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REVIEW 4 major objections 5 minor 70 references

The impact of applying black hole-host galaxy scaling relations to large galaxy populations

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Using a 400,000-galaxy sample, this paper shows that the choice of black hole scaling relation changes inferred supermassive black hole masses by factors of a few to orders of magnitude, and that the velocity-dispersion-based relation…

desk verdict A transparent, useful population comparison of scaling relations whose headline numbers need error bars and a sensitivity test on the virial factor transfer. read the letter →

arxiv 2506.08102 v1 pith:VKDYSWXH submitted 2025-06-09 astro-ph.GA

Maggie C. Huber (1) , Joseph Simon (1) , Julia M. Comerford (1) ((1) University of Colorado Boulder) This is my paper
classification astro-ph.GA
keywords supermassiveblackholesholescalingrelationsM_BH-sigmaM_BH-M_bulgesingle-epochvirialmassesgalaxymorphologySDSSgalaxiesgravitationalwavebackground
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

Most estimates of supermassive black hole masses in large galaxy samples rely on empirical relations calibrated on a few dozen nearby galaxies, and this paper asks how much the choice of relation matters for a large population. Using a mass-complete sample of about 400,000 SDSS galaxies with measured bulge masses and velocity dispersions, it shows that the $M_{BH}$--$M_{\rm bulge}$ and $M_{BH}$--$\sigma$ relations agree only for high-mass, bulge-dominated systems and diverge by factors of a few to orders of magnitude for disk-dominated or lower-mass galaxies. When both relations are compared against single-epoch virial H$\beta$ masses for 1,240 Type 1 AGN, the $M_{BH}$--$\sigma$ relation reproduces the virial masses much better (median ratios of about 1.2 versus 2.9 for McConnell & Ma 2013, and 2.7 versus 4.2 for Kormendy & Ho 2013). The paper also finds that velocity dispersions inferred from photometry are reliable, while bulge masses inferred from assumed bulge fractions can be off by orders of magnitude. These results imply that population-level black hole mass functions, and conclusions drawn from them about the gravitational wave background and black hole-galaxy coevolution, should be treated as scaling-relation-dependent outside the narrow subset used for calibration.

What carries the argument

The argument is carried by the two power-law scaling relations, $\log(M_{BH}/M_\odot) = \alpha + \beta \log X$ with $X = \sigma/200$ km/s or $M_{\rm bulge}/10^{11} M_\odot$, and by the single-epoch virial H$\beta$ mass estimator of Shen et al. (2024) -- a relation of the form $M_{SE} \propto L_{5100}^{0.5}\,\mathrm{FWHM}^2$ that calibrates its virial factor against dynamical masses of roughly 30 reverberation-mapped AGN, making it independent of the $M_{BH}$--$\sigma$ scaling relation. The sample of 413,538 SDSS galaxies with both measured bulge masses (from bulge-disk decompositions) and measured spectroscopic velocity dispersions is the enabling asset: it lets the paper separate the effect of the scaling relation itself from the effect of poorly inferred host properties. The comparison of inferred properties to measured properties, and the ratio diagnostics $M_{BH,\,M_{\rm bulge}}/M_{BH,\,\sigma}$ and $M_{BH,\,\mathrm{scaling}}/M_{SE,\,H\beta}$, are the quantitative tools that expose where the relations diverge.

What would settle it

Measure dynamical supermassive black hole masses (via spatially resolved gas or stellar kinematics) for a sample of disk-hosted and low-mass AGN spanning the same mass range as the 1,240 AGN studied here; if the dynamical masses correlate better with $M_{BH}$--$M_{\rm bulge}$ than with $M_{BH}$--$\sigma$, the paper's central ranking would be reversed. Alternatively, recomputing the comparison with the H$\alpha$ or MgII single-epoch recipes would show whether the 1.2-versus-3 result is an artifact of the H$\beta$ estimator.

