Test 39 — Primordial Black Hole Constraint Test — A Framework for a Theory of Everything

Test 39 — Primordial Black Hole Constraint Test

If early-universe density fluctuations were large enough to collapse directly into black holes, does the resulting population — across every mass window from evaporating micro–black holes to solar-mass-and-up LIGO-band objects — stay under the abundance, lensing, CMB-accretion, and gravitational-wave-merger-rate bounds that decades of astrophysical observation have built up? And does this framework supply any mechanism that would push fluctuations into PBH-forming territory, or is this purely an inherited consistency check on standard inflationary cosmology?

Agrees Primordial black holes

What we're checking

Observable: The fraction of dark matter locked in primordial black holes, f_PBH, across ~17 factors-of-ten in mass (from evaporating micro-holes to ~30-solar-mass objects).
Standard cosmology: No such population has ever been found. Every survey only sets a ceiling — primordial black holes are allowed, but not required, to exist.
Granularity (consistency reading): Introduces no mechanism that would boost small-scale ripples into black holes — so it, too, expects no primordial-black-hole population, and nothing that would breach the observed ceilings.
Measured: Zero primordial black holes detected; f_PBH held below 1 across the whole mass range (microlensing, CMB accretion, LIGO/Virgo/KAGRA, gamma-ray limits).
The epoch: Early — these objects, had they formed, would trace back to the first fractions of a second. This is the setting, not the compared quantity.
Verdict: Agrees — both roads reach the same conclusion: the data need no primordial black holes.

Some agreements are a triumphant collision at a single number. This one is quieter, and in its own way just as telling: two completely different ways of reading the early universe both look at the same evidence and answer the same question with the same word — none. Nobody has ever caught a primordial black hole. Decades of microlensing surveys, gamma-ray searches, and gravitational-wave catalogs have only ever drawn a ceiling and reported an empty room beneath it.

Standard cosmology arrives at that emptiness honestly: to make black holes in the newborn universe you need the tiny density ripples on small scales to be enormously amplified — hundreds of times stronger than the gentle ripples we actually see in the microwave sky — and nothing in the standard picture supplies that amplifier. This framework walks up to the same door and finds the same thing. It offers no mechanism that would kick those small-scale ripples up to black-hole strength, so it predicts no population and disturbs none of the ceilings.

That is the whole match — and it is a real one. Neither road is forced to invent a hidden population it can't see; both stay clear of every limit the telescopes have set. It is agreement on an absence, which is exactly the modest, honest kind of result that keeps a candidate theory believable: the framework does not claim to derive a number it hasn't earned, and it does not conjure black holes the sky refuses to show. If a future survey ever closes the last open window — the asteroid-mass gap the searches haven't reached yet — both pictures get tested together, on the same footing.

What this test checks: primordial black holes (PBHs) are a well-studied possible consequence of large primordial curvature perturbations — a way small patches of the very early universe could have collapsed straight into black holes instead of stars and galaxies. Decades of observation (evaporation limits, microlensing surveys, CMB accretion bounds, LIGO/Virgo/KAGRA merger rates) place tight ceilings on how many such black holes could exist at every mass, from evaporating micro-black-holes up to solar-mass LIGO-band objects. Standard cosmology's honest answer is: no PBH population has ever been confirmed, and none is required. This framework's answer agrees: it supplies no mechanism that pushes small-scale fluctuations into PBH-forming territory, so it predicts no excess population either — the same "nothing seen, nothing required" verdict. One thing is genuinely open, for everyone: the inflaton-slope calculation that would let either picture predict a specific small-scale spectrum (rather than just checking that nothing is being over-produced) hasn't been forced from geometry yet. Agreement here is two methods landing on the same honest "consistent, not required" answer — a good sign, not a proof of the framework.

1. Verdict

The number, both ways

Number we’re testing: The fraction of dark matter locked in primordial black holes, f_PBH(M), across ~17 factors-of-ten in mass (evaporating micro-holes to ~30-solar-mass objects)
Standard cosmology: No such population has ever been found; every survey only sets a ceiling — primordial black holes are allowed, but not required, to exist
This framework (granularity): Introduces no mechanism that would boost small-scale ripples into black holes — so it, too, expects no primordial-black-hole population and nothing that would breach the observed ceilings
Measured: Zero primordial black holes detected; f_PBH held below 1 across the whole mass range (microlensing, CMB accretion, LIGO/Virgo/KAGRA merger rates, gamma-ray limits; Carr et al. 2021 compilation)
Agreement: Both roads answer the same question with the same word — none: no population required, no ceiling breached; PBH formation would need P_ζ ~ 10⁻² at small scales, ~7 orders of magnitude above the measured A_s = 2.1×10⁻⁹, and neither picture supplies that boost (consistency check — shared inputs)

