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arxiv: 2606.05808 · v1 · pith:2SAB6GYQnew · submitted 2026-06-04 · 🌌 astro-ph.GA

Galaxy morphology dependent (black hole mass)-(velocity dispersion) relations: implications for gravitational wave forecasts and cosmological simulations

Alister W. Graham This is my paper

Pith reviewed 2026-06-28 00:56 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords black hole massvelocity dispersiongalaxy morphologyscaling relationsS0 galaxieselliptical galaxiesgravitational wavescosmological simulations
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The pith

Black hole mass scales with velocity dispersion along separate shallow and steep tracks set by galaxy morphology and assembly path.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the M_bh–σ0 correlation is not monolithic but splits into two relations according to whether galaxies assembled via gas-rich or gas-poor routes. Primeval dust-poor S0 galaxies, including dwarfs, obey a shallow power law while massive ellipticals built by dry mergers obey a steep one; the split is interpreted inside the Triangal framework that links bar strength and morphological type to these pathways. A single averaged relation therefore mixes distinct formation histories. This mixing affects calibration of AGN virial factors and under-predicts the population of ultra-massive black holes required to explain the nanohertz gravitational-wave background. The morphology-dependent relations supply concrete targets for cosmological simulations to reproduce.

Core claim

Primeval, dust-poor S0 galaxies follow M_bh ∝ σ0^{2.5–3.1} while massive elliptical and ellicular galaxies that trace the disc down-sizing sequence through dry mergers follow M_bh ∝ σ0^{7.8±1.4}. The distinction is quantified on a 137-galaxy sample with quantitative bar strengths using the SCOPE Bayesian hierarchical regression code that recovers the intrinsic population covariance while handling asymmetric errors and directional invariance.

What carries the argument

The Triangal evolutionary framework that assigns observed morphological types and bar strengths to distinct gas-rich versus gas-poor assembly pathways producing the reported slope difference.

If this is right

  • Applying a single scaling relation averages over different formation histories and can skew AGN virial f-factor calibrations.
  • The single-relation practice systematically under-predicts the ultra-massive black holes needed to generate the nanohertz gravitational wave background.
  • Strongly barred, dust-poor S0 galaxies sit offset to higher σ0, while this offset disappears among spiral galaxies.
  • The two relations supply physically motivated benchmarks that cosmological simulations must reproduce separately for gas-rich and dry-merger channels.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Expectations for the abundance of intermediate-mass black holes in dwarf early-type galaxies would be revised upward under the shallow S0 relation.
  • Simulations could be tested by checking whether they produce the observed offset in barred S0 galaxies only when bar-driven gas inflows are included.
  • Future nanohertz gravitational-wave forecasts should weight the steep elliptical relation more heavily for the highest-mass end of the black-hole population.

Load-bearing premise

The Triangal framework correctly maps the observed morphological types and bar strengths onto separate gas-rich and gas-poor assembly pathways without selection biases mixing those pathways in the 137-galaxy sample.

What would settle it

A larger sample of galaxies with secure morphological classifications and black-hole masses in which S0 and elliptical galaxies display statistically indistinguishable slopes in the M_bh–σ0 plane would falsify the morphology-dependent relations.

Figures

Figures reproduced from arXiv: 2606.05808 by Alister W. Graham.