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Extended reading notes

Core claim

The central claim is that the choice of black hole-host galaxy scaling relation changes inferred supermassive black hole masses by factors of a few to orders of magnitude depending on galaxy mass and morphology, and that $M_{BH}$--$\sigma$ reproduces masses from the single-epoch virial H$\beta$ method better than $M_{BH}$--$M_{\rm bulge}$ does. The paper establishes this by applying the two most widely used relations, $M_{BH}$--$M_{\rm bulge}$ and $M_{BH}$--$\sigma$, in the forms given by McConnell & Ma (2013) and Kormendy & Ho (2013), to 413,538 SDSS galaxies with well-defined bulge masses and aperture-corrected velocity dispersions, and by comparing the resulting masses against single-epoch virial H$\beta$ masses from Shen et al. (2024) for 1,240 Type 1 AGN. The median ratio of scaling-relation mass to single-epoch mass is 1.21 for $M_{BH}$--$\sigma$ versus 2.85 for $M_{BH}$--$M_{\rm bulge}$ (McConnell & Ma), and 2.65 versus 4.23 (Kormendy & Ho), so the bulge-based relations run systematically high. The two scaling relations agree only for galaxies with $\log(M_*/M_\odot) \gtrsim 11$ and prominent bulges; for disk-dominated systems $M_{BH}$--$M_{\rm bulge}$ predicts roughly three times lower masses than $M_{BH}$--$\sigma$. A separate result is that photometrically inferred velocity dispersions agree well with spectroscopic measurements, whereas bulge masses inferred from color-and-mass-based bulge fractions have large tails and can overestimate the decomposed bulge mass by orders of magnitude.

Load-bearing premise

The load-bearing premise is that the single-epoch black hole masses used as the comparison are an unbiased yardstick for these 1,240 active galaxies: the conversion factor linking broad-line width and luminosity to mass was calibrated on about 30 nearby galaxies, and it must transfer to the SDSS sample without a systematic offset for the ranking of the velocity-dispersion relation over the bulge-mass relation to hold.

Editorial extensions

If this is right

  • Black hole mass functions built from $M_{BH}$--$M_{\rm bulge}$ will be systematically offset from those built from $M_{BH}$--$\sigma$ for disk-hosted and low-mass galaxies, since bulge-based estimates run higher by factors of 2-4 even where they agree in ranking.
  • Gravitational wave background amplitude predictions that rely on bulge-based mass functions may be biased relative to sigma-based predictions, which matters for interpreting the pulsar timing array signal.
  • Photometric surveys without spectroscopy can still recover usable velocity dispersions from total stellar mass, effective radius, and Sersic index, but bulge masses inferred from assumed bulge fractions should be regarded as highly uncertain.
  • Simulations and empirical models that calibrate black hole growth against scaling relations must state which relation they use; results calibrated to $M_{BH}$--$M_{\rm bulge}$ versus $M_{BH}$--$\sigma$ are not interchangeable for disk-dominated galaxies.
  • The better performance of $M_{BH}$--$\sigma$ against virial masses supports the view that velocity dispersion is the more fundamental host-property correlate of black hole mass.

Reading between the lines

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

  • The comparison could be extended to the H$\alpha$, MgII, and CIV single-epoch estimators from Shen et al. (2024) to test whether the ranking of $M_{BH}$--$\sigma$ over $M_{BH}$--$M_{\rm bulge}$ is robust to the broad-line species and BLR geometry assumptions.
  • At higher redshifts, where only photometric inferences are possible, the paper's results imply that sigma-based mass estimates will capture the evolving mass-size relation while bulge-based estimates will not, so the divergence between the two scaling relations should grow with redshift.
  • For gravitational wave background modeling, the implication is that the choice of bulge-fraction prescription is a dominant systematic for bulge-based mass functions; the improved prescription proposed here could be propagated into pulsar timing array analyses to quantify that systematic.
  • A direct dynamical mass measurement campaign on a sample of disk-hosted AGN would turn the relative ranking into an absolute test: if dynamical masses track $M_{BH}$--$M_{\rm bulge}$ instead of $M_{BH}$--$\sigma$, the yardstick itself would need to be rethought.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper uses a sample of ~413,538 SDSS DR7 galaxies with bulge-disk decompositions and measured velocity dispersions to examine how the choice of black hole-host galaxy scaling relation affects SMBH mass estimates for large galaxy populations. The authors compare MBH-Mbulge and MBH-sigma masses from McConnell & Ma (2013) and Kormendy & Ho (2013), and for 1,240 Type 1 AGN they compare these scaling-relation masses to single-epoch virial Hβ masses from Shen et al. (2024). The central results are that (i) the two scaling relations agree only for high-mass, bulge-dominated galaxies, (ii) MBH-sigma reproduces the single-epoch Hβ masses better than MBH-Mbulge, and (iii) photometric inference of sigma is more accurate than inference of Mbulge. The authors are careful to note that single-epoch masses are not ground truth.

