Constraints on Quasi-dilaton Massive Gravity

Katarina Martinovic; Mairi Sakellariadou

REVIEW 3 major objections 3 minor 42 references

Constraints on Quasi-dilaton Massive Gravity

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

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Pith's one-line read Galaxy rotation curves cap the graviton mass in quasi-dilaton massive gravity at $10^{-31}$ eV.

desk verdict A sensible application of rotation-curve methods to QDMG, but the claimed bound rests on an unjustified quintic-dominant truncation, so m ≤ 10^-31 eV is not yet established. read the letter →

arxiv 1908.08247 v2 pith:QA5UBAOT submitted 2019-08-22 gr-qc astro-ph.GA

Katarina Martinovic , Mairi Sakellariadou This is my paper · ORCID

classification gr-qcastro-ph.GA PACS 04.50.Kd98.62.Dm04.30.-w

keywords quasi-dilatonmassivegravitygravitonmassboundgalaxyrotationcurvesSPARCcatalogueNavarro-Frenk-WhiteprofileBardeenpotentialsdecouplinglimitdarkenergy

topics Dark Energy Dark Matter

open problems Dark Matter Dark Energy

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

Quasi-dilaton massive gravity (QDMG), a scalar extension of ghost-free dRGT massive gravity, adds a quasi-dilaton field whose extra scalar degrees of freedom alter the Bardeen potentials of galaxies. The paper derives the resulting modification to dark-matter rotation curves and fits them to high-quality SPARC galaxies, obtaining an upper bound on the graviton mass of $m \leq 10^{-31}\,{\rm eV}$. If this bound holds, QDMG remains viable as a dark-energy candidate at $m \approx 10^{-33}\,{\rm eV}$, while ruling out heavier QDMG gravitons up to the previous $10^{-22}\,{\rm eV}$ limit. It also shows that tensor-to-scalar gravitational-wave decay is far too weak to deplete observable signals, so gravitational-wave observations cannot easily falsify the theory. The main claim is that QDMG's astrophysical predictions are consistent with existing rotation-curve data only if the graviton is extremely light.

What carries the argument

The central object is the parameter $\gamma$ defined in Eq. (27), which packages the graviton mass and the QDMG coupling coefficients into the single quantity controlling how much the dark-matter rotation curve deviates from the pure Navarro-Frenk-White/general-relativity prediction. It is derived from the decoupling-limit equations for the two Bardeen potentials and two galileon-type scalars; after assuming spherical symmetry the system reduces to a quintic for $x = \phi'/r$, and keeping the dominant $x^5$ term produces the modified velocity profile. A second piece of machinery is the $1\to 3$ decay vertex $h(\partial^2\pi)^3/\Lambda_3^6$ used to estimate tensor-to-scalar gravitational-wave depletion, with the phase-space integral evaluated following the standard three-body decay treatment.

What would settle it

Re-fit the SPARC galaxies used here (or a larger sample) with a Markov Chain Monte Carlo that varies $\gamma$, $r_s$, $\rho_s$, and the stellar mass-to-light ratios simultaneously; if the best-fit $\gamma$ corresponds to $m > 10^{-31}\,{\rm eV}$ while still matching the rotation curves, the paper's bound is false.

