Estimating the spins of supermassive black holes in distant ultraluminous quasars

M.Yu. Piotrovich; S.D. Buliga; T.M. Natsvlishvili

REVIEW 5 major objections 5 minor 1 cited by

Estimating the spins of supermassive black holes in distant ultraluminous quasars

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Ultraluminous quasars at redshift 6–7.5 harbor supermassive black holes spinning faster than 0.9.

desk verdict The paper's own mass-recalibration formula contradicts Table 2, so the headline spin>0.9 result is not supported by the described method. read the letter →

arxiv 2505.08310 v1 pith:EWD2EMP2 submitted 2025-05-13 astro-ph.HE

M.Yu. Piotrovich , S.D. Buliga , T.M. Natsvlishvili This is my paper · ORCID

classification astro-ph.HE PACS 97.10.Gz98.54.Aj

keywords supermassiveblackholesquasarspinsultraluminousquasarshigh-redshiftradiativeefficiencyaccretiondisksholegrowthepochofreionization

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 estimates the spins of 17 of the most distant known ultraluminous quasars, at redshifts 6–7.5, by converting each quasar's radiative efficiency into a black hole spin. The central result is that the inferred spins are on average greater than 0.9, with a distribution shaped like those of nearer active galaxies. The authors further find that spin increases with cosmic time over the short redshift window, that black hole mass grows while spin grows, and they interpret these correlations as evidence that the earliest supermassive black holes grew mainly by disk accretion at high accretion rate. If right, the work says the same spin-building mechanism seen in local active galaxies was already operating less than a billion years after the Big Bang.

What carries the argument

The operative object is the radiative efficiency $\varepsilon(a)$ of a thin accretion disk around a spinning black hole, obtained two ways: from Eq. (1), a relation connecting $\varepsilon$ to bolometric luminosity, optical luminosity, black hole mass, and the inclination angle through $\mu=\cos i$; and from the relativistic thin-disk expression for $\varepsilon$ in terms of the innermost stable circular orbit radius $R_{\rm ISCO}(a)$, together with the formula for $R_{\rm ISCO}$ itself. Setting the two equal lets each quasar's spin $a$ be solved numerically. The inclination angle enters both through the efficiency relation and through the virial mass estimate, and the paper handles the unknown angle by starting at $45^\circ$ and stepping by $5^\circ$ until a physically allowed spin emerges; masses are re-derived for self-consistency. That inversion is what converts observed luminosities and masses into spin values.

What would settle it

For a few of these quasars, estimate the accretion rate directly from a method independent of the efficiency formula, such as fits to the optical/UV continuum or far-infrared re-emission; if the resulting radiative efficiency $\varepsilon=L_{\rm bol}/(\dot{M}c^2)$ is systematically below about 0.15 for objects with reported spins above 0.9, the central claim is falsified.

Watch

Extended reading notes

Core claim

The paper claims that the supermassive black holes powering ultraluminous quasars at $6<z<7.5$ are fast rotators: most of the 17 objects examined receive spin estimates above 0.9, and the average exceeds 0.9. The spin is not measured directly; it is derived by estimating the radiative efficiency $\varepsilon(a)=L_{\rm bol}/(\dot{M}c^2)$ from a relation tuned to distant quasars (Eq. 1), then inverting the standard thin-disk efficiency formula that ties efficiency to the radius of the innermost stable circular orbit (Eqs. 8–9). The authors also find a moderate anti-correlation between spin and redshift, a strong anti-correlation between black hole mass and redshift, and a strong correlation between spin and mass. From these they argue that the black holes were built by disk accretion with high accretion rate, which raises spin efficiently, and that the same spin-building mechanism seen in nearer active galactic nuclei was already at work in the first billion years.

Load-bearing premise

Every spin value rests on the radiative-efficiency formula of Ref. 29 holding for these extreme high-redshift ultraluminous quasars, even though it was not calibrated in that regime.

