Slow and fat: low-spin SMBHs are more massive

Andrew R. King; Kastytis Zubovas

REVIEW 3 major objections 4 minor 68 references

Slow and fat: low-spin SMBHs are more massive

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper argues that supermassive black holes with low spin end up more massive than high-spin ones in the same host galaxy, because the mass a black hole must add to clear its host galaxy scales as the inverse square of the…

desk verdict A short analytic paper predicting that low-spin SMBHs end up more massive at fixed σ; the direction is robust but the claimed magnitude rests almost entirely on the least-constrained coupling parameter. read the letter →

arxiv 1908.02630 v1 pith:GRK7TXYV submitted 2019-08-07 astro-ph.GA

Kastytis Zubovas , Andrew R. King This is my paper · ORCID

classification astro-ph.GA

keywords supermassiveblackholesAGNfeedbackM-sigmarelationholespinradiativeefficiencygalacticoutflowsgrowthactivenuclei

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 a supermassive black hole (SMBH) keeps growing after it first reaches the critical mass that lets it drive a galaxy-scale outflow, and that this late growth is strongly controlled by the black hole's spin. Because the energy an AGN wind carries into the gas is proportional to the spin-dependent radiative efficiency $\eta$, the extra mass required to clear the galaxy scales as $\eta^{-2}$. Slowly spinning black holes therefore gain up to roughly 20 times more mass than rapidly spinning ones with the same host galaxy velocity dispersion $\sigma$. The result predicts that the most massive black holes at any $\sigma$ are the slowest spinners, so the residuals of the $M{-}\sigma$ relation should anti-correlate with spin, and the slow-spin population should trace the observed upper envelope of black hole masses. A reader should care because this turns the scatter in the $M{-}\sigma$ relation into a probe of black hole spin that upcoming surveys can test.

What carries the argument

The load-bearing object is the energy-balance equation between the binding energy of the galactic gas and the mechanical energy of the AGN wind that must expel it. Writing the wind energy as $E_w = (\eta/2) E_{\rm AGN} = (\eta^2/2) \Delta M_{\rm BH} c^2$, and equating $f E_w$ with $E_{\rm bind} \sim 2 f_g \sigma^4 R/G$ yields $\Delta M_{\rm BH} \propto \sigma^4 R/(f \eta^2)$; using $R_v \propto \sigma$ gives $\Delta M_{\rm BH} \propto \sigma^5$ and, relative to $M_{\rm crit} \propto \sigma^4$, an extra growth $\Delta M_{\rm BH}/M_{\rm crit} \propto \eta^{-2}$. The spin enters only through the average radiative efficiency $\eta$, defined as the mean of the prograde and retrograde efficiencies over many small accretion episodes, so the mechanism is deliberately parameter-free apart from the coupling efficiency $f$.

What would settle it

Measure spins and masses for a large sample of SMBHs in galaxies with a narrow range of $\sigma$. The prediction fails if high-spin black holes ($a > 0.9$) are found systematically above the $M{-}\sigma$ relation, or if the mass residual of a galaxy shows no anti-correlation with spin once selection effects are accounted for.

Watch

Extended reading notes

Core claim

The central claim is that SMBH growth continues by a factor of a few after the black hole reaches the critical mass $M_{\rm crit}$ at which it can drive a large-scale outflow, and that the mass increment $\Delta M_{\rm BH}$ depends on spin as $\Delta M_{\rm BH} \propto \eta^{-2}$, where $\eta$ is the average radiative efficiency of accretion. Since $\eta$ ranges from about 0.055 for a non-spinning black hole to about 0.23 for typical accretion on to a maximally spinning one, the increment can differ by a factor of up to 20 between slow and fast spinners. Combined with the observed size-velocity-dispersion relation, this gives $\Delta M_{\rm BH} \propto \sigma^5$ and steepens the overall $M{-}\sigma$ relation beyond $M \propto \sigma^4$. The paper therefore predicts that at fixed $\sigma$, the most massive black holes have the lowest spins, and that mass residuals from the $M{-}\sigma$ relation correlate strongly with spin.

