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Hiding Out at the Low End: No Gap and a Peak in the Black-Hole Mass Spectrum

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper finds that gravitational-wave data from GWTC-3 plus GW230529 place the first percentile of black-hole masses at $m_{1\%}=3.13^{+0.18}_{-0.04}\,M_{\odot}$, ruling out a lower mass gap.

desk verdict A clean, modestly novel population-inference paper whose central 'no gap' claim is more model-dependent than the abstract admits, but the percentile framing is useful and the paper deserves a serious referee. read the letter →

arxiv 2507.09099 v1 pith:IVRH3V2L submitted 2025-07-12 astro-ph.HE gr-qc

Anarya Ray , Will Farr , Vicky Kalogera This is my paper
classification astro-ph.HEgr-qc
keywords blackholemassfunctionlowergapgravitationalwaveastronomyGWTC-3populationinferencecompactobjectmassesGW230529spectrum
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

This paper asks whether a lower mass gap separates neutron stars from the lightest black holes, a question with direct bearing on supernova explosions and dense-matter physics. X-ray binary studies have reported a gap, with few black holes below roughly $4.5\,M_{\odot}$, but their selection effects are hard to model. The authors instead analyze 25 gravitational-wave events from GWTC-3 plus the mass-gap event GW230529, modeling the black hole mass function above a fixed lower boundary of $3\,M_{\odot}$. In their most flexible model the first percentile of black hole masses is $m_{1\%}=3.13^{+0.18}_{-0.04}\,M_{\odot}$, which they read as inconsistent with the existence of a lower mass gap. If the result holds, the apparent gap in X-ray binaries is likely a selection or formation-channel effect rather than a real empty interval in the mass spectrum.

What carries the argument

The central object is the first percentile $m_{1\%}$ of the 'common' black hole mass function $f(m)$, defined in Eq.~(9) after normalizing $f$ between a fixed lower boundary $m_{\mathrm{low}}=3\,M_{\odot}$ and an upper bound. The authors target this percentile rather than an explicit gap-shaped feature, so the data speak through the overall abundance of black holes just above the neutron-star limit. The mass function combines a Gaussian neutron-star component below $m_{\mathrm{low}}$ with a broken power law or broken power law plus Gaussian black-hole component above it, multiplied by a pairing function in total mass; selection effects are corrected with the public detector-injection set through hierarchical Bayesian inference. Because the model is normalized to have positive support at $m_{\mathrm{low}}$, the resulting $m_{1\%}$ posterior is influenced by that boundary, a property the authors examine by rerunning the analysis with $m_{\mathrm{low}}$ values from $2.5$ to $3.5\,M_{\odot}$.

What would settle it

A decisive test would be to fit the same events with a model in which the lower end of the black-hole mass function is a free cutoff, allowing an empty interval between roughly $2.5$ and $5\,M_{\odot}$, and compare the posterior for $m_{1\%}$; if the no-gap claim is robust, $m_{1\%}$ should stay below about $3.5\,M_{\odot}$, while a genuine gap would move it upward. A complementary check is to remove GW230529 from the sample: if $m_{1\%}$ then jumps above $4\,M_{\odot}$, the conclusion rests on that one event.

Watch

Extended reading notes

Core claim

The discovery is a measurement of the low-mass end of the black hole mass function from merging binaries, stated as a percentile rather than a cutoff. Working with two parameterized mass functions, a broken power law and a broken power law plus a Gaussian peak, the authors find that the more flexible model recovers a sharp excess of black holes near $m \simeq 9.3\,M_{\odot}$ and a decline toward lower masses that is less than an order of magnitude, reaching a rate density of about $19\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ per log mass squared at $3\,M_{\odot}$. The first percentile of the common mass function is $m_{1\%}=3.13^{+0.18}_{-0.04}\,M_{\odot}$, with a 90\% range of $[3.0, 3.5]\,M_{\odot}$, and the model gives no posterior support for a minimum black hole mass above $3.6\,M_{\odot}$. The authors state that this sample is therefore not consistent with a mass gap, while the less flexible broken-power-law model permits but does not require one.

