---
title: "Testing a spread of black hole masses against many observations"
description: "Derive the weighted ratio for one linear observation, combine several only under stated assumptions and see why overlaying limit curves fails."
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  zh-CN: "https://sandboxphysics.com/zh/stories/primordial-black-holes/test-a-mass-spread-against-many-limits/"
last_modified: "2026-11-12"
---

[Primordial black holes from early collapse to today’s limits](https://sandboxphysics.com/stories/primordial-black-holes/) L3

# Testing a spread of black hole masses against many observations

Sandbox Physics · 2026-11-12

For an observation whose signal is a sum over individual black holes, a mass function is allowed when the integral of $\psi/f_{\max}$ over $\ln M$ is at most one. Each observation gets its own ratio. Combining them, or drawing one envelope through all the curves, needs assumptions that must be stated; merger rates do not satisfy the linear rule at all.

## Before you begin

The previous two chapters, Taylor expansion and the idea of a chi-squared statistic.

## A limit curve caps a total, a mass function is a density

A published curve $f_{\max}(M)$ answers one question: if every black hole had mass $M$, how large could their total dark-matter fraction be? The mass function $\psi(M)=\mathrm{d}f/\mathrm{d}\ln M$ is a different quantity, the fraction per logarithmic mass interval. Drawing $\psi$ on the same axes and checking that it stays below every curve compares a density with a total. The 2026 review states this directly in its Sec. VI.

An example shows the size of the error. A lognormal with width $\sigma=2$ in $\ln M$ and $f=1$ has a peak height $1/(\sqrt{2\pi}\,2)=0.20$. It could sit below a curve at 0.3 everywhere and still hold more than three times the dark matter that curve allows at any single mass.

Reading: Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026) · Carr, Raidal, Tenkanen, Vaskonen & Veermäe · Primordial black hole constraints for extended mass functions · arXiv:1705.05567v3, Phys. Rev. D 96, 023514 (2017)

## Deriving the weighted ratio for one linear observation

Let an observable $\mathcal{A}$ be bounded by a measurement, $\mathcal{A}\le\mathcal{A}_{\rm exp}$. If each black hole adds to it independently of the others, it is linear in $\psi$: a background $\mathcal{A}_0$ plus an integral with a kernel $\mathcal{A}_1(M)$ that encodes the physics and the instrument.

For a single mass, $\psi=f\,\delta(\ln M-\ln M_c)$, the bound becomes $f\le(\mathcal{A}_{\rm exp}-\mathcal{A}_0)/\mathcal{A}_1(M_c)\equiv f_{\max}(M_c)$. The published curve therefore tells us the kernel, up to a constant. Substituting $\mathcal{A}_1=(\mathcal{A}_{\rm exp}-\mathcal{A}_0)/f_{\max}$ back into the general bound gives the weighted ratio. This is Eq. 12 of Carr et al. 2017 and Eqs. 25–28 of the 2026 review.

$$
\mathcal{A}[\psi]=\mathcal{A}_0+\int\mathrm{d}\ln M\,\mathcal{A}_1(M)\,\psi(M)\le\mathcal{A}_{\rm exp}
$$

Linearity means no term involves two black holes at once.

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

Apply it to each observation $j$ separately. Outside the mass range where a table has data, that observation contributes nothing, and the mass there stays untested by it.

Reading: Carr, Raidal, Tenkanen, Vaskonen & Veermäe · Primordial black hole constraints for extended mass functions · arXiv:1705.05567v3, Phys. Rev. D 96, 023514 (2017) · Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026)

## Does spreading the masses help or hurt?

Write $u=\ln(M/M_c)$ and $g(u)=1/f_{\max}$. For a narrow symmetric spread of width $\sigma$, expand $g$ to second order. The linear term averages to zero, leaving the correction below. Where $1/f_{\max}$ curves upward in $\ln M$, a spread makes the limit stronger; where it curves downward, weaker.

