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The population of NuSTAR Black Hole X-ray Binaries

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read X-ray binary black holes spin near maximum, a uniform NuSTAR sample shows.

desk verdict A valuable data release and transparent exploratory analysis, but the headline beta-distribution claim is under-specified and the XB/GW incompatibility is not yet demonstrated. read the letter →

arxiv 2506.12121 v2 pith:55NOSGV2 submitted 2025-06-13 astro-ph.HE

Paul A. Draghis , Jon M. Miller , Laura Brenneman , Elisa Costantini , Luigi C. Gallo , Mark Reynolds , John A. Tomsick , Abderahmen Zoghbi This is my paper · ORCID
classification astro-ph.HE
keywords blackholespinX-raybinariesrelativisticreflectionNuSTARbetadistributiongravitationalwavesnatalaccretiondisk
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 argues that the black holes in X-ray binaries almost all spin very fast, and that their measured spins follow a beta distribution with shape parameters alpha = 5.66 and beta = 1.09. The distribution is built from 245 uniform relativistic-reflection fits to NuSTAR spectra of 36 systems, so it is meant to be a systematic, source-independent characterization rather than a collection of heterogeneous measurements. The central stakes are formation physics: at the short orbital periods many of these systems have, accretion alone cannot have spun the holes up to the measured values, so the rotation must have been largely present at birth. That makes the observed spin distribution a direct constraint on stellar collapse and binary evolution, and it stands in sharp contrast to the low-spin distribution inferred from gravitational-wave mergers.

What carries the argument

The load-bearing object is the uniform sample itself: 245 NuSTAR spectra of 36 accreting black hole X-ray binaries, each fit with six flavors of the relxill relativistic-reflection model, selected by deviance information criterion and processed through MCMC posteriors. The named identity carrying the population claim is the $\beta$ distribution P(a) = Gamma($\alpha$+$\beta$)/(Gamma($\alpha$)Gamma($\beta$)) $a^{{alpha-1}}$(1-a)^{$\beta$-1} with $\alpha$ = 5.66 and $\beta$ = 1.09, which summarizes the observed spin sample. The comparison that makes the claim consequential is the same Bayesian-inference procedure applied to the GWTC-3 black-hole merger spins, yielding two distributions that barely overlap.

What would settle it

Take one high-quality NuSTAR spectrum from the selected Eddington range, leave the inner disk radius free instead of fixing it to the ISCO, and check whether the best fit prefers a radius significantly outside the ISCO; if it does, that spin is biased low and the beta(5.66, 1.09) distribution is not a clean natal-spin measurement. A second check would be to re-fit the faint spectra that currently yield low spin with absorption-line components and see whether the low-spin solutions disappear.

Watch

Extended reading notes

Core claim

The authors claim that the observed spin distribution of the 36 X-ray binary black holes is a $\beta$ distribution, P(a) proportional to $a^{{alpha-1}}$(1-a)^{$\beta$-1} with $\alpha$ = 5.66 and $\beta$ = 1.09, peaking near a approximately 0.97 and incompatible with the spin distribution inferred from GWTC-3, whose mode is around 0.18. The incompatibility is presented as the first such comparison built from X-ray measurements made with one uniform pipeline, and therefore not an artifact of mixing different model assumptions. The paper further claims that high spins in systems with short orbital periods exceed the maximum spin that accretion can deliver, so these black holes must have formed rotating near their maximum rate; that spin-measurement precision increases with black hole mass and decreases with distance and with lower reflection counts; and that low or negative individual spin fits are confined to faint spectra with low reflection strength, where the model cannot distinguish high-spin/high-emissivity from low-spin/low-emissivity solutions.

Load-bearing premise

The analysis assumes that in every selected observation the accretion disk reaches the innermost stable circular orbit, so the fitted inner radius can be read directly as spin; if some disks are truncated, those spins are biased low and the beta distribution would not describe true natal spins.

