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Inferences for f(R) Models from Late-Time Megamaser Observational Data

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Using megamaser angular-diameter distances, the paper finds three f(R) gravity models statistically indistinguishable from ΛCDM, with the deviation parameter b near zero and H0 near 73 km/s/Mpc.

desk verdict A legitimate but overstated application of P20 megamaser data to three f(R) models: the H0 anchor is fine, but the b posteriors do not support the claim that f(R) mimics ΛCDM. read the letter →

arxiv 2607.21129 v1 pith:LRO6A7GL submitted 2026-07-23 astro-ph.CO

Umang , Abha Dev Habib , Nisha Rani This is my paper · ORCID
classification astro-ph.CO
keywords f(R)gravitymodifiedmegamasersHubbleconstantcosmologicalparametersMarkovChainMonteCarlodarkenergyangulardiameterdistance
topics 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

The paper asks whether modified gravity of the f(R) type can leave a measurable imprint on the low-redshift expansion history when tested against purely geometric distances. Using angular-diameter distances and recession velocities of six water-megamaser host galaxies, it fits ΛCDM and three f(R) gravity models (Hu-Sawicki, Starobinsky, ArcTanh) with Markov Chain Monte Carlo. In all models the Hubble constant converges to about 73 km/s/Mpc, in line with other late-time measurements, while the f(R) deviation parameter b is consistent with zero. That means this dataset finds the f(R) models mimicking ΛCDM at the background level; AIC and BIC comparisons make them statistically indistinguishable. The matter density Ωm, however, is only weakly constrained, as expected for such low-redshift data.

What carries the argument

The load-bearing object is the deviation parameter b that enters the Hubble-rate expressions H(z; H0, b, Ωm) for the three f(R) models, each written as ΛCDM plus polynomial terms in b. Because b → 0 reduces each model to ΛCDM, the posterior on b measures how strongly data demand modified-gravity corrections. The analysis also leans on a combined likelihood that adds a peculiar-velocity uncertainty of 250 km/s in quadrature to each recession-velocity measurement, and on MCMC sampling to marginalize over the six galaxy velocities alongside H0, b, and Ωm.

What would settle it

Re-fit the six megamaser objects with the exact, untruncated H(z) expressions for each f(R) model; if the marginalized b then shifts away from zero by more than the quoted 1σ uncertainty, the paper's central conclusion is falsified. Alternatively, a future sample of ~30 megamaser hosts with 3% distance precision that rules out b = 0 at more than 2σ would overturn it.

Watch

Extended reading notes

Core claim

The central claim is that the megamaser sample — six galaxies with geometric angular-diameter distances and recession velocities — yields marginalized estimates of the f(R) deviation parameter b close to zero for all three models (b ≈ 0.011 ± 1.00 for Hu-Sawicki, 0.023 ± 1.04 for Starobinsky, −0.005 ± 1.01 for ArcTanh), while H0 is constrained near 73 km/s/Mpc in every model. The paper interprets b ≈ 0 as the signature that these f(R) models reduce to the ΛCDM expansion history at low redshift. Since the 1σ uncertainties on b are of order unity, the claim is not that b is tightly measured, only that the data do not prefer any departure from ΛCDM. The model-comparison statistics (ΔAIC, ΔBIC ≤

Load-bearing premise

The weakest point is the assumption that the H(z) formulas for the three f(R) models, imported from earlier work and truncated at low order in b, are the correct Hubble rates for those theories over the full prior range of b; if that mapping is wrong, the b ≈ 0 conclusion is an artifact.

Editorial extensions

If this is right

  • If correct, the megamaser geometry independently supports H0 ≈ 73 km/s/Mpc, reinforcing the disagreement with early-universe CMB-based estimates.
  • The b ≈ 0 result implies that, for background expansion, these f(R) models are not distinguishable from ΛCDM with current data; any discriminating power must come from structure growth or higher redshift.
  • The weak Ωm constraint (≈0.5 with σ≈0.4) demonstrates that low-redshift distance data alone cannot determine the matter density, so combined analyses with SNe Ia, BAO, or H(z) would be needed to sharpen all parameters.
  • AIC/BIC differences of order 2 mean that, by standard model-selection rules, the f(R) models are statistically equivalent to ΛCDM; the extra parameter b is not penalized into preference either way.