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Extended reading notes

Core claim

The paper's central claim is that quasi-dilaton massive gravity, taken in its decoupling limit up to cubic galileon order and with the quintic term dominant, predicts a dark-matter velocity profile of the form $v^2_{dm} = 4\pi G r_s^2 \rho_s \left[ \frac{1}{R}\left(\ln(1+R) - \left(1+\frac{1}{R}\right)^{-1}\right) - \gamma R^{1/5} \left(\ln(1+R) - \left(1+\frac{1}{R}\right)^{-1}\right)^{3/5} \right]$, where $\gamma$ encodes the graviton mass $m$ through $\gamma = (4\pi G\rho_s)^{-2/5} m^{4/5} (\alpha_3+4\alpha_4) \left[6(\alpha_3+4\alpha_4)^2 + \frac{3}{\omega}(\alpha_3-4\alpha_4)^2\right]^{-3/5}$. Fitting this profile to high-quality SPARC galaxies, using Navarro-Frenk-White halo parameters and stellar mass-to-light ratios taken from GR-based fits, yields consistency only for $m \leq 10^{-31}\,{\rm eV}$. This bound is stronger than the previous $10^{-22}\,{\rm eV}$ level from gravitational waves and Solar System tests, but it still permits the dark-energy-motivated value $m \approx 10^{-33}\,{\rm eV}$. The paper also computes that the $1\to 3$ decay of tensor to scalar modes has a width $\Gamma \approx 10^{-225}\,{\rm eV}$, far too small to observably deplete gravitational-wave signals.

Load-bearing premise

The argument assumes that the NFW halo parameters ($r_s$, $\rho_s$) and stellar mass-to-light ratios obtained from GR-based fits in ref. [39] can be used unchanged when fitting the QDMG parameter $\gamma$; if QDMG alters rotation curves, those parameters should be re-derived jointly, and the inferred graviton mass would change.

Editorial extensions

If this is right

If the bound is right, QDMG gravitons heavier than $10^{-31}\,{\rm eV}$ are ruled out, shrinking the allowed mass window to $m \lesssim 10^{-31}\,{\rm eV}$.
The dark-energy target $m \approx 10^{-33}\,{\rm eV}$ survives, so QDMG remains a viable explanation of cosmic acceleration.
Space-based gravitational-wave observatories with sensitivity up to $10^{-25}\,{\rm eV}$ will not reach this bound, so direct detection of the QDMG graviton mass is out of reach.
Depletion of gravitational-wave signals by tensor-to-scalar decay is negligible, so this decay channel cannot falsify QDMG observationally.
Since the Bardeen potentials satisfy $\Psi = \Phi$ as in general relativity, QDMG is indistinguishable from GR in gravitational lensing tests, making rotation curves the discriminating probe.

Reading between the lines

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

A joint fit that varies $\gamma$, the halo parameters, and the stellar mass-to-light ratios simultaneously could shift the inferred bound; if QDMG changes the effective halo, the bound might move by orders of magnitude.
Extending the calculation beyond the cubic galileon or including nonzero background values of the scalar fields could add terms that compete with the quintic, possibly altering the predicted velocity profile and the mass bound.
The same rotation-curve test could be applied to other scalar-extended massive-gravity theories with stable cosmological solutions, not just QDMG.
The bound is derived from high-mass, high-luminosity galaxies with the best-quality rotation curves; testing lower-mass or dwarf galaxies, where the NFW profile fits more poorly, could expose deviations or strengthen the constraint.

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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

The paper's novel bit is the QDMG-specific rotation-curve template: a modified NFW velocity profile with a characteristic R [ln(1+R)-R/(1+R)]^{3/5} correction and a parameter γ tied to the graviton mass. Applying the known beyond-Horndeski fitting method to QDMG is a legitimate extension, and the authors are upfront that a full MCMC treatment is missing. The GW decay estimate is a null result that agrees with earlier Horndeski work; that part is fine and not overclaimed.

The soft spot is load-bearing. In the appendix they give the full quintic for x (eq. 39), then keep only the x^5 term in eq. (22). The truncation is not justified. For a typical SPARC galaxy and m around the claimed bound, Λ3^3 and A are the same order, so the dimensionless source a is order one. In the dimensionless quintic all powers contribute comparably; there is no regime where the x^5 term dominates. I checked the scaling and it holds. That means eq. (26) and the subsequent γ fit are not the generic QDMG prediction in the galaxy regime. The bound m ≤ 10^-31 eV is derived by fitting a template that may not follow from the theory. The authors list several approximations but do not flag this one.