Editorial extensions

If this is right

  • If the beta distribution is correct, most stellar-mass black holes in X-ray binaries are born with near-maximal rotation, and supernova and binary-evolution models must reproduce that angular momentum before any accretion occurs.
  • Accretion spin-up cannot explain the fastest rotators at short orbital periods, so mass transfer is not the origin of the high spins; the observed values are close to natal values.
  • The X-ray and gravitational-wave spin distributions are genuinely different, or at least are not reconciled by uniform X-ray systematics, motivating searches for selection effects and formation-channel differences.
  • Spin measurements will be most precise for massive, nearby, bright systems with high reflection counts, while the low-spin tail of the distribution should be treated cautiously because it comes from faint spectra.
  • The published 245-fit dataset becomes a community resource for testing model degeneracies and future parameter correlations.

Reading between the lines

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

  • If the same pipeline were applied to a sample selected without the Eddington-fraction cut, or to AGN spins measured by reflection, the inferred beta parameters might shift, which would test whether the high-spin peak is a property of X-ray binaries or of the reflection method itself.
  • The tentative spin-emissivity degeneracy suggests that some low-spin measurements are algorithmic artifacts of faint spectra; high-resolution microcalorimeter spectra could distinguish real low spins from unmodeled absorption features.
  • A physical consequence the authors leave implicit is that if most X-ray binary black holes are born spinning fast while most merging black holes appear to spin slowly, the two populations may trace different mass or metallicity channels, not just different measurement techniques.
  • The density experiments imply that fitting with densities above log n = 20 lowers Fe abundance and inclination but not spin; extending this to a full sample could turn the bimodal Fe abundance into a diagnostic of unmodeled disk density.
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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 / 6 minor

Summary. Using the 245 NuSTAR spectra of 36 BH X-ray binaries previously analyzed in Draghis et al. (2024), this paper presents a population-level study of measured spins and spectral-fit parameters. It reports Spearman correlations between spin uncertainty and system properties (mass, distance, inclination), identifies parameter degeneracies in relativistic reflection fits (e.g., q1--a, R--Gamma, log xi--Gamma), and fits the observed XB spin distribution with a beta distribution alpha=5.66, beta=1.09, concluding that the XB distribution is incompatible with the low-spin distribution inferred from GWTC-3 and that most XB BHs must have formed with high natal spins. The full fitting table is released on Zenodo.

Significance. The paper's main value is as a resource: a uniformly reduced, publicly released table of 245 NuSTAR spectral fits and spin measurements, plus a set of cautionary examples about low-SNR degeneracies (low-q1/low-a vs high-q1/high-a) and the influence of absorption lines and disk density. The comparison of measured spins with the Fragos & McClintock (2015) accretion-spinup ceiling is an interesting and falsifiable argument for high natal spins. However, the two headline population claims--the beta-distribution characterization and the XB/GW incompatibility--are not yet established because the statistical inference is under-specified and the input sample is not independent. The paper is honest about many limitations and does not overstate the exploratory correlation analysis, but the central 'incompatible with GWTC-3' statement needs a properly specified and reproducible hierarchical fit before it can carry the weight placed on it.

