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The effect of the environment-dependent stellar initial mass function on the baryonic Tully Fisher relation

T0 review · 4 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read An environment-dependent stellar IMF, not a failure of MOND, can explain why the most massive disk galaxies sit off the baryonic Tully-Fisher relation.

desk verdict New IGIMF-to-BTFR application with a plausible qualitative story, but the claimed quantitative match needs error bars and a sensitivity test before it can be taken as demonstrated. read the letter →

arxiv 2507.12521 v1 pith:4RCRHCDD submitted 2025-07-16 astro-ph.GA

Akram Hasani Zonoozi , Hosein Haghi , Pavel Kroupa , Sara Yousefizadeh , Zhiqiang Yan , Tereza Jerabkova , Eda Gjergo This is my paper
classification astro-ph.GA
keywords galaxy-wideinitialmassfunctionIGIMFbaryonicTully-FisherrelationMONDmass-to-lightratiostarformationhistorystellarmetallicitySPARCgalaxies
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

The paper tests whether the observed scatter in the baryonic Tully-Fisher relation can be explained without altering gravity, by letting the galaxy-wide stellar initial mass function vary with star formation history and metallicity. It assumes MOND's predicted relation between baryonic mass and flat rotation velocity is exact, then computes model galaxies under the integrated galaxy-wide IMF (IGIMF) theory and asks what an observer using the standard fixed mass-to-light ratio $M_{\rm pop}/L_{[3.6]}=0.5$ would infer. The models show that high-mass galaxies have their stellar and remnant masses underestimated by the fixed ratio, while low-mass, gas-dominated galaxies are barely affected. For constant star formation the inferred apparent relation lands nearly exactly on the empirical SPARC fit, and a slowly declining star formation history produces even larger high-mass offsets. The conclusion is that population-dependent mass-to-light ratios, rather than a failure of MOND, can account for the observed deviation.

What carries the argument

The load-bearing object is the integrated galaxy-wide IMF (IGIMF), the composite stellar IMF obtained by summing optimally sampled stellar IMFs over the embedded clusters a galaxy forms, with the stellar IMF slopes depending on metallicity and embedded-cluster mass and the cluster mass distribution depending on the star formation rate; the paper adopts the most recent IGIMF formulation with the metallicity-slope coefficient $\Delta_\alpha = 63$. This object determines the mass-to-light ratio $M_{\rm pop}/L_{[3.6]}$ of living stars plus remnants, which is then used to convert each model galaxy's luminosity into the apparent baryonic mass an observer would record under the standard assumption $M_{\rm pop}/L_{[3.6]}=0.5$. The difference between the true and apparent baryonic masses is the entire source of the predicted BTFR offsets.

What would settle it

Measure the 3.6 micron stellar mass-to-light ratio in a sample of high-mass SPARC galaxies using a method that does not assume an invariant IMF, for example resolved stellar counts or full stellar-population fitting with metallicity and star-formation priors. If the true $M_{\rm pop}/L_{[3.6]}$ of these galaxies stays near $0.5$ regardless of mass, the predicted offsets vanish and the claimed resolution of the BTFR tension fails; a separate, equally direct check would be an independent measurement of $\Delta_\alpha$ from resolved stellar populations across a range of metallicities.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the mass-to-light ratio of a galaxy's stellar-plus-remnant population at 3.6 microns rises with galaxy mass when the galaxy-wide IMF is built from the IGIMF theory, because high star formation rates and high metallicities produce a bottom-heavy and remnant-rich population. Since observers convert 3.6 micron luminosity to stellar mass with a constant $M/L=0.5$, this rise means the true baryonic masses of the most massive disk galaxies are systematically underestimated. Placing the models on MOND's $V_f = (G a_0 M_b)^{1/4}$ relation and then re-deriving the apparent baryonic masses reproduces the offset seen in the SPARC data, most closely for a constant star formation history and more strongly for a slowly declining one. The offset is therefore presented as a stellar-population artifact rather than a gravitational anomaly.

Load-bearing premise

The quantitative size of the effect is set by the adopted IGIMF calibration, especially the coefficient $\Delta_\alpha = 63$ that makes the IMF's low-mass slope change steeply with metallicity; if that calibration is wrong, the high-mass offsets in the model would shift or disappear.

