Exploring the local black hole mass function below $10^6$ solar masses

Alberto Sesana; Elena Gallo

REVIEW 2 major objections 4 minor 44 references

Exploring the local black hole mass function below $10^6$ solar masses

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that the local number density of black holes between 100,000 and one million solar masses is pinned to within a factor of ten once galaxy counts and X-ray occupation constraints are combined.

desk verdict A transparent, honest first estimate of the low-mass black hole mass function with a real weakness—the occupation fraction is early-type only—but the authors say so plainly and the sub-dex claim should be read as conditional on that morphology assumption. read the letter →

arxiv 1909.02585 v1 pith:D6QMUALR submitted 2019-09-05 astro-ph.HE astro-ph.GA

Elena Gallo , Alberto Sesana This is my paper · ORCID

classification astro-ph.HEastro-ph.GA

keywords blackholemassfunctionoccupationfractiondwarfgalaxiesintermediate-massholesgalaxystellarX-raysurveysLISAgravitationalwavedetector

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the local number density of black holes with masses between $10^5$ and $10^6$ solar masses can be pinned down to within about a factor of ten, despite the absence of a clean way to weigh such black holes. The method combines the GAMA galaxy stellar mass function with an X-ray-based estimate of the fraction of galaxies that actually host a nuclear black hole, then convolves the two through a broad $M_{\rm BH}$–$M_\star$ scaling relation. If the estimate is right, LISA's most sensitive mass band has a known target density, which sharpens predictions for black hole merger rates and gives a local anchor for models of how the first supermassive black holes were seeded.

What carries the argument

The load-bearing element is the convolution in Equation 3, which builds the BHMF as $\Phi(M_{\rm BH}) = \int \Phi(M_\star)\,\lambda_{\rm occ}(M_\star)\,\frac{1}{\sqrt{2\pi\sigma^2}}\exp[-(M_{\rm BH}-(\alpha+\beta M_\star))^2/(2\sigma^2)]\,dM_\star$. Here $\Phi(M_\star)$ is the GAMA galaxy stellar mass function, $\lambda_{\rm occ}$ is the occupation fraction of Equation 1 — a smooth tanh ramp that is nearly zero below $10^7\,M_\odot$ and nearly one above $10^{10}\,M_\odot$, with a transition mass $M_{\star,0}$ constrained by X-ray observations of 326 nearby early-type galaxies — and the Gaussian carries the scatter in the $M_{\rm BH}$–$M_\star$ relation. This object does the work: it turns a galaxy census into a black hole census while suppressing low-mass galaxies that probably lack black holes, which is why the low-mass end of the BHMF is no longer set by the scaling relation alone.

What would settle it

Measure the nuclear black hole occupation fraction in a large, uniformly observed sample of late-type and dwarf galaxies with $M_\star$ between roughly $10^9$ and $10^{10}\,M_\odot$ (for example, a deep X-ray survey of about 5,000 such galaxies with sub-arcsecond resolution). If the occupation fraction at those masses differs substantially from the early-type-derived tanh ramp, the low-mass normalization of the BHMF shifts by the difference and the claimed sub-dex precision in the $10^5$–$10^6$ $M_\odot$ band would be contradicted.

Watch

Extended reading notes

Core claim

The paper's central claim is that the low-mass end of the local black hole mass function is empirically accessible despite the lack of a reliable black hole mass proxy. By folding an X-ray-derived occupation fraction into the convolution between galaxy stellar mass and black hole mass, the authors obtain a BHMF whose 68% normalization uncertainty is less than one dex across $10^5$–$10^6$ $M_\odot$, the band where LISA is most sensitive. The median best-guess mass function is approximated by $\log\Phi = -2.13 - 0.098\log(M_{\rm BH}/M_\odot) - 0.00011\,(M_{\rm BH}/M_\odot)$. The paper further reports that different choices of the $M_{\rm BH}$–$M_\star$ scaling relation change the inferred BHMF by only a factor of about three below $10^7$ $M_\odot$, so the occupation fraction, rather than the scaling relation, is the main lever on the low-mass normalization.

