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This paper argues that in f(R+αR²,T)=R+αR²+2βT gravity, strange stars built from the MIT bag equation of state can reach compactness above 1/3, form photon spheres, and emit kilohertz gravitational-wave echoes, which in turn tighten their m

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2026-08-01 12:49 UTC pith:BGL5ST4Y

load-bearing objection A new modified-gravity/strange-star echo scenario, but the central echo-time formula is dimensionally inconsistent and the parameter choices are tuned, so the quantitative results don't stand. the 3 major comments →

arxiv 2607.19298 v1 pith:BGL5ST4Y submitted 2026-07-21 gr-qc astro-ph.COhep-th

Gravitational wave echoes as probes of the maximum mass of strange stars in quadratic curvature-matter coupled gravity

Debadri Bhattacharjee , Pradip Kumar Chattopadhyay , Kazuharu Bamba This is my paper
classification gr-qc astro-ph.COhep-th
keywords gravitational wave echoesstrange starsMIT bag modelmodified gravityf(R,T) gravityTolman-Oppenheimer-Volkoff equationsphoton spheremaximum mass limits
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Strange quark stars in general relativity are not compact enough to trap gravitational waves: their compactness M/R is about 0.27, below the 1/3 threshold needed for a photon sphere. This paper claims that adding a quadratic curvature term and a matter-curvature coupling—the Lagrangian f(R+αR²,T)=R+αR²+2βT—lifts strange stars over that threshold, so they can host photon spheres and emit gravitational-wave echoes. Solving the modified TOV equations with the MIT bag model equation of state (the standard phenomenological quark-matter relation p=(ρ−4B_g)/3), the authors find new maximum mass-radius bounds (about 2.0–2.5 solar masses and 8.8–11.0 km depending on parameters) by imposing the photon-sphere and modified Buchdahl conditions. Their echo-time integral then puts echo frequencies in the kilohertz range, roughly 28–36 kHz for the tabulated cases, with higher bag constants giving softer stars, lower masses, shorter echo times, and higher frequencies. A sympathetic reader would care because it makes gravitational-wave echoes a possible simultaneous probe of the dense-matter equation of state and of modified gravity.

Core claim

Within this modified-gravity framework, the paper establishes that strange stars with the MIT bag equation of state can satisfy the echo compactness window 1/3 ≤ M/R ≤ 4/9 − a/6, where a=4πp_eff(R)R², which is impossible for the same stars in general relativity. The mass-radius curves intersect the photon-sphere condition at points the authors call the new maximum mass-radius limits of strange stars—for example, M≈2.481 solar masses, R≈10.97 km for α=10, β=−0.5, B_g=61 MeV/fm³. Using their echo-time integral, they obtain echo times of tens of microseconds and echo frequencies in the 28–36 kHz band, and show that these limits are tighter than the pure hydrostatic-equilibrium limits.

What carries the argument

The load-bearing objects are the modified Tolman-Oppenheimer-Volkoff (TOV) equations derived from f(R+αR²,T)=R+αR²+2βT, solved with the MIT bag model (the standard phenomenological quark-matter equation of state, p=(ρ−4B_g)/3). The echo condition is a compactness window: M/R ≥ 1/3 ensures a photon sphere at R=3M, and the modified Buchdahl bound M/R ≤ 4/9 − a/6, with a=4πp_eff(R)R², provides the upper limit. The echo-time integral τ_echo = ∫₀^{3M} dr r e^{2ν(r)} / (1 − 2M(r)/r), together with f_echo = π/τ_echo, converts each stellar profile into a predicted echo frequency, which is the observable that carries the argument.

Load-bearing premise

The numerical echo frequencies rest on Eq. (12), the paper's echo-time integral; it is not the standard round-trip delay formula used in the echo literature, and its integrand has mismatched physical dimensions, so if that formula is not the correct delay for the photon-sphere cavity, the quoted kilohertz predictions change even if the compactness result stands.