Significance. If the ranking result is robust, the paper provides a useful caution for GWB analyses, cosmological simulations, and SMBH mass function studies that rely on scaling relations for large populations. Strengths include the use of public catalogs, a transparent sample construction, the explicit choice of a single-epoch estimator calibrated without assuming MBH-sigma, and the recognition that single-epoch masses are not ground truth. The main limitation is that the headline ranking is not supported by statistical uncertainties or sensitivity tests, and the improved f_bulge prescription is claimed to be best without a quantitative comparison. These issues are fixable within the manuscript's scope.

major comments (4)
  1. [Section 3.2, Figure 4] The median ratios (1.21 vs 2.85 for McConnell & Ma 2013; 2.65 vs 4.23 for Kormendy & Ho 2013) are quoted without confidence intervals or significance tests, so the claim that MBH-sigma 'better reproduces' single-epoch Hβ masses is not statistically supported. Because all four median ratios exceed unity, a constant multiplicative offset in M_SE can in principle change which ratio is closer to 1; for the McConnell & Ma numbers, a systematic shift of about 0.3 dex in the virial factor would move the sigma ratio below unity and reverse the ranking. The authors should provide bootstrap uncertainties on the medians and a sensitivity analysis varying the virial factor normalization and allowing for luminosity/FWHM-dependent f, or soften the abstract claim accordingly.
  2. [Section 2.2 and Section 3.3] The text in Section 2.2 states that the proposed f_bulge prescription 'best recovers inferred Mbulge compared to the Mendel et al. (2013) results,' but no quantitative comparison to other prescriptions (e.g., Sesana 2013, Ravi et al. 2015, Arzoumanian et al. 2021) is shown. This claim is load-bearing for the conclusion in Section 3.3 that inferred Mbulge is less reliable than inferred sigma, since the conclusion depends on the adopted prescription being a fair representative of the method. A figure or table reporting median offsets and scatter for each prescription would substantiate the claim.
  3. [Section 3.3, Figure 6] The caption for Figure 6 says 'with 1,240 AGN in each panel,' but the density colorbar labeled 'Galaxies per pixel' and the surrounding text suggest the full 413,538-galaxy sample is plotted. The caption and text should be reconciled, and the comparison of inferred vs measured properties should report quantitative dispersions (e.g., median offset and scatter in dex) rather than only qualitative statements about being 'centered on one.'
  4. [Section 2.6 and Section 3.2] The Hellinger distance analysis restricts the representativeness of the AGN hosts to 10 < log(M*/M_sun) < 11.5, but the single-epoch comparison in Section 3.2 uses all 1,240 AGN, many of which lie outside this mass range according to Figure 2. Because Figure 3 shows that the offset between MBH-Mbulge and MBH-sigma depends strongly on total mass, the mass distribution of the AGN hosts could bias the medians in Figure 4. The analysis should be repeated or at least discussed for the representative mass range, or the caveat should be brought into the abstract and conclusion.
minor comments (5)
  1. [Section 2.2] The sentence 'We find that this prescription best recovers inferred Mbulge compared to the Mendel et al. (2013) results' is grammatically unclear; it should be reworded to indicate that the prescription best recovers the measured bulge masses from Mendel et al. (2013) relative to other prescriptions.
  2. [Section 2.6] The word 'statisticial' should be 'statistical' in the Hellinger distance sentence.
  3. [Section 2.1] The phrase 'Point-Spead-Function' should be 'Point-Spread-Function.'
  4. [Figure 4] The caption uses rounded factors ('offset by a factor of 3', 'offset by 4'); for consistency with the text, include the exact median values (1.21 and 2.85 for McConnell & Ma 2013; 2.65 and 4.23 for Kormendy & Ho 2013).
  5. [Table 2] The 2D Gaussian fit parameters are quoted without uncertainties, and the angle θ is not labeled as degrees or radians; please add uncertainties and define the angle convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the single-epoch Hβ masses are an external, independently calibrated benchmark, and the paper explicitly avoids the scaling-relation-calibrated virial factor.

