Do Pulsar Timing Datasets Favor Massive Gravity?

Chris Choi; Tina Kahniashvili

REVIEW 4 major objections 4 minor 106 references

Do Pulsar Timing Datasets Favor Massive Gravity?

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

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Pith's one-line read The paper claims that a massive-gravity overlap reduction function, keeping the pulsar-propagation exponentials and five gravitational-wave polarizations, fits the NANOGrav 15-year and CPTA DR1 angular correlations substantially better…

desk verdict New ORF calculation, but the data claim breaks on the dispersion relation. read the letter →

arxiv 2507.02059 v2 pith:PG43XGB7 submitted 2025-07-02 astro-ph.CO gr-qc

Chris Choi , Tina Kahniashvili This is my paper

classification astro-ph.COgr-qc

keywords massivegravitygravitonmassoverlapreductionfunctionHellings-DownscurvepulsartimingarraysstochasticgravitationalwavebackgroundNANOGrav15-yearCPTADR1

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 argues that the angular correlations seen by pulsar timing arrays — correlations usually read as evidence for a stochastic gravitational-wave background of tensor waves in general relativity, the pattern known as the Hellings-Downs curve — may actually be better described by massive gravity. It derives the overlap reduction function for a massive graviton, keeping the exponential pulsar-propagation factors that earlier analyses suppress, and fits the resulting curve to the NANOGrav 15-year and CPTA DR1 datasets with one free parameter, $|k|/k_0$. The claimed fits improve the reduced chi-square from about 1.71 to 0.55 for NANOGrav15 and from about 3.00 to 1.35 for CPTA DR1. If these fits are right, pulsar timing data already carry a propagation- and polarization-based signature of a nonzero graviton mass, independent of what produced the background, and future PTA data could constrain that mass.

What carries the argument

The load-bearing object is the effective overlap reduction function (the predicted angular correlation between two pulsar signals as a function of their sky separation), $\tilde\Gamma_T = \beta(\Gamma_T + \Gamma_V\,\Omega_V/\Omega_T + \Gamma_S\,\Omega_S/\Omega_T)$, which combines the tensor, vector, and scalar polarization overlaps of massive gravity under the assumption that the energy densities are equipartitioned, $\Omega_T = \Omega_V = 2\Omega_S$. Its new ingredient is retaining the factors $E_j(f,\hat{\Omega}) = e^{-i2\pi f L_j(1 + (|k|/k_0)\,\hat{\Omega}\cdot\hat{p}_j)} - 1$ in the integrand instead of replacing their product by 1; at pulsar distances $fL\sim 1$ these exponentials are not negligible and generate extra extrema in the ORF. The graviton mass enters only through the ratio $|k|/k_0$ via the dispersion relation $\omega^2 = |k|^2 + m_g^2$, so that single ratio is the only fit parameter.

What would settle it

Rebin the NANOGrav15 and CPTA DR1 data using the exact frequency integral of the massive-gravity ORF with published pulsar distances while keeping $\Omega_T=\Omega_V=2\Omega_S$; if the best-fit $|k|/k_0$ then moves to zero (the HD limit) or the $\chi^2/\mathrm{d.o.f.}$ no longer beats HD, the claimed preference is an artifact of the single-frequency, equipartition approximation. As a second check, fit each of CPTA's three frequency bins separately and test whether the inferred $|k|/k_0$ is constant across bins as the dispersion relation requires.

Watch

Extended reading notes

Core claim

The central discovery is that the effective massive-gravity overlap reduction function, computed without setting the exponential factors $E_j(f,\hat{\Omega})$ to unity, reproduces the observed pulsar-pair angular correlations better than the standard Hellings-Downs curve in both datasets analyzed. For NANOGrav15 the best fit has $|k|/k_0 = 0.61$ with $\chi^2/\mathrm{d.o.f.} \approx 0.55$, versus $1.71$ for HD; for CPTA DR1 the best fit has $|k|/k_0 = 0.01$ with $\chi^2/\mathrm{d.o.f.} \approx 1.35$, versus $3.00$ for HD. The massive-gravity curve also tracks the observed shift of the correlation minimum from the HD prediction and, in the CPTA case, an extra maximum near $\xi \sim 103^\circ$ that HD cannot produce. Because the modification enters through the dispersion relation and the additional vector and scalar polarization modes, the claimed preference does not depend on whether the background is astrophysical or cosmological.

Load-bearing premise

Everything hinges on modeling the detected correlation as the massive-gravity overlap reduction function evaluated at one representative frequency and pulsar distance $fL\approx1$ with tensor, vector, and scalar energy densities fixed in equipartition; if the real signal must be averaged over the full PTA band with actual pulsar distances, or the polarization mix differs, the reported chi-square improvements do not follow.

