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Bayesian Inference of Neutron Star Properties in $f(Q)$ Gravity Using NICER Observations

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Bayesian analysis of NICER pulsar data favors the exponential f(Q) gravity model and predicts neutron-star maximum masses near 3 solar masses, extending into the lower mass gap.

desk verdict First Bayesian NICER constraint on f(Q) neutron-star parameters, but the preferred nonlinear models violate the paper's own Eq. (28), so the headline claims are unsupported as written. read the letter →

arxiv 2601.16227 v2 pith:NYMCDWYD submitted 2026-01-17 gr-qc

Sneha Pradhan , N. K. Patra , Kai Zhou , P. K. Sahoo This is my paper
classification gr-qc
keywords f(Q)gravityneutronstarsBayesianinferenceNICERmass-radiusrelationtidaldeformabilitymassgapmodifiedTOVequations
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

This paper performs the first Bayesian inference of neutron-star structure in symmetric teleparallel f(Q) gravity, using NICER mass–radius measurements of four pulsars. It finds that the exponential form of f(Q) is statistically preferred over linear and logarithmic forms, and that all three constrained models predict maximum neutron-star masses reaching about 2.98 solar masses. Radii and tidal deformabilities at 1.4 solar masses remain consistent with current observations. If correct, this means f(Q) gravity can simultaneously explain NICER radii and predict heavy neutron stars in the mass gap, giving a testable alternative to general relativity.

What carries the argument

The central object is the nonmetricity scalar Q(r) = −2 e^{−λ}(ν′ + 1/r)/r built from the static spherical metric, along with the modified TOV system (Eqs. 25–34) that generalizes hydrostatic equilibrium to f(Q) gravity. A Bayesian sampler maps the f(Q) parameters (α, β, Q₀) to neutron-star observables through these equations, and Bayes factors rank the three functional forms.

What would settle it

For the best-fit exponential model parameters, compute the left-hand side of Eq. (28), (cotθ/2) Q′ f_QQ, along the stellar profile; if it does not vanish, the reported mass–radius solutions are not solutions of the field equations, and the statistical ranking would need to be recomputed with the full teleparallel connection.

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Extended reading notes

Core claim

Confronting f(Q)-gravity neutron-star models with NICER data does more than constrain free parameters—it yields a sharp prediction. The exponential model, f(Q) = −Q + αQ₀[1 − exp(−β√(Q/Q₀))], is favored over the linear and logarithmic models by Bayes factors, with its scale parameter Q₀ tightly constrained. All three models show 95% confidence bands of the mass–radius relation extending to nearly 3 M⊙, crossing into the lower mass gap (2.5–5 M⊙). The paper interprets this as evidence that nonmetricity-based gravity enhances high-density support, making mass-gap neutron stars a natural, testable outcome.

Load-bearing premise

The analysis assumes that the coincident-gauge, static spherically symmetric metric ansatz with a purely radial nonmetricity Q(r) satisfies the full f(Q) field equations; for the logarithmic and exponential models this is not the case, because Eq. (28) requires either f_QQ = 0 or Q′ = 0, and those models have neither.

Editorial extensions

If this is right

  • Exponential f(Q) is the statistically preferred model among the three, with well-constrained Q₀ and near-unimodal posteriors.
  • All three f(Q) models predict 95% confidence regions for the maximum mass that reach about 2.98 M⊙, placing stable configurations in the lower mass gap.
  • For the exponential model, the radius and tidal deformability at 1.4 M⊙ (R₁.₄ = 11.27 km, Λ₁.₄ = 156.95) are consistent with NICER and gravitational-wave constraints.
  • The mass-gap prediction is a clear, testable signature: precise mass measurements of neutron stars above 2.5 M⊙ would support f(Q) gravity.
  • The logarithmic model predicts the highest median maximum mass (2.36 M⊙), yet is less preferred statistically than the exponential model.