full rationale

The central comparison is not circular by construction. The reference masses M_SE,Hβ come from the Shen et al. (2024) single-epoch estimator (Eq. 8), whose mean virial factor was calibrated by comparing the virial product to dynamical SMBH masses of roughly 30 reverberation-mapped AGN, not by fitting to either M_BH-σ or M_BH-M_bulge. The paper even rejects the Liu et al. (2019) single-epoch masses for this analysis because that estimator's virial factor was obtained by assuming the M_BH-σ relation, and it explicitly states that the Shen et al. calibration 'avoids a circular comparison with M_BH-σ' (Section 2.6). The scaling relations themselves are external published fits (McConnell & Ma 2013; Kormendy & Ho 2013) with fixed slopes and intercepts; no parameter is refit to the 1,240 AGN to produce the ranking. The inferred-host-property comparison likewise employs external calibrations (Bezanson et al. 2011; Bertin et al. 2002) rather than fitting the same SDSS data, so the statement that σ can be inferred more accurately than M_bulge is a validation exercise, not a renaming of the input. The only residual concern is that the f-calibration anchor points and the local scaling relations share some dynamical mass measurements, and a constant offset in f would shift both median ratios; however, f is a single multiplicative constant fit on an external sample, so the ranking between the two scaling relations is not forced by construction. The paper also repeatedly disclaims that single-epoch masses are not ground truth (Sections 2.6 and 4), further showing that the authors are not presenting a circular argument as an independent prediction. Self-citations to Simon & Burke-Spolaor (2016) and Simon (2023) are motivational context for the GWB relevance and are not load-bearing for any derived result.

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

The analysis is almost entirely empirical and inherits its numerical content from literature-fitted scaling relations, a recently calibrated single-epoch estimator, and hand-chosen bulge fraction priors. No new entities are introduced. The main burden is the transferability of local calibrations to this sample and the active/quiescent universality assumption, both explicitly acknowledged in the paper.

free parameters (5)
  • McConnell & Ma (2013) scaling relation coefficients = Mbulge: alpha=8.46+/-0.08, beta=1.05+/-0.11; Msigma: alpha=8.32+/-0.05, beta=5.64+/-0.32
    Equation 6 applied to all 413,538 galaxies; offsets and the ranking against single-epoch masses depend on these externally fitted values.
  • Kormendy & Ho (2013) scaling relation coefficients = Mbulge: alpha=8.69+/-0.04, beta=1.16+/-0.06; Msigma: alpha=8.49+/-0.05, beta=4.38+/-0.29
    Cross-check set of scaling relations; same central dependence on literature-fitted values.
  • Shen et al. (2024) single-epoch Hbeta mass estimator constants = offset 0.85, L5100 slope 0.5, FWHM slope 2, intrinsic scatter 0.45 dex
    Equation 8 defines the reference masses for the main comparison; constants were fit to about 30 RM AGN dynamical masses.
  • Improved f_bulge prescription parameters = early type: 0.9 decreasing to 0.10 from 10^11 to 10^9 Msun; late type: 0.25 to 0.10; random interval +/-0.1
    Section 2.2; hand-chosen values based on Cappellari kinematic measurements drive the inferred Mbulge comparison in Section 3.3.
  • Aperture correction and virial constant parameters = aperture exponent 0.066; K_n(n)=73.32/(10.465+(n-0.94)^2)+0.954
    Equations 2 and 4 convert SDSS sigma to effective-radius sigma and infer sigma from photometry; values are inherited from Bezanson et al. 2011 and Bertin et al. 2002.
assumptions (5)
  • domain assumption Local MBH-host scaling relations apply to SDSS galaxies at 0.02<=z<=0.2 without significant redshift evolution.
    The pipeline assigns MBH using MM13 and KH13 relations calibrated on nearby dynamical samples; the paper notes evolution is still unclear in Section 2.4.
  • domain assumption Active and quiescent galaxies follow the same MBH-host scaling relations.
    Stated in Section 3.2; needed to extend the AGN single-epoch comparison to the full quiescent population.
  • domain assumption The Shen et al. (2024) average virial factor, calibrated on about 30 RM AGN dynamical masses, transfers to the 1,240 SDSS AGN without sample-dependent bias.
    Section 2.6; the reference masses in Equation 8 inherit this calibration.
  • domain assumption The Bezanson et al. (2011) virial-theorem sigma inference (Equation 5) is unbiased for all morphological types except a known 0.1 dex offset for Type 2 galaxies.
    Section 2.4; underpins the photometric sigma inference result in Section 3.3.
  • domain assumption Mendel et al. (2013) bulge-disk decomposition masses with the Chabrier IMF are sufficiently accurate to serve as measured Mbulge.
    Sections 2.1 and 2.3; the paper notes IMF sensitivity at the high-mass end but uses these masses as the measured reference.