Load-bearing premise

The calculation assumes a single constant feedback coupling efficiency $f$, estimated at about 0.15 from the observed offset between early- and late-type galaxy $M{-}\sigma$ relations; if the true $f$ is near the upper bound of 0.75, the predicted spin dependence of the extra mass growth nearly vanishes.

Editorial extensions

If this is right

At any fixed $\sigma$, the most massive black holes should have the lowest spins, and the slow-spinning population should trace the upper edge of the observed $M{-}\sigma$ scatter.
The residuals of the $M{-}\sigma$ relation should anti-correlate with SMBH spin, consistent with the observed anti-correlation between mass offset and Eddington ratio.
The $M{-}\sigma$ relation steepens because the late growth scales as $\sigma^5$, bringing the predicted slope closer to observed values around $\alpha \approx 5$.
Before reaching $M_{\rm crit}$, high-spin SMBHs grow more slowly by up to a factor of about 12, so at fixed galaxy mass and redshift there should be a mass-spin anti-correlation that strengthens at high redshift.
The effect is largest in massive, high-$\sigma$ galaxies and in small isolated galaxies least affected by mergers, which are the best laboratories for detecting the mass-spin correlation.

Reading between the lines

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

Editorial inference: If the spin distribution is bottom-heavy, the predicted effect would inflate the scatter of the $M{-}\sigma$ relation at the high-mass end; if it is top-heavy, the relation should show few outliers above the mean, so the shape of the scatter could constrain the spin distribution without direct spin measurements.
Editorial inference: The same energy argument implies that two galaxies with the same $M_{\rm BH}$ and $\sigma$ but different spins have released different amounts of feedback energy, so integrated outflow and star-formation signatures may offer an independent, qualitative spin diagnostic for quenched ellipticals.
Editorial inference: The prediction could be tested by combining reverberation-mapping masses with X-ray reflection spin measurements for a $\sigma$-selected sample; existing samples are too sparse, but a few dozen objects would separate the $a < 0.65$ and $a > 0.9$ predictions at the factor-of-2 level.
Editorial inference: A merger-dominated growth history would produce many high-spin SMBHs and thus many massive outliers above the $M{-}\sigma$ relation; the absence of such outliers would independently support chaotic accretion and slow spins.

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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

This is a short analytic extension of Zubovas & King's wind feedback model. The genuinely new piece is spin: after the black hole reaches the critical mass Mcrit, the extra mass needed to clear the galaxy scales as η^-2, so low-spin holes must grow more than high-spin ones. The derivation is simple, the paper is upfront about its parameter uncertainties, and the prediction of a spin–residual correlation in the M–σ relation is concrete and testable.

What the paper does well: it writes down the energy balance cleanly, gives an explicit range for the coupling efficiency f (0.05–0.75), and does not oversell current data. The observation that low-spin SMBHs should trace the upper envelope of the M–σ relation is a crisp, falsifiable statement. The discussion of how upcoming spin measurements could test it is sensible.

The soft spots are real but not fatal. First, the size of the effect, including the headline '20 times' claim, depends almost entirely on f = 0.15, which is itself calibrated from the same early-type versus late-type offset that the model later uses as a comparison data set in Figure 2. That is partly circular. At the paper's own upper limit f = 0.75, the spin contrast in final masses drops to roughly 60%—comparable to or below the intrinsic scatter of the M–σ relation—so the predicted correlation would be hard to detect. The authors acknowledge this in Figure 1 but still lead with the strong version in the abstract.

Second, there is an internal inconsistency about the σ scaling. The abstract and Section 3 claim ΔMBH ∝ σ^5 by taking Rv ∝ σ, but Figure 2 adopts Rv = 293 σ^2.19 kpc, which would give ΔMBH ∝ σ^6.19. That discrepancy matters because the steepening claim is central to the paper. Third, the calculation treats spin as fixed during the post-Mcrit growth, but if ΔMBH can be a few times Mcrit, the accreted mass itself can change the spin. This is not considered, and it could either strengthen or complicate the predicted correlation.