A second, smaller issue: they take halo parameters and stellar mass-to-light ratios from a GR-based fit in ref [39] and do not re-fit them jointly with γ. They acknowledge this, so it is not a hidden flaw, but it makes the bound conditional on the GR halo inference. It is a minor-to-moderate concern relative to the truncation problem.

What is good: the paper is honestly written, the derivation up to the full quintic is transparent, and the null GW decay result is consistent with prior literature. The citation pattern is reasonable; using best-fit α and ω coefficients from [25] is appropriate, though it inherits their uncertainties.

Who should read this: anyone working on massive gravity phenomenology. It is a plausible first step, not a definitive constraint. I would send it to a referee, but with the expectation that the truncation issue gets fixed or at least convincingly bounded. As it stands, I would not cite the bound in my own work.

Referee Report

3 major / 3 minor

Summary. The manuscript studies quasi-dilaton massive gravity (QDMG), a scalar-extended dRGT theory, and derives constraints on the graviton mass. The first part estimates the decay of gravitational-wave tensor modes into scalar modes and finds the decay width negligible. The second part works in the decoupling limit, derives equations for the Bardeen potentials and the scalar perturbations, truncates the full quintic equation for x to its x^5 term, and fits the resulting modified Navarro-Frenk-White rotation-curve template to SPARC data. The central claim is an upper bound m ≤ 10^-31 eV on the graviton mass in QDMG, with the conclusion that the dark-energy-motivated value m ~ 10^-33 eV remains viable.

Significance. If the bound were established, it would be a useful astrophysical constraint on QDMG and would complement existing LIGO and Solar System bounds. The paper is clearly written and makes a genuine attempt to connect the decoupling-limit equations to galaxy data; it also correctly notes several of its own approximations and the need for a future MCMC treatment. However, the main result depends on an uncontrolled truncation of the equation of motion, and the statistical treatment is too weak to give a robust quantitative bound. The gravitational-wave depletion argument is a reasonable order-of-magnitude null result, but it is not the main claim.

major comments (3)

[III B, Eq. (22), Appendix Eq. (39)] The replacement of the full quintic (39) by the x^5-only equation (22) is not controlled. The full equation contains terms proportional to x, x^2, x^3, x^4 and x^5, and no argument is given that the discarded terms are small at the relevant galactic radii. For a typical SPARC galaxy with M ~ 10^11 M_sun and r ~ 10 kpc, the dimensionless combination A/Lambda_3^3 is of order one at m ~ 10^-31 eV, so the coefficients of the lower powers are not parametrically suppressed relative to the quintic term. The root of the full quintic need not be close to the root of Eq. (22). Since the velocity profile (26), the fitted parameter gamma (27), and the bound (29) all follow from Eq. (22), the central claim is not established unless the full quintic is solved or a controlled dominance argument is supplied for the galaxies actually used.
[III B, after Eq. (28)] The bound relies on adopting Upsilon_disk, Upsilon_bulge, r_s and rho_s from the GR-based NFW fits of de Almeida et al. [39]. In QDMG the rotation curve is modified, so the inferred halo and stellar parameters should be re-derived jointly with the QDMG parameter gamma. Using GR-fit halo parameters can shift gamma and therefore the inferred graviton mass. The paper acknowledges this and defers a full MCMC, but as it stands the quoted inequality m <= 10^-31 eV is conditional on the GR halo model. In addition, no uncertainty is quoted for gamma or m, so the bound lacks a statistical definition.
[II and III B] The applicability of the decoupling limit to galactic rotation curves is asserted rather than demonstrated. The text states that the decoupling limit is valid for 'typical scales bigger than 1/m'; for the masses considered here, 1/m ~ 10^24 m (tens of megaparsecs), so galactic scales are not in that regime. If the intended criterion is instead r << 1/m, that should be stated explicitly and the ordering of scales checked. As written, the domain of validity of Eq. (3) does not clearly cover the SPARC data used in Section III B, which directly affects the derivation of the central bound.