Editorial extensions

If this is right

Longer PTA campaigns reaching lower minimum frequency would directly probe the optimistic mass $m_g \sim 1.31\times10^{-24}\,\mathrm{eV}$, since the dispersion relation ties the best-fit ratio $|k|/k_0$ to $m_g/k_0$.
A detectable frequency dependence of the overlap reduction function is predicted, because $|k|/k_0$ depends on frequency through the dispersion relation; CPTA's reported frequency-dependent correlations are a first test of this signature.
The preference for the massive-gravity curve is independent of the source of the stochastic background, so it holds for both supermassive-black-hole-binary and cosmological backgrounds; source modeling cannot remove the effect.
The concrete geometric signatures to look for in future data are the shifted minimum angle (near $\xi\sim95^\circ$-$97^\circ$ rather than HD's $82^\circ$) and, at small $|k|/k_0$, an extra maximum near $\xi\sim108^\circ$.

Reading between the lines

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

If future analyses integrate the ORF over each PTA's full frequency band with measured pulsar distances instead of evaluating at one $fL\sim1$ point, the best-fit $|k|/k_0$ values will shift; the equipartition assumption $\Omega_T=\Omega_V=2\Omega_S$ is the least physically motivated input and the easiest to break.
CPTA's three frequency bins (at $1/T$, $1.5/T$, and $2/T$ with $T\approx3.40$ yr) provide a ready-made three-point test: if the massive-gravity dispersion relation is correct, the best-fit $|k|/k_0$ inferred from each bin should be consistent, not just the fit at $1/T$.
Extending the same binned chi-square procedure to the other PTA datasets would show whether the improvement over HD is a common feature of pulsar timing data or specific to NANOGrav15 and CPTA DR1.
The same angular-correlation data could be fitted with an anisotropic-background model; comparing that against the massive-gravity curve would clarify whether the improved chi-square is specifically due to the extra polarizations or to any curve with extra freedom.

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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

Colleague—

The one thing to know: the ORF calculation in this paper is new and not sloppy, but the headline claim conflicts with the dispersion relation. They fix mg at 1.31×10^−24 eV and then fit |k|/k0 as a free parameter. Eq. (2) says those two are locked together once you pick a frequency. The fitted values are not compatible with the fixed mg.

What's genuinely useful: they compute the massive-gravity ORF without the usual approximation E=1, and they show the exponential factors matter at fL ~ 1. That is a legitimate extension of Ref. [67], which dropped them. The comparison to NANOGrav15 and CPTA DR1 is a fair exercise, and they make code and data available. They also flag their assumptions, including equipartition of the polarization energy densities.

The soft spot is load-bearing. For CPTA at f = 1/T_CPTA ≈ 9.3×10^−9 Hz, the dispersion relation with mg fixed gives |k|/k0 ≈ 0.99995, effectively the HD limit. Their best fit for CPTA is |k|/k0 = 0.01, which implies mg ≈ 2.4×10^−22 eV, about 180 times larger than the adopted mass. For NANOGrav, 0.61 implies a mass ~0.8 k0, also far from the fixed value. The two datasets therefore require two different graviton masses. The improved chi-square is a curve fit, not evidence for massive gravity with the stated mass. On top of that, they pick the CPTA frequency bin post hoc because it deviates most from HD, and they don't describe the normalization of the theoretical curves to the data, so the d.o.f. count is suspect.

The paper is a good candidate for a serious referee because the ORF calculation is new and could be reanalyzed correctly. But as written, the central claim is unsupported. A revision that fixes |k|/k0 via Eq. (2) for the adopted mg, or that treats mg as the single free parameter and reports the resulting constraints, would be worth reading.

I'd send it to peer review, expecting heavy revision. Not something I'd cite in its current form.

Referee Report

4 major / 4 minor

Summary. The paper studies the overlap reduction function (ORF) for a stochastic gravitational-wave background in ghost-free massive gravity, including the exponential factors usually dropped in the short-wavelength limit and the extra vector and scalar polarization modes. It compares the resulting effective ORF with angular-correlation data from NANOGrav 15-year and CPTA DR1 and reports that a massive-gravity ORF with a fitted ratio |k|/k0 = 0.61 (NANOGrav) and |k|/k0 = 0.01 (CPTA) gives substantially better chi-squared values than the Hellings-Downs curve. The authors conclude that massive gravity may be favored by current PTA data.

