Compact stars with gravitational wave echoes in $f(R,L_{m},T)$ gravitational theory

Meghanil Sinha; S. Surendra Singh

REVIEW 3 major objections 6 minor 70 references

Compact stars with gravitational wave echoes in $f(R,L_{m},T)$ gravitational theory

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Strange quark stars in f(R,L_m,T) gravity are predicted to emit gravitational-wave echoes at 7.5–11 kHz, with frequency set by the matter–geometry coupling γ.

desk verdict The M-R curves are routine, but the echo frequencies rest on an unjustified integral to 3M inside stars that are larger than 3M—and the field equations have algebraic errors. read the letter →

arxiv 2508.04736 v1 pith:TBABRAD2 submitted 2025-08-05 gr-qc hep-th

Meghanil Sinha , S. Surendra Singh This is my paper · ORCID

classification gr-qchep-th

keywords gravitationalwaveechoesstrangequarkstarsMITbagmodelcolor-flavor-lockedphasef(RLmT)gravitymodifiedTOVequationscompactstarstabilityechofrequency

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 strange quark stars—objects built from deconfined quark matter described by the MIT bag model or by the color-flavor-locked (CFL) superconducting phase—can emit gravitational-wave echoes when gravity is described by the modified theory $f(R,L_m,T)=R+\gamma T L_m$. The authors solve the modified Tolman-Oppenheimer-Volkoff equations, obtain mass-radius curves that overlap observed compact-star candidates, and verify stability through surface redshift and adiabatic index. Using the echo-time integral from the star's center to the photon-sphere radius $3M$, they find echo frequencies in the 7.5–11 kHz band, increasing as $\gamma$ becomes more negative and nearly linear in $\gamma$. If this is right, quark-star echoes become a measurable probe of both quark-matter equations of state and the matter–geometry coupling of modified gravity.

What carries the argument

Two pieces carry the argument. The first is the modified hydrostatic equilibrium system, equations (16)–(17), derived from $f(R,L_m,T)=R+\gamma T L_m$ with matter Lagrangian $L_m=-\rho$; solving it gives the mass, radius, and metric functions $e^{w(r)}$ and $e^{x(r)}$ for each EoS. The second is the echo-time formula $\tau_E=\int_0^{3M}e^{(x-w)/2}\,dr$ with $\omega_E\approx\pi/\tau_E$, which converts the accumulated metric combination between the center and the $3M$ photon sphere into a frequency. The linear-in-$\gamma$ frequency trend comes from $\gamma$ shifting the stellar structure and hence the integrand.

What would settle it

Compute the actual perturbative potential for axial gravitational waves in these $f(R,L_m,T)$ star solutions and locate its peak; if the peak lies near the surface ($R\approx11{-}12$ km) instead of at $3M$ ($\approx7{-}8$ km), the echo time changes by roughly $R/(3M)$ and the quoted band shifts by order unity. A dedicated search in 7–12 kHz with strain sensitivity near $10^{-23}\,\mathrm{strain}/\sqrt{\mathrm{Hz}}$ that sees no echoes from candidate quark stars would also count against the prediction.

Watch

Extended reading notes

Core claim

The central result is that this particular $f(R,L_m,T)$ model produces stable, horizonless strange quark stars whose gravitational-wave echo frequencies sit in a narrow kilohertz band. For the MIT bag model with bag constant $(168\,\mathrm{MeV})^4$ and $\gamma\in[-0.2,0.2]\times10^{-79}\,\mathrm{s^4/kg^2}$, the paper's Tables 2–4 give masses from about 1.59 to 2.03 solar masses, radii from about 10.4 to 12.1 km, and echo frequencies from 10.8 down to 7.5 kHz across the three EoS variants. The frequency falls almost linearly as $\gamma$ increases. Stability diagnostics—surface redshift below the isotropic-fluid bound $Z<2$ and adiabatic index above $4/3$ throughout the interior—hold for every

Load-bearing premise

The calculation assumes the echo-producing barrier sits at the photon-sphere radius $3M$, yet every tabulated star has its surface outside $3M$ (for the $\gamma=0$ MIT bag row, $R\approx11.6$ km versus $3M\approx7.7$ km), and the paper does not derive why the barrier lies inside the star at $3M$.

Editorial extensions

If this is right

If the model is right, a gravitational-wave echo detected in the 7.5–11 kHz band from a compact object would point to a horizonless quark star rather than a black hole, because the echo requires a reflecting surface outside an event horizon.
The near-linear dependence of echo frequency on $\gamma$ means a measured echo frequency can be inverted to constrain the matter–geometry coupling constant in $f(R,L_m,T)$ gravity.
The CFL configurations reach higher masses (up to about 2.0 solar masses) than the MIT bag configurations, so a confirmed high-mass quark star with an echo would favor the CFL phase over the simple bag model.
Within every parameter row, the surface redshift stays below 2 and the adiabatic index above 4/3, so the echo-producing stars are dynamically stable by the paper's stability criteria.