Load-bearing premise

The calculation assumes that the occupation fraction measured from 326 early-type galaxies applies to all galaxies at the same stellar mass, even though late-type and dwarf galaxies outnumber early types below $M_\star \sim 10^{10}\,M_\odot$; the authors themselves flag this as a serious drawback in Section 4.

Editorial extensions

If this is right

The local volume density of black holes near a few times $10^5\,M_\odot$ is bracketed to within about a factor of ten, so rate predictions for low-mass black hole mergers in LISA no longer float freely over this band.
Within $10^5$–$10^6\,M_\odot$ the 68% normalization uncertainty stays below one dex for each of the scaling relations considered, so the remaining disagreement among black hole mass proxies matters less than feared for the census.
Below $10^7\,M_\odot$ the occupation-corrected BHMFs built from different scaling relations agree to within a factor of about three, making the low-mass end comparatively stable against the proxy choice.
The median BHMF is approximated by $\log \Phi = -2.13 - 0.098\log(M_{\rm BH}/M_\odot) - 0.00011\,(M_{\rm BH}/M_\odot)$, a compact form that population-synthesis calculations can adopt directly.
A future X-ray survey of roughly 5,000 nearby galaxies with uniform, sub-arcsecond coverage could reduce occupation-fraction uncertainties to a few percent and tighten the BHMF further across the LISA band.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If late-type and dwarf galaxies turn out to have a lower nuclear occupation fraction at $M_\star \sim 10^9$–$10^{10}\,M_\odot$ than the early-type-based ramp, the claimed sub-dex normalization in the $10^5$–$10^6\,M_\odot$ band is likely optimistic, since a factor-of-two change in $\lambda_{\rm occ}$ shifts the BHMF normalization by roughly the same factor.
The same convolution could be run with the redshift-evolving galaxy stellar mass function to produce a $z>0$ BHMF, giving LISA a direct prediction for the redshift distribution of massive black hole mergers; the paper itself only builds the local function.
The paper's median BHMF, inserted into binary population models, yields concrete predictions for local merger and extreme-mass-ratio inspiral rates that LISA can test once it flies.
A cleaner test of the low-mass census would come from dynamical black hole mass measurements in dwarf galaxies, which would replace the broad $M_{\rm BH}$–$M_\star$ scatter with actual masses and directly validate the Gaussian convolution.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one new thing here is that they actually fold an observationally measured occupation fraction into a local BHMF calculation and push it down to 1e5–1e6 Msun. That is a first, and it matters for LISA event-rate forecasts and seeding models. The arithmetic is transparent: Eq. 3 is a standard convolution, the uncertainties are propagated from the posterior draws, and the comparison with earlier BHMFs (Shankar, Gair, Barausse) is useful. The paper also gives a simple Schechter-like fit to their median, which is convenient for people who want to plug it into something. Credit where due: the authors do not oversell the method, and they are explicit about the main caveats.

The soft spots are real but not hidden. The occupation fraction comes from 326 early-type galaxies only, while late types dominate the stellar mass range (Mstar ~ 1e9–1e10) that maps to the black hole masses of interest. The stress-test note is correct: a factor-of-3 lower late-type occupation shifts the median by ~0.5 dex, and a factor-of-10 lower occupation moves it outside the quoted 68% band. So the "<1 dex normalization uncertainty" in the abstract is not an uncertainty on the true local BHMF; it is a conditional statement about the fitted parameters given the early-type sample. The paper even calls this "a serious drawback" in Section 4, but does not propagate any morphology-dependent systematic into the yellow band. That is the one place where the claim in the abstract goes a bit further than what the error bars actually represent. The second soft spot is the MBH–Mstar relation: they fit at higher masses and extrapolate down, with scatter 0.61 dex, and they use a combined AGN+inactive sample that Reines & Volonteri themselves argued should not be fit with a single relation. The authors know this and justify it as bracketing uncertainty, which is a defensible choice, but it means the result is a plausible first constraint rather than a closed measurement.