Load-bearing premise

The no-gap conclusion assumes the black-hole mass function is normalized to have positive support starting at $m_{\mathrm{low}}=3\,M_{\odot}$, so the first percentile is constrained to sit near that bound rather than being measured freely, and the central estimate is heavily influenced by this imposed boundary and by the single event GW230529.

Editorial extensions

If this is right

  • If the no-gap measurement is correct, the compact-object mass spectrum is populated continuously from neutron stars into black holes, so supernova mechanisms that forbid black holes below about $4.5\,M_{\odot}$ are not needed to explain the gravitational-wave population.
  • The X-ray-binary lower mass gap would then be the product of transient selection effects or systematic mass errors, not a true property of the mass function.
  • The combination of a sharp peak near $9\,M_{\odot}$ and a shallow low-mass tail becomes a joint constraint on binary formation: no single stable-mass-transfer channel in the comparison model reproduces both features, pointing to additional formation pathways.
  • Ongoing and future observing runs should either populate the $3$--$5\,M_{\odot}$ range with more events, sharpening $m_{1\%}$, or reveal a sparse region that the current flexible model smooths away.

Reading between the lines

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

  • The fixed lower boundary at $3\,M_{\odot}$ and the single event GW230529 carry much of the weight of the central result; if a gap began just above $3\,M_{\odot}$, a model with a free lower cutoff could still recover it while the fixed-boundary analysis would not.
  • A direct extension would be to reanalyze the same events with the lower boundary as a free hyperparameter and to report the Bayes factor against the fixed-boundary model, testing whether 'no gap' is a property of the data or of the normalization.
  • If the no-gap result survives more data, electromagnetic searches for quiescent low-mass black holes in X-ray binaries should find a mass distribution that either disagrees with the merger population or reveals that X-ray-binary mass estimates are systematically biased upward.
  • The shallow low-mass tail predicted here is testable with O4 data alone, since additional events in the $3$--$5\,M_{\odot}$ range will directly populate the region of interest.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper analyzes 25 GWTC-3 binary black hole events plus GW230529, selecting events with primary mass > 3 Msun and chirp mass < 17.41 Msun, to infer the low-mass black hole mass function. Using hierarchical Bayesian inference with two flexible models (a broken power law, BPL, and a broken power law plus Gaussian peak, BPLG), the authors find a peak near 9.3 Msun, a merger-rate decline of less than an order of magnitude toward 3 Msun, and a first percentile m1% = 3.13+0.18-0.04 Msun (68%) for the BPLG model. They interpret this as evidence that the low-mass BH sample is not consistent with a lower mass gap, in contrast to X-ray binary studies, and discuss formation-channel and selection-effect explanations for the discrepancy.

Significance. The question of a lower mass gap in the compact-object mass spectrum is important for supernova physics and population synthesis. The paper brings a clean GW-based dataset and standard hierarchical inference to this question, and its characterization of the low-mass slope and peak is a useful contribution. If the no-gap conclusion were robust, it would materially challenge LMXB-based gap measurements. However, the central claim as stated is not supported by the analysis: m1% is a percentile of a model that cannot represent a gap, and the paper's own robustness tests (Fig. 5, Table 1) show the result tracks the imposed lower boundary and is model-dependent. The paper is therefore of interest but needs substantial revision to substantiate or appropriately qualify its headline claim.

