Properties of interacting quark star in light of Rastall gravity

Anirudh Pradhan; Archana Dixit; Ayan Banerjee; Izzet Sakalli

REVIEW 4 major objections 4 minor 63 references

Properties of interacting quark star in light of Rastall gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Quark stars in Rastall gravity can reach 2.81 solar masses, exceeding the general-relativistic limit and matching the GW190814 companion's mass.

desk verdict The interacting-quark EoS is new to Rastall TOV and the qualitative trend is plausible, but the paper never fixes which mass it reports, so the headline maximum masses and the GW190814 match are not well-defined. read the letter →

arxiv 2412.09306 v1 pith:X36ZRUKV submitted 2024-12-12 gr-qc

Ayan Banerjee , Anirudh Pradhan , Izzet Sakalli , Archana Dixit This is my paper · ORCID

classification gr-qc MSC 83C5583D05 PACS 04.40.Dg04.50.Kd

keywords quarkstarsRastallgravityinteractingmattermodifiedTOVequationsmass-radiusrelationGW190814colorsuperconductivitycompact

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 claims that quark stars built from a QCD-inspired interacting quark matter equation of state can be significantly more massive in Rastall gravity than in general relativity, reaching maximum masses up to 2.81 solar masses. If true, this would offer a modified-gravity explanation for massive compact objects like the GW190814 secondary without invoking exotic or unusually stiff matter. The authors derive the structure of these stars from modified Tolman-Oppenheimer-Volkoff equations, survey how the Rastall parameter, interaction parameter, and bag constant shift the mass-radius relation, and argue that the configurations satisfy standard stability and causality criteria.

What carries the argument

The load-bearing structure is the Rastall-gravity modification of the Tolman-Oppenheimer-Volkoff equations, in which the effective energy density ρ_eff and pressure p_eff are linear combinations of the ordinary fluid density and pressure weighted by the Rastall parameter η. The integration is performed for an effective mass m_eff defined through ρ_eff, while the exterior spacetime is taken as Schwarzschild with a mass m(r_s) that the paper treats as equal to the total mass; an explicit relation (Eq. 23) shows m_eff and m differ by a pressure integral, yet the paper does not clarify which mass is reported in the tables. The equation of state is the interacting quark matter model of Zhang and Mann, expressed in dimensionless form with parameters λ̄ and Beff, and used for the color-flavor-locked superconducting phase.

What would settle it

Recompute the mass-radius relation using the Schwarzschild mass m(r_s) obtained from Eq. (23), enforcing the exterior junction condition at p(r_s)=0; if the maximum mass at η=0.8 falls below the 2.5-solar-mass threshold or the λ̄=1.0 case drops below the GW190814 2.6-solar-mass bound, the central claim fails. Additionally, solving the full radial-oscillation equations could show an instability before the maximum mass, which would contradict the stability conclusion.

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Extended reading notes

Core claim

The paper's central claim is that inside Rastall gravity, quark stars described by an interacting quark matter equation of state can achieve higher maximum masses than their general-relativistic counterparts. Concretely, for an effective bag constant of 90 MeV/$fm^{3}$ and interaction parameter 0.6, lowering the Rastall parameter to 0.8 raises the maximum mass to 2.57 solar masses with a radius near 12 km, while the general-relativistic case (η = 1) gives only 2.22 solar masses. Raising the interaction parameter to 1.0 pushes the maximum to 2.81 solar masses at a radius of 12.91 km. The paper interprets these results as consistent with the ~2.6-solar-mass compact object in GW190814 and with massive pulsar measurements, and reports that static stability, adiabatic index, and sound-speed checks all support stability. The authors also constrain the model parameters η, λ̄, and Beff by comparing predicted masses with observed pulsars and the GW190814 event.

Load-bearing premise

The paper assumes that the mass it tabulates is the Schwarzschild mass that appears in the exterior spacetime, but the TOV integration computes an effective mass that differs from it by a pressure integral, and the paper never states which mass is actually reported.