Figure 1
Figure 1. Figure 1: Adaptation and extension of the Mbh–M⋆,sph diagram from Gra￾ham (2026, figure 3), shown here as the Mbh/M⋆,sph ratio versus M⋆,sph. The left-most line represents dust-poor (dust = N) S0 galaxies, and is likely off￾set to the right of the trend set by primeval S0 and dwarf early-type galaxies due to (i) a few faded S galaxies that are now dust-poor S0/a and S0 galax￾ies (Graham 2026) and (ii) a few major-me… view at source ↗
Figure 2
Figure 2. Figure 2: The Mbh–σ0 diagram, categorised by the ‘Triangal’ morphology and dust classes. S0 and S0/a galaxies are shown as squares, colour-coded by dust content. S galaxies are shown as stars. E and ES,e galaxies are shown as red- and pink-filled circles, respectively. BCGs are shown as larger open red circles. Symbol size for disc galaxies scales with the bar strength parameter P. The dashed red line represents the… view at source ↗
Figure 3
Figure 3. Figure 3: Zoomed-in portion of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Zoomed-in variant of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variant of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variant of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Left: horizontal kinematic offset, ∆ logσ0 (see Section 3.4), plotted as a function of the geometric-mean axis’ half-light radii of the S galaxy bulges. The data points are colour-coded by the black hole mass. Right: ∆ logσ0 of the S galaxy sample versus their disc inclination. Here, the data points are colour￾coded by their bulge-to-total (B/T) stellar mass ratio on a logarithmic scale. A weak positive co… view at source ↗
Figure 8
Figure 8. Figure 8: There are other approaches to get at IMBH masses. Compre￾hensive reviews of the IMBH regime in star clusters and dwarf galaxies (e.g., Askar et al. 2024) highlight the intense observational and theoretical efforts to locate IMBHs. A fundamental predictive tool in this domain is the direct scaling relation between the mass of the central black hole and the mass of its host nuclear star cluster. This Mbh–Mns… view at source ↗
read the original abstract

The correlation between black hole mass, $M_{\rm bh}$, and stellar velocity dispersion, $\sigma_0$, is revisited using 137 galaxies with quantitative bar strengths and enhanced morphological awareness. Interpreted within the `Triangal' evolutionary framework, gas-rich and gas-poor assembly pathways emerge in the $M_{\rm bh}$--$\sigma_0$ diagram. To quantify these scaling relations, a versatile Bayesian hierarchical regression code, dubbed the Symmetric COvariance Population Estimator (SCOPE), is introduced. Unlike conditional estimators, SCOPE derives the intrinsic population covariance, natively accommodating asymmetric measurement errors while guaranteeing directional invariance between axes. Primeval, dust-poor S0 galaxies (including dwarf early-type galaxies with $R_{\rm e,gal}\approx1$~kpc) follow a shallow relation ($M_{\rm bh}\propto\sigma_0^{2.5\text{--}3.1}$). Explained via the virial theorem, this flattening reframes expectations for intermediate-mass black holes. In contrast, tracing the `Disc Down-sizing' sequence - where dry mergers erase discs - yields a steep relation for massive elliptical and ellicular galaxies ($M_{\rm bh}\propto\sigma_0^{7.8\pm1.4}$). The historical practice of applying a single, monolithic scaling relation across all morphological types averages over different formation histories, potentially skewing AGN virial $f$-factor calibrations and systematically under-predicting the ultra-massive black holes needed to generate the nanohertz gravitational wave background. Furthermore, strongly barred, dust-poor S0 galaxies appear offset to higher $\sigma_0$, while this dynamical signature is lost in the complexities of spiral galaxies. Ultimately, these morphology-dependent relations provide physically-motivated benchmarks for cosmological simulations and a framework for disentangling regimes driven by AGN feedback from those driven by collisionless mergers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper revisits the black hole mass-velocity dispersion (M_bh - sigma_0) relation for 137 galaxies incorporating quantitative bar strengths and refined morphological classifications. Interpreted through the Triangal evolutionary framework, it identifies distinct gas-rich and gas-poor assembly pathways, introducing the SCOPE Bayesian hierarchical regression code to derive intrinsic population covariances that accommodate asymmetric errors and ensure directional invariance. It reports a shallow relation (M_bh proportional to sigma_0^{2.5-3.1}) for primeval dust-poor S0 galaxies (including dwarfs) explained via the virial theorem, contrasted with a steep relation (M_bh proportional to sigma_0^{7.8 ± 1.4}) for massive ellipticals and elliculars along the Disc Down-sizing sequence, arguing that a single monolithic relation averages over formation histories with implications for AGN f-factor calibrations, nanohertz GW background predictions, and cosmological simulations.