Editorial extensions

If this is right

If these estimates hold, the most distant quasars known are powered by near-maximally rotating black holes, with radiative efficiencies $\varepsilon \gtrsim 0.15$ for most objects.
The spin–redshift and mass–redshift correlations imply that, over the roughly $2.4\times 10^7$ years spanned by the sample, black hole mass and spin grew together, consistent with prolonged prograde disk accretion rather than chaotic or merger-dominated spin evolution.
The similarity of the spin distribution to that of lower-redshift quasars and Seyfert galaxies suggests the spin-building mechanism was already in place within the first billion years after the Big Bang.
Because the sample is ultraluminous, selection bias toward high radiative efficiency naturally favors high spin; the paper acknowledges that its average may overstate the typical $z>6$ black hole spin.
The estimated masses shift upward when the inclination-dependent virial factor is used, moving the mass distribution peak to slightly higher values than the literature values.

Reading between the lines

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

Editorial inference: the central result could be tested indirectly by measuring accretion rates independently for a few of the same quasars; if the implied radiative efficiencies fall below about 0.15, the >0.9 spins would not survive.
Editorial inference: the inclination-angle stepping procedure effectively chooses the lowest inclination that yields a valid spin, so the quoted spin values should be read as lower estimates on the spin for the assumed relation; a different angle-picking rule would change the distribution's shape.
Editorial inference: if high spins in the first billion years are confirmed, direct-collapse seed scenarios with sustained coherent accretion become more plausible than models where mergers or chaotic accretion dominate early growth.
Editorial inference: extending the same method to the larger samples of $z>6$ quasars now being found should sharpen the spin–redshift slope from its current uncertainty, which is larger than the claimed slope.

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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

Colleague,

The headline result here — z~6–7.5 quasars having spins mostly above 0.9 — does not survive contact with the paper's own method. In §3 they say that for each inclination they recompute the black hole mass using f≈1/(2 sin i), with the literature mass assumed to have been obtained at i≈35° (f=√3/2). That gives M*(i) = M_lit × (sin35°/sin i)². At i=45° that is a factor 0.66 (log −0.18), yet Table 2 lists log M* slightly above the input log M for every i=45° object. I checked one object: QSO J1007+2115, input log M=9.18, Table 2 gives 9.24; the stated recalibration would give 9.00. If you plug the correctly recalibrated mass into Eq (1), the radiative efficiency drops from 0.174 to about 0.10, and the spin falls below 0.9. So the central claim is an artifact of a table that does not follow the described algorithm.

That is the load-bearing flaw. There are other soft spots, minor by comparison: the Trakhtenbrot (2014) efficiency relation is extrapolated to higher redshift and higher luminosity than its calibration sample; the inclination is tuned per object in 5° steps until the spin becomes "physically meaningful," which invites selection bias; and two objects are excluded post hoc (VHS J0411 for high Eddington ratio, ULAS J1342 for large spin error). The authors do acknowledge the inclination choice and the selection bias toward ultraluminous objects, which is honest.

What the paper does well: it is short, the calculations are transparent enough to check (which is how the inconsistency was found), the sample is real and well-sourced from Zappacosta et al. (2023), and the statistical correlations are reported with their uncertainties. The idea of applying efficiency-based spin estimates to the first quasars is worth exploring; the execution just has a critical misstep.

For a reader: this is not a paper to cite for spin values. It could serve as a cautionary example in a reading group about how easy it is for a small calibration step to drive the entire result. I would not accept the conclusions as they stand, but the underlying question — whether the earliest SMBHs grew by coherent disk accretion — remains interesting, and the authors could fix the mass-recalibration and reissue the table. My recommendation: send it to a referee who will check Table 2 against §3; the authors need to show the correct M*(i) and redo the spins from there.

Referee Report

5 major / 5 minor

Summary. The paper estimates the spins, inclination angles, and black-hole masses of 18 quasars at 6 < z < 7.5 by combining the Trakhtenbrot (2014) radiative-efficiency relation (Eq. 1) with the Bardeen ISCO efficiency formula (Eqs. 8-9). The inclination angle is first set to 45 degrees and then varied in 5-degree steps until a physically allowed spin is obtained, and the virial black-hole masses are supposedly recalibrated for the adopted inclination using Eqs. (3)-(6). After excluding two objects post hoc, the authors report that the estimated spins are on average greater than 0.9, and they interpret the spin-redshift and spin-mass correlations as evidence for rapid disk accretion in the early Universe.