Editorial extensions

If this is right

If the maximum-mass results hold, Rastall gravity offers a way to accommodate compact objects above 2.5 solar masses without resorting to unusually stiff hadronic equations of state.
The study provides a parameter-constraint scheme (η, λ̄, Beff) that can be tested against future pulsar mass measurements and gravitational-wave detections of compact object mergers.
The modified TOV framework could be applied to other equations of state, such as hybrid stars or hyperon-inclusive matter, to see whether the mass enhancement persists.
The stability analysis suggests that static stability, adiabatic index, and causality checks are satisfied in the Rastall setting, supporting the viability of these configurations.
The mass-radius relations for different η values indicate that the deviation from general relativity is most pronounced at high central densities, where quark matter is stiffest.

Reading between the lines

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

The reported maximum masses may be sensitive to which mass definition is tabulated: if the effective mass m_eff is used instead of the Schwarzschild mass m(r_s), the GW190814 compatibility could weaken, because Eq. (23) shows m_eff differs from m by a pressure-volume term.
A full radial-oscillation analysis (Sturm-Liouville eigenmode problem) is needed to confirm the claimed stability; the static criteria alone are necessary but not sufficient for dynamical stability in modified gravity.
The same interacting quark matter EoS could be tested in other modified gravity theories, such as f(R) or scalar-tensor models, to see whether the mass enhancement is a generic feature of non-minimal matter-geometry coupling.
If future gravitational-wave observations refine the mass of the GW190814 secondary or discover a heavier compact object, the Rastall parameter η would be more tightly constrained than the current range [0.8, 1.2] allows.

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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 thing to know: this is a straightforward TOV shootout for quark stars in Rastall gravity, and the qualitative message—smaller η gives larger maximum masses—is probably right. But the quantitative headline is compromised because the paper never says whether the tabulated masses are meff(rs) or m(rs)=∫4πr²ρ dr, and these differ for η≠1. The junction to the Schwarzschild exterior is therefore ambiguous, which makes the 2.57 and 2.81 M⊙ numbers and the GW190814 comparison unfixed.

What is new is the combination: the interacting quark EoS from Zhang and Mann has not been run through Rastall TOV before. The paper does that cleanly, gives M-R and compactness curves, and reports the expected trends (max mass rises as η drops, as λ¯ rises, as Beff drops). The numerical work is transparent enough that someone could reproduce it. The limitations the authors state—no crust, no radial oscillations—are acknowledged openly, which is honest.

The soft spots are real. The mass ambiguity is the load-bearing one, and it's exactly as the stress test says: Eq. (23) shows meff and m differ by a pressure integral, and the boundary condition p(rs)=0 doesn't identify which one appears in Tables I–III. If the tables list meff, the text's definition of M as m(rs) is wrong; if they list m, the interior metric 1−2meff/r is matched to the wrong exterior. Either way the central numbers need re-deriving. Secondary issues: the stability conclusions import GR criteria (γ>4/3, dM/dρc) without deriving Rastall versions; the parameter "constraints" are really a fit to the very pulsars and GW events used for validation, so calling it consistent is circular in a weak sense; and there are internal inconsistencies (GW190425 in the abstract vs. GW190814 in the body, Table II caption says η=0.8 while §IV.C uses η=0.6, Fig. 5 described as M−ρc when it shows the adiabatic index). None of these alone would sink the paper, but together with the mass issue they leave the viability claim unsupported.

Who is this for? People working on modified-gravity neutron/quark star models will want to see the M-R curves and the qualitative trend. They should not cite the numbers until the authors fix the mass definition. I would not use this as a citable result, but it deserves a serious referee because the underlying question is legitimate and the fix is plausibly short.

Recommendation: send to peer review with a request for major revision; the authors need to state which mass is integrated, enforce the junction condition consistently, and re-run the numbers. If they do, the qualitative conclusion likely survives.

That's my take.

