-08-20 | Meaning |
|---|---|
| Question status | Partially resolved: a useful recast formalism and curated artifacts exist, while a public full-likelihood synthesis does not |
| Evidence status | Supported with explicit conditions for the integral method and class-separated picture; insufficient for a universal combined envelope |
| Confidence | Moderate; high for domain and provenance requirements, moderate for numerical combinations without raw likelihoods and covariance |
| Domain | PBH evaporation, lensing, gravitational-wave, dynamical, accretion, and structure constraints under declared population and response assumptions |
| What could change the conclusion | A public likelihood-level synthesis, a dominant-curve correction or withdrawal, newly identified data overlap, or a versioned artifact tied to source likelihoods |
A starting project: test one extended-mass recast
Section titled “A starting project: test one extended-mass recast”This is a good entry point for someone who can normalize probability densities, perform numerical quadrature, and read a survey selection function. Read the linked observational background first, then derive the single-observable recast in Carr et al. 2017, §III, Eqs. (8)–(12), revised v3. Extract the linear-response assumption explicitly. Read Appendix A only after that derivation; its combination rule has additional statistical assumptions.
Begin with an analytic positive response function whose mass integral you can evaluate independently. Compare a direct expected-signal calculation with the recast for a normalized lognormal distribution. Recover the monochromatic limit, vary the quadrature range, and deliberately add a quadratic response term to expose the approximation’s failure. This synthetic test validates your implementation. Next select one real constraint with a documented response and interval convention, preserving the original input and any digitization uncertainty. Do not start by combining a whole compilation.
Produce a reusable notebook, a comparison of direct and recast predictions, and a one-page account of where the approximation is valid. A laptop suffices; interpreting the real constraint benefits from a collaborator familiar with its observational mechanism. The useful candidate extension is to quantify recast error for a particular published observable or replace it with its full response. That becomes a research proposal only after checking whether the comparison already exists and whether the required inputs are available.
Pivot to a different constraint if its curve cannot be traced to a confidence convention or response model. If discrepancies persist as the mass grid is refined, investigate the response approximation before combining more curves. A reproducible demonstration that a commonly reused input is inadequate can be a useful result; a smoother exclusion envelope alone is not a statistical advance.
When the extended-mass recast is valid
Section titled “When the extended-mass recast is valid”Suppose a monochromatic analysis provides a response or upper fraction as a function of mass. If the expected signal is linear in PBH abundance, additive across mass bins, and evaluated with nuisances that remain applicable, an extended distribution can be integrated against that response. Carr et al. derive this construction in §III, Eqs. (8)–(12), and demonstrate examples in §§III–IV 2017, §III, Eqs. (8)–(12), revised v3. Their v3 correction to the power-law plot matters; earlier reproductions should not silently retain the superseded figure.
The method can fail when the observable depends nonlinearly on abundance or population history. PBH binary merger rates, accretion feedback, disruption, and structure formation can have such dependence. It can also fail statistically if curves use overlapping events or share a halo, background, or cosmology nuisance but are treated as independent. In those cases one needs the source likelihood or a new forward model, not a rescaled envelope.
The current 2026 synthesis is valuable because it separates major physical classes and supplies digitized tables and a notebook. Its §IV.A–IV.E, Eq. (29), Figs. 4–6, and data-availability statement make both the utility and assumptions visible Carr et al. 2026, §IV and data availability. The artifact is curated from heterogeneous vintages and statistics; it is not raw covariance proof.
Results and open questions at 2026-08-20
Section titled “Results and open questions at 2026-08-20”| Claim or evidence | What it supports | What it does not establish |
|---|---|---|
| Linear-response extended-mass formula | A defensible recast for eligible independent constraints | Universal treatment of nonlinear or correlated observables |
| 2026 class-separated compilation | Current orientation, digitized tables, and a reusable plotting artifact | A new joint likelihood or uniform confidence semantics |
| Critical multi-probe reviews | A taxonomy of model and astrophysical dependence | Current numerical limits without inspecting newer primary releases |
| Microlensing likelihood anatomy | Need for efficiency, cadence, source-size, wave-optics, halo-density, and velocity inputs | Transfer of one survey curve to arbitrary lens populations |
| Formation and evaporation reviews | Nonlinear population mapping and particle-model dependence | A unique mass function or shared nuisance model across messengers |
The formal derivation receives high weight inside its assumptions. Review curves receive less weight for numerical combination than primary likelihoods, but they are useful for identifying domains and missing nuisance variables. Multiple messenger classes are not automatically independent merely because their instruments differ.