Editorial extensions

If this is right

  • If the central claim is right, the empirical BTFR slope near $3.85$ is a population-weighted average: it falls below MOND's predicted slope of $4$ because high-mass galaxies are assigned too little stellar mass, and correcting the mass-to-light ratio restores the slope.
  • Stellar mass estimates from 3.6 micron photometry with a fixed mass-to-light ratio are systematically low for massive disk galaxies, by a factor that grows with mass and with a more steeply declining star formation history.
  • The small observed scatter of the BTFR becomes a constraint on star formation histories, favoring near-constant star formation rates over cosmic time rather than rapidly declining ones.
  • Low-mass, gas-dominated galaxies remain close to the canonical relation, so the tightness of the BTFR at low masses does not by itself discriminate between this population effect and MOND.

Reading between the lines

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

  • Editorial inference: the same mass-dependent mass-to-light correction should leave imprints in other relations that assume constant mass-to-light ratios, such as stellar mass functions and stellar-mass-to-halo-mass comparisons; independent dynamical mass estimates for massive disk galaxies would test this.
  • Editorial inference: the argument takes MOND's BTFR to be the underlying true relation, but the correction is really a statement about stellar population modeling, so in a dark-matter picture the same underestimate of stellar mass would also shift galaxies relative to the baryonic relation; the test is therefore not unique to MOND.
  • Editorial inference: a direct way to tighten the argument would be to measure the IMF in a handful of the most massive SPARC galaxies using gravity-sensitive spectral features or resolved stellar populations; the paper does not carry out such a check, leaving its calibration as the main open link.
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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 / 7 minor

Summary. The paper tests whether the environment-dependent integrated galactic IMF (IGIMF) can explain the apparent high-mass offset in the baryonic Tully-Fisher relation (BTFR) under the assumption that MOND's prediction Vf = (G a0 Mb)^{1/4} is correct. The authors build a grid of idealized galaxy models with the public stellar population synthesis code SPS-VarIMF, using either a constant canonical IMF or the IGIMF with metallicity- and SFR-dependent slopes. They compute present-day luminosities and masses (including remnants), assign gas masses from the empirical MHI-L[3.6] relation of Lelli et al. (2016), and then compare the 'observed' baryonic masses derived with a constant mass-to-light ratio 0.5 against the MOND prediction and the Lelli et al. (2019) BTFR fit. The main reported result is that, for a constant SFR, the apparent masses of high-mass galaxies are underestimated, shifting the model relation close to the observed BTFR; a declining SFR amplifies the offset. The paper concludes that IGIMF variations can resolve the MOND-BTFR tension at high masses without dark matter.

Significance. The paper addresses a well-known tension: the observed BTFR has a shallower slope than the MOND prediction at high masses, and the authors propose a concrete baryonic explanation via a varying IMF rather than modified gravity or dark matter. The study uses a publicly available code (SPS-VarIMF), explicit analytic formulations for the IMF variation, and two idealized SFHs, and it makes a falsifiable prediction linking the BTFR offset to SFH. If the claimed agreement with Lelli et al. (2019) can be made quantitative and robust to the adopted IMF calibration, this would be an important contribution to the interpretation of galaxy scaling relations. However, in its current form the central claim rests on visual agreement in Fig. 3, a single inherited calibration parameter (Delta_alpha), and an unverified assertion that the closed-box models reproduce the observed mass-metallicity relation. These issues must be resolved before the result can be considered established.