The method tutorial checks this with the Subaru microlensing table. Centred at $10^{-8}\,\mathrm{M}_\odot$, where $1/f_{\max}$ is falling away from its peak near $10^{-9}\,\mathrm{M}_\odot$ and bends downward, widening to $\sigma=0.5$ raises the allowed fraction from 0.00949 to 0.00964. Centred at $10^{-6}\,\mathrm{M}_\odot$, where $1/f_{\max}$ bends upward, the same width lowers it from 0.169 to 0.160. Neither direction can be assumed.

$$
R\approx f\left[g(0)+\frac{\sigma^2}{2}\,g''(0)\right],\qquad g(u)=\frac{1}{f_{\max}(M_ce^{u})}
$$

Valid while $\sigma$ is small compared with the scale on which $g$ changes curvature.

Reading: Carr, Raidal, Tenkanen, Vaskonen & Veermäe · Primordial black hole constraints for extended mass functions · arXiv:1705.05567v3, Phys. Rev. D 96, 023514 (2017)

## Several observations: separate ratios first

The safe statement is that every ratio $R_j$ must be at most one. Carr et al. 2017 also derive a combined test in their appendix. It assumes independent observables with Gaussian errors, each linear in $\psi$, each with its expected value at zero black holes, and every bound quoted at the same confidence level. Under those assumptions the chi-squared statistic adds the ratios in quadrature.

The same appendix shows why a single envelope curve is not a shortcut. Folding all observations into one curve $f_{\max}=(\sum_j f_{\max,j}^{-2})^{-1/2}$ and then applying the one-observation rule overestimates the constraint, by the triangle inequality. Taking the lowest curve at each mass has the same flaw. The Lab therefore shows each ratio on its own and puts the quadrature combination in a collapsed panel, for the linear tables only.

In the Lab’s stellar-mass preset, the OGLE ratio is 3.06 and the CMB accretion ratio is 3.58. Each alone excludes the spread. Their quadrature combination is 4.71, which holds only under the assumptions above.

$$
\sum_{j}\left(\int\mathrm{d}\ln M\,\frac{\psi}{f_{\max,j}}\right)^2\le1
$$

Carr et al. 2017 Eq. 13 and Appendix; Carr et al. 2026 Eq. 29.

Reading: Carr, Raidal, Tenkanen, Vaskonen & Veermäe · Primordial black hole constraints for extended mass functions · arXiv:1705.05567v3, Phys. Rev. D 96, 023514 (2017) · Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026) · Mróz, Udalski et al. · No massive black holes in the Milky Way halo · arXiv:2403.02386v3, Nature 632, 749 (2024) · Serpico, Poulin, Inman & Kohri · Cosmic microwave background bounds on primordial black holes including dark matter halo accretion · arXiv:2002.10771v2, Phys. Rev. Research 2, 023204 (2020)

## Where the linear rule breaks

Mergers involve two black holes, so the merger rate contains a term with $\psi$ twice and the rule above does not apply. Clustering, mass segregation in galaxies and structure formation can also make an observable depend on more than one black hole at a time. The review names Lyman-alpha, dynamical and, to a lesser degree, lensing and accretion limits as affected.

For that reason the Lab draws the LIGO-Virgo-KAGRA table for reference and greys out its ratio. In the stellar-mass preset that number is 443, but it is not a valid application of the rule. The O3 analysis itself fits several mass functions directly with a hierarchical method and finds at most about $10^{-3}$ of the dark matter in black holes of 1 to 200 solar masses in every scenario it considers.

Masses can also change after formation. Accretion can grow heavy black holes and weaken limits above about ten solar masses, while clustering can strengthen accretion limits. The review suggests reading its mass functions as effective ones that already include such changes, possibly different for each observation. The Lab uses the formation mass function throughout.