Reading between the lines

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

Not in the paper: the echo-time integral is cut at $3M$ even though every tabulated surface radius exceeds $3M$; if the reflecting barrier sits at the surface instead, the frequencies would shift by roughly the ratio $R/(3M)$, i.e., by order unity.
A direct next step would be to solve the axial perturbation equations for these $f(R,L_m,T)$ stars and locate the peak of the effective potential; that replaces the assumed $3M$ reflector with a derived one.
The same TOV solutions with $L_m=p$ instead of $L_m=-\rho$ give different mass–radius relations, so the echo band would also change; future echo detections could therefore discriminate between matter Lagrangian choices.
The paper itself notes current detectors target roughly 20 Hz–4 kHz, so a consequence it leaves implicit is that testing the 7.5–11 kHz prediction requires a dedicated high-frequency gravitational-wave search.

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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

First thing to know: this paper applies the f(R,Lm,T)=R+γT Lm model to strange quark stars, solves the TOV equations with MIT bag and CFL EoSs, and computes gravitational wave echo frequencies from τ_E = ∫_0^{3M} e^{(x-w)/2} dr. The M-R curves and stability analyses are standard and look plausible; the new ingredient is the near-linear dependence of echo frequency on γ in the 7.5–11 kHz band.

That central claim is unsupported. For every star in Tables 2–4, the radius R is larger than 3M (e.g., γ=0, MIT bag: M=1.745 M☉, 3M≈7.7 km, R≈11.6 km). So the integral (23) stops inside the fluid, not at an exterior photon sphere. The paper states that a photon sphere at R_PH=3M is needed, but for these stars there is no such sphere; the exterior vacuum region begins at R>3M. The authors do not demonstrate that the interior potential has a reflective barrier at 3M. As a result, the computed echo times and frequencies are not physically meaningful. This is the load-bearing flaw, not a cosmetic issue.

Compounding it, the field equations are internally inconsistent. Eq. (7) does not reduce to Eq. (11) for L_m=-ρ; the coefficient of T_ηχ in (11) is off by a factor of 3 if (7) is meant literally. The non-conservation equation (14) disagrees with the TOV expression (17) by factors of 2. These are not typos in the abstract; they are in the equations that generate the numbers. The central density for each tabulated star is not reported, so the calculation is not reproducible.

The paper engages the literature responsibly, citing the f(R) and f(R,T) echo papers and the earlier model [37]. The self-citation is fine because the derivation here is independent.

In short, the qualitative idea—quark stars in this modified gravity might echo GWs—is plausible, but the quantitative results are not established. The paper needs a proper scattering calculation to locate the barrier, a corrected set of field equations, and full input data. I would not cite the frequencies as they stand, but the paper might be useful as a cautionary example in a reading group.

Referee Report

3 major / 6 minor

Summary. This paper studies static, spherically symmetric strange quark stars in the f(R,Lm,T) = R + γ T Lm gravitational theory. The authors derive formal field equations and a non-conservation equation, choose Lm = -ρ, and solve TOV equations for the MIT bag and CFL equations of state. They compute mass-radius relations, compactness, surface redshift, and the adiabatic index, and then use Eq. (23), τ_E = ∫_0^{3M} e^{(x-w)/2} dr, together with Eq. (24), ω_E ≈ π/τ_E, to predict gravitational wave echo frequencies in the range 7.5–11 kHz. They conclude that these compact stars can generate GWEs and that the predicted frequencies are accessible to current detectors.

Significance. If the central claim were established, the paper would provide a concrete, falsifiable prediction connecting f(R,Lm,T) gravity, strange quark matter, and gravitational wave echoes. The paper is genuinely useful in assembling the modified TOV equations for this gravity model and presenting M, R, and frequency tables for several values of γ. However, the derivation contains serious algebraic inconsistencies, and the echo-time calculation is not physically applicable to the tabulated stellar configurations. The central quantitative claim is therefore not supported in the present manuscript.

major comments (3)