Bottom line: this is a useful, clearly stated Letter for anyone working on low-mass black hole demographics or LISA extreme-mass-ratio inspiral rates. It deserves a serious referee—the central idea is sound, the weaknesses are acknowledged, and the result is likely to be cited as the first occupation-corrected BHMF below 1e6. I would accept it to refereeing, with the main request being that the morphology caveat be explicitly reflected in the abstract's uncertainty claim. For my own work, I would cite it with the caveat noted.

Referee Report

2 major / 4 minor

Summary. This Letter estimates the local black hole mass function (BHMF) below 10^6 solar masses by convolving the GAMA galaxy stellar mass function with a black hole occupation fraction derived from Chandra X-ray observations of 326 nearby early-type galaxies and with a log MBH–log Mstar scaling relation fit to the combined AGN plus inactive galaxy sample of Reines & Volonteri (2015). The central quantitative claim is that the 68% normalization uncertainty of the resulting BHMF is less than 1 dex over the range [10^5, 10^6] M_sun, the range most relevant for LISA. The paper also compares with AGN-only and inactive-only scaling relations, provides a Schechter-like fitting formula for the median BHMF, and illustrates how future X-ray surveys with about 5,000 galaxies could shrink the occupation-fraction uncertainty.

Significance. If the result holds, this is one of the first observationally grounded estimates of the BHMF below 10^6 M_sun, with direct implications for LISA event-rate predictions and for seeding models of supermassive black holes. The convolution framework in Equation (3) is transparent, and the use of an empirically constrained occupation fraction rather than an assumed 100% occupation is a genuine step forward. The paper also deserves credit for propagating the posterior uncertainties of the fitted parameters and for providing a simple analytic approximation to the median BHMF. However, the headline uncertainty claim is conditioned on an assumption that the authors themselves describe as a serious drawback, namely that the early-type occupation fraction applies to the full galaxy population used in the convolution; this systematic is not included in the reported uncertainty band.

major comments (2)

[Section 4, Figure 2] In Section 4 the authors write that using the early-type occupation fraction 'represents a serious drawback for our estimates,' and yet the 68% band shown in Figure 2 and the less-than-1-dex claim in the abstract are computed from Equation (3) using only the lambda_occ posterior derived from 326 early-type galaxies. In the stellar mass range that maps to MBH in [10^5, 10^6] M_sun for the adopted scaling relation (roughly Mstar from 10^9.2 to 10^9.8 M_sun), late-type galaxies dominate the GAMA mass function, so a systematically lower late-type occupation fraction would shift the median Phi by about 0.5 dex for a factor-of-3 difference and by more than the quoted band for a factor-of-10 difference. Because no morphology-dependent systematic is included in the yellow band, the headline normalization uncertainty is conditional on the early-type occupation fraction rather than being an uncertainty on the true local BHMF.
[Section 3, Equation (3)] Equation (3) adopts a single linear log MBH–log Mstar relation fitted to the Reines & Volonteri sample and then applies it down to the GAMA mass limit, Mstar approximately 10^7.5 M_sun, with no allowance for a change of slope, intercept, or scatter in the dwarf-galaxy regime. The posterior range of alpha, beta, and sigma is propagated, but the unvalidated extrapolation itself is not included in the systematic budget of Figure 2. Since the central result is the normalization in the 10^5–10^6 M_sun range, where this extrapolation is directly relevant, the paper should either justify the extrapolation with low-mass black hole mass estimates and upper limits or add an explicit systematic term.

minor comments (4)