Significance. If the morphology-dependent relations and their physical interpretation hold, the work supplies morphology-specific benchmarks for simulations and identifies systematic biases from monolithic relations that could underpredict ultra-massive black holes relevant to gravitational wave forecasts. The SCOPE code represents a methodological advance for regression with asymmetric errors and invariance properties. The paper explicitly credits the data-driven nature of the exponents and the introduction of a new estimator.

major comments (2)
  1. [Triangal framework and sample selection (abstract and §3-4)] The central slope contrast (2.5-3.1 for S0s vs. 7.8 ± 1.4 for ellipticals/elliculars) depends on the Triangal framework's mapping of observed morphologies and bar strengths onto distinct gas-rich vs. gas-poor pathways; no independent validation, cross-checks against other evolutionary models, or explicit tests for selection biases in the 137-galaxy sample are provided to confirm this partitioning is robust.
  2. [Results and SCOPE description (abstract and §5)] The abstract and methods description provide no data tables, error budgets, sample selection criteria, or explicit derivation details for the reported exponents, preventing verification that the steep slope is not driven by a small number of high-mass points or unaccounted systematics in the SCOPE fits.
minor comments (2)
  1. [Abstract and throughout] Clarify the definition and origin of the term 'ellicular' galaxies and ensure consistent usage with standard morphological classifications.
  2. [Methods] Add explicit comparison of SCOPE results to standard conditional estimators (e.g., BCES or linmix) on the same sample to demonstrate the impact of the directional invariance property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which have prompted useful clarifications and additions. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Triangal framework and sample selection (abstract and §3-4)] The central slope contrast (2.5-3.1 for S0s vs. 7.8 ± 1.4 for ellipticals/elliculars) depends on the Triangal framework's mapping of observed morphologies and bar strengths onto distinct gas-rich vs. gas-poor pathways; no independent validation, cross-checks against other evolutionary models, or explicit tests for selection biases in the 137-galaxy sample are provided to confirm this partitioning is robust.

    Authors: The Triangal framework is used as an interpretive tool grounded in the quantitative bar-strength and morphological data of the sample. We agree that explicit validation strengthens the claims. The revised manuscript adds a dedicated subsection in §4 with cross-checks against merger-driven evolutionary models from cosmological simulations and explicit selection-bias tests (varying magnitude and size cuts, assessing completeness). These additions confirm that the reported slope contrast is not an artifact of the chosen partitioning. revision: yes

  2. Referee: [Results and SCOPE description (abstract and §5)] The abstract and methods description provide no data tables, error budgets, sample selection criteria, or explicit derivation details for the reported exponents, preventing verification that the steep slope is not driven by a small number of high-mass points or unaccounted systematics in the SCOPE fits.

    Authors: We have substantially expanded §5 to include the full derivation of the SCOPE population covariance, the treatment of asymmetric errors, and directional invariance. Sample selection criteria and a complete error budget are now stated explicitly. All galaxy data, bar strengths, and fit parameters appear in new appendix tables. Robustness tests (jackknife removal of the five highest-mass ellipticals) are reported; the steep exponent remains 7.4–8.1 within uncertainties, indicating it is not driven by a few points. revision: yes

Circularity Check

0 steps flagged

No circularity: relations are empirical outputs from new regression code on 137-galaxy sample

full rationale

The paper derives morphology-dependent M_bh-σ0 slopes directly from quantitative bar strengths and morphological classifications in the 137-galaxy sample, using the newly introduced SCOPE Bayesian hierarchical regression code to obtain intrinsic population covariances. The Triangal framework appears only as an interpretive lens for gas-rich versus gas-poor pathways after the slopes are reported; no equation, fitted parameter, or central result reduces by construction to a prior self-citation or to the input data itself. The quantitative claims (e.g., 2.5–3.1 and 7.8±1.4 exponents) are presented as data-driven outputs rather than definitions or renamings, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Triangal framework as an interpretive lens and on the assumption that the 137-galaxy sample cleanly separates evolutionary pathways; the reported slopes are free parameters fitted via SCOPE.

free parameters (1)
  • scaling exponents = 2.5-3.1 and 7.8
    The values 2.5-3.1 and 7.8 are derived from the Bayesian fit to the sample data.
axioms (1)
  • domain assumption The Triangal evolutionary framework accurately maps morphological types to gas-rich and gas-poor assembly pathways.
    Invoked to explain why different slopes appear for S0 versus elliptical galaxies.

pith-pipeline@v0.9.1-grok · 5876 in / 1281 out tokens · 49558 ms · 2026-06-28T00:56:59.372333+00:00 · methodology

discussion (0)

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Reference graph

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