[Sec. 2, Eqs. (7) and (11)] Equation (7) does not follow algebraically from Eq. (2) for f = R + γ T Lm, and the printed term '3γ/2 γ' is ambiguous. Direct substitution into Eq. (2) gives extra terms not present in Eq. (7), e.g., γ Lm^2 T_{ηχ} and -γ Lm^3 g_{ηχ}. For Lm = -ρ, T = -ρ + 3p, the coefficient in front of T_{ηχ} becomes 8π + (3γ/2)p - (γ/2)ρ + γρ^2, whereas Eq. (11) uses 8π + (3γ/2)(p - ρ). These differ by γρ + γρ^2, and the g_{ηχ} terms also differ (γρ^3 vs. -γρ^2). Since Eqs. (12)-(13) and the TOV system are based on Eq. (11), the stellar-structure equations are not presently derived from the stated action.
[Sec. 4, Eq. (23) and Tables 2-4] The echo-time integral in Eq. (23) is evaluated from r=0 to r=3M, but every tabulated stellar model has radius R > 3M. For example, the γ=0 MIT bag row has M=1.745 M☉ (so 3M ≈ 7.7 km) and R=11.57 km. Thus the entire integration domain lies inside the fluid star, and the exterior photon sphere at 3M does not exist for these configurations. The paper provides no derivation that the interior effective potential for gravitational perturbations has a reflecting barrier at r=3M. Consequently, τ_E and the quoted echo frequencies in Tables 2-4 and the figures are unsupported. This is the central quantitative claim of the paper.
[Sec. 2, Eqs. (14) and (17)] The non-conservation equation used to construct the TOV equation contains a factor-of-2 discrepancy. Equation (14) has denominator 16π + 3γ(p - ρ), while Eq. (8) or its equivalent from Eq. (5) yields 8π + (3γ/2)(p - ρ) for Lm = -ρ. This factor of 2 propagates into Eq. (17). In addition, the denominator in Eq. (17), [1 + γ(3p(1-dρ/dp)-4ρ(dρ/dp))/(16π+3γ(p-ρ))], is not derived in the text. Since all numerical mass-radius results depend on these equations, the results need to be rederived and recomputed.

minor comments (6)

[Title and throughout] There are typographical errors, e.g., 'thoery' in the title, 'adaibatic', and 'redshift analyss'. These should be corrected.
[Sec. 2, Eq. (7)] The expression '3γ/2 γ(T + 2Lm)' is ambiguous; presumably a factor γ is spurious. This should be clarified.
[Sec. 3, Eqs. (20)-(22)] The CFL EoS contains missing or poorly formatted symbols (β², Σ), making it difficult to verify the expressions. Please provide a clean typeset version.
[Sec. 6, last paragraph] The paper states that 7.5–11 kHz is within the range of advanced LIGO, Virgo, and KAGRA, whose quoted band is ~20 Hz–4 kHz. The predicted band lies above the quoted detection band, so the detectability claim is not supported by the cited sensitivities.
[Sec. 3, Tables 2-4] The central energy density (or pressure) used to start each TOV integration is not listed. Without this information, the mass-radius profiles are not reproducible.
[Sec. 4, Figs. 2 and 5] Figure 5's caption says 'MIT Bag model' but the corresponding text and axes suggest a CFL phase with ms=100 MeV; please check. Also, the photon-sphere line in Fig. 2 is plotted for radii that are all below the stellar radii, which visually underscores the issue raised in Major Comment 2.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: echo frequencies are derived from TOV outputs via a standard echo-time formula; the only self-citation is not load-bearing.

full rationale

The central echo-frequency claim (Eq. 24, Tables 2-4) is a derived output, not a fitted input. The paper solves the modified TOV equations (Eqs. 16-17) for given EoS and γ, obtains M and R, and then applies the standard echo-time formula τ_E = ∫_0^{3M} e^{(x-w)/2} dr and ω_E ≈ π/τ_E. No echo datum is used to tune γ, B, △, or ms; no parameter is fitted to the 7.5-11 kHz band; the near-linear variation with γ is a consequence of the TOV solutions, not an input. The authors' self-citation [37] supplies the f(R,Lm,T)=R+γ T Lm model, but the field equations and TOV equations are re-derived in this paper (Eqs. 11-17), so the citation is not load-bearing. The main weakness is a non-circular physical-applicability gap: the paper states 'For echoes of GWs to be generated, a photon sphere at RPH = 3M(total mass M) is needed' and then integrates Eq. (23) to r=3M, yet all tabulated stars have radii larger than 3M (e.g., MIT bag γ=0: R≈11.57 km vs 3M≈7.7 km). No barrier or reflecting surface at r=3M is shown to exist inside these stars, and no scattering calculation is performed; this undermines the quantitative echo prediction but does not reduce the derivation to its inputs. Score 2 reflects only the minor non-load-bearing self-citation; the derivation itself is not circular.

