Primordial Black Holes and Dark Matter Limits

If black holes formed before stars, how would we find them today? Look for what their mass and radiation would do today. Black holes from rare dense patches in the early universe would act as dark matter. Light ones would still be evaporating, heavier ones would bend starlight or accrete gas, and the heaviest would merge and emit gravitational waves. Around asteroid masses, current limits still allow them to be all of the dark matter, but none has been detected.

Move a peak in the early density spectrum and the collapse threshold. See how many black holes form, at which masses, and which observations already rule them out.

Share of dark matter in these black holes $0.986$

Most common black hole mass $3.16\times10^{-11}\,\mathrm{M}_\odot$

Fraction of horizon patches that collapse $2.74\times10^{-13}$

Tightest single limit (above 1 excludes) $R_{\mathrm{SUB}}=0.08$

Shape the early peak

M_c

Black hole mass at the spectrum peak

3.98\times10^{-11}\,\mathrm{M}_\odot

A

Peak amplitude

0.0608

\Delta

Peak width in wavenumber

0.5

\delta_c

Collapse threshold

0.45

Drag the gold grips in the scene to move the mass or the threshold.

Conventions and the combined ratio

$M=\gamma M_H,\quad \gamma=0.2,\quad M_c=2\gamma M_*$

The horizon mass is fixed by the wavenumber that re-enters the horizon. A narrow peak gives the largest variance at twice that horizon mass.

$\psi(M)=\gamma\,\beta(M_H)\,\frac{(\Omega_r F^2)^{3/4}}{\Omega_{\rm DM}}\left(\frac{M_{H,0}}{M_H}\right)^{1/2},\qquad f_{\rm PBH}=\int\psi\,\mathrm{d}\ln M$

Radiation density, the present Hubble mass and the dark matter density are fixed teaching values; the relativistic species count is held at 106.75.

$\Big(\sum_{j\in\mathrm{lin}} R_j^2\Big)^{1/2}=0.08$

This quadrature sum assumes linear, independent, Gaussian observables. It is not applied to the gravitational-wave table.

Model assumptions

This teaching model starts from a lognormal peak in the primordial curvature spectrum. It smooths the density contrast at horizon re-entry with a Gaussian window in a radiation-dominated universe, with no transfer function, and uses the Press–Schechter Gaussian tail with a fixed threshold. Every black hole receives one fifth of the horizon mass. Critical collapse, non-Gaussian tails, clustering, accretion, evaporation and mergers after formation are omitted, and the number of relativistic species is fixed at 106.75. The observational curves are published monochromatic limits digitized in the MIT-licensed PBHconstraints repository at a fixed commit. Each linear limit is applied separately through its weighted ratio; the curves are not combined into an envelope. The gravitational-wave curve is shown only as a monochromatic reference, because a merger rate depends nonlinearly on the mass function. The density slice in the scene is schematic: a constrained Gaussian field drawn from the same correlation function, with one rare patch placed at the centre. Primordial black holes remain candidates, not a detected population.

Allowed by each linear table taken separately.

Model: the collapsing tail at the peak scale

Model: mass function today $\psi(M)$

Observations: published limits for a single mass (digitized)

EGR: Extragalactic photon background from evaporation
SUB: Subaru HSC microlensing of M31
OGLE: OGLE long-timescale microlensing
acc: CMB limits on accretion
LIGO: LIGO–Virgo–KAGRA O3 merger rate

Model weighed by each limit $R_j=\int\mathrm{d}\ln M\,\psi/f_{\max,j}$

Values and why the curves are not combined

$R_{\mathrm{EGR}}<10^{-4}$
$R_{\mathrm{SUB}}=0.08$
$R_{\mathrm{OGLE}}<10^{-4}$
$R_{\mathrm{acc}}<10^{-4}$
$R_{\mathrm{LIGO}}<10^{-4}$

Each limit carries its own assumptions, so the curves are not merged into one envelope. The gravitational-wave table is a monochromatic reference only: a merger rate depends on pairs of black holes, not linearly on their number.

Data: PBHconstraints repository (MIT), commit ef2fb03. The scene is a schematic field slice, not data.

A rare tail decides how many black holes form

BackgroundIf the early universe had a region much denser than its surroundings, that region could collapse into a black hole as soon as it came inside the horizon. Such primordial black holes would form long before any star.

Why it mattersOnly an extremely rare tail of the density distribution collapses, so a small change in the early spectrum changes the number of black holes enormously. Observations today then test that number one mass range at a time.

