---
title: "When a dense patch of the early universe becomes a black hole"
description: "Derive the horizon mass in the radiation era, the collapse threshold and the black hole mass, then follow two worked examples from asteroid to stellar mass."
canonical_url: "https://sandboxphysics.com/stories/primordial-black-holes/how-an-early-overdensity-becomes-a-black-hole/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/primordial-black-holes/how-an-early-overdensity-becomes-a-black-hole/"
last_modified: "2026-11-12"
---

[Primordial black holes from early collapse to today’s limits](https://sandboxphysics.com/stories/primordial-black-holes/) L1

# When a dense patch of the early universe becomes a black hole

Sandbox Physics · 2026-11-12

A patch collapses when it re-enters the horizon with a density contrast above a threshold of about 0.4 to 0.6, and the black hole takes a fixed fraction of the horizon mass at that moment. Because the horizon mass grows in proportion to time, the formation time alone fixes the mass.

## Before you begin

Newtonian gravity, the Schwarzschild radius, logarithms and powers of ten. The Friedmann equation is quoted, not derived.

## The horizon mass grows in proportion to time

Work in the radiation era, when the expansion rate is $H=1/(2t)$. The distance over which regions can influence one another is the Hubble radius $c/H$. The Friedmann equation gives the mean density $\rho=3H^2/(8\pi G)$, where $\rho$ is energy density divided by $c^2$. The mass inside a sphere of radius $c/H$ is the horizon mass.

The result grows linearly with time. Its coefficient $c^3/G$ is about $2.03\times10^{5}$ solar masses per second, which the Particle Data Group rounds to $2\times10^{5}$. A second line of the same calculation shows that the Schwarzschild radius of the horizon mass equals the Hubble radius. A patch that collapses as soon as it enters the horizon therefore makes a black hole comparable in size to the horizon itself.

$$
M_H=\frac{4\pi}{3}\rho\left(\frac{c}{H}\right)^3=\frac{c^3}{2GH}=\frac{c^3t}{G}
$$

Substitute $\rho=3H^2/(8\pi G)$ and then $H=1/(2t)$.

$$
r_s(M_H)=\frac{2GM_H}{c^2}=\frac{c}{H}
$$

The horizon mass sits exactly at its own Schwarzschild radius.

Reading: Particle Data Group · Review 27, Dark Matter (2025 edition, 1 December 2025) · Sasaki, Suyama, Tanaka & Yokoyama · Primordial black holes: perspectives in gravitational wave astronomy · arXiv:1801.05235v1

## Which patches collapse

Describe a patch by its density contrast $\delta=(\rho-\bar\rho)/\bar\rho$, smoothed over the horizon scale at the moment it re-enters. While the patch is larger than the horizon it simply expands with everything else. After re-entry, pressure resists collapse. Only patches with $\delta$ above a threshold $\delta_c$ collapse before pressure wins.

The value of $\delta_c$ depends on the shape of the density profile and on exactly how $\delta$ is defined. Harada, Yoo and Kohri give an analytic estimate of about 0.41 for radiation in the measure used by numerical simulations. The 2026 review uses 0.55 for one of its examples. The Lab lets you move $\delta_c$ between 0.35 and 0.65 and starts at 0.45.

Reading: Harada, Yoo & Kohri · Threshold of primordial black hole formation · arXiv:1309.4201v4, Phys. Rev. D 88, 084051 (2013) · Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026)

## The black hole keeps a fraction of the horizon mass

The black hole mass is written $M=\gamma M_H$. The Particle Data Group quotes $\gamma\simeq(1/\sqrt3)^3\approx0.19$ during radiation domination, and Sasaki and collaborators use $\gamma=0.2$. This series uses 0.2. In detailed collapse simulations the mass also depends on how far above threshold the patch was, which spreads the masses formed at one time. The Lab keeps one mass per formation time, so its spread of masses comes only from the spread of scales in the early density peak.

$$
M=\gamma M_H\approx\gamma\,\frac{c^3t}{G}\approx4\times10^{4}\,\mathrm{M}_\odot\left(\frac{t}{1\,\mathrm{s}}\right)\quad(\gamma=0.2)
$$

The formation time fixes the mass. Heavier black holes formed later.

Reading: Particle Data Group · Review 27, Dark Matter (2025 edition, 1 December 2025) · Sasaki, Suyama, Tanaka & Yokoyama · Primordial black holes: perspectives in gravitational wave astronomy · arXiv:1801.05235v1 · Carr, Raidal, Tenkanen, Vaskonen & Veermäe · Primordial black hole constraints for extended mass functions · arXiv:1705.05567v3, Phys. Rev. D 96, 023514 (2017)

## Worked examples: an asteroid-mass and a stellar-mass black hole

Take the Lab’s default peak mass, $M=3.16\times10^{-11}\,\mathrm{M}_\odot$, about $6\times10^{19}$ kilograms. The horizon mass is $M/\gamma=1.58\times10^{-10}\,\mathrm{M}_\odot$. Dividing by $2.03\times10^{5}\,\mathrm{M}_\odot\,\mathrm{s}^{-1}$ gives a formation time of $7.8\times10^{-16}$ seconds. The Schwarzschild radius of the black hole is $2.95\,\mathrm{km}\times3.16\times10^{-11}\approx93$ nanometres, one fifth of the Hubble radius $2ct\approx470$ nanometres at that time.