Referee Report

4 major / 4 minor

Summary. The paper studies static, spherically symmetric interacting quark stars in Rastall gravity using a pQCD-inspired equation of state. The authors derive modified TOV equations, solve them numerically for variations of the Rastall parameter η, the interaction parameter ¯λ, and the effective bag constant B_eff, and report maximum masses up to 2.57 M⊙ (η=0.8) and 2.81 M⊙ (¯λ=1.0). They compare these results with observed pulsar masses and the GW190814 secondary object, and assess stability via the static stability criterion, the adiabatic index, and the causality condition v_s²<1. The central claim is that Rastall gravity can accommodate stable quark stars with masses exceeding those allowed in general relativity, and that the model is consistent with massive compact object observations.

Significance. If the central result were established, it would be a useful contribution to the literature on modified-gravity neutron-star and quark-star models: the paper combines a contemporary interacting quark matter equation of state with the Rastall formalism and provides a systematic parameter scan. The derived relations (14)-(19) are a clear extension of the standard TOV equations, and the comparison with several pulsar mass measurements is a potentially valuable falsifiable setup. However, the numerical headline masses and all astrophysical comparisons rest on an unresolved definition of the total mass, and the stability conclusions rely on general-relativistic criteria that are asserted, not derived, for Rastall gravity. These issues are central and must be fixed before the results can be evaluated.

major comments (4)

[§III Eq. (23)] The paper never states which mass is reported. The TOV integration solves for m_eff via Eq. (17), and the metric in Eq. (19) contains m_eff, but the exterior is matched with M=m(r_s) where m is defined in Eq. (20) as the integral of the physical density ρ. Equation (23) shows that m_eff and m differ by a pressure integral whenever η≠1; for η=0.8, m_eff = (7/6)m − (1/2)I_p, which is not a negligible correction. If Tables I-III list m_eff(r_s), then the exterior matching condition in Eq. (25) is inconsistent; if they list m(r_s), then the interior metric used in the TOV equation is not the same quantity that is matched. The maximum masses 2.57 M⊙ and 2.81 M⊙, and hence the claimed consistency with GW190814 and massive pulsars, are not well-defined until the authors specify which mass is tabulated, enforce the correct junction condition, and recompute the relevant quantities.
[§V] Stability is claimed on the basis of dM/dρ_c > 0, γ > 4/3, and v_s² < 1, but these criteria are imported from general relativity without derivation. The static stability criterion is derived in GR from the properties of equilibrium sequences and radial oscillation modes; the adiabatic-index threshold γ > 4/3 is likewise derived from the GR pulsation equation. In Rastall gravity, the field equations and the effective fluid variables differ, so the standard criteria do not automatically apply. The manuscript itself states that radial oscillations are not analyzed, but it still uses these criteria to assert dynamical stability. The authors should either derive the Rastall radial perturbation equations and show that the GR criteria remain valid, or soften the stability claim to a necessary-but-not-sufficient check that is explicitly conditioned on the validity of the GR criteria.
[§IV.A] The parameter constraints in Section IV.A are obtained by varying η, ¯λ, and B_eff until the predicted maximum masses match the observed masses of PSR J0348+0432, PSR J0740+6620, PSR J0952-0607, and GW190814. The abstract and Section VI then present the agreement with these same observations as confirmation that the model is viable. This is a circular validation procedure: the same data are used both to set the parameters and to test the model. The authors should reframe this as calibration, and provide an independent check—for example, comparing predicted radii or tidal deformabilities with independent measurements—or explicitly state that the observations are used only to select parameters, not to validate the model.
[§IV.B-§IV.C, Table II, Fig. 5] There are several internal inconsistencies that make the results difficult to interpret. The abstract and introduction mention GW190425 as the gravitational-wave constraint, but the numerical sections use GW190814 (§IV.B and Fig. 1). Table II has a caption stating η=0.8 while the text of §IV.C says η=0.6, and the figures are generated with η=0.6. In §V.A, the static stability discussion refers to Fig. 5 as the M−ρ_c plot, but Fig. 5 in the manuscript shows the adiabatic index and the M−ρ_c curves are in Fig. 4; §V.B also refers to Fig. 5 for the adiabatic index. These inconsistencies must be resolved because they prevent the reader from associating the tabulated maximum masses with the stated parameter values.