Significance. If the central claim were correct, it would be a notable result: PTA angular correlations are currently one of the few observables that could distinguish polarization content and dispersion of gravitational waves, and a demonstration that the massive-gravity ORF fits both NANOGrav and CPTA better than the standard Hellings-Downs curve would be of broad interest. The paper has some strengths: it makes source code and data publicly available, it explicitly states its assumptions (equipartition of polarization energy densities, a fixed graviton mass, and a representative frequency for each dataset), and it is careful to use an estimator not biased toward the HD curve. However, the central quantitative claim is unsupported because the fitted quantity |k|/k0 is not a free parameter of the model once the graviton mass is fixed, and several additional choices (the post hoc selection of the CPTA frequency bin, the treatment of pulsar distances, and the assumed energy partition) are load-bearing for the reported chi-squared improvements.

major comments (4)

[Observational constraints and Results; Eq. (2) and Table I] The ratio |k|/k0 is not a free parameter of massive gravity once the graviton mass and observation frequency are specified. Equation (2), ω² = |k|² + m_g², with k0 = ω = 2πf, gives |k|/k0 = sqrt(1 - m_g²/k0²). The paper fixes m_g = 1.31×10⁻²⁴ eV in the Observational constraints section but then, in Results, states it is 'keeping m_g fixed and the ratio |k|/k0 as a free parameter' and fits |k|/k0. This is internally inconsistent. For CPTA at f = 1/T_CPTA ≈ 9.3×10⁻⁹ Hz, the fixed mass gives |k|/k0 ≈ 0.999, not the fitted 0.01 of Table I; for NANOGrav at f ≈ 1/(15 yr) ≈ 2.1×10⁻⁹ Hz, it gives |k|/k0 ≈ 0.989, not 0.61. The reported chi-squared improvements in Table I therefore do not test the massive-gravity model with the adopted m_g; they are fits of a quantity that the model fixes. If the ratio is instead treated as a free graviton-mass parameter, the two datasets imply mutually incompatible masses, and the claim that massive gravity 'predicts' better fits is no longer supported.
[Results, CPTA DR1 paragraph] The frequency bin for the CPTA comparison is chosen post hoc. The text states: 'We use the data for 1/T_CPTA, since it diverges from HD to the greatest extent.' This selection is made after inspecting the three frequency bins provided by CPTA, and no trials factor or simultaneous analysis of the three bins is presented. The reported improvement at the most favorable bin is therefore not a valid measure of the model's performance; the paper should either analyze all three frequency bins consistently or apply a correction for the selection.
[Overlap reduction function, Eq. (5), and Results, Fig. 2] The exponential factors E_j(f, Ω̂) in Eq. (5) depend on the pulsar distance L_j, and the paper emphasizes that these factors are important for fL ~ 1. However, the data comparison in Fig. 2 and Table I does not describe how the distances of the actual pulsars in NANOGrav15 and CPTA DR1 are used. PTA arrays contain pulsars over a wide range of distances, many with fL >> 1 for the frequencies considered, which would change the binned ORF shape. The authors should state the distances assumed for the pairs in Fig. 2 and show that the reported chi-squared values are robust to the actual distance distribution.
[Eq. (6) and Observational constraints] The effective ORF in Eq. (6) relies on the equipartition assumption Ω_T = Ω_V = 2Ω_S, which is introduced without derivation or observational justification. The shape of the effective ORF—and hence the fitted values of |k|/k0 in Table I—depends on this assumption. Since the paper's central claim is that massive gravity fits the data better, the analysis should either justify the partition from a specific model or show how the chi-squared comparison changes when the polarization energy-density ratios are varied. Without this, the reported improvement is conditional on an untested assumption.

minor comments (4)

[Abstract and Discussion] The abstract and Discussion state that massive gravity 'predicts better fits' for the observed correlations, but the Results section obtains those fits by adjusting |k|/k0 to the data. The wording should be softened to 'can describe' or 'can fit' unless the parameter is fixed a priori.
[Figure 1] Figure 1 does not state the value of fL used for the plotted ORFs; since the behavior depends strongly on fL, the caption should give the parameter values explicitly.
[Footnote 4] The phrase 'Fischer formalism' appears to be a typo for 'Fisher formalism.'
[Results, NANOGrav paragraph] The description of how the 13 bins were constructed to match the CPTA bin means is not detailed enough for reproducibility; the paper should specify the bin edges or the exact method used.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed MG 'prediction' of better PTA fits reduces to fitting the ratio |k|/k0, which Eq. (2) fixes once mg is specified; the fitted ratios are incompatible with the adopted mg.

fitted input called prediction [Eq. (2); Observational constraints; Results; Table I]
"ω2 = |k|2 + m2g , (2) where ω ≡ k0 = 2 πf , the frequency of the GW. ... This corresponds to mg ∼ 1.31 × 10−24 eV. ... Going forth, this is the mass that we will take mg to be in our analysis. ... We compute ˜ΓT , keeping mg fixed and the ratio |k|/k0 as a free parameter. ... the best-fit ˜ΓT has a ratio |k|/k0 = 0.61 ... the best-fit ˜ΓT has a ratio |k|/k0 = 0.01"