Reading between the lines

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

  • If the mass-gap prediction holds, it offers a rare direct probe of symmetric teleparallel gravity in the strong-field regime: a population of 2.5–3 M⊙ neutron stars would be difficult to explain in general relativity with conventional equations of state.
  • The paper fixes the DDME2 equation of state to isolate gravity effects; a joint inference over both EoS and f(Q) parameters, or the inclusion of GW170817 tidal constraints as an additional likelihood, could tighten or overturn the model ranking.
  • The Bayesian methodology could be extended to other f(Q) functional forms or to torsion-based teleparallel gravity, enabling a systematic observational ranking of teleparallel gravity models.
  • The reported preference for the exponential model might be sensitive to the chosen priors; a prior-sensitivity analysis would clarify whether the conclusion is robust.
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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 / 5 minor

Summary. The manuscript performs a Bayesian inference for three f(Q) gravity models (linear, logarithmic, and exponential), with the DDME2 equation of state fixed, using NICER mass–radius posteriors for PSR J0030+0451, PSR J0740+6620, PSR J0437+4715, and PSR J0614+3329. It derives modified TOV equations, computes mass–radius relations, tidal deformabilities, and maximum masses, and reports that the exponential model is statistically preferred while all models allow maximum masses extending toward ~2.98 M_sun and into the lower mass gap. The central technical problem is that the nonlinear logarithmic and exponential solutions do not satisfy the full field equations as written, because Eq. (28) requires either f_QQ=0 or Q'=0, neither of which is true for these models. The reported posterior distributions, mass–radius bands, and Bayes factors therefore are not supported by the stated theory.

Significance. If the results were valid, this would be a useful first systematic Bayesian constraint on f(Q) gravity from neutron-star observations, with a falsifiable mass-gap prediction. The paper addresses a currently underexplored comparison between symmetric teleparallel gravity and NICER data, and the Bayesian setup is standard. However, the main conclusions rest on nonlinear f(Q) solutions that are not admitted by the coincident-gauge ansatz used in the derivation. The exponential-model preference and the mass-gap prediction are precisely the parts of the analysis that are invalidated by this inconsistency. The paper also contains internal inconsistencies in the reported Bayes factors. These are not merely presentational issues; they affect the central claims.

major comments (4)
  1. [Sec. III, Eq. (28)] Equation (28), an independent component of the field equations for the metric (23), states 0 = (cotθ/2) Q' f_QQ. For the logarithmic model (36), f_QQ = -α/Q^2, and for the exponential model (37), f_QQ is nonzero in the posterior region. The numerical Q(r) from Eq. (29) is not constant, so Q' ≠ 0. Thus the nonlinear solutions do not satisfy the full field equations. The paper integrates the reduced system (25)-(27)/(33)-(34) and uses the resulting M-R curves in the NICER likelihood and Bayes factors, but these curves are not solutions of the stated theory. The linear model (35) is consistent because f_QQ=0, but the headline claims concern the logarithmic and exponential models. The standard remedy, a non-trivial teleparallel connection as in Ref. [23], changes the modified TOV system; this is not implemented or discussed. This is a load-bearing gap.
  2. [Table IV and Sec. VIII] The Bayes-factor evidence is internally inconsistent. Table IV lists ln(BF_12) = -1.81 for linear versus exponential and -0.43 for logarithmic versus exponential, with exponential as H2. The concluding section states 'values of −0.35 and −1.28' and describes the evidence as strong or decisive. On the paper's own scale in Sec. VI B 2, ln(BF) = -0.43 is only weak-to-moderate evidence for the exponential model, not 'definitely the best.' Please correct the numbers and re-evaluate the model-ranking claim.
  3. [Table I and Sec. IV B-C] The prior for Q0 is given as uniform on [-1,1], which includes Q0=0. Both the logarithmic model, f(Q) = -Q + α ln(β Q/Q0), and the exponential model, f(Q) = -Q + α Q0 [1 - exp(-β sqrt(Q/Q0))], are singular or undefined at Q0=0. No exclusion, regularization, or reparameterization is described. As written, the Bayesian models are not well defined on the stated prior support. If the implementation de facto excludes Q0=0, that restriction must be stated explicitly and motivated.
  4. [Sec. III, Eq. (33)] Equation (33) is printed as λ' = -kr(ρ+P)e^λ - ν'/f(Q). In the units used throughout the paper, f(Q) has dimension length^{-2}, so the term ν'/f(Q) has dimension length, while λ', r(ρ+P), and the left-hand side have dimension length^{-1}. This equation is therefore dimensionally inconsistent as it stands, and the GR limit f(Q) = -Q does not reproduce the standard TOV equation. If this is a typographical error, the corrected form of the modified TOV system must be given explicitly, since this is the system that is numerically integrated to produce all the paper's results.
minor comments (5)
  1. [Abstract] The sentence 'and exhibits well-constrained' is incomplete; the quantity that is well constrained should be stated.
  2. [Sec. III, Eq. (30)] Equation (30) contains two equal signs and is not a well-formed equation as printed. Please correct the displayed expression so that the pressure-gradient equation is unambiguous.
  3. [Sec. VII B and Fig. 4] The statement that the M-R curves 'pass through the centers of the observational bands' is an in-sample consistency check, because the f(Q) parameters were fitted to exactly those NICER posteriors. It is not by itself evidence of model success. A posterior predictive check or out-of-sample test would be more informative.
  4. [Sec. VII B / Table III] The abstract and Sec. VII B refer to 95% confidence regions, while Table III reports 68%(90%) intervals. Please clarify the coverage used in the quoted maximum-mass values and in Fig. 4.
  5. [Sec. V] The boundary condition λ(0)=0 is non-standard for the coordinate r; in Schwarzschild-like coordinates one usually has e^{-λ(0)}=1 but λ(0) itself can be set to zero by redefinition. The shooting condition e^{ν(R)}=e^{-λ(R)} should be stated in terms of the asymptotic Schwarzschild mass, which is not explicitly defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central inference chain is self-contained, though it has a separate field-equation consistency issue.