Significance. If the derived spins were reliable, this would be a valuable constraint on supermassive black-hole growth at z > 6, a regime where direct spin measurements are extremely difficult. The paper is transparent in presenting its samples, formulas, and full result tables, and the use of the standard Kerr ISCO efficiency mapping is appropriate. However, the central claim rests on an extrapolated empirical relation, a per-object inclination tuning, and, most importantly, an internal inconsistency between the stated mass-recalibration procedure and the tabulated masses. As written, the headline result is not reproducible from the described method.

major comments (5)

[Section 3, Eqs. (3)-(6), and Table 2] Table 2 is not consistent with the mass-recalibration described in Section 3. Equations (3)-(6) imply M*_BH(i) = M_lit [f(i)/f(35 deg)]^2 = M_lit (sin 35 deg / sin i)^2. For i = 45 deg this factor is 0.66, i.e. log M* = log M_lit - 0.18. Yet every i = 45 deg entry in Table 2 is about 0.06 dex above the input log M_BH in Table 1; for example QSO J1007+2115 has input log M_BH = 9.18 and tabulated log M* = 9.24, while the stated recalibration gives 9.00. Because epsilon in Eq. (1) is proportional to M* mu^1.5, the tabulated efficiencies are too high by roughly a factor 1.7 if the paper's own recalibration is applied. For QSO J1007+2115, epsilon = 0.174 would become about 0.10, corresponding to a about 0.7 rather than 0.934. The 'average spin > 0.9' claim is therefore not a consequence of the described algorithm; either the recalibration step is mis-specified or Table 2 was computed with a different, undocumented method.
[Section 3, inclination-angle choice] The algorithm starts from i = 45 deg and changes i in 5-deg steps until a physically meaningful spin is obtained. Since Eq. (1) scales as cos^1.5 i, this is effectively a per-object free parameter that guarantees a spin in the allowed range 0.039 < epsilon < 0.324. The claim that the resulting angles are lower estimates is not derived from any independent observational constraint on the inclination. The tuning matters for the objects assigned i = 50, 55, or 60 deg in Table 2, whose efficiencies are lowered relative to what they would be at i = 45 deg. The paper should justify why this procedure does not simply select the largest inclinations that keep the spin high.
[Sections 2 and 4, sample selection] Two objects are excluded after examining their derived properties: VHS J0411-0907 is removed because of its unusually high Eddington ratio, and ULAS J1342+0928 is removed after Table 2 because its spin uncertainty is large. These post hoc cuts reduce the sample from 18 to 16 objects, and all reported correlations and the claim of an average spin above 0.9 refer to the reduced sample. The paper should state an a priori criterion for exclusion and show the results with and without these objects, since the correlations in Figs. 10-12 may depend on the cuts.
[Equation (1) and Section 3] The central input Eq. (1) is taken from Trakhtenbrot (2014) without evidence that the relation, calibrated on lower-redshift AGN, remains valid for z > 6 quasars with L_bol ~ 10^47 erg/s and M_BH ~ 10^9 M_sun. The additional bolometric correction of Eq. (2) from Hopkins et al. (2007) introduces further systematic uncertainty. The quoted +/-0.10 dex errors therefore do not capture the dominant uncertainties. A concrete robustness test would be to repeat the calculation with the alternative bolometric corrections cited in Refs. 32, 34-36 and to check whether any of the objects with a > 0.9 becomes a < 0.9.
[Table 2, spin uncertainties] Even taking Table 2 at face value, the spin errors are large and asymmetric, and for several objects the lower 1-sigma bound extends below a = 0.9: for example QSO J1007+2115 has a = 0.934^+0.058_-0.182 and DES J0252-0503 has a = 0.964^+0.034_-0.128. Given these uncertainties, the statement that the spin values are 'on average greater than 0.9' is not supported at the 1-sigma level for a majority of the sample. The paper should report the fraction of objects whose lower error bars remain above 0.9.

minor comments (5)