Reading: Carr, Iovino, Perna, Vaskonen & Veermäe · Primordial black holes: constraints, potential evidence and prospects · arXiv:2601.06024v2 (1 April 2026) · Andrés-Carcasona, Iovino, Vaskonen & Veermäe · Constraints on primordial black holes from LIGO-Virgo-KAGRA O3 events · arXiv:2405.05732v2, Phys. Rev. D 110, 023040 (2024) · Carr, Raidal, Tenkanen, Vaskonen & Veermäe · Primordial black hole constraints for extended mass functions · arXiv:1705.05567v3, Phys. Rev. D 96, 023514 (2017)

## [Read each limit as its own weighted ratio](https://sandboxphysics.com/labs/primordial-black-hole-mass-function/)

1. Choose “Ruled out by microlensing” and predict which ratio exceeds one before reading the bars.
2. Choose “Stellar-mass black holes”, open the conventions panel, and compare the largest single ratio with the quadrature combination.
3. Drag the mass grip slowly across the Subaru curve and note where the ratio first passes one; compare that mass with where the curve itself looks lowest.

## Exercises with explanations

1. Show that using the lowest curve at each mass, $f_{\rm env}(M)=\min_j f_{\max,j}(M)$, in the one-observation rule gives a ratio at least as large as every individual $R_j$. Why is that not a derived bound?

   Read the explanation

   At each mass $1/f_{\rm env}\ge1/f_{\max,j}$ for every $j$, and $\psi\ge0$, so the integral is at least each $R_j$. The rule was derived for one observable with one kernel. The envelope switches kernels from mass to mass, which corresponds to no single measurement and no stated likelihood, so its value has no probabilistic meaning.

2. The Subaru table gives $1/f_{\max}=120.5,\ 105.4,\ 86.9$ at $\log_{10}(M/\mathrm{M}_\odot)=-8.2,\ -8.0,\ -7.8$, and $8.27,\ 5.92,\ 4.04$ at $-6.2,\ -6.0,\ -5.8$. Use the second-order formula to predict whether widening a spread weakens or strengthens the limit at each centre.

   Read the explanation

   The second differences are $120.5-2(105.4)+86.9=-3.4$ and $8.27-2(5.92)+4.04=+0.47$. So $g''(0)<0$ at $10^{-8}\,\mathrm{M}_\odot$: a spread lowers $R$ per unit $f$ and the limit weakens. At $10^{-6}\,\mathrm{M}_\odot$, $g''(0)>0$ and the limit strengthens. The tutorial’s full integrals agree in both cases.

3. Two linear observations give $R_1=3.06$ and $R_2=3.58$. What is the quadrature combination, and what changes if one of them was quoted at 95% confidence and the other at 99%?

   Read the explanation

   $\sqrt{3.06^2+3.58^2}=4.71$. The derivation needs every bound at the same confidence level, because each $f_{\max,j}$ is the kernel scaled by the same number of standard deviations. With mixed levels one must rescale to a common level before combining; otherwise only the separate tests are justified.

[Previous: 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/) [Next: 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/)

## 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. [Mróz, Udalski et al. · No massive black holes in the Milky Way halo · arXiv:2403.02386v3, Nature 632, 749 (2024)](https://arxiv.org/abs/2403.02386v3)

   Abstract only. Twenty years of OGLE data on nearly 80 million LMC stars: no events longer than one year, so compact objects from $1.8\times10^{-4}$ to 6.3 solar masses make up at most 1% of the dark matter.

4. [Serpico, Poulin, Inman & Kohri · Cosmic microwave background bounds on primordial black holes including dark matter halo accretion · arXiv:2002.10771v2, Phys. Rev. Research 2, 023204 (2020)](https://arxiv.org/abs/2002.10771v2)

   Abstract only. Accretion changes the ionization history seen in the CMB; the bound reaches a fraction of $3\times10^{-9}$ near $10^{4}$ solar masses and depends on accretion physics.

5. [Andrés-Carcasona, Iovino, Vaskonen & Veermäe · Constraints on primordial black holes from LIGO-Virgo-KAGRA O3 events · arXiv:2405.05732v2, Phys. Rev. D 110, 023040 (2024)](https://arxiv.org/abs/2405.05732v2)

   Abstract only. A hierarchical analysis of O3 mergers with several mass functions limits PBHs to at most about $10^{-3}$ of the dark matter between 1 and 200 solar masses.

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