For fixed mg and frequency, Eq. (2) gives |k|/k0 = sqrt(1 − mg^2/k0^2), so the ratio is determined by the model, not free. With the adopted mg ≈ 1.31×10^(−24) eV and CPTA at f = 1/T_CPTA ≈ 9.3×10^(−9) Hz, one has mg/k0 ≈ 0.0054 and hence |k|/k0 ≈ 0.99998, essentially the HD limit. The fitted values 0.61 (NANOGrav15) and 0.01 (CPTA DR1) imply graviton masses about 30–180 times larger than the adopted value, and mutually inconsistent masses between the two datasets. The improved χ2 values in Table I are therefore obtained by fitting a parameter that the model has already fixed; presenting this as 'massive GWs predict better fits' reduces the central prediction to the fitted value rather than to the theory.

full rationale

The overlap-reduction-function derivation itself (Eqs. (4)–(6)) is self-contained and built on standard PTA formalism; the self-citations [87,89] are contextual and not load-bearing. The circularity is localized to the central statistical claim. After fixing mg ≈ 1.31×10^(−24) eV, the paper treats |k|/k0 as a free fit parameter, even though Eq. (2) fixes it once mg and the observation frequency are chosen. At PTA frequencies the fixed mass gives |k|/k0 ≈ 1, so the fitted ratios 0.61 and 0.01 do not correspond to the stated massive-gravity model. The reported chi-square improvement is thus an artifact of fitting a quantity that the model has already determined, making the main 'prediction' partially circular. No additional circularity from self-citation or imported uniqueness theorems was found.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper's central claim depends on a fixed graviton mass chosen as a 'best case', an equipartition assumption for polarization energy, a single-frequency ORF approximation, and a fitted ratio |k|/k0. No new entities are introduced; the extra polarizations are standard massive-gravity content.

free parameters (1)

|k|/k0 = 0.61 (NANOGrav15), 0.01 (CPTA DR1)
Introduced as the free shape parameter in the effective ORF and fit to the same PTA data used to claim support for MG. The claim of a better fit depends entirely on this fitted value.

assumptions (5)

domain assumption The SGWB is isotropic and the two-point correlation factorizes as in Eq. (3).
Stated in the Discussion: 'An assumption made in this Letter is the isotropy of the SGWB.' This is needed to define the ORF.
ad hoc to paper Energy densities of polarizations are equipartitioned: Omega_T = Omega_V = 2 Omega_S.
Stated after Eq. (6) with no physical justification. Directly determines the relative weights of tensor, vector, and scalar ORFs in the effective curve.
ad hoc to paper Graviton mass is fixed at mg = 1.31e-24 eV, the 'best-case' lower bound from a ~100-year PTA observation.
Stated in 'Observational constraints'. This is not the actual PTA band for NANOGrav15 or CPTA DR1, and the paper does not tie mg to the frequencies at which the data were taken.
domain assumption The ORF is evaluated at a single representative frequency (f = 1/T_CPTA for CPTA) rather than integrated over the PTA band.
Used in the Results and Fig. 2. Real PTA correlations combine contributions across a frequency band; this approximation is not justified quantitatively.
domain assumption Ghost-free dRGT massive gravity with five polarizations and dispersion relation omega^2 = k^2 + mg^2.
Standard framework from prior literature, used in Eq. (2) and the polarization decomposition. Not derived here.

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Cite this review

Pith. "Pith review of Do Pulsar Timing Datasets Favor Massive Gravity?." pith.science (2026). https://pith.science/paper/PG43XGB7

@misc{pith2026250702059,
  author       = {Pith},
  title        = {Pith review of: Do Pulsar Timing Datasets Favor Massive Gravity?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PG43XGB7}},
  note         = {Machine review of arXiv:2507.02059}
}

read the original abstract

Several observational phenomena suggest that the standard model of cosmology and particle physics requires revision. To address this, we consider the extension of general relativity known as massive gravity (MG). In this Letter, we explore the imprints of MG on the propagation of gravitational waves (GWs): their modified dispersion relation and their additional (two vector and one scalar) polarization modes on the stochastic GW background (SGWB) detected by pulsar timing arrays (PTAs). We analyze the effects of massive GWs on the Hellings-Downs curve induced by modification of the overlap reduction function. Our study consists of analyzing observational data from the NANOGrav 15-year dataset and the Chinese PTA Data Release I, and is independent of the origin of the SGWB (astrophysical or cosmological). By considering the bound on the graviton mass imposed through the dispersion relation, we scrutinize the possibility of detecting traces of MG in the PTA observational data. We find that massive GWs predict better fits for the observed pulsar correlations. Future PTA missions with more precise data will hopefully be able to detect the GW additional polarization modes and might be effectively used to constrain the graviton mass.

Figures

Figures reproduced from arXiv: 2507.02059 by the authors.

**Figure 2.** FIG. 2: The effective ORF plotted as a function of the angular separation [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: The [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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