major comments (4)
  1. [Sec. 2.1, Eq. (9); Sec. 3] The definition of m1% in Eq. (9) normalizes the mass function on the fixed interval [mlow, mhigh] with mlow = 3 Msun, and the models in Eqs. (4)-(5) have positive density at mlow with no mechanism to represent an interval of zero density. Consequently, m1% is structurally anchored near mlow whenever any low-mass event contributes, rather than being an independent measurement of a gap. Figure 5 confirms this: the m1% posterior peak tracks the chosen mlow across 2.5-3.5 Msun. The abstract's statement that the sample 'is not consistent with the existence of a mass gap' therefore does not follow from the m1% posterior. To support the claim, the paper needs to fit an explicit gap model (e.g., a mass function with a zero-density or strongly depleted interval) and compare it with the no-gap model, or restrict the conclusions to 'consistent with no gap under the assumed model family.'
  2. [Sec. 3, Table 1] The BPL model yields m1% with a 90% range [3.1, 4.8] Msun, meaning it permits a gap up to 4.8 Msun. The text acknowledges this in the discussion, but the abstract and the phrase 'BPLG model rules out the existence of a lower mass gap' (Sec. 3) overstate the evidence. The no-gap conclusion is only valid for the BPLG model, whose low-mass tail is a power law anchored at mlow; the less flexible BPL model does not require a gap but allows one. This model dependence should be stated prominently in the abstract and conclusions, not relegated to the discussion.
  3. [Sec. 2.1, Sec. 3] The text says the analysis is performed 'both with and without GW230529' (Sec. 2.1), but no results excluding GW230529 are reported. Given that the m1% inference is largely driven by GW230529 (as the Fig. 5 discussion indicates: for mlow > 3.25 Msun, posterior samples of GW230529 start getting excluded), the absence of this robustness check is a significant omission. Please report the posterior for m1% (and the mass function) without GW230529, and quantify how much the no-gap conclusion depends on this single event.
  4. [Sec. 2.3] The selection correction treats catalog events and injections asymmetrically: catalog events are selected by requiring 50% of posterior samples to lie within the mass cuts, while injections are rejected if their true parameter values lie outside the cuts. The authors argue chirp mass is well measured so this is approximately equivalent, but this is an approximation that could bias the inferred low-mass slope and m1%. The robustness to the 50% vs 90% threshold is shown in Fig. 4, but the full asymmetry is not validated. Please either apply the same posterior-sample criterion to injections (or a reweighting that accounts for the measurement uncertainty of injection parameters) or demonstrate explicitly that the asymmetry does not affect the low-mass inference.
minor comments (5)
  1. [Sec. 1] Typo: 'GWCT3' should be 'GWTC-3'.
  2. [Sec. 2.1] Typo: 'rate denisty' should be 'rate density'.
  3. [Sec. 4] The sentence 'posterior distribution of the first percentile of the BH mass function to be 90% within 3.13+0.18-0.04 M⊙' mixes the 68% interval with a 90% statement; the 90% range from Table 1 is [3.0, 3.5] M⊙ for the BPLG model.
  4. [Fig. 4 caption] The caption says 'our three models' but the figure appears to show two models (BPL and BPLG) with two selection cuts; please clarify what the three models are.
  5. [Sec. 3] The phrase 'rules out the existence of a lower mass gap in our sample' is too strong given the model dependence; consider 'rules out a gap in this model' or similar.

Circularity Check

3 steps flagged · score 6.0 of 10

The 'no gap' claim largely restates the chosen mass-function support and the m1% definition rather than testing an independent gap hypothesis.

  1. self definitional [Sec. 2.1, Eq. (9) and following sentence]
    "Given measurements of these hyper-parameters, the “common” mass function f can be normalized to obtain a probability distribution for mlow ≤ m ≤ mhigh which can be written as: p(m) ≡ f (m)R mhigh mlow dm′ f (m′) . (9) Denoting m1% to be the first percentile of this distribution, we can rely on its measurements to infer the minimum BH mass and hence the existence of a lower mass-gap (Farr et al. 2011)."

    Eq. (9) makes m1% a functional of the fitted f after normalizing on [mlow, mhigh] with mlow fixed at 3 M⊙. Because both model families (Eqs. 4-5) are strictly positive on that interval, the distribution never has a zero-density gap; the only way to infer 'no gap' is through the location of the first percentile. A low m1% therefore restates the input support (plus the presence of low-mass events) rather than testing an independent gap hypothesis.

  2. self definitional [Sec. 3, discussion of Fig. 5]
    "We find that the peak of the posterior closely follows the location of mlow, which indicates the absence of a lower mass-gap for all mlow values chosen."

    Since m1% is defined as the first percentile of a distribution whose support starts at mlow, moving mlow moves the lower endpoint of the support and hence shifts the quantile. The posterior peak following mlow is the near-tautological consequence of this normalization, not a new empirical fact. Interpreting this tracking as evidence against a gap makes the verdict a function of the chosen boundary by construction.