Focus question: Why does lowering the peak by a few percent remove almost all the black holes?
One-sentence intuition: The collapsed fraction sits far out in a Gaussian tail, where its logarithm scales as the inverse variance; each observation must then be weighed against the whole mass function.

Core mathematical model

Variance at horizon re-entry

\sigma^2(M_H)=\frac{16}{81}\int\mathrm{d}\ln k\,(kR)^4e^{-(kR)^2}\mathcal{P}_\zeta(k)

The curvature spectrum sets the density variance smoothed on the horizon scale. For a narrow peak the variance is largest at twice the horizon mass of the peak wavenumber.

The collapsed fraction

\beta=\operatorname{erfc}\!\left(\frac{\delta_c}{\sqrt2\,\sigma}\right)\approx\sqrt{\frac{2}{\pi}}\frac{\sigma}{\delta_c}\,e^{-\delta_c^2/2\sigma^2}

In the Press–Schechter estimate, every patch whose smoothed density contrast exceeds the threshold collapses. The approximation holds when the threshold lies many standard deviations out.

From a mass function to one limit

R_j=\int\mathrm{d}\ln M\,\frac{\psi(M)}{f_{\max,j}(M)}\le1

A limit derived for black holes of one mass constrains a spread of masses through this weighted integral, if the observable is linear in the black hole number. Each observation gets its own ratio.

Common difficulties

Reading a limit off the plot

Typical misconceptionIf the mass function stays below every limit curve, the population is allowed.

Better mental modelThe mass function is a density per logarithmic mass; a monochromatic limit is a total. Compare them only through the weighted ratio for each observation.

Allowed is not detected

Typical misconceptionA mass window with no limit means primordial black holes are there.

Better mental modelAn open window only means these observations cannot exclude them. No primordial black hole has been identified.

Run the experiment

01
Lower the peak
Choose All dark matter, asteroid mass, then Lower the peak by 15%.
What to observe: The share of dark matter falls by more than a factor of a hundred, while the mass where black holes form barely moves.
02
Slide into a limit
Drag the mass grip on the ruler toward heavier masses.
What to observe: The Subaru microlensing ratio passes 1 once the mass function overlaps its curve, and the verdict changes to excluded.
03
Raise the threshold
Drag the gold threshold plane upward a little.
What to observe: The patch must reach a higher contrast, so the collapsing fraction drops steeply and the tail in the first chart shrinks.

Sources and scope

Read the course: from early collapse to mass-function limits

The variance and tail follow Young, Byrnes and Sasaki; the abundance follows Sasaki et al.; the weighted ratio and its quadrature combination follow Carr et al. (2017). The 2026 review by Carr, Iovino, Perna, Vaskonen and Veermäe frames the current limits and the warning against reading a mass function off monochromatic curves.

Independent checks cover a direct tail integral, the flat and narrow spectrum limits, a printed reference amplitude and the narrow-distribution limit of the weighted ratio. They test the numerics, not the omitted physics.

Primordial Black Holes and Dark Matter Limits

Model: the collapsing tail at the peak scale

Model: mass function today $\psi(M)$

Observations: published limits for a single mass (digitized)

Model weighed by each limit $R_j=\int\mathrm{d}\ln M\,\psi/f_{\max,j}$

A rare tail decides how many black holes form

Core mathematical model

Variance at horizon re-entry

The collapsed fraction

From a mass function to one limit

Common difficulties

Reading a limit off the plot

Allowed is not detected

Run the experiment

Lower the peak

Slide into a limit

Raise the threshold

Sources and scope

Related experiments and articles

Articles

Black holes from the first second

Model: the collapsing tail at the peak scale

Model: mass function today ψ(M)\psi(M)ψ(M)

Observations: published limits for a single mass (digitized)

Model weighed by each limit Rj=∫dln⁡M ψ/fmax⁡,jR_j=\int\mathrm{d}\ln M\,\psi/f_{\max,j}Rj​=∫dlnMψ/fmax,j​

Core mathematical model

Variance at horizon re-entry

The collapsed fraction

From a mass function to one limit

Common difficulties

Reading a limit off the plot

Allowed is not detected

Run the experiment

Lower the peak

Slide into a limit

Raise the threshold

Sources and scope

Related experiments and articles

Articles

Black holes from the first second

Model: mass function today $\psi(M)$

Model weighed by each limit $R_j=\int\mathrm{d}\ln M\,\psi/f_{\max,j}$