full rationale

The paper's derivation chain is not circular. The f(Q) model parameters are fitted to NICER mass-radius posteriors through the likelihood in Eq. (39), and the reported R_1.4, Lambda_1.4, M-R curves, and M_max are posterior outputs of that same Bayesian procedure. The statement that M-R curves "pass through the centers of the observational bands" is an in-sample fit description, not an independent prediction, and the paper does not disguise it as a forecast. The headline forward-looking claim, that "all three constrained models predict maximum neutron star masses reaching M_max ~ 2.98 M_sun" and that this "mass-gap prediction emerges naturally from the Bayesian-constrained parameter space," is a posterior extrapolation: no mass-gap object was used as a fitting target, so the result does not reduce to the inputs by construction. Self-citations such as refs. [25], [26], and [64] are used only to motivate functional forms and cosmological viability; they are not load-bearing for the TOV derivation or the Bayesian ranking. The main substantive weakness noted by the reader—Eq. (28), 0 = (cotθ/2) Q' f_QQ, requiring either f_QQ = 0 or Q' = 0 while the logarithmic and exponential models have f_QQ ≠ 0 and r-dependent Q—is a potential internal inconsistency or omitted consistency condition, not a circularity; it belongs to correctness assessment rather than to this circularity pass. Therefore, no circular step can be exhibited under the required standard.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on three fitted gravity parameters per model, the fixed DDME2 EoS, the coincident-gauge spherical TOV ansatz (questionable for non-linear f(Q), per Eq. 28), an assumed Schwarzschild exterior, and the treatment of published NICER posteriors as likelihoods. No new physical entities are invented; Q0 is a scale parameter inherited from the model definitions. The most fragile inputs are the gauge/connection assumption and the unquantified EoS contribution to the mass-gap result.

