Realistic neutron stars in the constant-Q branch of f(Q) gravity

Debadri Bhattacharjee; P. H. R. S. Moraes; Pradip Kumar Chattopadhyay

REVIEW

Realistic neutron stars in the constant-Q branch of f(Q) gravity

Debadri Bhattacharjee, Pradip Kumar Chattopadhyay, P. H. R. S. Moraes

General Relativity and Quantum Cosmology · Submitted 27 Oct 2026

Pith's verdict: Competent but overreaching: the Tsujikawa model's claim of explaining GW190814 is unsupported because the paper uses different α values for low-mass and high-mass fits, and no joint parameter region exists.

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arXiv ID2610.36814 (version 2)

Pith ID4P7BUHBD, Pith's permanent record of this paper and its review

CitationDebadri Bhattacharjee, P. H. R. S. Moraes, Pradip Kumar Chattopadhyay. "Realistic neutron stars in the constant-Q branch of f(Q) gravity." arXiv:2610.36814 (2026). https://pith.science/paper/4P7BUHBD

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Editor's note

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Here's my quick take on Bhattacharjee et al. on neutron stars in constant-Q f(Q) gravity. The bottom line: it's a competent phenomenological study that's upfront about its limitations, but its central claim about GW190814 doesn't hold up under scrutiny.

What's done well: The authors clearly derive the modified TOV equations from the covariant f(Q) formalism, then numerically solve them for three new models (Born-Infeld, Hu-Sawicki analogue, Tsujikawa-inspired) with two realistic EOSs (DD2, SLy4). The mathematics looks correct, and the paper honestly acknowledges that this is a constant-Q branch (b248-b249) and that results should not be interpreted as a complete exploration (b250). The comparison between models and the complementarity between max mass and radius constraints is a useful pedagogical point.

Where it goes soft: The parameter fitting is circular. The Born-Infeld and Hu-Sawicki models reproduce low-mass radii well only because the parameters (Q0, μ, m) are fine-tuned post hoc to match those radii. The Tsujikawa model is worse: to get low-mass radii the authors use α=0.1, but to reach the GW190814 mass they switch to α=0.15 (DD2) or α=0.63 (SLy4). No single α can simultaneously satisfy both constraints. The paper notes the overprediction of low-mass radii qualitatively but does not quantify the tension or attempt a joint fit. So the phrase 'offering a viable physical origin for the … secondary object in GW190814' (b6) is not supported. The authors themselves say the Tsujikawa results 'are not compatible' with low-mass radii (Section IV), yet still claim viability. That's inconsistent. The constant-Q branch itself is a strong restriction that eliminates the derivative terms characteristic of f(Q), and its physical relevance is not proven.

Who should read this: People working on neutron stars in modified gravity who want a clear example of how parameter tuning can produce agreement with data, but who will also see that without independent constraints the exercise remains exploratory. It could serve as a cautionary example in a reading group.

Recommendation: Accept for peer review with major revision. The paper needs to either (a) perform a joint parameter estimation showing a single α that satisfies all constraints, or (b) drop the GW190814 claim entirely and reframe as a survey of possible behaviors within the constant-Q branch. The qualitative claims about Born-Infeld and Hu-Sawicki models are less problematic but would also benefit from a statistical comparison. The paper is worth discussing but in its current form the central result is overstated.

Referee report

Major objections (2)

[Section IV, Table VII] The Tsujikawa-inspired model is presented as offering a viable physical origin for the GW190814 secondary, but the analysis uses different α values for low-mass stars (α = 0.1) and for GW190814 (α = 0.15 for DD2, α = 0.63 for SLy4). No demonstration is given that a single value of α can simultaneously satisfy both the low-mass radius constraints (e.g., EXO 1745-248, GW170817) and the high-mass requirement (M > 2.59 M⊙). The paper acknowledges fine-tuning for GW190814 but does not quantify the tension or perform a joint parameter survey. Without such an analysis, the claim of a viable physical origin is not supported; the model is instead fitted separately to two different mass ranges. This issue must be addressed explicitly, either by finding an α that passes all constraints within observational uncertainty or by substantially tempering the viability claim.
[Section V (Discussion)] The paper clearly states that the constant-Q branch suppresses all derivative contributions from f'' and f''', and that the results should not be interpreted as a complete exploration of f(Q) gravity. However, the central phenomenological claims—especially the viability of the Tsujikawa model for GW190814—are drawn entirely within this restricted branch. The paper does not provide an estimate of how the inclusion of non-constant Q (and the associated gradient terms) might alter the mass-radius predictions. A discussion of possible systematic effects or a justification that the constant-Q branch is a good approximation for the conclusions would strengthen the paper. This is particularly important because the branch choice directly eliminates the degrees of freedom that could otherwise modify the radius predictions.