Check the source → the calculation shown on this page (Data Used · Calculation Summary)

Agrees. The current PBH abundance constraint curve \(f_{\rm PBH}(M)\) across roughly 17 decades of mass, from evaporation limits below \(\sim10^{15}\,\mathrm{g}\) through microlensing, CMB-accretion, and gravitational-wave bounds at higher masses, is standard astrophysics: no PBH population has been detected, and observation only sets ceilings on how many could exist. This framework supplies no mechanism that forces a small-scale curvature-perturbation boost, so it predicts no PBH population that would conflict with those ceilings. The two pictures agree because neither one currently requires PBHs to exist at all — this is a real, if modest, point of agreement rather than a coincidence.

2. Tested Claim

The precise claim under test: if some epoch of this framework's history (inflation, reheating, a phase transition, or any other high-density event) produced curvature perturbations large enough at some comoving scale to trigger gravitational collapse into black holes at horizon re-entry, then the resulting PBH population, expressed as the fraction of dark matter in PBHs \(f_{\rm PBH}(M) \equiv \Omega_{\rm PBH}(M)/\Omega_{\rm DM}\) as a function of PBH mass \(M\), must lie below the current combined envelope of: (a) evaporation/extragalactic-gamma-ray constraints for \(M \lesssim 10^{17}\,\mathrm{g}\), (b) microlensing constraints (Subaru HSC, EROS/MACHO, OGLE) for \(10^{20}\text{--}10^{28}\,\mathrm{g}\) (\(\sim10^{-13}\text{--}10^{-6}\,M_\odot\)), (c) CMB spectral-distortion/accretion constraints for \(M\gtrsim1\text{--}100\,M_\odot\), and (d) LIGO/Virgo/KAGRA compact-binary merger-rate constraints for \(M\sim1\text{--}100\,M_\odot\). The framework's interpretation treats a hypothetical successful PBH-formation event as a "high-density record-collapse": a region where the local overdensity becomes large enough that a new, stable, gravitationally-bound distinction (a black hole) is recorded rather than dispersing.

3. Data Used

Comprehensive PBH constraint review: Carr, Kohri, Sendouda & Yokoyama, "Constraints on primordial black holes," Rep. Prog. Phys. 84, 116902 (2021) (arXiv:2002.12778, updated through 2021) — the standard reference compiling \(f_{\rm PBH}(M)\) exclusion curves across evaporation, lensing, dynamical, CMB, and large-scale-structure probes.
Evaporation / extragalactic gamma-ray bound: PBHs with \(M \lesssim 5\times10^{14}\, \mathrm{g}\) would have evaporated via Hawking radiation before today (Hawking 1974/1975 evaporation timescale \(\tau \propto M^3\)); PBHs in the range \(\sim10^{15}\text{--}10^{17}\,\mathrm{g}\) are excluded as the dominant dark-matter component by the extragalactic gamma-ray background (Carr et al. 2021, §3; also Coogan, Morrison & Profumo 2021, arXiv:2010.04797, tightening this window via Voyager-1 electron/ positron data).
Microlensing constraints: Subaru Hyper Suprime-Cam Andromeda microlensing survey (Niikura et al. 2019, Nature Astronomy 3, 524, arXiv:1701.02151) excludes \(f_{\rm PBH}\approx1\) for \(M\sim10^{20}\text{--}10^{24}\,\mathrm{g}\); EROS-2 (Tisserand et al. 2007, A&A 469, 387) and OGLE (Niikura et al. 2019b, Phys. Rev. D 99, 083503, arXiv:1901.07120) extend microlensing exclusion up through \(\sim10^{28}\,\mathrm{g}\) (\(\sim10^{-6}\,M_\odot\)), together excluding \(f_{\rm PBH}\gtrsim0.1\text{--}1\) over most of the \(10^{20}\text{--}10^{28}\,\mathrm{g}\) window depending on mass function assumed.
CMB accretion constraint: Ali-Haïmoud & Kamionkowski (2017), Phys. Rev. D 95, 043534 (arXiv:1612.05644), and Planck 2018 (Planck Collaboration VI, A&A 641, A6, arXiv:1807.06209) limits on CMB spectral distortions and anisotropies from PBH accretion during the pre-recombination era exclude \(f_{\rm PBH}\gtrsim10^{-3}\text{--}10^{-2}\) for \(M\gtrsim1\text{--}100\,M_\odot\), depending on the assumed accretion disk model (disk vs. spherical Bondi accretion span roughly two orders of magnitude in the resulting bound).
Gravitational-wave merger-rate constraint: LIGO/Virgo/KAGRA O3 compact-binary-coalescence catalog (GWTC-3, Abbott et al. 2023, Phys. Rev. X 13, 011048, arXiv:2111.03606) merger-rate density for \(10\text{--}100\,M_\odot\) black holes, compared against PBH-binary merger-rate predictions (e.g. Sasaki, Suyama, Tanaka & Yokoyama 2016, Phys. Rev. Lett. 117, 061101, arXiv:1603.08338), constrains \(f_{\rm PBH}\lesssim10^{-3}\) for \(\sim30\,M_\odot\) PBHs if PBH binaries formed in the early universe are the dominant channel for the observed merger rate — current LIGO/Virgo/KAGRA data are consistent with the observed compact-object mergers being ordinary stellar-remnant black holes, so no PBH detection is claimed.
Induced gravitational waves from large curvature perturbations: if the curvature power spectrum is boosted enough at small scales to produce an observable PBH abundance, second-order scalar-induced gravitational waves are generically produced at the time of horizon re-entry (Saito & Yokoyama 2009, Phys. Rev. Lett. 102, 161101); current pulsar-timing-array data (NANOGrav 15-yr, Agazie et al. 2023, ApJ Lett. 951, L8, arXiv:2306.16213) show a stochastic common-spectrum signal consistent with (among other explanations) a supermassive-black-hole-binary background, and set complementary constraints on any PBH-forming curvature spectrum peaked near nanohertz-corresponding scales — this is cited as context only; no claim is made that NANOGrav requires or confirms PBHs.