Who is this for? Anyone working on the origin of M–σ scatter or on indirect spin constraints. It is a useful, clearly written think-piece, and it deserves a serious referee. A referee should, however, push on the f calibration, the inconsistency between the two Rv relations, and the spin-evolution assumption. I would send it to review with major revisions expected, even though I would not cite the quantitative claims without more thought.

Referee Report

3 major / 4 minor

Summary. The paper presents an analytic model for supermassive black hole (SMBH) growth after the black hole reaches the critical mass Mcrit at which its wind can drive a large-scale outflow. Equating the energy required to unbind the host galaxy's gas to the fraction of AGN wind energy absorbed by that gas, the authors derive an additional mass growth ΔMBH ∝ f^{-1} η^{-2} σ^4 R, where f is the wind coupling efficiency, η is the spin-dependent radiative efficiency, and R is the radius to which gas must be expelled. Since η depends on spin, slowly spinning black holes accumulate more mass while clearing their hosts; combining this with a galaxy size–velocity dispersion relation yields a steepening of the M–σ relation. The paper predicts that at fixed σ the most massive black holes have the lowest spins, that M–σ residuals anti-correlate with spin, and that this becomes more pronounced at high σ. The predictions are compared with the observed M–σ relations of McConnell & Ma (2013) and with current, sparse spin estimates, and future observational tests are discussed.

Significance. If the central claim holds, the paper offers a simple physical mechanism linking SMBH spin to the scatter and slope of the M–σ relation, with a falsifiable prediction that can be tested with large samples of spin and mass measurements. The derivation is transparent, uses standard general-relativistic radiative efficiencies, and yields explicit scaling relations rather than a purely numerical fit. The model also correctly identifies that the direction of the effect is robust: because a constant f multiplies all spin-dependent terms, slower spins always produce larger ΔMBH. However, the quantitative strength of the effect and its observability depend heavily on the poorly constrained coupling efficiency f and on the adopted galaxy size scaling, and the current observational comparison is partly circular. These issues do not invalidate the qualitative direction of the prediction, but they materially affect the strength of the claims as written.

major comments (3)

[Section 3, eqs. (6), (7), and Fig. 2] The abstract and Section 3 state that ΔMBH ∝ σ^5, but the size–velocity dispersion relation actually used for Figure 2, Rv = 293 σ_{200}^{2.19} kpc, gives ΔMBH ∝ σ^4 Rv ∝ σ^{6.19}, not σ^5. The σ^5 result follows only from the alternative relation Rv ∝ σ mentioned in the text, which is not the relation used in the comparison plot. Please reconcile this inconsistency: either justify and consistently adopt Rv ∝ σ, or revise the scaling claim to match the adopted Rv(σ) relation.
[Section 3, f calibration and Fig. 2] The feedback coupling efficiency f ≈ 0.15 is estimated from the observed offset between the M–σ intercepts of early-type and late-type galaxies (McConnell & Ma 2013), and the same observed early-type and late-type relations are then used as the comparison data in Figure 2. This makes the apparent agreement in Figure 2 partly a consistency check rather than an independent test of the spin-dependent predictions. The paper should state this explicitly, and ideally show the model predictions for a range of f values against the data so that the sensitivity of the comparison to the calibrated parameter is transparent.
[Equations (10), (11) and Fig. 1; abstract] The headline claim that 'slowly-spinning SMBHs gain potentially 20 times more mass' refers to the ratio of incremental masses ΔM, not the ratio of final black hole masses. At the paper's own upper-limit value f = 0.75, eqs. (10) and (11) give ΔM/Mcrit ≈ 0.67 for a = 0 and ≈ 0.037 for a = 1, so the final masses differ by only about 60%, which is comparable to or smaller than the observed intrinsic scatter of the M–σ relation. Even at the adopted f = 0.15, the final mass contrast is only a factor of about 3.6. The observability of the predicted mass–spin correlation therefore rests almost entirely on the least-constrained parameter f. The authors should quantify this sensitivity in the abstract and discussion, and temper the '20 times' phrasing so that it is not read as a prediction for the final mass ratio.

minor comments (4)

