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REVIEW 4 major objections 4 minor 59 references

The Fundamental Relation between Supermassive Black Holes and Their Host Galaxies

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

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Pith's one-line read This paper argues that a supermassive black hole's mass is set by its host bulge's velocity dispersion alone, making the M_BH-sigma_e relation fundamental and the other scaling relations secondary.

desk verdict The analytic projection method is genuinely new, but the paper's central claim that M-sigma is fundamental flips under its own covariance test, so the abstract oversells it. read the letter →

arxiv 1909.01749 v2 pith:IRGRFIDD submitted 2019-09-04 astro-ph.GA

Stefano de Nicola , Alessandro Marconi , Giuseppe Longo This is my paper · ORCID
classification astro-ph.GA
keywords supermassiveblackholesscalingrelationsFundamentalPlanevelocitydispersiongalaxybulgesAGNfeedbackholemass
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

Using 83 spatially resolved supermassive black hole mass estimates, this paper asks which galaxy property fundamentally determines black hole mass. It confirms the known correlations, finding that the $M_{\rm BH}$--$\sigma_e$ relation has the lowest intrinsic scatter, and that adding effective radius or luminosity does not significantly improve it. The paper then analytically combines black hole masses with the Fundamental Plane, treating the plane as a trivariate Gaussian, and finds that $\sigma_e$ alone predicts the other scaling relations better than any competing variable or plane. The conclusion is that $M_{\rm BH}$--$\sigma_e$ is the fundamental relation, with other scaling laws arising through the Fundamental Plane, and that pseudobulges do not participate because their secular evolution rarely triggers black hole accretion. This matters because it isolates the single observable from which black hole masses can be predicted and points to a physical mechanism, AGN feedback switching from momentum-driven to energy-driven, behind the $M_{\rm BH}$--$\sigma_e$ relation.

What carries the argument

The load-bearing mechanism is the Fundamental Plane modelled as a trivariate Gaussian distribution of luminosity $L$, effective radius $R_e$ and velocity dispersion $V$, with a covariance matrix taken from an optical survey sample. The paper's analytic projection equations express the slope and intrinsic scatter of each monovariate $M_{\rm BH}$ relation (e.g. $M_{\rm BH}$--$L$, $M_{\rm BH}$--$R_e$) as functions of the hyperplane coefficients $A,B,C,\Sigma$ and of the covariance matrix of the trivariate Gaussian. A $\chi^2$ statistic then compares the slopes and scatters predicted by assuming each candidate relation is fundamental with the values measured by direct regression, and the candidate with the lowest $\chi^2$ is identified as fundamental.

What would settle it

Rerun the chi-squared comparison after measuring a full 3x3 covariance matrix of $V$, $L$, $R_e$ from a homogeneous sample of about 100 early-type galaxies in the K/3.6 micron bands; if the early-type variances from the paper's own sample (Table 10) hold, the $\sigma_e$--$R_e$ plane wins, so a definitive test is whether a large infrared sample supports those variances or the SDSS z-band ones.

Watch

Extended reading notes

Core claim

The central claim is that the effective velocity dispersion $\sigma_e$ of a bulge, not a combination of $\sigma_e$ and effective radius $R_e$, is the fundamental link between supermassive black holes and their hosts. Starting from a four-dimensional hyperplane $M_{\rm BH} = A L + B R_e + C V + g_0\Sigma$ and the Fundamental Plane described as a trivariate Gaussian, the paper derives analytic formulas for the slope and intrinsic scatter that each candidate fundamental relation would predict for the others. For the early-type sample with the adopted SDSS z-band covariance matrix, assuming $M_{\rm BH}$--$V$ as fundamental gives the best predictions, with $\chi^2 \simeq 1.4$, far below the alternatives; the Hopkins-type plane $M_{\rm BH}\propto \sigma_e^4 R_e^{0.4}$ also reproduces the relations acceptably when the Fundamental Plane is included, but the canonical $M_{\rm BH}$--$\sigma_e$ relation wins. The authors caution that the ranking depends critically on the adopted covariance matrix: with variances estimated from their own early-type sample, the $\sigma_e$--$R_e$ plane achieves $\chi^2=0.2$ while $\sigma_e$ alone gives $7.7$, reversing the main conclusion.

Load-bearing premise

The analysis assumes that the z-band SDSS covariance matrix of Bernardi et al. (2003b) describes the variances and correlations of $V$, $L$ and $R_e$ for the paper's infrared-selected sample; if the variances are instead estimated from the paper's own early-type sample, the ranking reverses.

