---
title: "How a rare Gaussian tail sets the number of primordial black holes"
description: "Go from a curvature spectrum to the density variance, the collapsed fraction and today’s mass function, and see why a 15% change matters."
canonical_url: "https://sandboxphysics.com/stories/primordial-black-holes/how-a-rare-gaussian-tail-sets-the-abundance/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/primordial-black-holes/how-a-rare-gaussian-tail-sets-the-abundance/"
last_modified: "2026-11-12"
---

[Primordial black holes from early collapse to today’s limits](https://sandboxphysics.com/stories/primordial-black-holes/) L2

# How a rare Gaussian tail sets the number of primordial black holes

Sandbox Physics · 2026-11-12

The collapsed fraction is the area of a Gaussian tail lying seven or more standard deviations out. Its logarithm scales as the inverse of the variance, so a 15% lower spectrum peak removes about 99% of the black holes. Each collapsed patch then gains weight relative to radiation as the universe expands, which converts a fraction near $10^{-13}$ into all of the dark matter.

## Before you begin

The previous chapter (horizon mass and threshold), the Gaussian distribution, and integrals in logarithmic variables.

## From the curvature spectrum to the density variance

Inflation-type models describe early fluctuations by the curvature power spectrum $\mathcal{P}_\zeta(k)$: the variance of the curvature perturbation per logarithmic interval of wavenumber. The Lab uses a lognormal peak of amplitude $A$ and width $\Delta$ at wavenumber $k_*$.

What decides collapse is the density contrast smoothed over the horizon scale $R$ at re-entry. In the radiation era the density contrast is $(4/9)(kR)^2$ times the curvature perturbation near horizon crossing, so the density power carries a factor $(16/81)(kR)^4$. With a Gaussian window the variance follows the integral below (Young, Byrnes and Sasaki). Modes much longer than $R$ are suppressed by $(kR)^4$; modes much shorter are cut off by the window.

Horizon entry happens when $k=aH$. Because $a\propto t^{1/2}$ and $H=1/(2t)$, $aH\propto t^{-1/2}$, so the horizon mass scales as $M_H\propto t\propto k^{-2}$. Writing $M_*$ for the horizon mass when $k_*$ enters gives $k_*R=(M_H/M_*)^{1/2}$.

$$
\mathcal{P}_\zeta(k)=\frac{A}{\sqrt{2\pi}\,\Delta}\exp\!\left[-\frac{\ln^2(k/k_*)}{2\Delta^2}\right]
$$

Integrating over $\ln k$ gives $A$: the amplitude is the total power in the peak.

$$
\sigma^2(M_H)=\frac{16}{81}\int\mathrm{d}\ln k\,(kR)^4e^{-(kR)^2}\,\mathcal{P}_\zeta(k),\qquad k_*R=\left(\frac{M_H}{M_*}\right)^{1/2}
$$

The factor $16/81$ is $(4/9)^2$. The Gaussian window $e^{-(kR)^2}$ is a convention; other windows change the answer by order-unity factors in the variance.

Reading: Young, Byrnes & Sasaki · Calculating the mass fraction of primordial black holes · arXiv:1405.7023v4, JCAP 1407 (2014) 045 · Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026)

## Two limits you can check by hand

For a very narrow peak, replace the spectrum by $A\,\delta(\ln k-\ln k_*)$. The integral collapses to one point, and with $r=M_H/M_*$ the variance becomes $(16/81)A\,r^2e^{-r}$. Setting the derivative of $r^2e^{-r}$ to zero gives $r=2$: the variance, and hence collapse, peaks at twice the horizon mass of the peak scale. That is why the Lab writes the most common black hole mass as $M_c=2\gamma M_*$.

For a very broad peak, $\mathcal{P}_\zeta$ is nearly constant over the window. The remaining integral is $\int_0^\infty x^3e^{-x^2}\mathrm{d}x=1/2$, so $\sigma^2=8\mathcal{P}_\zeta/81$. The reference code reproduces the flat limit to 6 parts in $10^5$ and the narrow limits to 3 parts in $10^5$.

$$
\sigma^2_{\rm narrow}=\frac{16}{81}A\,r^2e^{-r},\qquad \sigma^2_{\rm flat}=\frac{8}{81}\mathcal{P}_\zeta
$$

Both limits follow from the same integral with no further assumption.