minor comments (3)

[III B, Eq. (17)] The quantity A in Eq. (17) is defined with M(r), but in Eq. (22) it appears as a source term; please clarify whether M(r) is the total enclosed mass or only the dark matter mass entering the NFW profile.
[III B, Figure 1] Only two representative galaxies are shown, and no error bars are visible on the data points. The text states that 'the galaxies that we take show consistency' in gamma, but a table or plot reporting gamma for every galaxy used, with uncertainties, is needed for reproducibility.
[Notation] The paper oscillates between 'MPl' and 'M_Pl' for the Planck mass; please unify the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the graviton-mass bound is obtained by fitting a derived parameter to external rotation-curve data, not by inserting the target result.

full rationale

The paper's derivation chain is: QDMG decoupling-limit action, field equations (13)-(16), spherical reduction to the algebraic system (18)-(21), full quintic for x (Appendix eq. 39), dominant-quintic approximation (eq. 22), NFW-based velocity profile (eq. 26) with gamma(m) given by eq. (27), fit of gamma to SPARC rotation curves, and finally the bound m <= 1e-31 eV (eq. 29). I find no step in which a predicted quantity is defined in terms of the target or in which a fitted parameter is renamed as a prediction. gamma is an explicit function of m and the theory parameters (eq. 27); fitting gamma to rotation-curve data and inverting to obtain m is ordinary parameter inference, not a circular construction. The theory coefficients alpha3, alpha4, omega are taken from the independent cosmological fit [25] by Gannouji et al., not from the present authors, and the NFW parameters come from [39] by de Almeida et al.; these are external inputs whose validity affects the bound but does not make the derivation circular. The quintic-dominant truncation (22) is a real approximation concern, because the full quintic (39) contains linear through quartic terms whose relative size is not established; however, that is a correctness risk rather than circularity. The paper also explicitly flags its own limitations: no MCMC, background fields set to zero, and truncation at the cubic galileon. There are no self-citations carrying the argument. I therefore assign a circularity score of 0.

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

The central bound rests on the QDMG action, the decoupling-limit reduction, the zero-background-field choice, the NFW halo assumption, and external best-fit values for the theory and halo parameters. No fundamentally new entities are introduced.

free parameters (4)

α3, α4, ω (QDMG action coefficients) = best-fit values from Gannouji et al. 2013 [25]
The QDMG velocity profile and the mapping from γ to m depend on these coefficients; the bound is therefore conditional on their values.
NFW halo parameters r_s, ρ_s = values from Table 4 of de Almeida et al. 2018 [39]
The dark-matter velocity profile and the fit of γ are computed using these fixed halo parameters.
Stellar mass-to-light ratios Υ_disk, Υ_bulge = values from Table 4 of de Almeida et al. 2018 [39]
Used to subtract baryonic contributions from observed rotation curves to isolate the dark-matter component.
γ (QDMG modification parameter) = not reported
Fit to SPARC data to set the bound; the fitted values and uncertainties are not given in the paper.

assumptions (5)

domain assumption The quasi-dilaton massive gravity action (Eq. 1) is the correct theory to test; the graviton mass m is the parameter to bound.
The paper takes QDMG as the theoretical framework without deriving it from a deeper theory.
domain assumption The decoupling limit action (Eq. 3) is valid on galactic scales and the background values of the scalar fields π0 and σ0 can be set to 0.
The authors state in the discussion that these are approximations; their validity on galactic scales is not established.
domain assumption The NFW profile describes the dark-matter halo of the selected galaxies.
The rotation curve model is based on the NFW profile; deviations could bias the fitted γ.
ad hoc to paper The quintic term in the equation for x (Eq. 22) dominates, reducing the problem to a simple power law.
The approximation is used to obtain the analytic velocity profile (Eq. 26) but its domain of validity is not quantified.
domain assumption Only terms up to the cubic galileon are relevant; quartic and higher galileon terms, time derivatives, and FLRW background dynamics are negligible.
The paper states it considers only terms up to the cubic galileon and ignores a(t), H(t), and time derivatives.

