---
title: "Primordial Black Holes and Dark Matter Limits Interactive Lab | Sandbox Physics"
description: "If black holes formed before stars, how would we find them today? Look for what their mass and radiation would do today."
canonical_url: "https://sandboxphysics.com/labs/primordial-black-hole-mass-function/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/primordial-black-hole-mass-function/"
last_modified: "2026-11-14"
---

# 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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** If 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 matters** Only 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 misconception** If the mass function stays below every limit curve, the population is allowed.

**Better mental model** The 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 misconception** A mass window with no limit means primordial black holes are there.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

Primordial black holes trace a rare Gaussian tail; limits test their whole mass function one observation at a time, and so far allow them only in some mass ranges.

## Sources and scope

[Read the course: from early collapse to mass-function limits](https://sandboxphysics.com/stories/primordial-black-holes/)

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.

- [Young, Byrnes & Sasaki (2014), arXiv:1405.7023v4](https://arxiv.org/abs/1405.7023v4)
- [Sasaki, Suyama, Tanaka & Yokoyama (2018), arXiv:1801.05235v1](https://arxiv.org/abs/1801.05235v1)
- [Carr, Raidal, Tenkanen, Vaskonen & Veermäe (2017), arXiv:1705.05567v3](https://arxiv.org/abs/1705.05567v3)
- [Carr, Iovino, Perna, Vaskonen & Veermäe (2026), arXiv:2601.06024v2](https://arxiv.org/abs/2601.06024v2)
- [Particle Data Group (2025), Dark Matter review](https://pdg.lbl.gov/2025/reviews/rpp2025-rev-dark-matter.pdf)
- [PBHconstraints digitized limits (MIT), commit ef2fb03](https://github.com/vianvask/PBHconstraints/tree/ef2fb03aa0f6c3fab58e9b23ee64331e88c75a86)

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.

## Related experiments and articles

Black holes from the first second · Reading path

### Black holes from the first second

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How to look for black holes that formed before the first stars**](https://sandboxphysics.com/stories/primordial-black-holes/how-to-find-black-holes-older-than-stars/)
2. [Article **When a dense patch of the early universe becomes a black hole**](https://sandboxphysics.com/stories/primordial-black-holes/how-an-early-overdensity-becomes-a-black-hole/)
3. [Article **How a rare Gaussian tail sets the number of primordial black holes**](https://sandboxphysics.com/stories/primordial-black-holes/how-a-rare-gaussian-tail-sets-the-abundance/)
4. [Article **Testing a spread of black hole masses against many observations**](https://sandboxphysics.com/stories/primordial-black-holes/test-a-mass-spread-against-many-limits/)
5. [Article **Extend a microlensing limit to a spread of black hole masses**](https://sandboxphysics.com/stories/primordial-black-holes/extend-a-microlensing-limit-to-a-mass-spread/)
6. [Article **How to read the 2026 primordial black hole review**](https://sandboxphysics.com/stories/primordial-black-holes/read-the-2026-primordial-black-hole-review/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)