minor comments (4)

[Eqs. (21), (23)] There are typographical issues: both integrals contain a stray comma before dr′ (e.g., "4πr′2ρ(r′), dr′"), and the notation would be clearer if written as 4πr′²ρ(r′) dr′.
[§IV.D] The text says "massive QSs exits with masses 2.20 to 2.65 M⊙"; this should be "exist". Also, in the same paragraph the statement that the maximum compactness "does not affected by increasing or decreasing values of B_eff" is grammatically awkward and should be rephrased.
[Fig. 6] The caption of Fig. 6 and the text in §V.C describe v_s² as a function of radial coordinate r, but the discussion of the "brinjal" (purple) line and the divergence in the outer layers would be easier to follow if the figure included a clear legend matching the parameter values used in the three panels.
[References] Several references are arXiv preprints without journal or DOI information (e.g., Refs. [1], [4], [5], [6], [42]). If the manuscript is intended for journal publication, these should be updated to published versions where available.

Circularity Check

1 steps flagged · score 6.0 of 10

The model parameters are tuned so that maximum masses match the observed pulsar/GW masses, so the GW190814 'consistency' is a fitted input, not an independent prediction; the underlying TOV/EoS computation itself is not circular.

fitted input called prediction [Section IV.A (Constraints on Model Parameters), with the conclusion echoed in Section VI]
"The constraints on this parameter come from ensuring that the predicted maximum masses of quark stars match the observed masses of massive pulsars. Higher values of ¯λ, such as 1.0, correspond to more massive and larger-radius quark stars, fitting well within the range of observed pulsar data. ... In conclusion, the constraints on η, ¯λ, and Beff were derived by varying these parameters and comparing the corresponding mass-radius relations with observational data."

The parameters are selected by matching the model maximum masses to the same pulsar/GW masses that are later quoted as independent support. η=0.8 and λ̄=1.0 are specifically retained because they make Mmax=2.57 and 2.81 M⊙, i.e., reach the GW190814 2.6 M⊙ threshold; the final claim that the stars are 'consistent with observations of massive pulsars, such as those seen in the GW190814 event' restates that selection criterion. The maximum masses are genuine TOV outputs, so this is a fitted-input validation loop rather than a fully self-definitional derivation; but it is not an independent prediction.

full rationale

The numerical core of the paper — the Rastall-modified TOV equations (16)-(17) integrated with the IQM EoS (6) and the CFL-phase choices — is a self-contained calculation; no step of that derivation reduces to the target conclusion. The GR limit η=1 and the EoS limits (λ̄→0, λ̄→∞) are internally consistent, and the energy-condition and stability citations [54, 55, 59] are external, not a self-citation chain. I therefore do not rate the whole paper circular. The circularity is confined to the validation loop: Sec. IV.A explicitly says the constraints on η, λ̄, Beff were obtained by varying them until predicted maximum masses match observed pulsar/GW masses; Sec. VI then presents agreement with GW190814 etc. as a confirmation. That is a fitted input called a prediction (pattern 2), so the headline consistency is not independent. The authors' self-citations [7, 42] are not load-bearing. Separately, the meff/m ambiguity in Eqs. (19)-(25) is serious — the text defines M via m(rs) in Eq. (25) while the integration solves for meff and Eq. (23) shows they differ by a pressure term — but this is a definitional/consistency flaw rather than a circularity, so it does not further raise the circularity score. Overall score 6 reflects one fitted-input validation loop; the underlying TOV integration and M-R computation still contain independent physical content.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the Rastall field equations with free coupling eta, the Zhang-Mann interacting quark equation of state, and two unproved modeling assumptions: GR stability criteria and the mass-matching procedure. No new entities are postulated. The free parameters eta, lambda_bar and B_eff are hand-picked to reproduce observed mass ranges, so they carry much of the predictive weight.