[Abstract and Conclusion] The phrase 'on average greater that 0.9' should read 'greater than 0.9'.
[Section 4, discussion of Fig. 10] The text around Fig. 10 states that redshifts 6 < z < 7.5 correspond to times from the Big Bang of 9.4 x 10^7 years > t > 7 x 10^7 years. This is off by about an order of magnitude: the corresponding cosmic ages are roughly 0.7-0.9 Gyr, i.e. 7-9 x 10^8 years.
[Section 4, first paragraph] The statement that the estimated SMBH mass distribution peaks at a slightly higher mass than the input distribution because 'we took larger average inclination angles' is inconsistent with Eq. (6): for a fixed FWHM, the virial mass scales as f^2 = 1/(4 sin^2 i), so larger i gives a smaller mass, not a larger one. This wording further highlights the discrepancy between the stated method and the values in Table 2.
[Figures 10-12] The reported Pearson correlation coefficients (r = -0.43, -0.63, 0.67) are given without p-values or confidence intervals. With only 16 objects, these correlations would be usefully quantified by a p-value or a bootstrap interval.
[Table 2] The inclination angles are assigned without uncertainties, as the paper acknowledges. However, since the spin depends strongly on i through Eq. (1), a sensitivity table showing how a and epsilon change when i is varied by +/-5 deg for representative objects would be helpful.

Circularity Check

2 steps flagged · score 6.0 of 10

The spin-mass correlation is built into Eq. (1) (ε ∝ M*), and the inclination angle is scanned until the model output is physical, so the headline 'average a > 0.9' and the spin-mass relation are largely method-generated rather than measured.

self definitional [Section 4, Fig. 12 discussion; Eq. (1) in Section 3]
"The dependence of estimated spin on SMBHs mass demonstrates strong correlation of the parameters in the form of a = (0.17 ± 0.05) log(M*BH/M⊙) − (0.67 ± 0.49), which confirms our assumption that the growth of SMBHs mass should occur mainly due to disk accretion."

Eq. (1) sets ε proportional to M8 μ^1.5 (with Lbol and Lopt factors that are nearly fixed across this sample), and Eq. (8) then maps ε monotonically to the spin a. Therefore a is a deterministic function of the input M* and chosen i. Plotting the derived a against the same M* that was inserted into Eq. (1) cannot provide independent evidence of a spin-mass correlation; the reported 'strong correlation' is generated by the construction ε ∝ M*, not discovered from the data. The conclusion that spin grows with mass is thus a restatement of the input dependence of Eq. (1).
fitted input called prediction [Section 3, paragraph on inclination angle choice; Table 2]
"for each object, we took the average value of the angle i = 45◦ and if the numerical method did not give a physically meaningful result, then we changed the angles to smaller and larger values with a step size of 5◦ until achieving a meaningful outcome."

The inclination angle is not measured but scanned until the output is 'physically meaningful.' Because a is a monotonic function of ε (Eq. 8), and ε depends on i through μ^1.5 and through the recalibrated M*, choosing i is effectively choosing where a lands in the allowed range. The paper later concedes: 'we set the exact value ourselves.' The headline result that spins are on average greater than 0.9 is therefore not a prediction from the data alone; it is a by-product of the allowed parameter scan combined with the normalization of Eq. (1), which for these ultraluminous objects places most allowed solutions at high ε and hence high spin.

full rationale

The central spin estimates are obtained by applying the externally published Trakhtenbrot (2014) relation, Eq. (1), to the input Lbol, Lopt, M_BH, and an assumed inclination, then inverting the Bardeen–Press–Teukolsky ε(a) relation. That step alone is not circular: the calibration is an external empirical relation and no load-bearing self-citation is used; Refs. [44,45] appear only for comparison of the spin-mass slope. However, two parts of the derivation are circular or fitted. First, Eq. (1) makes ε proportional to M*, and since a is a monotone function of ε, the reported a–M* correlation in Fig. 12 is mathematically inherited from the input masses rather than an independent empirical result; this is a prediction that reduces by construction. Second, the inclination angle is a free parameter stepped in 5° increments until the model returns a physical result, with the paper explicitly stating the authors set the value themselves; the reported 'average spin > 0.9' is thus partly a product of this scanning procedure and of the high normalization of Eq. (1) for ultraluminous sources. There is also a serious internal inconsistency, noted for correctness but not counted as circularity: with f ≈ 1/(2 sin i), the stated recalibration should make M* decrease for i > 35°, yet Table 2 lists M* values above the input masses and the text says larger angles make masses 'slightly larger'; this makes Table 2 non-reproducible from the described method and further undermines the robustness of the a > 0.9 claim. Overall, because one central correlation is built into the equations and the headline spin values depend on a fitted inclination scan, the paper merits a partial circularity score of 6.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on adopting Trakhtenbrot's radiative efficiency formula, thin-disk virial scaling, and per-object inclination tuning. No new entities are introduced. The main free parameters are the inclination angles and adopted 0.1 dex uncertainties.