Reading: Young, Byrnes & Sasaki · Calculating the mass fraction of primordial black holes · arXiv:1405.7023v4, JCAP 1407 (2014) 045

## The collapsed fraction is a far tail

If the smoothed density contrast is Gaussian with variance $\sigma^2$, the Press–Schechter estimate of the mass fraction collapsing at horizon mass $M_H$ is twice the probability above threshold. The factor two is a convention from Press–Schechter theory. Young, Byrnes and Sasaki compare it with other choices.

At the Lab’s default peak, $\sigma=0.0616$ and $\delta_c=0.45$, so the threshold is $\nu=\delta_c/\sigma=7.31$ standard deviations out and $\beta=2.74\times10^{-13}$. The asymptotic form on the right is 1.8% high at this $\nu$; the Lab and the reference code both evaluate the full function.

$$
\beta(M_H)=2\int_{\delta_c}^{\infty}\frac{e^{-\delta^2/2\sigma^2}}{\sqrt{2\pi}\,\sigma}\,\mathrm{d}\delta=\operatorname{erfc}\!\left(\frac{\delta_c}{\sqrt2\,\sigma}\right)\approx\sqrt{\frac{2}{\pi}}\,\frac{1}{\nu}\,e^{-\nu^2/2}
$$

An independent check integrates the tail directly at 40 digits for $\nu$ from 1 to 25 and agrees with the closed form to better than $10^{-30}$.

Reading: Young, Byrnes & Sasaki · Calculating the mass fraction of primordial black holes · arXiv:1405.7023v4, JCAP 1407 (2014) 045 · Sasaki, Suyama, Tanaka & Yokoyama · Primordial black holes: perspectives in gravitational wave astronomy · arXiv:1801.05235v1

## Worked example: why 15% removes 99%

Lower the amplitude from $A=0.0608$ to $0.0516$, a factor 0.849. The peak variance falls by the same factor, so $\sigma$ drops from 0.0616 to 0.0568 and $\nu$ rises from 7.31 to 7.93.

The exponent $\nu^2/2$ goes from 26.7 to 31.4. The difference, 4.74, is 2.06 decades; the prefactor $1/\nu$ adds 0.04 decades. The predicted drop is 2.10 decades and the full calculation at the peak gives 2.09. Integrated over all masses, the dark-matter share falls from 0.986 to 0.0074, a factor of 133. A modest change in the early spectrum moves the answer by orders of magnitude, so a claimed abundance is only as good as the threshold, window and statistics behind it.

$$
\frac{\mathrm{d}\ln\beta}{\mathrm{d}\ln\sigma^2}\approx\frac{\nu^2}{2}
$$

At $\nu\approx7$, a 1% change in the variance changes $\beta$ by about 27%.

Reading: Young, Byrnes & Sasaki · Calculating the mass fraction of primordial black holes · arXiv:1405.7023v4, JCAP 1407 (2014) 045 · Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026)

## From the collapsed fraction to today’s dark-matter share

At formation the black holes carry a fraction $\gamma\beta$ of the energy density. Afterwards their density falls as $a^{-3}$ while radiation falls as $a^{-4}$, so their share grows in proportion to $a$ until matter domination. A patch formed earlier has more growing to do. Since $a\propto t^{1/2}\propto M_H^{1/2}$ at formation, the share today scales as $\gamma\beta M_H^{-1/2}$.

Collecting the constants gives the mass function $\psi$, the dark-matter fraction per logarithmic mass interval. Its integral is the total fraction $f$. With $g_*=106.75$ relativistic degrees of freedom, $\Omega_{\rm DM}=0.264$ and $H_0=67.4\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, the coefficient is $C=3.77\times10^{8}$. Evaluated at their reference value $g_*=10.75$ and rewritten in the form of Sasaki et al.’s Eq. 27, it comes out 11% above their rounded $2.7\times10^{8}$. The difference comes from the cosmological parameters; it is reported, not tuned away.