major comments (4)
  1. [Sec. 4.1.1, Fig. 3] The central quantitative claim that the red solid line is 'nearly exactly in agreement' with the Lelli et al. (2019) fit is not supported by any statistical measure. The paper does not report the fitted slope and intercept of the red line, the residuals, the Vf range used for the fit, or the weighting of the model grid points. Since the observed fit has A=1.99±0.18 and B=3.85±0.09, please provide the model fit parameters with uncertainties (e.g., from bootstrap or by varying the grid range and spacing) and a goodness-of-fit metric such as the RMS offset in log Mb or a chi-square relative to the observed relation over the stated mass range.
  2. [Sec. 2.2.2, Eq. (6)] The amplitude Delta_alpha = 63 is adopted from Yan et al. (2020, 2021) and is not constrained by the BTFR data used in this paper. The predicted mass dependence of Mpop/L[3.6] (Fig. 1), which produces the offset in Fig. 3, is highly sensitive to this coefficient: at Z*=0.02 it raises alpha1 to about 1.67, and halving it substantially reduces the M/L enhancement. A sensitivity test is needed: recompute the red and dashed lines in Fig. 3 for Delta_alpha values spanning the plausible range (e.g., 30 and 90) and show that the agreement with the Lelli et al. fit is not a calibration artifact.
  3. [Sec. 2.1 and Sec. 2.4] The main IGIMF models adopt a closed-box chemical evolution with fst = 0.3, and the text asserts that the resulting galaxies align with the observed mass-metallicity relation (Eq. 3). Because the mass-dependent M/L variation is driven by the metallicity trend, this assertion is load-bearing but is not demonstrated. Please include a figure or table comparing the model final stellar (or gas) metallicities with Eq. (3) for both the constant and declining SFH cases, and quantify any offset or scatter.
  4. [Sec. 4, Eq. (12) and Eq. (4)] The model assigns gas masses using the empirical MHI-L[3.6] relation (Eq. 4 from Lelli et al. 2016), which is derived from the same SPARC sample used in the BTFR comparison. This means the 'apparent' baryonic masses in Fig. 3 are partially constructed from the same gas scaling as the observed fit. The authors should discuss this potential coupling and test the sensitivity of the model offset to the amplitude and slope of Eq. (4) (e.g., vary the normalization by a typical ±0.5 dex uncertainty in MHI at fixed L) to verify that the agreement in the gas-dominated low-mass regime is not an artifact of this shared input.
minor comments (7)
  1. [Sec. 2, opening paragraph] Typographical errors: 'reffer' should be 'refer' and 'asuumed' should be 'assumed'.
  2. [Sec. 2.2.2] Typographical error: 'soloar neighborhood' should be 'solar neighborhood'.
  3. [Fig. 3 caption] The caption attributes the blue dashed line to Lelli et al. (2016), while the text in Sec. 4 attributes the fit A=1.99, B=3.85 to Lelli et al. (2019); please make the reference consistent.
  4. [Sec. 2.1] The star formation efficiency is defined as fst = Mtot/Mg, which ratios total formed stellar mass to final gas mass; please add a sentence explaining this choice, since it is not the standard definition used in the literature and it directly sets the closed-box metallicity.
  5. [Sec. 2.4] The statement that 'assuming a constant SFR is well justified for these galaxies' cites Kroupa et al. (2020a), but the SPARC sample likely contains galaxies with a range of SFHs; please qualify the statement or provide evidence that the BTFR sample is dominated by galaxies with near-constant SFHs.
  6. [Abstract and Conclusions] The phrasing 'IGIMF theory offers a solution to the observed offsets' should be softened to reflect that the calculation is conditional on the combined assumptions of MOND and the IGIMF theory; suggest 'if both MOND and the IGIMF are assumed, the offsets can be explained'.
  7. [Data Availability] The Data Availability statement says 'data are available in the article' but does not mention the SPS-VarIMF code repository given in Sec. 2.1; please include the URL there for reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the BTFR offset is a forward-model prediction conditional on externally calibrated IGIMF parameters, not a fit to the BTFR.

full rationale

The paper does not tune any parameter to the BTFR. For each model galaxy it computes Mpop/L[3.6] from stellar population synthesis with a fixed IGIMF prescription (Eq. 6, Δα = 63 inherited from Yan et al. 2020/2021), derives Vf from the assumed MOND relation (Eq. 1), and then evaluates the apparent baryonic mass an observer would infer with a constant M/L = 0.5 (Eq. 12). The comparison in Fig. 3 is therefore a genuine conditional prediction. The Δα calibration comes from prior work by overlapping authors, but that work used independent constraints (star-forming dwarfs, ultra-compact dwarfs, ellipticals) and not the BTFR; the paper explicitly emphasizes that the IGIMF theory was developed independently of any MOND application. The only input derived from the same galaxy data is the gas-luminosity fit (Eq. 4, Lelli et al. 2016), used to assign gas masses; the authors flag this limitation, and it affects gas-dominated low-mass galaxies, not the high-mass offsets that carry the main claim. Self-citations are present but are not used to forbid alternatives or to define the result; no equation here reduces to its own input by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim depends on a chain of external inputs: the IGIMF calibration, the adopted star formation histories, the closed-box enrichment model, the stellar evolution and remnant prescriptions, and the empirical gas and metallicity scaling relations. None are derived in this paper; the free parameters listed are the ones most directly affecting the M/L variation.