major comments (4)
  1. [Section 3.1, Eq. (1)] The Bayesian inference that produces alpha=5.66 and beta=1.09 is not described. The text does not state the likelihood, the prior on (alpha, beta), the input data (36 source-level values from Table 1, 36 full posteriors, or the 245 per-spectrum fits), how asymmetric 1-sigma intervals are propagated, or how the posterior draws shown as thin blue lines in Figure 4 are generated. The beta distribution is only defined on [0,1], whereas individual fits in Figure 6 extend to negative a, so the treatment of negative/retrograde values must be stated. Without these details, and without credible intervals or a goodness-of-fit statistic for alpha and beta, the claim that the XB distribution 'is well approximated' by beta(5.66,1.09) cannot be evaluated or reproduced; this is the load-bearing step for the XB/GW comparison.
  2. [Section 4, Figures 6 and 8] The correlation analysis treats the 245 spectra as independent samples even though they are repeated observations of 36 sources. The quoted Spearman coefficients and their +/- uncertainties therefore ignore clustering; for example, Figure 8 panels (e) and (f) report rho=0.32+/-0.01 and 0.57+/-0.01 on 245 points, while Figure 9 shows clear source-level structure in the same parameter combinations. A source-resampling or mixed-effects analysis is needed to establish whether the trends are within-source or between-source and whether the reported significance survives. The same issue applies to the spin-uncertainty correlations in Figure 3, although those use one value per source.
  3. [Section 3.1, Figure 4] The paper states that the two distributions are 'clearly distinct', but the XB distribution is explicitly the observed distribution with no selection-function correction, while the GWTC-3 distribution is selection-corrected. This asymmetry is acknowledged in the text but not accounted for in the conclusion. Because the XB sample is selected by outburst activity, Eddington fraction, detection of reflection, and successful spin constraint, the observed high-spin excess could be partly a selection artifact. The authors should either model the XB selection function or restrict the claim to the observed sample with the selection caveat carried through the abstract and conclusions.
  4. [Sections 2 and 4.1] The assumption that the inner disk radius equals the ISCO in every fitted observation is the physical link between the reflection fits and the spin a. The defense in Section 4.1, based on hardness-intensity diagrams for nine sources (Figure 10), is indirect and relies on the absence of an obvious hardness trend in the spin constraints rather than a direct test of Rin/ISCO. If some hard-state disks are truncated, the inferred spins are biased low, and the beta distribution in Eq. (1) describes biased measurements rather than true spins. A direct test, such as letting Rin vary in a subset of spectra and quoting the change in fit statistic or the posterior on Rin/ISCO, would make the population claim much stronger.
minor comments (6)
  1. [Eq. (1)] The definition of the gamma function contains a typo: 'r^{-t}' should be 'e^{-t}'.
  2. [Sections 2 and 4.6] Section 2 states that 36 of the 245 spectra required the zxipcf component for complex obscuration, while Section 4.6 states that 96 spectra required an absorption Gaussian line and 149 did not; these numbers need to be reconciled or explicitly described as different diagnostics.
  3. [Section 3.1, Figure 4] The phrase 'modes and +/-1 sigma of the mean distributions' is confusing; please specify whether the vertical lines are the mode and central credible interval of the population distribution itself.
  4. [Figures 6 and 8] Several panels use the placeholder symbol 'square' as the y-axis label (e.g., Figure 8 panels e, f, i, o); these should be replaced with the actual parameter name (Gamma or another label).
  5. [Table 1 footnote] The footnote marker '3 represents an inclination estimate based on dips' should be typeset as a superscript to match the table entries and to avoid confusion with the numeric value 3.
  6. [Figure 2, panel (g)] The caption for the theoretical curves from Fragos & McClintock (2015) does not state the assumed accretion efficiency or spin-up prescription; adding one sentence would help readers interpret the comparison.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the beta distribution is an explicitly fitted description of the authors' own published spin measurements, not a prediction derived from itself, and the GW comparison uses an external catalog.

full rationale

The paper's central quantitative claim is that the observed NuSTAR reflection-measured spins of 36 X-ray binaries are described by a beta distribution with alpha=5.66 and beta=1.09 (Section 3.1, Eq. 1). This is presented explicitly as a Bayesian characterization of the observed sample, not as a prediction generated from first principles. The input data are the 36 source-level spin measurements published in Draghis et al. (2024), and those measurements were obtained by spectral fitting with relxill models; the population-level beta parameters are then fit to those measurements. That is an empirical summary, and no equation in the paper reduces to another by construction. The comparison to gravitational-wave spins uses the external GWTC-3 catalog, so the claimed incompatibility is an external benchmark rather than a self-referential statement. The self-citations to Draghis et al. (2023b, 2023c, 2024) are legitimate reuse of the authors' own published data and methods; the spin table is accompanied by a Zenodo release (doi:10.5281/zenodo.15801174), making the input data independently accessible, and the cited prior work contains the detailed fitting and MCMC procedures. The ISCO assumption (Section 2, defended in Section 4.1) is a modeling assumption that can bias spin values if disks are truncated, but it is not a circular argument: the spins are fitted from spectra, not assumed by the population analysis. The remaining concerns about Section 3.1 are statistical transparency issues (e.g., whether the population likelihood uses 36 source-level posteriors or per-spectrum values, and how asymmetric credible intervals are propagated), which affect robustness and reproducibility but do not constitute circularity. Accordingly, the analysis is self-contained with respect to its own derivation chain, and the only mild concern is the heavy reliance on the authors' own prior dataset, which is published and reproducible rather than load-bearing in a circular sense.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central results rest on the prior spin measurements, which depend on the ISCO assumption and relxill model; the new beta-distribution fit adds two fitted parameters. No new physical entities are introduced.