Reading between the lines

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

  • Editorial inference: Because the b posterior is only constrained to |b| ≲ 1, the polynomial truncation in the H(z) expressions could matter; recomputing with higher-order terms would test whether b ≈ 0 is robust.
  • Editorial inference: The paper works at the background level; one natural extension is to add growth-rate data (e.g., fσ8 measurements) to see whether the same f(R) models, with b consistent with zero from background, remain viable when perturbations are included.
  • Editorial inference: The fixed 250 km/s peculiar-velocity uncertainty is conservative; a full marginalization over per-object peculiar velocities with realistic priors could tighten or widen the quoted errors, though the paper's stochastic check suggests little change.
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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

3 major / 5 minor

Summary. The paper uses six megamaser angular-diameter distances and recession velocities from the Megamaser Cosmology Project (Pesce et al. 2020) to constrain H0, Omega_m, six recession-velocity nuisance parameters, and, for the three f(R) models, a deviation parameter b. The likelihood combines velocity and distance terms with a peculiar-velocity uncertainty of 250 km/s. MCMC posteriors give H0 around 73.5–73.7 km/s/Mpc for all four models, Omega_m near 0.5 with errors around 0.35–0.4, and b near 0 with errors around 1.0. AIC/BIC comparisons are used to claim that the f(R) models are statistically indistinguishable from LambdaCDM, and the paper concludes that this late-time dataset predicts that f(R) models mimic LambdaCDM.

Significance. The H0 constraint, if correct, is an interesting independent late-time geometric probe favoring the local value near 73 km/s/Mpc, and the paper demonstrates a clean likelihood treatment of the P20 sample. However, the advertised f(R) result is not supported by the reported errors: the b posteriors have 1-sigma widths comparable to the prior, so the data do not constrain b. The model-comparison section also contains an internal inconsistency with its own table. The paper is therefore more valuable for its H0 result than for its f(R) discrimination, which is effectively non-constraining at the current sample size.

major comments (3)
  1. [Abstract; Table 3] The claim that the P20 dataset 'predicts that f(R) models mimic LambdaCDM' is not supported by the quoted constraints. Table 3 gives b = 0.011 +/- 1.00, 0.023 +/- 1.04, and -0.005 +/- 1.01, while the prior is U(-1.5, 1.5). These 1-sigma intervals cover most of the prior; the data do not constrain b away from the prior. The abstract and Section 4 should be softened to state that the data are consistent with b=0 but provide no significant evidence either way.
  2. [Sec. 2.3, Eqs. (7)-(9)] The Hubble rates for the three f(R) models are imported from Sultana et al. (2022) with only a parenthetical reference. The paper does not define the f(R) actions, the parameter b, or the expansion used. Eqs. (7)-(9) are truncated series in b (up to order b^2 or b^4), yet the prior allows |b| up to 1.5 and the posteriors have errors of order 1, so the validity of the truncation is not established. The authors must state the model actions and b definition, justify the truncation, or use exact expressions; otherwise the b constraints are not interpretable.
  3. [Sec. 3.1, Table 5] The text states that '0 <= |Delta X| <= 2' and uses this to conclude that the f(R) models are statistically indistinguishable from LambdaCDM. Table 5 lists Delta AIC = 2.15-2.16 and Delta BIC = 2.64-2.65, which contradict the stated inequality. The comparison should be rephrased using a defined evidence scale, and the table values must be reconciled with the text.
minor comments (5)
  1. [Abstract] Typo: 'predict' should be 'predicts'.
  2. [Introduction] Typo: 'Active Galctic Nucleus' should be 'Active Galactic Nucleus'.
  3. [Sec. 4] The notation 'b- -> 0' and 'b- -> infinity' is unclear; presumably this should be the single parameter b.
  4. [References] Several references have minor formatting issues, e.g., 'V ol.' in Schinckel et al. and the Hogg (1999) entry is incomplete.
  5. [Sec. 2.2] The robustness check with sigma_pec drawn from U(150,250) is only described qualitatively. Please report the resulting parameter shifts or remove the claim.