free parameters (3)

eta (Rastall coupling parameter) = varied in [0.8,1.2]; eta=0.6 used in Fig. 2; no quoted best fit
Controls deviation from GR; M_max increases from 2.07 to 2.57 solar masses as eta decreases; chosen so the model reaches observed high masses.
lambda_bar (interacting quark parameter) = varied in [0.1,1.0]
Encodes pQCD and color-superconductivity corrections (Delta=100 MeV, m_s=150 MeV); M_max ranges 2.04 to 2.81 solar masses; treated as free to fit the observed mass range.
B_eff (effective bag constant) = varied in [60, 90] MeV/fm^3
Nonperturbative QCD vacuum parameter; M_max decreases with B_eff; chosen by hand and not inferred from data.

assumptions (4)

domain assumption Rastall field equations with non-conserved energy-momentum and free coupling eta (Eq. 8)
Adopted from Rastall (1972) and refs [46,47]; the paper does not derive or independently constrain eta.
domain assumption Zhang-Mann interacting quark matter equation of state (Eq. 6) applies throughout the star, without a hadronic crust
Adopted from refs [44,45]; the authors note a crust would change the radius and slightly affect the mass, so this choice is load-bearing.
ad hoc to paper GR stability criteria (static stability, gamma>4/3, v_s^2<1) remain valid in Rastall gravity
Section V states the criteria apply without deriving Rastall-specific radial oscillation equations; this supports the stability conclusion.
ad hoc to paper Exterior Schwarzschild matching with m(r_s)=M while integrating m_eff
Eqs. (21)-(25): m and m_eff differ by a pressure integral, so the mass boundary condition is ambiguous.

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Pith. "Pith review of Properties of interacting quark star in light of Rastall gravity." pith.science (2026). https://pith.science/paper/X36ZRUKV

@misc{pith2026241209306,
  author       = {Pith},
  title        = {Pith review of: Properties of interacting quark star in light of Rastall gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X36ZRUKV}},
  note         = {Machine review of arXiv:2412.09306}
}

abstract

This study explores the properties of quark stars (QS) formulated with an interacting quark matter equation of state (EoS) within the framework of Rastall gravity, a modified theory of gravity. We derive the mass-radius relationships and calculate the maximum gravitational masses and their corresponding radii, comparing these results under both Rastall gravity and general relativity. Our analysis incorporates recent observational data, including the GW190425 event, to constrain the model parameters ($\bar{\lambda}, \eta, B_{\rm eff}$). We also assess the stability of these quark stars by evaluating their static stability, adiabatic index, and sound velocity profiles, thus confirming their viability within the modified gravitational framework.

Figures

Figures reproduced from arXiv: 2412.09306 by the authors.

**Figure 3.** FIG. 3. We display the mass-radius and mass-compactness re [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** FIG. 4. Profiles of the mass versus central density relations. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 6.** Figure 6: FIG. 6. The squared speed of sound [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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1 and Table I)

Parameter η (Rastall’s Coupling Constant) : We vary η within the range [0.8, 1.2] as illustrated in Sec- tion IV B (see Fig. 1 and Table I). By comparing the mass-radius relations derived from the model with observed data from massive pulsars, including PSR J0348+0432, PSR J0740+6620, and the compact object in FIG. 2. We display the mass-radius and mass-c...

work page
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(7) or, equivalently, p = ρ − 2Beff after scaling back using Eq. (4). However, for negative values of λ, taking the EoS to the ¯λ → ∞ limit, Eq. (6) does not yield a finite form. As expected, the positive increasing values of ¯λ correlate with the maximum mass of QS, as detailed in Refs. [44, 45]. It is worth noting that the coefficients ξ seen in Eq. 3 c...