free parameters (3)
  • alpha (beta distribution shape) = 5.66
    Fitted to the observed spin values of the 36 X-ray binaries in Section 3.1.
  • beta (beta distribution shape) = 1.09
    Fitted to the observed spin values of the 36 X-ray binaries in Section 3.1.
  • Eddington fraction selection window = 10^-3 to 0.3
    Hand-chosen sample selection window in Section 2 intended to ensure the disk reaches the ISCO; directly shapes which spectra are included in the analysis.
assumptions (4)
  • domain assumption The accretion disk extends to the ISCO during the selected observations.
    Assumed in Section 2 when selecting Eddington fractions 1e-3 to 0.3 and when interpreting the inner radius as ISCO. If false, spins are biased low.
  • domain assumption The relxill family of reflection models accurately describes the reflected spectra and the relevant parameter relationships.
    All spin and parameter values come from fitting six relxill flavors (Section 2). Model systematics are discussed but not independently validated here.
  • domain assumption Literature values for BH masses, distances, companion masses, and independent inclinations are accurate enough for the correlation analysis.
    Table 1 relies on published measurements; errors in these propagate into the Spearman correlations in Section 3.
  • ad hoc to paper The 245 spectra can be treated as independent samples for correlation analysis.
    Sections 4 and Figures 6-8 pool all spectra from all sources, ignoring that multiple spectra from one source are correlated, which affects the reported significance.

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Cite this review

Pith. "Pith review of The population of NuSTAR Black Hole X-ray Binaries." pith.science (2026). https://pith.science/paper/55NOSGV2

@misc{pith2026250612121,
  author       = {Pith},
  title        = {Pith review of: The population of NuSTAR Black Hole X-ray Binaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55NOSGV2}},
  note         = {Machine review of arXiv:2506.12121}
}
abstract

The spin of a black hole (BH) encodes information about its formation and evolution history. Yet the understanding of the distribution of BH spins in X-ray binaries (XBs), of the models used to measure spin, and of their impact on systematic uncertainties remains incomplete. In this work, we expand on previous analyses of the entire NuSTAR archive of accreting BH XBs. Prior work compiled a sample of 245 spectral fits using the relativistic reflection method for NuSTAR observations of 36 BH systems. Here, we aim to probe two aspects: the connection between BH spin and binary system properties, and the relationships between parameters in the spectral fits. We identify moderate negative correlations between spin uncertainty and both BH mass and system inclination, and a moderate positive correlation with distance. We also point out tentative multidimensional degeneracies between inclination, disk density, Fe abundance, ionization, and the presence or absence of absorption features from ionized outflows linked to disk winds. Lastly, we provide a comprehensive view of the observed distribution of BH spins in XBs, in comparison to spins inferred from gravitational waves. We find that the distribution of BH spins in XBs can be described by a beta distribution with $\alpha=5.66$ and $\beta=1.09$. This data set is highly complex, and the analysis presented here does not fully explore all potential parameter correlations. We make the full data set available in Zenodo to the community to encourage further exploration.

Figures

Figures reproduced from arXiv: 2506.12121 by the authors.