Independence, covariance, and population evolution
Section titled “Independence, covariance, and population evolution”Microlensing illustrates why a curve is compressed information. Detection efficiency, cadence, finite source size, wave optics, halo density and velocities, and lens clustering enter before a mass limit is produced. Green and Kavanagh detail these dependencies in §III.2 and discuss combined constraints in §III.7 2021, §§III.2 and III.7. Digitizing the final curve cannot restore nuisances that were fixed or marginalized upstream.
Formation adds another layer. Critical scaling, collapse threshold and profile, non-Gaussian tails, and thermal history map a primordial perturbation model to a time-dependent mass function Escrivà, Kühnel, and Tada 2022, §§2–10. Hawking constraints further depend on particle content, spin, greybody factors, secondary spectra, propagation, and backgrounds Auffinger 2023, §§2–9. A synthesis that changes the population but retains every published curve unchanged is not self-consistent.
Counterexamples, null results, and failed routes
Section titled “Counterexamples, null results, and failed routes”A merger-rate bound is a counterexample to applying the linear integral recipe universally: changing the abundance can change binary formation and disruption nonlinearly. Two curves derived from the same event catalog are a counterexample to treating visual separation as statistical independence. A monochromatic curve with a fixed halo is not a likelihood marginalized over arbitrary clustering.
The failed practical route is to declare the lower envelope of published curves a global confidence limit. Confidence levels and one- versus two-sided semantics can differ; old and new constraints can overlap; and some rows are forecasts or disputed translations. The absence of a public likelihood and covariance means the result must be presented class by class or with sensitivity analyses, not with false precision.
What would change the assessment
Section titled “What would change the assessment”A release would materially advance the field if it provided machine-readable source likelihoods or validated surrogate likelihoods, a common population and evolution model, nuisance priors, a data-overlap graph, nonlinear response tests, and leave-one-class-out results. A correction or withdrawal of a dominant curve could alter an apparent window immediately. A tagged PBHconstraints release tied point-by-point to immutable source products would strengthen artifact provenance even before full likelihoods become available.
Research checklist
Section titled “Research checklist”- Define the population at formation and today, including normalization, tails, spin, clustering, accretion, and merger evolution.
- For every constraint, record DOI, data release, estimator or likelihood, interval convention, nuisances, and validity domain.
- Decide whether the response is linear and separable in mass; demonstrate the approximation when using an integral recast.
- Build an overlap and covariance graph for shared events, sky, halo, backgrounds, and theory inputs.
- Harmonize confidence or credibility semantics rather than relabeling them.
- Pin digitization provenance, units, interpolation, commit hashes, and file checksums.
- Perform leave-one-class-out, nuisance, prior, and mass-grid sensitivity tests.
- Publish both class-level results and any justified combination.
Source and data map
Section titled “Source and data map”Carr et al. 2017 provide the foundational extended-mass method. Carr et al. 2026 provide the current compilation and the PBHconstraints repository. Carr et al. 2021 supply a critical constraint taxonomy §§3–9; Green and Kavanagh expose microlensing and combination assumptions; Escrivà et al. and Auffinger delimit formation and evaporation translations.
No notebook or raw likelihood was rerun. Before reusing PBHconstraints, pin a commit, checksums, units, curve labels, confidence conventions, and the primary source for every point. No signed or tagged archive was verified that links the exact 2026 review tables to an immutable commit. Reproducing its plotted envelope checks plotting and formula application, not the originating experiments or a joint statistical construction.
References
Section titled “References”- Auffinger, J. 2023, “Primordial Black Hole Constraints with Hawking Radiation—A Review,” Progress in Particle and Nuclear Physics 131, 104040, DOI 10.1016/j.ppnp.2023.104040.
- Carr, B., Raidal, M., Tenkanen, T., Vaskonen, V., and Veermäe, H. 2017, “Primordial Black Hole Constraints for Extended Mass Functions,” Physical Review D 96, 023514, arXiv:1705.05567v3, DOI 10.1103/PhysRevD.96.023514. Open PDF.
- Carr, B., Kohri, K., Sendouda, Y., and Yokoyama, J. 2021, “Constraints on Primordial Black Holes,” Reports on Progress in Physics 84, 116902, DOI 10.1088/1361-6633/ac1e31.
- Carr, B., Iovino, A. J., Perna, G., Vaskonen, V., and Veermäe, H. 2026, “Primordial Black Holes: Constraints, Potential Evidence and Prospects,” Rivista del Nuovo Cimento 49, 225–274, arXiv:2601.06024v2, DOI 10.1007/s40766-026-00080-z.
- Escrivà, A., Kühnel, F., and Tada, Y. 2022, “Primordial Black Holes,” arXiv:2211.05767v5, arXiv.
- Green, A. M., and Kavanagh, B. J. 2021, “Primordial Black Holes as a Dark Matter Candidate,” Journal of Physics G 48, 043001, DOI 10.1088/1361-6471/abc534.