[General] There are several typographical and formatting issues, including 'et a l.' in the affiliation line, 'approximatey' in the discussion of Figure 2, and repeated uses of '∼−' where a single approximation sign is intended (e.g., 'f ∼− 0.75' and '∆MBH ∼− 6.5Mcrit').
[Section 3, after eq. (7)] The sentence 'Substituting this relation into eq. (6) gives ∆MBH ∝ σ^5' implicitly refers to Rv ∝ σ, but the preceding paragraph does not give the normalization or scatter of that relation; adding the explicit relation would help the reader reproduce the scaling.
[Section 5] The statement that available data show 'a general trend of more massive black holes spinning more slowly' is supported only by a list of references without quantitative scatter or sample definitions; a brief statement of the evidence strength would be useful, especially since the next sentence notes selection effects.
[Figure 2] The figure caption and text use fg = 0.05 for the predicted curves while earlier equations use fg = 0.16; the switch is explained in the text but should be reiterated in the caption to avoid confusion.

Circularity Check

1 steps flagged · score 4.0 of 10

Spin ordering is a genuine derivation from GR efficiencies, but the absolute M-σ offset used as evidence is calibrated to the same McConnell & Ma data it is compared against.

fitted input called prediction [Section 3, after Eq. (7); Figure 2 and surrounding text]
"considering the difference in the intercepts of M − σ relation for elliptical and spiral galaxies (McConnell & Ma 2013) leads to ∆MBH ∼− Mcrit and f ∼ 0.15. ... We also adopt f = 0.15. ... the predicted relations for black holes with different spins agree with observations rather well. In particular, the relation for the slowest-spinning SMBHs approximately traces the upper edge of the locus of data points."

The coupling efficiency f is not independently predicted; it is calibrated so that the model reproduces the observed early-type/late-type intercept offset of McConnell & Ma (2013). Figure 2 then adopts this same f = 0.15 and compares the resulting absolute M−σ curves with that same McConnell & Ma sample, claiming the slow-spin curve traces the upper envelope. The amplitude of the predicted offset is therefore partly built into the model from the comparison data. The spin ordering and the factor ∼20 in ΔMBH between a=0 and a≈1 are independent of f, since f cancels in the efficiency ratio, so the circularity is partial rather than total.

full rationale

The core claim, ΔMBH ∝ η^{-2}, follows from the paper's energy balance (Eqs. 5-7) together with standard GR radiative efficiencies (η_a=0 = 0.055 and ⟨η⟩_max = 0.23), neither of which is fitted to the spin data. The often-quoted factor of ~20 is the square of the ratio of these efficiencies, not a fitted parameter. No uniqueness theorem is invoked, and the self-citations (King 2010; Zubovas & King 2012a) anchor the outflow model with the relevant equations presented in the text. The one partially circular element is the calibration of f: it is chosen to match the McConnell & Ma (2013) early/late-type offset, and the same sample and value are then used to claim that the a=0 curve traces the upper envelope of the data. This affects the normalization of the predicted M−σ offset, but not the spin ordering, so the score is 4 rather than higher. Whether the effect remains observable at f = 0.75 is a robustness concern, not circularity.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on a small set of parameters: the coupling efficiency f, the gas fraction fg, and the empirical Rv-sigma scaling. The spin dependence uses standard radiative efficiencies from general relativity and an assumed equal mixture of prograde and retrograde accretion episodes. No data or code are released.

free parameters (3)

Feedback coupling efficiency f = ~0.15
Used in equations (6), (7), (10), and (11). Estimated in Section 3 by matching the inferred ΔM/Mcrit ~ 1 offset between elliptical and spiral M-sigma relations from McConnell & Ma (2013), so it is fitted to data later used for comparison.
Gas fraction fg = 0.16 in analytic estimates; 0.05 in Figure 2
Chosen to represent cosmological or observed baryon fractions. Affects the normalization of Mcrit and ΔM_BH but not the spin ordering.
Virial radius scaling Rv = 293 sigma^2.19 kpc = normalization 293, exponent 2.19
Adopted from Jorgensen & Chiboucas (2013) and Kravtsov (2013) for Figure 2. It drives the claimed steepening, while the text's ΔM ∝ sigma^5 uses the simpler Rv ∝ sigma approximation.