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Cite this review

Pith. "Pith review of Constraints on Quasi-dilaton Massive Gravity." pith.science (2026). https://pith.science/paper/QA5UBAOT

@misc{pith2026190808247,
  author       = {Pith},
  title        = {Pith review of: Constraints on Quasi-dilaton Massive Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QA5UBAOT}},
  note         = {Machine review of arXiv:1908.08247}
}

abstract

The last decade has seen great advancements in the field of modified gravity, motivated by the dark energy problem, or by the search for a fundamental quantum gravity theory. With a phenomenologically-driven approach, we consider dRGT theory and its extension, quasi-dilaton massive gravity (QDMG). When looking for ways to constrain the theory, a promising direction appeared to be astrophysical tests. The scalar gravitational degree of freedom and quasi-dilaton degree of freedom alter the evolution of Bardeen potentials, which in turn affects the galaxy rotation curves. We find an upper bound on graviton mass in QDMG to be $m \leq 10^{-31} {\rm eV}$. This result agrees with bounds from LIGO and numerous Solar System tests. However, the extremely small mass of the graviton remains a detection out of reach, with LISA's sensitivity exploring the parameter space up to $m \leq 10^{-25} {\rm eV}$.

Figures

Figures reproduced from arXiv: 1908.08247 by the authors.

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Pith tools

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

[1] [39]

Navarro, Carlos S

Julio F. Navarro, Carlos S. Frenk, and Simon D. M. White. The Structure of cold dark matter halos. Astro- phys. J. , 462:563–575, 1996

work page 1996

[2] [25]

Tate Deskins, John T

Furqan Dar, Claudia De Rham, J. Tate Deskins, John T. Giblin, and Andrew J. Tolley. Scalar Gravitational Radiation from Binaries: Vainshtein Mechanism in Time- dependent Systems. Class. Quant. Grav. , 36(2):025008, 2019

work page 2019

[3] [1]

Tests of general relativity in the solar system

Serge Reynaud and Marc-Thierry Jaekel. Tests of general relativity in the solar system. Proc. Int. Sch. Phys. Fermi, 168:203–217, 2009

work page 2009

[4] [2]

Constraints on Quasi-dilaton Massive Gravity

and strong ﬁeld regimes (e.g. merger events [ 3] and pulsars [ 4]), in recent years. While the success of General Relativity strongly suggests that this theory is indeed a good description of gravity, exploring modiﬁcations to it is an important test of the theory itself. Moreover, a modiﬁcation of GR may provide a natural explanation of the current accel...

work page Pith review arXiv 1908

[5] [3]

Collett, Lindsay J

Thomas E. Collett, Lindsay J. Oldham, Russell J. Smith, Matthew W. Auger, Kyle B. Westfall, David Bacon, Robert C. Nichol, Karen L. Masters, Kazuya Koyama, and Remco van den Bosch. A precise extragalactic test of General Relativity. Science, 360:1342, 2018

work page 2018

[6] [4]

B. P. Abbott et al. Tests of general relativity with GW150914. Phys. Rev. Lett. , 116(22):221101, 2016. [Erratum: Phys. Rev. Lett.121,no.12,129902(2018)]

work page 2018

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M. Kramer et al. Tests of general relativity from timing the double pulsar. Science, 314:97–102, 2006

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Carroll, Antonio De Felice, Vikram Duvvuri, Damien A

Sean M. Carroll, Antonio De Felice, Vikram Duvvuri, Damien A. Easson, Mark Trodden, and Michael S. Turner. Cosmology of generalized modiﬁed gravity models. Phys. Rev. D , 71:063513, Mar 2005

work page 2005