Minor objections (4)

[Tables V, VI, VII] The parameter values (Q0, μ, m, α) are tuned separately for each observed star; this is essentially a per-object fit. The paper should note that the models are not predictive across the population unless a fixed set of parameters is used. For the Born-Infeld and Hu-Sawicki models this is less problematic because the claim is limited, but it should be clarified.
[General] The term 'control parameter' is used for Q0, μ, m, and α interchangeably. Consider distinguishing between model parameters that are fixed (μ, m, α) and parameters that can be varied to recover GR (Q0).
[Table I caption] The caption says 'maximum mass and corresponding radius obtained from Fig. 1' but Figure 1 shows multiple curves; specify which parameter values correspond to each row.
[p. 4, Eq. (10)] The notation ˜Gij is introduced but not used again; it may be clearer to write the left-hand side as the full geometric tensor directly.

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The paper

The paper claims

A constant-Q branch of f(Q) gravity produces neutron stars massive enough to match the GW190814 secondary.

The reading

This paper investigates neutron stars in f(Q) gravity, a modified theory where gravity is mediated by non-metricity rather than curvature. The authors focus on the constant-Q branch, where the non-metricity scalar Q is set to a constant value throughout the star, simplifying the equations while retaining the model's functional form. They explore three new f(Q) models: a Born-Infeld-inspired model, a Hu-Sawicki analogue, and a Tsujikawa-inspired model. Using realistic equations of state (DD2 and SLy4), they solve the modified Tolman-Oppenheimer-Volkoff equations. The key result is that the Tsujikawa-inspired model, with f(Q)=Q+α Q₀ tanh(Q/Q₀), can support neutron stars up to about 3.36 solar masses, potentially explaining the ~2.6 solar mass compact object in GW190814. However, this model predicts radii for low-mass neutron stars that are too large compared to observations. The other two models match low-mass radius constraints but cannot reach masses above about 2.43 solar masses. The authors conclude that within the constant-Q branch there is a complementarity between maximum mass and radius bounds, emphasising the importance of multimessenger observations to constrain modified gravity.

What carries the argument

The constant-Q branch (Q(r)=Q_*, ∂_r Q=0) of f(Q) gravity, which suppresses derivative terms involving f″(Q) and f‴(Q) from the field equations, while retaining the model dependence through f(Q_*) and f′(Q_*). Three specific nonlinear forms are proposed: (i) a Born-Infeld-inspired square-root model, (ii) a Hu-Sawicki analogue with a rational function, and (iii) a Tsujikawa-inspired model with a hyperbolic tangent. The modified Tolman-Oppenheimer-Volkoff equations are solved with unified realistic equations of state (GPPVA(DD2) and SLy4), and stability is checked via the Harrison-Zel'dovich-Novikov criterion.

Extended reading notes

Core claim

Within the constant-Q branch of f(Q) gravity, the Tsujikawa-inspired model (f(Q)=Q+α Q₀ tanh(Q/Q₀)) stiffens the effective gravitational coupling, producing maximum neutron star masses of approximately 3.36 M⊙ with the DD2 equation of state and 2.85 M⊙ with SLy4. This provides a viable physical origin for the 2.59(+0.08/−0.09) M⊙ secondary object in GW190814. In contrast, the Born-Infeld-inspired and Hu-Sawicki analogue models remain below 2.43 M⊙ and 2.06 M⊙ respectively, but they reproduce observed radii for low-mass (<2 M⊙) neutron stars. The paper demonstrates a phenomenological complementarity: models that reach high maximum masses tend to overpredict radii for ordinary neutron stars, and vice versa.