[Section 3, Equation (3)] Equation (3) writes MBH in the exponent, but the scaling relation is defined for log MBH; please make the logarithm explicit (for example, log MBH - (alpha + beta log Mstar)) to avoid ambiguity.
[Section 2, Equation (1)] The typeset argument of the tanh in Equation (1) is missing a division: it should be 2.5 |8.9 - log Mstar,0| log(Mstar/Mstar,0), not 2.5 |8.9 - log Mstar,0| log Mstar Mstar,0.
[Figure 2 caption] The caption of Figure 2 contains a typo ('Gair et at. 2010' should be 'Gair et al. 2010') and also uses 'BHFM' instead of 'BHMF'; both should be corrected.
[Section 4, right panel of Figure 3] In the discussion of the right panel of Figure 3, the phrase 'the corresponding BHMF uncertainty would shrink to less than about 2' lacks units; please specify whether this is 2 dex or another quantity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: BHMF is an error-propagated convolution of independent inputs; the late-type caveat is a stated systematic limitation, not a circular reduction.

full rationale

The central BHMF, Eq. (3), is an explicit convolution of three independent empirical inputs: the GAMA galaxy stellar mass function (Wright et al. 2017), the occupation fraction posterior from Chandra X-ray data on 326 early-type galaxies (Miller et al. 2015 plus Gallo et al. 2019), and an MBH-Mstar scaling relation refit from the Reines & Volonteri (2015) sample using Kelly (2007). None of these inputs is the target BHMF itself, and the paper does not fit Phi(MBH) to any data. The claimed less-than-1-dex normalization uncertainty is obtained by propagating posterior distributions through Eq. (3), not by inverting or reabsorbing a measurement of the BHMF. The self-citations to Gallo et al. (2019) and Lee et al. (2019) supply separate X-ray observations and contamination assessment routines; they are data and methodology references, not unverified theorems invoked to force the result. The serious drawback that occupation constraints come only from early-type galaxies, while late types dominate the relevant stellar mass range, is explicitly acknowledged in Section 4 and is a legitimate external-validity/systematic concern, but it does not make the derivation circular: the calculation honestly reports the uncertainty conditional on the adopted early-type occupation posterior rather than claiming that posterior was derived from the BHMF. No equation reduces by construction to its own input, and no fitted parameter is renamed as a prediction. The paper is therefore self-contained as an error-propagation exercise, with its main caveat lying outside the scope of circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on fitted or hand-chosen parameters, Mstar,0, alpha, beta, sigma, and the tanh shape constants, plus several unvalidated modeling assumptions about occupation fraction transferability, log-normal scatter, and scaling relation extrapolation. No new physical entities are postulated.

free parameters (5)

log Mstar,0 (occupation fraction transition mass) = 8.17 (median of posterior)
Sets the stellar mass at which the probability of hosting a nuclear black hole transitions from near zero to near one in Eq 1. Fitted to the Chandra active fraction data for 326 early-type galaxies. The 68 percent posterior range is broad and drives part of the BHMF normalization uncertainty.
alpha (intercept of log MBH versus log Mstar relation) = 8.13 +/- 0.09
Fitted via LINMIX_ERR to the combined AGN and inactive galaxy sample of Reines & Volonteri. Sets the normalization of the black hole to stellar mass mapping in Eq 3.
beta (slope of log MBH versus log Mstar relation) = 1.72 +/- 0.14
Fitted in the same regression. Controls how black hole mass scales with host stellar mass, and therefore how the galaxy mass function maps onto the BHMF.
sigma (intrinsic scatter in log MBH at fixed Mstar) = 0.61 +/- 0.05
Fitted in the same regression. This is the width of the log-normal kernel in Eq 3. The assumption that the scatter is constant across stellar mass is arbitrary.
lambda_occ shape parameters (2.5 and 8.9 in Eq 1) = chosen by hand
The 2.5 factor sets the steepness of the tanh transition and 8.9 fixes the pivot scale in Eq 1. They are not fitted to data in this paper and influence the sharpness of the occupation fraction rise, which affects the low-mass BHMF shape.