free parameters (2)

Inclination angle i per object = 45, 50, 55, 60 deg depending on object
Chosen as the smallest angle yielding a physically meaningful spin; acts as a tuning parameter.
Adopted uncertainty 0.1 dex for log Lbol, log lE, log MBH = 0.1
Assigned uniformly because Ref 17 does not give uncertainties; affects error bars on spin.

assumptions (3)

domain assumption Shakura-Sunyaev thin disk model and Novikov-Thorne radiative efficiency apply
Required for Eq (1) and the epsilon(a) mapping.
ad hoc to paper Trakhtenbrot (2014) relation (Eq 1) is valid for z>6 ultraluminous quasars
This is the central formula; extrapolated to a new regime without recalibration.
standard math H/R << 1 and f = 1/(2 sin i) for virial mass estimates
Used in Eq (6) to rescale masses with inclination; assumes geometrically thin BLR.

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Cite this review

Pith. "Pith review of Estimating the spins of supermassive black holes in distant ultraluminous quasars." pith.science (2026). https://pith.science/paper/EWD2EMP2

@misc{pith2026250508310,
  author       = {Pith},
  title        = {Pith review of: Estimating the spins of supermassive black holes in distant ultraluminous quasars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EWD2EMP2}},
  note         = {Machine review of arXiv:2505.08310}
}

read the original abstract

We estimated spin, inclination angle and corresponding SMBH mass values for sample of extremely distant (6 < z < 7.5) ultraluminous quasars. The estimated spin values are on average greater that 0.9 and the spin distribution has a characteristic appearance, similar to ones obtained for other types of AGNs and quasars. The dependence of estimated parameters on each other shows strong correlations between them, from which we can assume that in this early quasars the growth of SMBHs mass should occur mainly due to disk accretion with high accretion rate, which very effectively increases the spin.

Figures

Figures reproduced from arXiv: 2505.08310 by the authors.

**Figure 2.** Distributions of objects by SMBH mass MBH We took a sample of 18 distant (6 < z < 7.5) ultraluminous QSOs from Ref. 17 ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Distributions of objects by Eddington ratio [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Distributions of objects by cosmological redshift [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Dependence of the radiative efficiency ε on the spin a (see Eqs.(8,9)). We can estimate the spin of a black hole by determining the radiative efficiency ε(a) of its accretion disk, which depends on the black hole’s spin21–24 (see [PITH_FULL_IMAGE:figures/full_fig_p005…

**Figure 6.** Figure 6: Dependence of the estimated radiative efficiency [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Distribution of the estimated spin a [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Distribution of the estimated SMBH mass M∗ BH 45 50 55 60 0 2 4 6 8 1 0 1 2 N i [ deg] [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Distribution of the estimated inclination angle [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: Dependence of the estimated spin a on cosmological redshift z 0.84 0.8 6 0.88 0. 90 0. 92 0. 94 9. 0 9. 5 1 0. 0 l o g ( M * B H /Msu n ) log(1 + z) log(M* BH /Msun ) = (−6. 91 ± 2. 27) log(1 + z) + (1 5. 53 ± 2. 00) [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: Dependence of the estimated SMBH mass M∗ BH on cosmological redshift z parameters [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 12.** Figure 12: Dependence of the estimated spin a on SMBH mass M∗ BH error margins) lie in the area corresponding to a > 0.9 [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

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

Spins of SMBHs in distant low luminosity AGNs
astro-ph.HE 2025-06 reject novelty 4.0 of 10

Spin estimates for 33 high-redshift low-luminosity AGNs show a spin-mass correlation that the authors attribute to disk accretion.

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