Editorial extensions

If this is right

  • If $\sigma_e$ is fundamental, black hole masses can be estimated from a velocity dispersion measurement alone without loss of accuracy; photometric parameters add no predictive power.
  • The $M_{\rm BH}$--$L$ and $M_{\rm BH}$--$R_e$ correlations and the apparent bivariate relations are secondary, produced through the Fundamental Plane rather than reflecting independent physical couplings.
  • The slope near 4 in the $M_{\rm BH}$--$\sigma_e$ relation is consistent with AGN feedback switching from momentum-driven to energy-driven, giving the relation a physical rather than empirical status.
  • Pseudobulges, which grow by secular processes and do not follow the relations, should be excluded when calibrating the fundamental scaling laws.

Reading between the lines

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

  • Editorial inference: because the ranking reverses when the covariance matrix is estimated from the paper's own sample, the claim that $\sigma_e$ is fundamental is not yet settled; the adopted covariance matrix is doing real work in the comparison.
  • Editorial inference: the same projection machinery could be applied to other proposed drivers of black hole growth, such as bulge mass or dark matter halo mass, to test whether any of them beats $\sigma_e$.
  • Editorial inference: if the fundamental relation really is $\sigma_e$ alone, its intrinsic scatter and slope should be invariant across environment and redshift; measuring it in high-redshift early-type galaxies would be a direct test of the feedback explanation.
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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 / 4 minor

Summary. The paper compiles a sample of 83 supermassive black hole masses with host-galaxy parameters (σe, L, Re), fits mono- and bivariate scaling relations, and introduces an analytic framework that combines the BH-host relation with the Fundamental Plane through a trivariate Gaussian covariance model. The central claim is that MBH–σe is the fundamental relation, rather than a bivariate 'BH Fundamental Plane' of the form MBH ∝ σ^4 Re^β. The evidence for this claim is a χ² comparison (Eq. 17) between observed slopes/scatters and those predicted by assuming each candidate relation to be fundamental. With the z-band SDSS covariance of Bernardi et al. (2003b), the velocity dispersion V gives the lowest χ² for early types (Table 6). However, when the variances are instead fitted to the authors' own sample (Tables 8 and 10), the ranking reverses: M_Hop gives χ²=0.2 while V gives χ²=7.7 for early types (Table 11). The paper itself concludes that the result 'critically depends on the covariance matrix one chooses out for the analysis,' yet the abstract states that MBH–σe 'appears to be the fundamental relation.' The central claim is therefore not supported by the paper's own evidence.

Significance. If the analytic projection framework were robust, it would offer a useful way to connect BH scaling relations to the Fundamental Plane and to test candidate fundamental relations. The compiled sample and the regression tables are useful resources, and the paper is transparent about the covariance dependence in its conclusions. However, the main scientific claim—that MBH–σe is fundamental and the BHFP is not—is directly contradicted by the authors' own self-consistency test in Section 4.4. Because the preference flips with a plausible change in the assumed covariance matrix, the paper as written does not establish its central conclusion.

major comments (4)
  1. [Abstract and Section 4.4, Table 11] The abstract's claim that MBH–σe 'appears to be the fundamental relation rather than a putative BH Fundamental Plane' is contradicted by the paper's own alternative covariance test. For the early-type sample, using variances fitted to the authors' data (Table 10) yields χ²(M_Hop)=0.2 and χ²(U_grav)=2.0, while χ²(V)=7.7 (Table 11). This reverses the ordering obtained with the Bernardi z-band covariance (Table 6: χ²(V)=1.4, χ²(M_Hop)=7.4). Since the central conclusion depends on which covariance matrix is adopted, the abstract overstates the result and should either be revised to report the covariance dependence or be supported by a justification for preferring one covariance matrix.
  2. [Section 4.4, Tables 8 and 10] The alternative covariance test is incomplete and therefore cannot rescue the central claim. The authors refit only the variances while keeping the correlations fixed at the Bernardi et al. (2003b) values from Table 4. A full maximum-likelihood covariance matrix estimated from their own early-type sample is never tested. The paper itself cautions that 'using a covariance matrix estimated from a biased and heterogeneous sample can significantly alter the results of the analysis,' but the abstract does not carry this caveat. The reader is left with the ordering of fundamental-relation candidates depending on an untested part of the covariance model.
  3. [Section 4.3, Eq. (17)] The χ² comparison in Eq. (17) includes only slopes and intrinsic scatters, not zero-points. The model predictions from Eqs. (12)–(16) likewise predict only α and ε, so a candidate 'fundamental relation' that predicts the right slope and scatter but the wrong normalization would be treated as successful. Since the paper's stated goal is to identify which relation 'is able to reproduce their slopes and intrinsic scatters' (Section 4), the omission of zero-points may be intentional, but it is load-bearing and should be explicitly justified or the χ² should be extended to include zero-point agreement.
  4. [Conclusions, Section 5] The Conclusions explicitly state that the result 'critically depends on the covariance matrix one chooses out for the analysis' and that 'the whole picture is still uncertain.' This is not merely a caveat; it directly undermines the abstract's definitive-sounding claim. A reader of the abstract alone cannot know that the paper's own favored conclusion is sensitive to a modeling choice in the way quantified by Table 11. The paper should either present the covariance-dependent result as the main finding or provide a criterion for selecting among covariance matrices.
minor comments (4)
  1. [Section 4.1] In the sentence 'All the variables of appearing in this equation (MBH, L, V and Re) are logarithmic,' the word 'of' appears to be a typo and should be removed.
  2. [Section 4.2] The sentence 'we speculate that these six parameters might be so well constrained' appears to mean 'might not be so well constrained'; the missing negation makes the methodological motivation unclear.
  3. [Figure 5 caption] The phrase 'with respect to to the new best-fit line' contains a duplicated 'to' and should be corrected.
  4. [Section 4.3] The linear combinations W = aV + bRe are stated with 'a and b are integers,' but the example M_Hop uses b=0.4, which is not an integer; this should be clarified as 'integers or fixed real coefficients.'