assumptions (6)

domain assumption The GAMA galaxy stellar mass function of Wright et al. (2017) is accurate and complete down to about 10^7.5 solar masses.
Used in Eq 3 to provide Phi(Mstar). The low-mass end of the mass function is treated as a single power law below the roughly 10^9.5 solar mass upturn, which is an extrapolation of the fitted form.
domain assumption Nuclear X-ray luminosity, after probabilistic X-ray binary subtraction, traces accreting massive black holes through a single LX-Mstar relation with constant intrinsic scatter.
Underpins the occupation fraction inference in Section 2. The relation is fit to early-type galaxy data and assumed to hold across the range used for lambda_occ and luminosity assignment.
domain assumption The Bayesian formalism of Miller et al. (2015) correctly maps the observed X-ray detection fraction into the occupation fraction lambda_occ.
Invoked in Section 2 to convert detections and upper limits into occupation constraints. Any systematic error in the X-ray binary contamination modeling propagates into Mstar,0.
domain assumption The distribution of MBH at fixed Mstar is log-normal with constant scatter sigma in Eq 3.
The kernel in Eq 3 is chosen for convenience; there is no direct low-mass data validating log-normality or constant scatter.
ad hoc to paper The combined AGN plus inactive MBH-Mstar relation is linear and can be extrapolated to stellar masses well below the fitted sample.
The authors fit the full sample of Reines & Volonteri (2015) despite those authors disfavoring a single relation. It is justified as a bracket for uncertainties, not as a physically preferred relation, and is central to populating the 10^5 to 10^6 solar mass range.
ad hoc to paper Early-type galaxy occupation fraction applies to the entire galaxy population used in Eq 3.
The occupation fraction is measured for early types only but convolved with the total galaxy stellar mass function. The authors flag this as a serious drawback in Section 4.

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Pith. "Pith review of Exploring the local black hole mass function below $10^6$ solar masses." pith.science (2026). https://pith.science/paper/D6QMUALR

@misc{pith2026190902585,
  author       = {Pith},
  title        = {Pith review of: Exploring the local black hole mass function below $10^6$ solar masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D6QMUALR}},
  note         = {Machine review of arXiv:1909.02585}
}

abstract

The local black hole mass function (BHMF) is of great interest to a variety of astrophysical problems, ranging from black hole binary merger rates to an indirect census of the dominant seeding mechanism of supermassive black holes. In this Letter, we combine the latest galaxy stellar mass function from the Galaxy And Mass Assembly survey with X-ray-based constraints to the local black hole occupation fraction to probe the BHMF below $10^6$ $M_{\odot}$. Notwithstanding the large uncertainties inherent to the choice of a reliable observational proxy for black hole mass, the resulting range of BHMFs yields a combined normalization uncertainty of $\lesssim$1 dex over the $[10^5-10^6]$ $M_{\odot}$ range, where upcoming, space-based gravitational wave detectors are designed to be most sensitive.

Figures

Figures reproduced from arXiv: 1909.02585 by the authors.

**Figure 1.** Updated constraints on the black hole occupation fraction via the M?,0 parameter in Equation 1, based on 326 nearby early type galaxies with uniform Chandra coverage. Shown here is the posterior distribution of M?,0 vs. the slope of the LX:M? relation, with the median values shown by a cross. assessment, the efficiency with which an actively accreting, massive black hole is detected is set by the intrinsic fraction … view at source ↗

**Figure 2.** The occupation-corrected BHMF defined by Equation 3. The solid red line traces the median; the dark and light yellow areas trace the 68 and 95% confidence regions, respectively, with all systematic errors included (whereas the long-dashed dark and light yellow lines trace the same confidence level regions ignoring systematic errors). This BHFM is derived from Equation 3, where the adopted observational proxy for b… view at source ↗

**Figure 3.** Left: Illustration of how different choices of black hole mass scaling relations affect the inferred BHFM. The yellow curve covers the same 68% confidence region as shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reviewed August 14, 2026 · model on record in the stance chip above.

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