Now take $M=30\,\mathrm{M}_\odot$, a typical mass in the gravitational-wave catalogues. The horizon mass is $150\,\mathrm{M}_\odot$ and the formation time $7.4\times10^{-4}$ seconds. The two examples differ by about $10^{12}$ in mass and in formation time; a black hole mass function spanning many decades corresponds to a long stretch of the first second.

Reading: Particle Data Group · Review 27, Dark Matter (2025 edition, 1 December 2025) · Sasaki, Suyama, Tanaka & Yokoyama · Primordial black holes: perspectives in gravitational wave astronomy · arXiv:1801.05235v1

## Evaporation sets the lightest survivors

Black holes lose mass by Hawking radiation, faster the lighter they are. According to the 2026 review, a non-spinning black hole formed with about $5\times10^{14}$ grams, or $2.5\times10^{-19}\,\mathrm{M}_\odot$, would be completing its evaporation today. The Lab’s mass ruler starts at $10^{-18}\,\mathrm{M}_\odot$, above this edge, and ignores the small mass loss of survivors.

Reading: Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026) · Particle Data Group · Review 27, Dark Matter (2025 edition, 1 December 2025)

## [Watch one patch enter the horizon and collapse](https://sandboxphysics.com/labs/primordial-black-hole-mass-function/)

1. Predict: if you move the black hole mass from $10^{-11}$ to $10^{-5}$ solar masses, how much later did the patch form?
2. Drag the mass grip along the ruler and read the most common black hole mass. Check your prediction with $t=M/(\gamma\,c^3/G)$.
3. Drag the threshold plane up a little and watch the fraction of horizon patches that collapse fall.

## Exercises with explanations

1. At what time after the beginning did a black hole of one solar mass form, with $\gamma=0.2$?

   Read the explanation

   The horizon mass is $M/\gamma=5\,\mathrm{M}_\odot$. Dividing by $2.03\times10^{5}\,\mathrm{M}_\odot\,\mathrm{s}^{-1}$ gives $t\approx2.5\times10^{-5}$ seconds.

2. Show that $r_s(M_H)=c/H$ using only $M_H=c^3/(2GH)$.

   Read the explanation

   $r_s=2GM_H/c^2=2G\,c^3/(2GH\,c^2)=c/H$. The result does not depend on the radiation era; it holds for any flat Friedmann universe, where $M_H=c^3/(2GH)$ follows from the critical density.

3. If the true efficiency were $\gamma=0.1$, when would a $3.16\times10^{-11}\,\mathrm{M}_\odot$ black hole have formed, and what would change in the Lab’s mapping from spectrum peak to mass?

   Read the explanation

   The horizon mass doubles to $3.16\times10^{-10}\,\mathrm{M}_\odot$, so the formation time doubles to $1.6\times10^{-15}$ seconds. For the same early density peak every black hole would be half as heavy, and the present abundance per collapsed patch would change too, because it scales as $\gamma\,M_H^{-1/2}$; that scaling is derived in the next chapter.

[Previous: 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/) [Next: 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/)

## Sources and how much of each was read

1. [Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026)](https://arxiv.org/abs/2601.06024v2)

   Review; arXiv lists the journal reference Riv. Nuovo Cim. (2026). Read in full: Sec. II.A.2 on non-Gaussianity, Sec. IV.C (gravitational waves), IV.E (accretion), V (asteroid-mass window), VI (Eqs. 25–29) and the captions of Figs. 2–6; parts of Sec. II and III.

2. [Carr, Raidal, Tenkanen, Vaskonen & Veermäe · Primordial black hole constraints for extended mass functions · arXiv:1705.05567v3, Phys. Rev. D 96, 023514 (2017)](https://arxiv.org/abs/1705.05567v3)

   Full text. The linear rule for one observable (Eq. 12) and the quadrature combination of several (Eq. 13), with the assumptions under which they hold.

3. [Sasaki, Suyama, Tanaka & Yokoyama · Primordial black holes: perspectives in gravitational wave astronomy · arXiv:1801.05235v1](https://arxiv.org/abs/1801.05235v1)

   Section 2.3, Eqs. 21–31: horizon mass, collapse efficiency and the present abundance per formation fraction.

4. [Harada, Yoo & Kohri · Threshold of primordial black hole formation · arXiv:1309.4201v4, Phys. Rev. D 88, 084051 (2013)](https://arxiv.org/abs/1309.4201v4)

   Abstract only. For radiation it gives a threshold amplitude of about 0.41 in the measure used by numerical simulations.

5. [Particle Data Group · Review 27, Dark Matter (2025 edition, 1 December 2025)](https://pdg.lbl.gov/2025/reviews/rpp2025-rev-dark-matter.pdf)

   Sections 27.1, 27.3 and 27.7: formation time and mass, the main classes of limits and the window of roughly $10^{17}$ to $10^{21}$ grams.

## Keep reading

Black holes from the first second · Reading path

### Black holes from the first second

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. [You are here **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/)