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the model-comparison test uses an external FP covariance and is explicitly acknowledged to be covariance-dependent.

full rationale

The paper's central comparison is a model-selection exercise rather than a derivation that reduces to its own inputs. The authors adopt the Bernardi et al. (2003b) z-band FP covariance matrix as an external input, fit the four-dimensional hyperplane (eq. 4) to their MBH data, then assume each candidate relation in turn to be fundamental, set the other slopes to zero, and project through eqs. 12, 14, and 16 to predict the slopes and intrinsic scatters of the other monovariate relations. These predicted values are not identical to the fitted inputs: the candidate's fitted slope and scatter are used to obtain different relations' slopes and scatters, and the covariance matrix is not derived from the MBH fits. The formulas are analytic projections, not tautologies. The self-citations present (e.g., Marconi & Hunt 2003; Savorgnan et al. 2013) are background references and are not load-bearing in the central chain. The paper itself explicitly flags the main limitation in Sec. 4.4 and the Conclusions: when the variances are instead fitted to the authors' own sample, the ordering reverses, with M_Hop and U_grav outperforming V (Tabs. 9 and 11). The text states that the result 'critically depends on the covariance matrix one chooses out for the analysis.' This is a genuine robustness/correctness concern about sensitivity to an assumed input, but it is not circularity: the input is independent external data, and the conclusion is not enforced by definition or by a self-referential fit. Because no prediction is equivalent to its inputs by construction and no load-bearing self-citation chain is invoked, the appropriate circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The derivation's core input is the joint distribution of V, L, Re encoded in the covariance matrix; changing it flips the main conclusion (Sec. 4.4), so the covariance parameters are the most important free components. The projection formulas then propagate the fitted hyperplane slopes. No new physical entities are introduced.