Circularity Check

1 steps flagged · score 6.0 of 10

The claim that P20 predicts f(R) mimics ΛCDM rests on a fit of the deviation parameter b whose posterior equals the prior and whose zero value is ΛCDM by construction; the 'prediction' is therefore inherited from the ansatz.

  1. fitted input called prediction [Abstract; Section 2.3, Eqs. (7)–(9); Table 2; Table 3; Section 4]
    "The marginalized estimates of the deviation parameter, b, for the three f(R) gravity models lie close to zero. This late-time dataset, thus predicts that f(R) models mimic ΛCDM. ... In the limit b−→0, these models are constructed to reproduce the ΛCDM to ensure consistency with current observational constraints."

    Each of Eqs. (7)–(9) is written as the ΛCDM expression 1−Ωm+(1+z)^3 Ωm plus polynomial terms multiplied by powers of b, so b=0 returns ΛCDM by construction. Table 3 reports b=0.011±1.00, 0.023±1.04, −0.005±1.01, while Table 2 sets the prior b∼U(−1.5,1.5); the 1σ widths are essentially the full prior width, so the P20 data do not constrain b. The central value near zero is the mean of a prior-dominated, nearly flat posterior. Calling this a 'prediction' that f(R) mimics ΛCDM is thus a restatement of the model definition and prior, not an independent, data-driven result.

full rationale

The P20 angular-diameter-distance and velocity data are external to the f(R) constructions, and the H0 constraint (~73 km/s/Mpc) is largely model-independent, so that part of the analysis is not circular. The paper's central 'f(R) mimics ΛCDM' claim, however, is not independently established: the Hubble rates in Eqs. (7)–(9) are imported without derivation from Sultana et al. (2022), and b is constructed so that b=0 is exactly ΛCDM. Because the fitted b posteriors (0.011±1.00, 0.023±1.04, −0.005±1.01) are as wide as the prior U(−1.5,1.5), the data carry no information about b, and the abstract's 'predicts' overstates what is a prior/ansatz-derived central value. The model-comparison section is also internally inconsistent (Section 3.1 says '0≤|ΔX|≤2' while Table 5 lists ΔAIC≈2.15 and ΔBIC≈2.64), which is a correctness issue rather than circularity. Overall, the core inference about b and ΛCDM mimicry reduces, at least in part, to the construction of b and the choice of prior; hence a moderate circularity score is warranted.

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

No new theoretical entities are introduced: the f(R) models, the b parameter, and the P20 data are all from prior literature. The free parameters are fit to P20 data, while the main imported assumption is the b-expanded H(z) taken from Sultana et al. (2022), which carries the model dependence of the analysis.