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2 and Table II), ¯λ is varied between [0.1, 1.0]

Parameter ¯λ (Interaction Parameter): As shown in Section IV C (Fig. 2 and Table II), ¯λ is varied between [0.1, 1.0]. The constraints on this parameter come from ensuring that the predicted maximum masses of quark 6 FIG. 3. We display the mass-radius and mass-compactness re- lations for QSs in Rastall gravity. Here, we vary Beff ∈ [60, 90] MeV/fm3, and o...

work page
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3 and Table III), we explore the effect of vary- ing Beff within the range [60, 90] MeV/fm 3

Parameter Beff (Effective Bag Constant): In Section IV D (Fig. 3 and Table III), we explore the effect of vary- ing Beff within the range [60, 90] MeV/fm 3. Our results indicate that increasing Beff decreases both the maxi- mum mass and the radius of the quark stars. This en- ables us to rule out values ofBeff that produce quark star masses below the 2 M⊙...

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Furthermore, we gather the data for maximum compact- ness in Table III depending on the parameters and is found to be 0.297

It can be seen that the maximum compactness does not affected by increasing or decreasing values of Beff. Furthermore, we gather the data for maximum compact- ness in Table III depending on the parameters and is found to be 0.297. 8 FIG. 4. Profiles of the mass versus central density relations. The parameters used are the same as in Figs. 1 to 3. FIG. 5. ...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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1 and Table I)

Parameter η (Rastall’s Coupling Constant) : We vary η within the range [0.8, 1.2] as illustrated in Sec- tion IV B (see Fig. 1 and Table I). By comparing the mass-radius relations derived from the model with observed data from massive pulsars, including PSR J0348+0432, PSR J0740+6620, and the compact object in FIG. 2. We display the mass-radius and mass-c...

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[2] [2]

(7) or, equivalently, p = ρ − 2Beff after scaling back using Eq. (4). However, for negative values of λ, taking the EoS to the ¯λ → ∞ limit, Eq. (6) does not yield a finite form. As expected, the positive increasing values of ¯λ correlate with the maximum mass of QS, as detailed in Refs. [44, 45]. It is worth noting that the coefficients ξ seen in Eq. 3 c...

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[3] [3]

2 and Table II), ¯λ is varied between [0.1, 1.0]

Parameter ¯λ (Interaction Parameter): As shown in Section IV C (Fig. 2 and Table II), ¯λ is varied between [0.1, 1.0]. The constraints on this parameter come from ensuring that the predicted maximum masses of quark 6 FIG. 3. We display the mass-radius and mass-compactness re- lations for QSs in Rastall gravity. Here, we vary Beff ∈ [60, 90] MeV/fm3, and o...

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[4] [4]

3 and Table III), we explore the effect of vary- ing Beff within the range [60, 90] MeV/fm 3

Parameter Beff (Effective Bag Constant): In Section IV D (Fig. 3 and Table III), we explore the effect of vary- ing Beff within the range [60, 90] MeV/fm 3. Our results indicate that increasing Beff decreases both the maxi- mum mass and the radius of the quark stars. This en- ables us to rule out values ofBeff that produce quark star masses below the 2 M⊙...

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[5] [5]

Furthermore, we gather the data for maximum compact- ness in Table III depending on the parameters and is found to be 0.297

It can be seen that the maximum compactness does not affected by increasing or decreasing values of Beff. Furthermore, we gather the data for maximum compact- ness in Table III depending on the parameters and is found to be 0.297. 8 FIG. 4. Profiles of the mass versus central density relations. The parameters used are the same as in Figs. 1 to 3. FIG. 5. ...

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X. L. Zhang, Y. F. Huang and Z. C. Zou, [arXiv:2404.00363 [astro-ph.HE]]

arXiv

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L. Q. Su, C. Shi, Y. F. Huang, Y. Yan, C. M. Li, W. L. Yuan and H. S. Zong, Astrophys. Space Sci. 369, 29 (2024)

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E. E. Kolomeitsev and D. N. Voskresensky, [arXiv:2404.09875 [nucl-th]]

work page Pith review arXiv