Load-bearing premise

The entire analysis hinges on the constant-Q branch, where the non-metricity scalar Q is taken to be constant throughout the star, suppressing all radial derivatives of Q and thus eliminating contributions from f″(Q) and f‴(Q) in the field equations; this branch is a simplifying assumption motivated by regularity at the centre, but it is not proven to be the only physically realisable branch, and the results do not capture the full dynamical degrees of freedom of f(Q) gravity.

Editorial extensions

If this is right

If the Tsujikawa-inspired model is correct, the GW190814 secondary is a neutron star rather than a low-mass black hole, with a radius around 12.5 km (from the model fits).
The complementarity shown means that any single f(Q) model in the constant-Q branch cannot simultaneously explain both the highest masses and the smallest radii; multimessenger observations are essential to break the degeneracy.
Future radius measurements of low-mass neutron stars (e.g., from NICER or future X-ray missions) can directly rule out or support the Tsujikawa model if they fall below ~12 km for a 1.4 M⊙ star.
The constant-Q branch serves as a controlled testbed, but the paper acknowledges that full f(Q) dynamics including radial variation of Q must be explored to draw definitive conclusions.
The parameter space of the models can be constrained further by combining mass-radius data from gravitational wave events (GW170817, GW190814) with X-ray binary observations.

Reading between the lines

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

The failure of the Tsujikawa model to match low-mass radii may indicate that the constant-Q branch is insufficient, and a radially varying Q profile is needed to reconcile both endpoints of the mass-radius curve.
If future observations pin down the radius of a ~2.6 M⊙ compact object (e.g., through a gravitational wave event with tidal deformability constraints), the Tsujikawa model could be decisively tested.
The paper's results suggest that within the constant-Q branch, the functional form of f(Q) itself (not just its first derivative) determines the high-mass behavior, pointing to a need for systematic classification of viable models.
A non-constant-Q extension might restore compatibility with low-mass radius constraints while preserving the high maximum mass, possibly through additional degrees of freedom that soften the equation of state at low densities.