assumptions (7)

domain assumption Momentum-driven outflow criterion gives Mcrit ∝ sigma^4 (equation 1).
Imported from King (2010) and the authors' earlier work; not re-derived here, but parameter-free and widely used in this model.
domain assumption Isothermal sphere distribution for gas and dark matter with binding energy ~ Mg sigma^2 (equations 2 and 3).
Used to compute the gas binding energy. The authors note an NFW profile changes the estimate by about 40%, which is absorbed into f.
domain assumption Wind energy is Ew = (eta/2) EAGN = eta^2 ΔM c^2 / 2 (equation 5).
Assumes a fixed fraction of radiated energy is converted into kinetic wind energy; this factor enters the ΔM ∝ eta^-2 result.
domain assumption Average radiative efficiency over many accretion episodes is the arithmetic mean of prograde and retrograde efficiencies (equation 9).
Assumes equal numbers of prograde and retrograde episodes and similar mass increments, citing King et al. (2005) and King & Pringle (2006). If prograde events dominate, the spread in eta changes.
ad hoc to paper Coupling efficiency f is constant across spin and galaxy type.
The paper assigns f ~ 0.15 everywhere. This is the main calibration parameter and controls whether the spin signal is observable.
domain assumption Black hole spin stays roughly constant during the extra growth phase.
Based on episodic accretion with ΔM1 ~ 10^-3 MBH (King & Pringle 2006; King & Nixon 2015). Required for treating eta as fixed in ΔM ∝ eta^-2.
domain assumption Gas must be expelled to the virial radius to halt further SMBH growth.
For spirals only bulge clearing is needed, lowering the required energy by a factor of about 460; this choice determines whether ΔM is significant.

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Pith. "Pith review of Slow and fat: low-spin SMBHs are more massive." pith.science (2026). https://pith.science/paper/GRK7TXYV

@misc{pith2026190802630,
  author       = {Pith},
  title        = {Pith review of: Slow and fat: low-spin SMBHs are more massive},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRK7TXYV}},
  note         = {Machine review of arXiv:1908.02630}
}

abstract

Active galactic nuclei (AGN) probably control the growth of their host galaxies via feedback in the form of wide-angle wind-driven outflows. These establish the observed correlations between supermassive black hole (SMBH) masses and host galaxy properties, e.g. the spheroid velocity dispersion $\sigma$. In this paper we consider the growth of the SMBH once it starts driving a large-scale outflow through the galaxy. To clear the gas and ultimately terminate further growth of both the SMBH and the host galaxy, the black hole must continue to grow its mass significantly, by up to a factor of a few, after reaching this point. The mass increment $\Delta M_{\rm BH}$ depends sensitively on both galaxy size and SMBH spin. The galaxy size dependence leads to $\Delta M_{\rm BH} \propto \sigma^5$ and a steepening of the $M-\sigma$ relation beyond the analytically calculated $M \propto \sigma^4$, in agreement with observation. Slowly--spinning black holes are much less efficient in producing feedback, so at any given $\sigma$ the slowest--spinning black holes should be the most massive. Current observational constraints are consistent with this picture, but insufficient to test it properly; however, this should change with upcoming surveys.

Figures

Figures reproduced from arXiv: 1908.02630 by the authors.

**Figure 1.** Fractional growth of the SMBH after beginning to drive a large-scale outflow, as function of the SMBH spin parameter a. Two lines correspond to different feedback coupling efficiency, f = 0.75 is very efficient feedback, f = 0.15 is our estimate of a typical value. We use a relation Rv = 293σ 2.19 200 kpc, derived from a combination of the Re − σ (where Re is the galaxy effective radius; Jørgensen & Chiboucas 201… view at source ↗

**Figure 2.** The M − σ relation predicted for SMBHs with different spins. Solid black line shows Mcrit with fg = 0.05 (eq. 1), dashed and dot-dashed black lines show the predicted relation for SMBHs with maximal and zero spin, respectively. Green solid line is the observed M −σ for the whole sample from McConnell & Ma (2013), while the red dashed and blue dash-dotted lines show the observed relation for early- and late- type gal… view at source ↗

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