At the default peak, $M_H=1.58\times10^{-10}\,\mathrm{M}_\odot$, so $\gamma C M_H^{-1/2}=6.0\times10^{12}$. Multiplying by $\beta=2.74\times10^{-13}$ gives $\psi\approx1.6$ at the peak. The spread of masses has an effective width of about 0.6 in $\ln M$, so $f\approx1$.

$$
\psi(M)=\frac{\mathrm{d}f}{\mathrm{d}\ln M}=\gamma\,\beta(M_H)\,C\left(\frac{M_H}{\mathrm{M}_\odot}\right)^{-1/2},\qquad M=\gamma M_H,\qquad f=\int\psi\,\mathrm{d}\ln M
$$

The rare tail $\beta$ and the growth factor $M_H^{-1/2}$ together set how much dark matter one decade of mass holds.

Reading: Sasaki, Suyama, Tanaka & Yokoyama · Primordial black holes: perspectives in gravitational wave astronomy · arXiv:1801.05235v1 · Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026)

## What this model leaves out

Every step above assumes Gaussian statistics and a linear map from curvature to density. The 2026 review stresses that both fail in realistic cases: the nonlinear relation between curvature and density, and curvature that is non-Gaussian from the start, both change the tail exponentially. The threshold also depends on the profile, and critical collapse spreads the masses formed at one time. Use this chapter’s formulas to see how the pieces fit, then read the review before trusting a number.

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

## [Feel how sharp the tail is](https://sandboxphysics.com/labs/primordial-black-hole-mass-function/)

1. Predict how much the dark-matter share will fall if the peak amplitude drops by 15%.
2. Choose “Lower the peak by 15%” and compare the collapsed fraction and the share of dark matter with your prediction.
3. Return to the default, then raise the threshold from 0.45 to 0.47 and explain why the change is about as large as an 8% change in amplitude.

## Exercises with explanations

1. For a flat spectrum with $\delta_c=0.5$, what amplitude gives $\beta=10^{-20}$? Compare with the bound printed by Young, Byrnes and Sasaki.

   Read the explanation

   Solve $\operatorname{erfc}(0.5/\sqrt{2}\sigma)=10^{-20}$ with $\sigma^2=8\mathcal{P}_\zeta/81$. This needs $\nu=9.34$, so $\sigma=0.0535$ and $\mathcal{P}_\zeta=0.0290$, matching their printed 0.029.

2. At $\sigma=0.0616$, raise the threshold from 0.45 to 0.47. By how much does $\beta$ change?

   Read the explanation

   $\nu$ goes from 7.31 to 7.63 and $\log_{10}\beta$ from $-12.56$ to $-13.63$: a factor of 12. A 4% change in the threshold acts like an 8% change in the variance, because $\nu^2$ depends on $\delta_c^2/\sigma^2$.

3. Why is the peak of $\psi(M)$ slightly lighter than the black hole mass $M_c=2\gamma M_*$ set by the narrow-peak limit?

   Read the explanation

   $\psi$ multiplies $\beta$ by $M_H^{-1/2}$, which favours lighter masses, and a finite width $\Delta$ changes where the variance peaks. At the default the slider reads $3.98\times10^{-11}\,\mathrm{M}_\odot$, while on the Lab’s 0.05-decade grid the mass function peaks at $3.16\times10^{-11}\,\mathrm{M}_\odot$, about 0.1 decade lighter.

[Previous: 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/) [Next: Testing a spread of black hole masses against many observations](https://sandboxphysics.com/stories/primordial-black-holes/test-a-mass-spread-against-many-limits/)

## 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. [Young, Byrnes & Sasaki · Calculating the mass fraction of primordial black holes · arXiv:1405.7023v4, JCAP 1407 (2014) 045](https://arxiv.org/abs/1405.7023v4)

   Sections 1–4.2: the smoothed density variance with a Gaussian window, the Press–Schechter tail and the printed bound of Eq. 28 used as an independent check.

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.

## 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. [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. [You are here **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/)