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Pith. "Pith review of The impact of applying black hole-host galaxy scaling relations to large galaxy populations." pith.science (2026). https://pith.science/paper/VKDYSWXH

@misc{pith2026250608102,
  author       = {Pith},
  title        = {Pith review of: The impact of applying black hole-host galaxy scaling relations to large galaxy populations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VKDYSWXH}},
  note         = {Machine review of arXiv:2506.08102}
}
abstract

Supermassive black holes (SMBHs) with dynamically measured masses have shown empirical correlations with host galaxy properties. These correlations are often the only method available to estimate SMBH masses and gather statistics for large galaxy populations across a range of redshifts, even though the scaling relations themselves are derived from a small subset of nearby galaxies. Depending on the scaling relation used, estimated SMBH masses can vary significantly. The most widely used scaling relations are the M$_{BH}-$M$_{\mathrm{bulge}}$ and M$_{BH}- \sigma$ relations, where M$_{\mathrm{bulge}}$ is galaxy bulge mass and $\sigma$ is the bulge velocity dispersion. In this paper, we determine how severely the choice of scaling relation impacts SMBH mass estimates for different subsets of a large galaxy population. For this analysis we use a sample of $\sim$ 400,000 galaxies, including 1,240 Type 1 AGN from the Sloan Digital Sky Survey. We calculate SMBH masses from M$_{BH}-$M$_{\mathrm{bulge}}$ and M$_{BH}- \sigma$ and compare to single-epoch virial SMBH masses from broad-line H$\beta$, which are derived independently of black hole-host galaxy scaling relations. We find that SMBH masses derived from the single-epoch virial relation for H$\beta$ are better reproduced by M$_{BH}- \sigma$ than M$_{BH}-$M$_{\mathrm{bulge}}$. Finally, in cases where $\sigma$ and M$_{\mathrm{bulge}}$ cannot be measured directly, we show that it is possible to infer $\sigma$ from photometry with more accuracy than we can infer M$_{\mathrm{bulge}}$.

Figures

Figures reproduced from arXiv: 2506.08102 by the authors.

Figure 1
Figure 1. Here, our sample of 413,538 galaxies is visualized as a 2D histogram on a total mass vs. redshift plot. The colorbar indicates the galaxy count in each bin. Laurikainen et al. 2005). However, Simard et al. (2011) note that they deliberately choose two-component fits to allow interfacing between their results and higher-redshift galaxy popu￾lations, where three-component fits are largely unavailable. This is especial… view at source ↗
Figure 2
Figure 2. Here, our sample of 1,240 Type 1 AGN are red points overplotted on the entire sample of 413,538 galaxies on a total mass vs. redshift plot (same as in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The SMBH mass ratio for total mass and galaxy type bins, where the types are spaced out on the x-axis for clarity. Type 1 galaxies are single-component bulges, Type 2 are disk-dominated galaxies, Type 3 are bulge+disk systems, and Type 4 are irregular galaxies. Figure (a) shows the offset between the McConnell & Ma (2013) scaling relations, where MBH−Mbulge predicts more massive SMBHs than MBH −σ for low mass Type 1… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Normalized histogram of the ratio between the SMBH mass scaling relations (Equation 6) and the single-epoch mass estimator from Hβ (Equation 8) on a log scale. The red and blue lines represent the medians of the mass ratio with MBH − σ and MBH−Mbulge respectively. The …
Figure 5
Figure 5. Figure 5: Scatter plot showing the SMBH mass from scaling relations (Equation 6) vs. the SMBH mass from the single-epoch mass estimator from Hβ (Equation 8) on a log scale. The contours are the 2D Gaussian models with parameters in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Scatter plot showing the offset between inferred vs. directly measured galaxy properties with 1,240 AGN in each panel. Figure (a) has the Mbulge from bulge-disk decomposition on the x-axis and Mbulge inferred from total stellar mass on the y-axis. Figure (b) has σ meas…

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