1 more flagged steps
  1. renaming known result [Sec. 3, discussion of Fig. 4 and Table 1]
    "The BPLG model yields a much tighter constraint on m1% and has no posterior support for the minimum BH mass to be above 3.6M⊙. In other words, BPLG model rules out the existence of a lower mass gap in our sample of low mass BHs with 90% posterior probability while the BPL model does not necessitate a gap but may permit one."

    The paragraph equates 'no posterior support for the minimum BH mass above 3.6' with 'rules out a lower mass gap.' But the model's minimum mass is the fixed mlow; the posterior quantity is m1%. Since the model family contains no gap parameter or zero-density interval, the statement is a re-description of a fitted percentile, not a comparison against a gap model. The BPL model's wider m1% range shows the conclusion is model-dependent, while no explicit gap model is fit to compare.

full rationale

Most of the analysis is self-contained and rests on public LIGO/Virgo/KAGRA data and injections; the hierarchical likelihood, selection functions, and mass-function fits are standard and not circular. Self-citations (Farr et al. 2011 for the percentile approach; Roy et al. 2025 for the mass cut) are not load-bearing in the sense of smuggling in the result. The circularity is concentrated in the interpretive step from a fitted quantile to 'no mass gap.' Because m1% is defined by normalizing the fitted f on [mlow, mhigh], and the adopted models cannot represent an interval of zero density, the no-gap conclusion is substantially fixed by the support choice and the inclusion of GW230529, not by an explicit comparison with a gap model. The paper itself shows the m1% posterior peaks track mlow (Fig. 5), which is the signature of this definitional anchoring. This is partial, not total, circularity: the BPLG model does genuinely pin down a low first percentile, and a complete gap over [3, 3.5] is empirically excluded by GW230529's existence; but the abstract's 'in other words, no gap' overstates what the construction can establish.

Assumptions & free parameters 9 free parameters · 7 assumptions · 0 invented entities

The model contains no invented physical entities. The central quantity m1% is a deterministic function of the fitted mass function and the fixed lower bound, so the no-gap result is not independent of the modeling choices. Key drivers are the lower cutoff mlow = 3 M⊙, the requirement that f(m) > 0 at the cutoff, the functional forms of Eqs. (4)-(5), and the inclusion of GW230529.

free parameters (9)
  • mlow (lower mass bound) = 3 M⊙, fixed by hand
    Lower support of the BH mass function in Eq. (9); Fig. 5 shows m1% tracks this value.
  • Mmax,chirp (upper chirp-mass cut) = 17.41 M⊙, fixed by hand
    Event-selection upper cut; removes high-mass events that do not inform the low-mass BH population.
  • mb or μ (break/peak mass) = ≈ 9.3 M⊙ (inferred)
    Peak of the BH mass function; fitted from the same events.
  • α1 and α2 (power-law slopes) = inferred
    Control the rise and fall around the break and the low-mass tail.
  • β (pairing-function index) = inferred
    Power-law pairing in Eq. (6); affects mass-ratio weighting.
  • fg and σ (Gaussian fraction and width) = inferred
    Shape parameters of the BPLG model; produce the 8-10 M⊙ peak.
  • rNS, μNS, σNS (NS component) = inferred
    Parameters of the neutron-star mass function in Eq. (3); not central to the BH result.
  • R0 (rate normalization) = inferred
    Overall merger rate scale; fitted but not central to m1%.
  • event selection threshold = 50% posterior samples (90% in robustness check)
    Events included when half their posterior mass lies inside the mass cuts; affects the inclusion of borderline events.
assumptions (7)
  • domain assumption The merger rate factorizes as R(z) f(m1) f(m2) g(m1,m2) with a common component mass function and a pairing function.
    Used in Eq. (1); this standard ansatz assumes component masses are drawn from the same distribution, which may not hold for NSBH systems.
  • domain assumption The BH mass function is either a broken power law or a broken power law plus a Gaussian peak, with f(m) > 0 throughout [mlow, mhigh].
    Eqs. (4)-(5); these functional forms prevent a sharp gap from being represented and force positive density at the low-mass cutoff.
  • domain assumption Redshift evolution of the merger rate is taken from Madau-Dickinson with zp = 1.9, kappa = 5.6, lambda = 2.7.
    Eq. (8); fixed from prior literature, reasonable for low redshift but an input assumption.
  • domain assumption The selection function can be computed by reweighting LVK detectable injections and excluding injections whose true parameters fall outside the mass cuts.
    Secs. 2.2-2.3; asymmetric with the event selection, which uses posterior samples, so the normalization is approximate.
  • domain assumption GW230529_181500 and all selected events with m1 > 3 M⊙ are black holes drawn from the same BH mass function.
    Sec. 2.3; this single event contributes most of the low-mass tail and drives the no-gap inference.
  • domain assumption Component spins are not modeled; they are fixed to the isotropic uniform-magnitude priors used in single-event PE.
    Sec. 2.2; mass-spin correlations in the population are ignored.
  • standard math The hierarchical likelihood and Monte Carlo selection normalization follow Mandel et al. (2019) and converge.
    Sec. 2.2; standard machinery, but convergence is only flagged, not demonstrated in detail.