Figure 1
Figure 1. shows all the 36 measurements presented in Draghis et al. (2024). As highlighted in the article, 86% of the measurements allow a ≥ 0.95, and 100% allow a ≥ 0.7 (at the 1 σ level). Furthermore, 28% of the spins have a lower bound of their 1 σ credible interval greater than a ≥ 0.9. At the lower end, only ∼ 17% of the measurements allow a ≤ 0.6 at the 1 σ level. AT 2019wey LMC X-3 LMC X-1 MAXI J0637-430 MAXI J1348-630… view at source ↗
Figure 2
Figure 2. The legend in the top-left panel shows the markers that were used in the all panels except for the top right to identify the 36 sources analyzed in this work, including in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Uncertainty in spin measurements vs. BH mass (panel a), distance to the system (panel b), the ratio of BH mass to distance to the system squared (panel c), and the measured inclination (panel d). The ρ values in the individ￾ual panels represent the Spearman correlation coefficients for the samples. 0.0 0.2 0.4 0.6 0.8 1.0 a 0 1 2 3 4 5 6 7 8 P ( a ) GWTC-3: a = 0.18+0.2 −0.14 XB: a = 0.97+0.02 −0.18 [PITH_FULL_IMAG… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Probability distributions for the distributions of BH spins inferred based on GWTC-3 (red) and the measure￾ments in this work (blue). The solid curves represent the mean and central 90% credible bounds inferred on the distri￾butions. The thin lines represent individual…
Figure 5
Figure 5. Figure 5: Correlation matrix for parameters of interest. Redder colors represent stronger positive correlations, while bluer colors represent stronger negative correlations. See Ap￾pendix A for the full correlation matrix, together with the numerical values of the correlation co…
Figure 6
Figure 6. Figure 6: Results of the spectral analysis of the 245 spectra of the 36 sources treated in this work. The x-axis shows the spin measured from each spectrum, while the y-axis represents a few parameters of interest. The transparency of the points represents the strength of the re…
Figure 7
Figure 7. Figure 7: Statistic (χ 2 /ν), BH spin (a), viewing inclination (θ), and Fe abundance (AFe) measured by fitting the NuS￾TAR spectra of MAXI J1820+070 from ObsID 90401309023 with the reflionx HD model, with disk density fixed at mul￾tiple values across the entire allowed parameter…
Figure 8
Figure 8. Figure 8: Interesting parameter combinations emerging from the analysis of the 245 spectra treated in Draghis et al. (2024). The transparency of the points is proportional to the strength of reflection during each of the observations that produce the individual points. The trend…
Figure 9
Figure 9. Figure 9: The nine panels show the log(ξ) and Γ values measured in each individual observation in our analysis for nine sources that have multiple observations. The colors of the points represent the 10–79 keV flux during each individ￾ual observation. While it is appealing to tr…
Figure 10
Figure 10. Figure 10: Hardness-intensity diagrams for nine sources with multiple observations. The color of the points repre￾sents the measured value of the BH spin, and the size of the points represents the uncertainty in the spin measurement, as shown in the central panel, with larger ma…
Figure 11
Figure 11. Figure 11: Left: 1 σ uncertainty in spin measurement in all observations in this study vs. the number of counts in the reflection component, calculated as the difference be￾tween the total flux during the observation and the flux in the underlying continuum, multiplied by the ex…
Figure 12
Figure 12. Figure 12: The measured inner emissivity index q1 vs. BH spin a (top) and ionization parameter log(ξ) vs. in￾clination of the inner accretion disk θ (bottom) produced from the models that include an absorption gaussian com￾ponent around 7 keV (left) and that do not include an ab…
Figure 13
Figure 13. Figure 13: Complete correlation matrix of all the parameters in the spectral models used to fit the 245 NuSTAR spectra presented in this work. The numbers in each cell represent the Spearman correlation coefficient for the given combination of parameters. Redder colors represent…

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Spin Constraints on 4U 1630-47 via combined Continuum Fitting and Reflection methods: a comparative study using Frequentist and Bayesian statistics

    astro-ph.HE 2025-09 conditional novelty 5.0 of 10

    Combined continuum fitting and reflection modeling of NICER/NuSTAR spectra yield spin a*=0.93(+0.05,-0.04), mass about 9 solar masses, distance about 10.5 kpc, and inclination about 54 degrees for 4U 1630-47.

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