free parameters (4)
  • Hyperplane coefficients A, B, C, Sigma (eq. 4) = A=-0.12±0.33, B=0.56±0.33, C=4.18±0.48, Sigma=0.35±0.04 (Sec. 4.1)
    These fitted coefficients define the four-dimensional relation MBH = A L + B Re + C V + g0 Sigma that is then projected onto each axis to derive predicted slopes and scatters.
  • FP covariance parameters (sigma_V, sigma_L, sigma_Re, rho_VL, rho_VRe, rho_ReL) = Bernardi et al. (2003b) z-band values in Tab. 4; alternate ML-fitted variances in Tabs 8 and 10
    The joint distribution of V, L, Re is an input that determines the projected slopes. The paper shows the central conclusion flips when these parameters are changed, so they are effective free parameters of the analysis.
  • Fitted slope and scatter of the assumed fundamental relation = e.g. C=4.32, epsilon=0.41 for MBH-V (Tab. 2); similar for L, Re, M_Hop, Mvir, Ugrav
    When testing a candidate fundamental relation, its fitted slope and intrinsic scatter are fed directly into eqs. 12, 14, 16 to predict the other relations.
  • Weights a, b in linear combinations W = a V + b Re = a=2,b=1 (Mvir); a=4,b=0.4 (M_Hop); a=4,b=1 (Ugrav)
    These integer weights are chosen by hand to represent physically motivated combinations, affecting which candidate relations are compared.
assumptions (5)
  • standard math The Fundamental Plane of early-type galaxies is well described by a trivariate Gaussian distribution of V, L, Re.
    Adopted from Bernardi et al. (2003b) in Sec. 4.2, eqs. 6-10; this is the statistical backbone of the projection formulas.
  • domain assumption The z-band SDSS covariance matrix applies to the authors' K-band / 3.6 micron photometry.
    Explicitly stated in Sec. 4.2: 'assuming that both variances and correlations do not change significantly'. The paper's main conclusion relies on this untested transferability.
  • domain assumption Spatially resolved BH mass estimates from the literature (stellar dynamics, gas dynamics, masers) are reliable and approximately unbiased.
    Sec. 2.1 uses 83 published MBH values, excluding virial/reverberation methods. The authors acknowledge the sample is biased toward high masses (Shankar et al. 2016), which weakens the slopes and scatters.
  • domain assumption Classical bulges from decomposed spirals follow the same Fundamental Plane as early-type galaxies.
    Sec. 4.3 states 'classical bulges behave in the same way as early-types', allowing reuse of the Bernardi covariance matrix for the ClBul sample.
  • domain assumption The momentum-driven to energy-driven AGN feedback model is the correct physical explanation of the MBH-sigma relation.
    Invoked in Sec. 5 and cited to King & Pounds (2015); used as interpretation, not as part of the statistical derivation.

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Cite this review

Pith. "Pith review of The Fundamental Relation between Supermassive Black Holes and Their Host Galaxies." pith.science (2026). https://pith.science/paper/IRGRFIDD

@misc{pith2026190901749,
  author       = {Pith},
  title        = {Pith review of: The Fundamental Relation between Supermassive Black Holes and Their Host Galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRGRFIDD}},
  note         = {Machine review of arXiv:1909.01749}
}
read the original abstract

We study the correlations between Supermassive Black Holes (BH) and their host galaxies, using a sample of 83 BH masses collected from the most recent and reliable spatially resolved estimates available from the literature. We confirm the mono- and bivariate correlations between SMBHs and the bulges of their host galaxies, confirming that the correlation with the effective velocity dispersion is not significantly improved by higher dimensionality. Instead, pseudobulges do not seem to correlate with their SMBHs, probably because their secular evolution is often unable to trigger accretion onto the central BH. We then present a novel approach aimed at finding the fundamental relation between SMBHs and their host galaxies. For the first time, we analytically combine BH masses with the Fundamental Plane (FP), showing that Mbh-sigma_e appears to be the fundamental relation rather than a putative "BH Fundamental Plane" of the kind Mbh-sigma_e-R_e. These results can be explained by a picture which sees the Mbh-sigma_e relation as a natural outcome of the change in AGN feedback from momentum- to energy-driven. The other scaling relations are then established through the FP.

Figures

Figures reproduced from arXiv: 1909.01749 by the authors.

Figure 1
Figure 1. Comparison of the K-band and the Spitzer photometries for the galaxies of our sample for which both measurements are avilable. The red lines are the 1:1 lines. The values are in good agreement with the Spitzer radii being on average slightly larger, which can be expected since Spitzer data are deeper. slope. Since the intrinsic scatter embeds all factors not ac￾countable with measurement errors, it appears that SMBH… view at source ↗
Figure 2
Figure 2. Monovariate correlations between BH masses and galaxy parameters (upper row: σe, lower row, left: L, lower row, right: Re). Galaxies are colored according to the T flag defined in Tab. 1 of Saglia et al. (2016) (Col. 2 of Tab. 1). Red points are omitted from the regressions (see App. A). The intrinsic scatter and the Spearman’s coefficient are printed on the bottom-right of the plot. The dashed lines delimit the ran… view at source ↗
Figure 3
Figure 3. Bivariate correlations between BH masses and galaxy parameters (left: MBH -σe-L, right: MBH -σe-Re). Symbols and color coding are the same as in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The 3D representation of the putative “BH hy￾perplane”. Different symbols denote different galaxy subsets, as shown in the legend. Points are colored according to their MBH values. where the g’s are Gaussian random numbers with zero mean and unit variance, MNRAS 000, 1…
Figure 5
Figure 5. Figure 5: Left: Effective radius Re as a function of the FP relation found by Bernardi et al. (2003c). Our galaxies follow that FP, whose equation is reported on the x-axis, but are larger than expected. Right: Distribution of the residuals (normalized to unity) with respect to …

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    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

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