4. Calculation Summary

Window definition (per README Minimum Calculation 1): \(W = \{\)any epoch from the end of inflation through the present in which a region of comoving scale \(k^{-1}\) re-enters the horizon while carrying a curvature perturbation \(\delta\) of order unity; formation is essentially instantaneous (within a Hubble time) at horizon re-entry\(\}\).

PBH formation mass (per test-suite Step 2): a PBH forms with a mass of order the horizon mass at the time of formation,

\[ M_{\rm PBH} \;\sim\; \gamma\, M_H(t_f) \;=\; \gamma\,\frac{4\pi}{3}\rho(t_f)\,H^{-3}(t_f), \]

with the standard collapse-efficiency factor \(\gamma \approx 0.2\) (Carr 1975). Numerically, for radiation-domination this gives the well-known mass–time relation

\[ M_{\rm PBH} \;\approx\; 10^{15}\ \mathrm{g}\ \left(\frac{t_f}{10^{-23}\ \mathrm{s}}\right), \]

so a PBH forming at \(t_f\sim10^{-23}\,\mathrm{s}\) (temperature \(T\sim10^{13}\,\mathrm{GeV}\)) would have a mass around the evaporation-lifetime threshold (\(\tau_{\rm evap}\sim\) age of universe for \(M\sim5\times10^{14}\,\mathrm{g}\)); a PBH forming at the QCD confinement epoch (\(t_f\sim10^{-5}\,\mathrm{s}\), \(T\sim150\,\mathrm{MeV}\), see Test 19) would have \(M_{\rm PBH}\sim M_\odot\), in the LIGO-relevant window — this is why the QCD-epoch equation-of-state softening is a commonly cited (standard, non-distinctive) mechanism for boosting solar-mass PBH formation probability (Byrnes, Franciolini, Hindmarsh, Jedamzik, Pieroni, Sakellariadou et al. 2018, JCAP 08, 041, arXiv:1801.06138).

Abundance threshold (per test-suite Step 3 / README Minimum Calculation 3): PBH formation requires the curvature/density perturbation at horizon crossing to exceed a threshold \(\delta_c\approx0.4\text{--}0.5\) (radiation domination; Musco 2019, Phys. Rev. D 100, 123524, arXiv:1809.02127), which in turn requires the primordial curvature power spectrum at the relevant scale to be boosted to \(\mathcal{P}_\zeta(k)\sim10^{-2}\) — roughly seven orders of magnitude above the CMB-scale-measured amplitude \(A_s = 2.1\times10^{-9}\) (Planck 2018, \(n_s=0.9649\pm0.0042\) at \(k_*=0.05\,\mathrm{Mpc}^{-1}\), Planck Collaboration X, A&A 641, A10, 2020, arXiv:1807.06211). No mechanism in this framework computes or forces such a boost at any particular small scale — per the GAP08 constraint, this framework's inflaton-slope candidate (\(\lambda^2=1/6\)) is not geometrically forced and the derivation-honest first-principles radion slope is \(8/3\); neither number, nor any feature producing a scale-dependent spectral boost, is derived here. This step is therefore reported as a generic requirement from the PBH-formation literature, not as a framework computation.