free parameters (6)
  • star formation efficiency fst = 0.3
    Chosen in Sec. 2.1 to set the closed-box chemical enrichment; affects metallicities and hence M/L ratios.
  • initial metallicity Z0 = 0.0002
    Assumed in Sec. 2.1; sets the starting point for enrichment and the low-metallicity remnant masses.
  • SFH timescale tau = 10 Gyr declining, or constant
    Two SFH prescriptions in Sec. 2.4 bracket the M/L ratios and the BTFR offsets.
  • IGIMF metallicity-slope coefficient Delta_alpha = 63
    Adopted from Yan et al. (2020, 2021); controls how strongly IMF slopes vary with Z and therefore how much M/L departs from 0.5 at high mass.
  • mass-metallicity relation coefficients (Eq. 3) = 0.40, 0.67, -1.04
    External fit from Ma et al. (2016) used to assign metallicities in one family of models.
  • gas mass-luminosity relation amplitude and slope (Eq. 4) = 10^3.9 and 0.54
    External fit from Lelli et al. (2016) used to estimate Mg from L[3.6]; most relevant for low-mass galaxies.
assumptions (5)
  • domain assumption MOND's BTFR prediction (Eq. 1) is exact for all baryonic masses
    Used throughout Sec. 4 as the true relation from which Vf is computed for each model galaxy.
  • domain assumption IGIMF theory as formulated by Yan et al. (2021) correctly describes IMF variation
    The M/L variations, and hence the offsets, are direct outputs of this prescription (Sec. 2.2.2).
  • domain assumption Closed-box chemical evolution with instantaneous recycling
    Adopted in Sec. 2.1; limits metal enrichment to self-enrichment with no inflows or outflows; the paper flags the omission of SNe Ia.
  • domain assumption PARSEC tracks and Spera et al. (2015) initial-final mass relation are accurate
    Stellar evolution and remnant masses in Sec. 2.3 determine Mpop and L[3.6].
  • domain assumption The SPARC gas-luminosity relation (Eq. 4) applies to all model galaxies
    Used to compute Mg in the baryonic mass budget; from the same SPARC sample used for the comparison.

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Cite this review

Pith. "Pith review of The effect of the environment-dependent stellar initial mass function on the baryonic Tully Fisher relation." pith.science (2026). https://pith.science/paper/4RCRHCDD

@misc{pith2026250712521,
  author       = {Pith},
  title        = {Pith review of: The effect of the environment-dependent stellar initial mass function on the baryonic Tully Fisher relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4RCRHCDD}},
  note         = {Machine review of arXiv:2507.12521}
}
read the original abstract

We investigate the impact of an environment-dependent galaxy-wide stellar initial mass function (gwIMF) on the baryonic Tully-Fisher relation (BTFR). The integrated galaxy-wide IMF (IGIMF) theory, which incorporates variations in stellar populations due to star formation history (SFH) and metallicity, provides a more accurate framework for understanding systematic deviations in galaxy scaling relations than that given by an invariant gwIMF. By considering how the mass-to-light ratio of the stellar population is influenced by metallicity and SFH, we show that high-mass galaxies have their masses in stars and remnants underestimated under the assumption of a constant mass-to-light ratio. In contrast, low-mass, gas-dominated galaxies are less affected. Our results suggest that the discrepancies between the true and observed BTFR are primarily driven by the evolving nature of the stellar IMF, particularly in galaxies with slowly declining SFHs. The IGIMF theory offers a solution to the observed offsets in the BTFR, especially for high-mass galaxies, where the rotational velocities are higher than predicted by MOND. We conclude that incorporating the IGIMF provides a more accurate description of galaxy dynamics, revealing the importance of stellar population characteristics in refining our understanding of the baryonic mass-velocity relationship. This study underscores the necessity of accounting for the variation of the gwIMF when interpreting the BTFR, particularly in the context of alternative gravitational theories like MOND.

Figures

Figures reproduced from arXiv: 2507.12521 by the authors.

Figure 2
Figure 2. Mpop/L[3.6] in dependence on the present-day stellar mass assuming a constant SFR and time-evolving metallicities, but with different stellar remnant prescriptions. Circle symbols represent models with an invariant canonical IMF, while the square symbols correspond to the IGIMF case. The black squares and filled red circles are identical to [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 1
Figure 1. Mpop/L[3.6] in dependence on the present-day stellar plus remnant mass (Mpop top panel) and luminosity in the [3.6]- band (lower panel) assuming constant and slowly declining SFHs and constant and time-evolving metallicities. Circle symbols rep￾resent models with an invariant canonical IMF, while the plus signs correspond to the IGIMF case with a declining SFR, and square symbols represent the IGIMF with a constant … view at source ↗
Figure 3
Figure 3. Effect of a varying stellar IMF on the BTFR for different SFHs. The black solid line represents the MOND prediction re￾lating the baryonic mass of galaxies to their flat rotation velocity (Eq. 1). The blue dashed line (Eq. 10) shows the slope of the BTFR derived from the SPARC sample by Lelli et al. (2016), assuming an invariant gwIMF, which deviates from the MOND prediction. The Mb,cML refers to the derived baryoni… view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.