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

The central claim rests on an assumed modified-gravity action, a specific choice of matter Lagrangian (Lm=-ρ), the standard static perfect-fluid stellar model, and an echo-time formalism imported from the black-hole/ultra-compact-object literature. No new particle or entity is introduced. The unstated central density is the main 'free' choice that determines the tabulated results.

free parameters (5)

gamma (matter-geometry coupling) = varied in [-0.2,0.2] x 10^-79 s^4/kg^2
Free parameter of the f(R,Lm,T)=R+γT Lm model; explored, not fitted to specific observables.
central energy density (rho_c) = not reported
The TOV integration requires a central density/pressure; different choices give different M, R and hence different echo frequencies. The paper does not state the selection rule for the tabulated stars.
bag constant B = (168 MeV)^4
Standard MIT bag constant used for both EoS; from prior literature.
CFL pairing gap Delta = 350 MeV
EoS input for CFL phase.
strange quark mass m_s = 0 or 100 MeV
EoS input for CFL phase.

assumptions (5)

domain assumption The f(R,Lm,T)=R+γT Lm gravitational action is the theory under test.
The paper adopts this model from ref [37] (same authors) without testing alternative forms.
domain assumption The matter Lagrangian is chosen as Lm = -ρ.
Section 2; the paper notes Lm = p would give different results.
domain assumption The star is a static, spherically symmetric perfect fluid.
Section 2, metric (10) and stress tensor (9).
domain assumption The echo time formula τ_E = ∫_0^{3M} e^{(x-w)/2} dr with ω≈π/τ_E applies to these compact stars.
Section 4, Eqs. (23)-(24); taken from prior echo literature without derivation for this setup.
domain assumption The modified Buchdahl bound from ref [50] applies.
Section 4, used to argue the stars avoid collapse.

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Cite this review

Pith. "Pith review of Compact stars with gravitational wave echoes in $f(R,L_{m},T)$ gravitational theory." pith.science (2026). https://pith.science/paper/TBABRAD2

@misc{pith2026250804736,
  author       = {Pith},
  title        = {Pith review of: Compact stars with gravitational wave echoes in $f(R,L_m,T)$ gravitational theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBABRAD2}},
  note         = {Machine review of arXiv:2508.04736}
}

abstract

This work explores the gravitational wave echoes (GWEs) from the compact stellar configurations in the backdrop of $f(R,L_{m},T)$ gravity within static and spherically symmetric framework. Our study has utilized the MIT Bag model and color-favour-locked (CFL) phase equations of state (EoS) for matter description. Mass-radius profiles were determined by solving the hydrostatic equilibrium equations. Model parameter variations were used to assess the configuration stability here. TOV solutions helped to evaluate compactness. Our results indicate that MIT bag model and CFL EoS in $f(R,L_{m},T)$ modified gravitational theory are capable of producing GWEs. The calculated wave frequencies lie within the range of $ 7.5-11 $ kHz range. We have also demonstrated that how different gravitational theory parametrization within $f(R,L_{m},T)$ theory affect our star structure and echo frequency characteristics. Surface redshift and adiabatic index analysis confirm the stability of our stellar model here.

Figures

Figures reproduced from arXiv: 2508.04736 by the authors.

**Figure 2.** Plot for the photon sphere, Black hole and Buchdahl limit fo [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Graph of GWE frequency within allowed γ for MIT bag model γ=0.2 γ=0.1 γ=0 γ=-0.1 γ=-0.2 8.5 9.0 9.5 10.0 0.0 0.5 1.0 1.5 R (km) M/M☉ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: M-R plot for the CFL phase state, ms = 0, △ = 350MeV 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: M-R plot for the MIT Bag model , ms = 100MeV, △ = 350MeV CFL ms = 0 CFL ms = 100 -0.2 -0.1 0.0 0.1 0.2 8.0 8.5 9.0 9.5 10.0 10.5 GWE Frequency (kHz) [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Graph of GWE frequency for the CFL phases with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Variation in the compactness of the stellar interior for MIT [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: Mass dependency on radial co-ordinate for MIT bag, [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: Mass dependency on radial co-ordinate for CFL phase, [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: Mass dependency on radial co-ordinate for CFL phase, [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: For the MIT bag model, the nature of the redshift param [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 12.** Figure 12: For the CFL phase, the nature of the redshift paramet [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 13.** Figure 13: For the CFL phase, the nature of the redshift paramet [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 14.** Figure 14: Adaibatic index behavior against radius is shown for the MI [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]

**Figure 15.** Figure 15: Adaibatic index behavior against radius is shown for CFL ph [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]

**Figure 16.** Figure 16: Adaibatic index behavior against radius is shown for CFL ph [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]

Discussion (0). Continue with ORCID to comment.

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