Original abstract

We investigate the hydrostatic equilibrium and global structure of neutron stars within symmetric teleparallel gravity, where gravitational interactions are mediated by the non-metricity scalar $Q$. Working within the metric-affine formulation, we adopt the regular, constant-$Q$ branch ($Q = Q_*$) of $f(Q)$ gravity, which preserves regularity at the stellar centre while maintaining explicit dependence on the non-linear functional form through $f(Q_*)$ and its derivatives. We propose three novel non-linear $f(Q)$ extensions: (i) a Born-Infeld-inspired model, (ii) a Hu--Sawicki analogue, and (iii) a Tsujikawa-inspired model. Incorporating state-of-the-art realistic unified equations of state, namely GPPVA(DD2) and SLy4, we numerically solve the modified Tolman-Oppenheimer-Volkoff equations and evaluate stellar stability via the Harrison-Zel'dovich-Novikov criterion. The Born-Infeld and Hu-Sawicki analogues smoothly recover the general relativistic limit as their control parameters increase, predicting stellar radii for low-mass objects ($M < 2.0\,M_\odot$) that match observational constraints from X-ray binaries and GW170817; however, their maximum masses remain capped below $2.43\,M_\odot$ (DD2) and $2.06\,M_\odot$ (SLy4). Conversely, the Tsujikawa-inspired model significantly stiffens the effective gravitational coupling, accommodating massive compact objects up to $\sim 3.36\,M_\odot$ and offering a viable physical origin for the $2.59_{-0.09}^{+0.08}\,M_\odot$ secondary object in GW190814, albeit at the expense of overpredicting radii for typical low-mass neutron stars. These contrasting behaviours demonstrate a key phenomenological complementarity between maximum mass and radius bounds within the constant-$Q$ framework, highlighting the utility of multimessenger observations in constraining non-linear non-metricity gravity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-28: 1 request, 6,247 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.92 · human participants 0.01 · controlled comparison 0.44 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.85 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We investigate the hydrostatic equilibrium and global structure of neutron stars within symmetric teleparallel gravity, where gravitational interactions are mediated by the non-metricity scalar $Q$. Working within the metric-affine formulation, we adopt the regular, constant-$Q$ branch ($Q = Q_*$) of $f(Q)$ gravity, which preserves regularity at the stellar centre while maintaining explicit dependence on the non-linear functional form through $f(Q_*)$ and its derivatives. We propose three novel non-linear $f(Q)$ extensions: (i) a Born-Infeld-inspired model, (ii) a Hu--Sawicki analogue, and (iii) a Tsujikawa-inspired model. Incorporating state-of-the-art realistic unified equations of state, namely GPPVA(DD2) and SLy4, we numerically solve the modified Tolman-Oppenheimer-Volkoff equations and evaluate stellar stability via the Harrison-Zel'dovich-Novikov criterion. The Born-Infeld and Hu-Sawicki analogues smoothly recover the general relativistic limit as their control parameters increase, predicting stellar radii for low-mass objects ($M < 2.0\,M_\odot$) that match observational constraints from X-ray binaries and GW170817; however, their maximum masses remain capped below $2.43\,M_\odot$ (DD2) and $2.06\,M_\odot$ (SLy4). Conversely, the Tsujikawa-inspired model significantly stiffens the effective gravitational coupling, accommodating massive compact objects up to $\sim 3.36\,M_\odot$ and offering a viable physical origin for the $2.59_{-0.09}^{+0.08}\,M_\odot$ secondary object in GW190814, albeit at the expense of overpredicting radii for typical low-mass neutron stars. These contrasting behaviours demonstrate a key phenomenological complementarity between maximum mass and radius bounds within the constant-$Q$ framework, highlighting the utility of multimessenger observations in constraining non-linear non-metricity gravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the hydrostatic equilibrium and global structure of neutron stars within symmetric teleparallel gravity, where gravitational interactions are mediated by the non-metricity scalar $Q$.	contribution 0.73	0.77	0.03	asserted 0.96
s1	Working within the metric-affine formulation, we adopt the regular, constant-$Q$ branch ($Q = Q_$) of $f(Q)$ gravity, which preserves regularity at the stellar centre while maintaining explicit dependence on the non-linear functional form through $f(Q_)$ and its derivatives.	method 0.95	0.79	0.03	asserted 0.98
s2	We propose three novel non-linear $f(Q)$ extensions: (i) a Born-Infeld-inspired model, (ii) a Hu--Sawicki analogue, and (iii) a Tsujikawa-inspired model.	contribution 0.98	0.92	0.03	asserted 0.98
s3	Incorporating state-of-the-art realistic unified equations of state, namely GPPVA(DD2) and SLy4, we numerically solve the modified Tolman-Oppenheimer-Volkoff equations and evaluate stellar stability via the Harrison-Zel'dovich-Novikov criterion.	method 1.00	0.94	0.04	asserted 0.97
s4	The Born-Infeld and Hu-Sawicki analogues smoothly recover the general relativistic limit as their control parameters increase, predicting stellar radii for low-mass objects ($M < 2.0\,M_\odot$) that match observational constraints from X-ray binaries and GW170817; however, their maximum masses remain capped below $2.43\,M_\odot$ (DD2) and $2.06\,M_\odot$ (SLy4).	result 1.00	0.93	0.16	asserted 1.00
s5	Conversely, the Tsujikawa-inspired model significantly stiffens the effective gravitational coupling, accommodating massive compact objects up to $\sim 3.36\,M_\odot$ and offering a viable physical origin for the $2.59_{-0.09}^{+0.08}\,M_\odot$ secondary object in GW190814, albeit at the expense of overpredicting radii for typical low-mass neutron stars.	result 0.97	0.92	0.37	asserted 0.74
s6	These contrasting behaviours demonstrate a key phenomenological complementarity between maximum mass and radius bounds within the constant-$Q$ framework, highlighting the utility of multimessenger observations in constraining non-linear non-metricity gravity.	interpretation 1.00	0.74	0.70	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.57 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: The maximum mass-radius and mass vs central density diagrams for the Born-Infeld model using realistic

**Figure 2.** FIG. 2: Mass-radius and mass vs central density diagrams for Hu-Sawicki model using realistic neutron star EoS,

**Figure 3.** FIG. 3: Mass-radius and mass vs central density diagrams for Hu-Sawicki model using realistic neutron star EoS,

**Figure 4.** FIG. 4: Mass-radius and mass vs central density diagrams for Tsujikawa model using realistic neutron star EoS,

Figures reproduced from arXiv:2610.36814 by the authors.

Discussion

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