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Cite this review

Pith. "Pith review of Hiding Out at the Low End: No Gap and a Peak in the Black-Hole Mass Spectrum." pith.science (2026). https://pith.science/paper/IVRH3V2L

@misc{pith2026250709099,
  author       = {Pith},
  title        = {Pith review of: Hiding Out at the Low End: No Gap and a Peak in the Black-Hole Mass Spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IVRH3V2L}},
  note         = {Machine review of arXiv:2507.09099}
}
abstract

In recent years, the existence of a gap in the mass spectrum of compact objects formed from stellar collapse, between the heaviest neutron stars and the lightest black holes, has been a matter of significant debate. The presence or absence of a mass gap has implications for the supernova mechanism, as well as being a fundamental property of the compact object mass function. In X-ray binaries containing black holes a gap is observed, but it is not known whether this is representative of a true gap in the mass function or due to selection effects or systematic biases in mass estimation. Binary black hole mergers detected from gravitational waves in the GWTC-3 transient catalog furnish a large sample of several tens of low-mass black holes with a well-understood selection function. Here we analyze the \nevts{} GWTC-3 merger events (along with GW230529\_181500) with at least one black hole ($3 \, M_\odot < m_1$) and chirp masses below those of a $20\,M_\odot$--$20\,M_\odot$ merger ($\mathcal{M} < 17.41 M_{\odot}$) to uncover the structure of the low-mass black hole mass function. Using flexible parameterized models for the mass function, we find, similar to existing studies, a sharp peak in the mass function at $m \simeq (8-10 M_{\odot})$. We observe a steady decline in the merger rate to lower masses, but by less than an order of magnitude in total, and find that the first percentile of black hole masses in our most flexible model is $m_{1\%} =3.13^{+0.18}_{-0.04}$. In other words, this sample of low-mass black holes is not consistent with the existence of a mass gap.

Figures

Figures reproduced from arXiv: 2507.09099 by the authors.

Figure 1
Figure 1. Contour plot of the likelihood functions for the primary and secondary black hole masses in the events considered in this analysis. The contours show credible regions containing 50% and 90% of the likelihood for each event. The dashed lines show our selection cuts, with m1 > 3 M⊙ and M < 17.41 M⊙. 3. RESULTS In this section, we present our results obtained from the 25 GWTC-3 events that satisfy our detection thresho… view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Inferred common mass distribution, p(m), (see Eq. (9)) for both models considered in this analysis. Dark lines show the posterior mean mass distribution; light lines are individual draws from the posterior over mass distributions. At high masses, m ≫ µ, mb, the mass function falls steeply in both models; the broken power law plus Gaussian slope α2 = −2.5 +1.4 −3.4 . Toward lower masses from the peak, both power law … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The posterior distribution for m1%, the first percentile of the “common” mass function for our three models. The broken power law plus Gaussian model with 50% selection cut has m1% = 3.13+0.18 −0.04 M⊙ at 1σ (68%) credibility. The solid lines are the result for the 50%…
Figure 5
Figure 5. Figure 5: The posterior for m1%, corresponding to different values of mlow for the BPLG model in the form of violin plots. models yield more constrained and peaked posteriors for smaller values of mlow. We see the posteriors start to grow broader with increasing mlow. The broade…

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