free parameters (3)
  • α = linear 0.28, logarithmic 1.55, exponential −5.18 (posterior medians)
    Amplitude of the f(Q) correction; uniform prior [−15, 15] (Table I).
  • β = linear −1.03, logarithmic 1.11, exponential 2.64 (posterior medians)
    Scale/departure parameter controlling the nonmetricity dependence; uniform prior [−3, 5] (Table I).
  • Q0 = linear 0.02, logarithmic −0.26, exponential −0.35 (posterior medians)
    Nonmetricity scale with dimension [length]^{-2}; uniform prior [−1, 1]. The prior includes Q0 = 0, where Q/Q0 is singular, and Q0 > 0, where the logarithmic and exponential functions are complex-valued inside the star (Q < 0). Posteriors are consistent with Q0 = 0.
assumptions (6)
  • domain assumption The coincident-gauge static spherically symmetric ansatz (metric Eq. 23, Q from Eq. 29) is a valid description of non-linear f(Q) neutron stars.
    Sec. III (Eqs. 23–34). Contradicted by the paper's own Eq. (28) unless f_QQ = 0 or Q′ = 0, which the logarithmic/exponential solutions violate. The non-trivial teleparallel connection required for spherical symmetry in non-linear f(Q) (see ref [23]) is never introduced.
  • domain assumption The exterior vacuum is the Schwarzschild metric and the matching e^ν(R) = e^{−λ}(R) is sufficient.
    Sec. V (shooting method). No f(Q) vacuum solution is derived; for non-linear f(Q) the exterior differs from Schwarzschild and matching conditions on Q are nontrivial.
  • domain assumption The DDME2 EoS captures dense-matter physics adequately.
    Sec. V. Fixing this EoS isolates gravity effects, but M_max inherits EoS stiffness; no GR+DDME2 baseline run is shown, so the f(Q)-induced boost is unquantified.
  • domain assumption The hypermomentum satisfies ∇_μ ∇_ν H^{μν}_α = 0.
    Sec. II, after Eq. (19); taken from ref [60].
  • domain assumption Published NICER mass–radius posteriors can be used directly as likelihoods.
    Sec. VI A, Eq. (39). Published posteriors embed the original analyses' priors; no prior re-weighting is discussed, and the choice between Miller- and Riley-team posteriors for J0030/J0740 is not stated.
  • ad hoc to paper The prior ranges in Table I are broad enough and well defined on the support of the models.
    α ∈ [−15,15], β ∈ [−3,5], Q0 ∈ [−1,1]. Q0 = 0 is singular in Q/Q0, and for Q0 > 0 the logarithmic/exponential terms are complex for Q < 0, so part of the prior volume lies outside the real-model domain.

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Pith. "Pith review of Bayesian Inference of Neutron Star Properties in $f(Q)$ Gravity Using NICER Observations." pith.science (2026). https://pith.science/paper/NYMCDWYD

@misc{pith2026260116227,
  author       = {Pith},
  title        = {Pith review of: Bayesian Inference of Neutron Star Properties in $f(Q)$ Gravity Using NICER Observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYMCDWYD}},
  note         = {Machine review of arXiv:2601.16227}
}
abstract

In this work, we investigate neutron stars (NSs) in the strong field regime within the framework of symmetric teleparallel $f(Q)$ gravity, considering three representative models: linear, logarithmic, and exponential. While Bayesian studies of NS observations are well established in general relativity and curvature based modified gravity theories, such analyses in $f(Q)$ gravity remain largely unexplored. For the first time we perform a Bayesian inference analysis by confronting theoretical NS mass-radius predictions with NICER observations of PSR J0030+0451, PSR J0740+6620, PSR J0437+4715, and PSR J0614+3329 in the background of nonmetricity based gravity. The dense matter equation of state is fixed to DDME2 in order to isolate the effects of modified gravity on NS structure. Our results show that the exponential $f(Q)$ model is statistically preferred over the linear and logarithmic cases, as confirmed by Bayes factor comparisons, and exhibits well-constrained. For this model, we obtain a radius and tidal deformability at $1.4\,M_\odot$ of $R_{1.4} = 11.27^{+0.53}_{-0.36}\,\mathrm{km}$ and $\Lambda_{1.4} = 156.95^{+84.02}_{-41.73}$, respectively, consistent with current observational constraints. Remarkably, all three constrained models predict maximum neutron star masses reaching $M_{\max} \simeq 2.98\,M_{\odot}$, with the $95\%$ confidence regions extending into the lower mass gap ($\sim 2.5$--$5\,M_{\odot}$). This mass-gap prediction emerges naturally from the Bayesian-constrained parameter space. These results highlight the potential of NSs as powerful probes of symmetric teleparallel gravity in the strong field regime.

Figures

Figures reproduced from arXiv: 2601.16227 by the authors.

Figure 1
Figure 1. FIG. 1: The energy density ( [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The marginalized posterior distributions of the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Corner plots for the marginalized posterior distributions (salmon) of the tidal deformability Λ [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The 95% confidence interval distributions for the radius [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The Pearson’s correlation coefficients among parameters and selected NS properties for linear (left panel), [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Violin plots of the posterior distributions for the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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