free parameters (4)
  • H0 = 73.583±2.99 (ΛCDM); 73.726±4.79 (Hu-Sawicki); 73.519±3.46 (Starobinsky); 73.458±4.96 (ArcTanh) km/s/Mpc
    Hubble constant fitted with a broad flat prior U(50,100); central to the H0 claim and to the comparison with other late-time measurements.
  • Ωm = 0.511±0.39 (ΛCDM); 0.514±0.38 (Hu-Sawicki); 0.515±0.39 (Starobinsky); 0.505±0.34 (ArcTanh)
    Matter density fitted with U(0,1); the posterior is close to the prior midpoint, so the data provide almost no constraint.
  • b (f(R) deviation parameter) = 0.011±1.00 (Hu-Sawicki); 0.023±1.04 (Starobinsky); -0.005±1.01 (ArcTanh)
    Fitted with U(-1.5,1.5); the posterior width is comparable to the prior, so this parameter is not actually constrained by P20 data.
  • v_i (six galaxy recession velocities, i=1..6) = Table 4: v1≈3514, v2≈10190, v3≈7813, v4≈8535, v5≈7088, v6≈559 km/s
    Nuisance velocity parameters for each megamaser galaxy, fitted with U(500,12000) km/s and constrained by the observed velocities plus a common peculiar-velocity uncertainty σ_pec=250 km/s in the likelihood.
assumptions (5)
  • standard math The FLRW angular-diameter distance formula (Eq. 5) applies for all models.
    Standard cosmological distance integral; no new derivation is provided.
  • domain assumption Equations (7)-(9) correctly give the Hubble rates of the Hu-Sawicki, Starobinsky, and ArcTanh f(R) models.
    Imported from Sultana et al. (2022) without derivation; the expressions are expansions around the ΛCDM Hubble rate in powers of b, so the 'mimics ΛCDM' result is partly built into the ansatz.
  • domain assumption The low-redshift relation z_i = v_i/c (Eq. 1) is accurate for the megamaser sample.
    Used to convert recession velocities to redshifts for the distance computation; only a first-order approximation and may introduce small systematic errors at z≈0.03.
  • domain assumption A single peculiar-velocity uncertainty σ_pec=250 km/s applies to all six galaxies.
    Follows Pesce et al. (2020) and is acknowledged as conservative; the authors test random σ_pec∈U(150,250) in a few runs but do not report quantitative changes.
  • domain assumption The combined Gaussian likelihood (Eq. 4) with flat priors is a valid statistical model for the data.
    Assumes independent Gaussian distance and velocity errors, no covariance between galaxies, and treats the recession velocities as free nuisance parameters.

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Pith. "Pith review of Inferences for f(R) Models from Late-Time Megamaser Observational Data." pith.science (2026). https://pith.science/paper/LRO6A7GL

@misc{pith2026260721129,
  author       = {Pith},
  title        = {Pith review of: Inferences for f(R) Models from Late-Time Megamaser Observational Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LRO6A7GL}},
  note         = {Machine review of arXiv:2607.21129}
}
abstract

In this work, we study three widely used models of f(R) gravity, namely HuSawicki, Starobinsky and ArcTanh along with the standard cosmology model ($\Lambda$CDM). For this, we employ the megamaser angular diameter distance and velocity measurements from the Megamaser Cosmology Project, which provide a purely geometric determination of the Hubble constant. We constrain the parameters using the Markov Chain Monte Carlo method. Our results show that values of the Hubble Constant, $H_{0}$, obtained for all four models are in concordance with its value obtained from other late-time observational data such as SNe Ia. The constraints on $H_{0}$ in all the models under study are restrictive and the marginalized estimates lie close to 73 $\mathrm {km s^{-1} Mpc^{-1}}$. The marginalized estimates of the deviation parameter, b, for the three f(R) gravity models lie close to zero. This late-time dataset, thus predicts that f(R) models mimic $\Lambda$CDM. However, the matter density, $\Omega_m$, remains weakly constrained for all the models with its marginalized estimate close to 0.5. Further, comparison of the four models (f(R) models and $\Lambda$CDM) using information criteria such as Akaike Information Criterion and Bayesian Information Criterion shows that within current uncertainties, the dataset finds f(R) models statistically indistinguishable from $\Lambda$CDM. This is consistent with the fact that the favoured value of b for each of the f(R) models lies close to zero.

Figures

Figures reproduced from arXiv: 2607.21129 by the authors.

Figure 1
Figure 1. Constraints on the parameters using emcee for the combined likelihood. Darker to lighter color [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. 1σ and 2σ confidence contours and posterior distributions for the ΛCDM parameters, i.e., H0 and Ωm [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Constraints on the parameters using emcee for the combined likelihood. Darker to lighter color [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Constraints on the parameters using emcee for the combined likelihood. Darker to lighter color [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Constraints on the parameters using emcee for the combined likelihood. Darker to lighter color [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Marginalized 1σ and 2σ credible intervals for the model parameters H0, b and Ωm for the three f(R) models. To bring model parameters in focus, we slice Figures 3, 4 and 5 to produce sub-plots 6a, 6b and 6c respectively [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Plots to verify the convergence of MCMC chains using auto-correlation time function, [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: A comparative study of H0 constraints obtained in this work using the P20 dataset (labeled in black color) with other late-time datasets as mentioned in [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

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