Constraint comparison (per test-suite Step 3): across the full mass range \(10^{-18}\text{--}10^{3}\,M_\odot\) (\(\sim10^{15}\text{--}10^{35}\,\mathrm{g}\)), the Carr, Kohri, Sendouda & Yokoyama (2021) compilation shows no mass window in which \(f_{\rm PBH}=1\) (PBHs as all of dark matter) is currently allowed without additional model-dependent assumptions (extended mass functions, non-Gaussianity, or clustering); the closed asteroid-mass window \(\sim10^{17}\text{--}10^{21}\,\mathrm{g}\) remains the least-constrained region where \(f_{\rm PBH}\) up to order unity is not yet excluded by current data.

Cross-epoch consistency (per README Minimum Calculation 5): because this framework asserts no specific curvature-spectrum boost, it introduces no PBH abundance that could conflict with BBN (Test 23), \(N_{\rm eff}\) (Test 22), or CMB acoustic structure (Test 31) — there is nothing to check for tension because nothing is being asserted beyond the generic, standard-cosmology possibility that PBHs might exist at some undetermined abundance below current bounds.

5. Granularity Interpretation

On the framework's reading, a successful PBH-formation event would be the most extreme case of a "high-density record-collapse": a region's local density perturbation grows so large that it separates from the surrounding smooth expansion and becomes a permanently bound, causally-disconnected object — an irreversible, maximally stable distinction. This is a coherent extension of the "increasing recordable distinction" narrative in the abstract. But the framework supplies no mechanism that forces, predicts, or even motivates a small-scale curvature-spectrum boost at any specific scale — the interpretive label describes what a PBH would mean if it existed, without supplying any calculation of whether, when, or how many would form. Treating this interpretive compatibility as evidence for the framework would be exactly the "qualitative story substituted for a numerical constraint" failure mode the guardrails warn against; it is not treated that way here.

6. Gate Routing

This test informs the high-density record-collapse gate, which asks what a runaway local overdensity means on this framework's reading of the early universe. It is logically downstream of Test 07 (inflation scalar spectrum), Test 09 (inflation non-Gaussianity), and Test 19/20 (QCD confinement / equation-of-state), since any credible PBH-formation mechanism in the literature draws on one of those ingredients (a spectral feature or bump, non-Gaussian tails, or an equation-of-state softening). The inflaton-slope calculation that would let this gate make a forced, scale-dependent prediction is not yet in hand, so for now this test simply flags the PBH constraint set as a standing check any future small-scale spectrum computation from this program would have to pass.

7. Failure Mode

This test did not fail — no contradiction between this framework and PBH observations exists, because the framework makes no forced small-scale-spectrum claim to contradict them. The honest gap: no calculation in this framework currently produces a curvature-spectrum amplitude, shape, or feature at any PBH-relevant scale, so there is no distinctive abundance \(f_{\rm PBH}(M)\) to check against the bounds compiled above. Per the guardrails: a measured/derived observational constraint set (the \(f_{\rm PBH}(M)\) exclusion curves) is being correctly treated as an external, measured-input input; no relic is asserted without an abundance calculation (none is asserted at all); and no qualitative "increasing distinction" narrative is substituted for the numerical threshold-and-bounds comparison above.

8. Next Action

Derivation, if ever pursued — not yet performed: (a) if this program's technical inflation or reheating sector is ever extended to compute a scale-dependent curvature power spectrum (rather than only the single forced-or-not tilt/tensor-ratio numbers addressed in Tests 07–09), re-run this exact threshold-and-bounds comparison against the then-current \(f_{\rm PBH}(M)\) exclusion curves before claiming any PBH-related result; (b) track future tightening of the asteroid-mass window (\(10^{17}\text{--}10^{21}\,\mathrm{g}\)) via proposed femtolensing, gamma-ray-burst, and Galactic-center microlensing surveys; (c) track LIGO/Virgo/KAGRA O4/O5 and pulsar-timing-array (NANOGrav, EPTA, PPTA) results for any statistically robust PBH-binary or induced-gravitational-wave signature; (d) no dead-end routing is warranted at this time — this is an ordinary, well-posed open question in standard cosmology that this framework has not yet supplied any mechanism to engage with quantitatively.

Bottom line

This is a case of honest agreement: the PBH formation mechanism, mass-function estimate, and the full stack of abundance/microlensing/CMB/gravitational-wave constraints belong to standard inflationary cosmology and astrophysics, and this framework imports them wholesale rather than deriving its own number. Because it forces no small-scale curvature-spectrum boost, it predicts no PBH population — matching the plain observational fact that none has ever been seen. Nothing here is contradicted, and nothing here is claimed as a novel derivation. The constraint set stands as an external empirical boundary any future computation from this program would have to respect.

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