What would settle it

Compute the echo delay for the same TOV profiles using the standard round-trip integral between the stellar surface and the photon-sphere barrier; if the resulting frequencies are far from the quoted 28–36 kHz band, the paper's quantitative echo claim is refuted. Alternatively, search archival post-merger gravitational-wave data for a repetitive echo train with the predicted spacing; a null result would not overturn the compactness analysis but would constrain the allowed (α, β, B_g) parameter space.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • In this theory, strange stars can be ultracompact enough to be genuine gravitational-wave echo sources, something the same equation of state cannot achieve in general relativity.
  • Echo constraints place stricter maximum mass-radius limits on strange stars than hydrostatic equilibrium alone, so a detected echo would directly bound the stellar mass.
  • Because increasing the bag constant lowers the mass and raises the echo frequency, measuring the echo frequency would constrain the dense-matter equation of state.
  • A kilohertz echo signal would be a signature of quadratic curvature-matter coupled gravity, distinguishing it from general-relativistic post-merger interpretations.
  • The predicted kilohertz band falls in the range targeted by planned high-frequency detectors, making the prediction observationally testable in the near term.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural next calculation, not performed in the paper, is to recompute the echo delay with the standard round-trip cavity integral for the same stellar profiles; the compactness conclusion would likely survive, but the exact kilohertz tuning could shift, which would sharpen the observational prediction.
  • The same mass-radius solutions could be used to predict other post-merger observables, such as tidal deformability or quasi-normal-mode frequencies, giving independent cross-checks of the echo-based mass limits.
  • The tabulated values show compactness sitting just above the 1/3 threshold; if future observations find strange-star candidates clustered close to this threshold, that would support the modified-gravity picture over the general-relativistic one.
  • If the predicted kilohertz echoes are not seen in next-generation high-frequency searches, the combination (α, β, B_g) would be constrained rather than the whole framework ruled out, since the compactness increase has several contributing parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper investigates whether strange stars in f(R+αR²,T) gravity can be sufficiently compact to generate gravitational-wave echoes. Using the MIT bag model equation of state, the authors derive modified TOV equations, solve them for selected α>0 and β<0, and report maximum-mass configurations with compactness M/R ≈ 1/3 (Tables I-II). Intersecting the M-R curves with the photon-sphere and Buchdahl conditions yields what they call new maximum mass-radius limits (Tables III-IV). Adopting the echo-time formula τ_echo = ∫_0^{3M} dr r e^{2ν}/(1−2M(r)/r) and f_echo=π/τ_echo, they obtain echo times ~88-113 μs and kHz frequencies (Tables V-VI), concluding that future high-frequency detectors can probe these objects and that echo conditions tighten the strange-star mass bound.

Significance. The proposal to use the compactness requirement for photon spheres as a probe of both modified gravity and the strange quark matter EoS is timely and could be interesting. The paper is explicit about its model and provides tables and figures, and the predicted kHz echo frequencies are falsifiable by future detectors. However, as it stands, the quantitative claims are not supported: the echo-time formula in Eq. (12) is dimensionally inconsistent and not the standard round-trip delay, and the modified TOV system (10)-(11) is not reduced to a closed, first-order ODE system. The free parameters α, β, and the lower bound of B_g are tuned to produce the desired compactness, so the 'new maximum mass limits' are not robust constraints. These issues are central, not cosmetic.

major comments (3)
  1. [Sec. III, Eq. (12)] The echo-time integral is dimensionally inconsistent and not the standard cavity round-trip time. With G=c=1, τ_echo, r, and M all have dimension length while ν is dimensionless; the integrand has dimension length², so the right-hand side has dimension length². Moreover, the standard echo delay for a static spherical metric is (up to a factor of two) ∫ e^{-ν}(1−2m/r)^{-1/2} dr between the reflecting surface and the photon-sphere barrier, not ∫ r e^{2ν}(1−2m/r)^{-1} dr from r=0 to 3M. Since Tables V-VI and Figs. 7-10 are computed from Eq. (12), the quoted echo frequencies and the resulting 'new limits' are unsupported.
  2. [Sec. II, Eqs. (10)-(11)] The modified TOV system is not closed. Eq. (10) contains ν'' and R''; Eq. (11) contains p_eff with R_r^r, □R, and ∇_r∇_r R. The text says the equations are integrated with only m(0)=0 and ρ(0)=ρ_c, but no evolution equations or boundary conditions are given for ν, ν', R, and R'. Without a closed ODE system, the M-R curves in Figs. 1-2 and all downstream results cannot be reproduced or checked. A derivation showing how the higher derivatives are eliminated, or an explicit first-order system, is essential.
  3. [Sec. III, parameter choice] The theory parameters are not independently constrained; they are scanned specifically to force compactness ≥1/3. The bullet in Sec. III states 'Suitable combinations of β and α are considered based on the physically acceptable TOV solutions,' and bullet (iv) raises the lower bound of B_g from 57.55 to 61 MeV/fm³ for the same purpose. This makes the 'new maximum mass-radius limits' in Tables III-IV and the associated echo predictions a consequence of the parameter choice rather than a test of the theory. To support claims of revised mass bounds, the authors need either external constraints on α and β or a clear statement that the results are a proof-of-principle with no predictive constraint.
minor comments (4)
  1. [Sec. III, Tables III-IV] The role of the modified Buchdahl bound is unclear. With the tabulated negative values of a, the bound M/R ≤ 4/9 − a/6 is looser than the GR value, so it does not exclude the M/R≈1/3 solutions. The discussion should clarify that the selection is driven by the compactness threshold, not by the Buchdahl bound.
  2. [Sec. III, Eq. (13)] The factor π in f_echo = π/τ_echo is not derived and is in tension with the factor 1/2 used by Abedi and Afshordi, which the authors cite. Please justify the factor.
  3. [Sec. II] The conversion of B_g from MeV/fm³ to the geometrized units used in the numerical integration is not specified, making the results difficult to reproduce.
  4. [Sec. III, Tables III-IV] No stability or causality check is presented for the configurations with M/R≥1/3. If these lie on the unstable branch of the M-R curve, the echo-based 'maximum mass' would not correspond to a viable astrophysical object. A stability criterion is needed.

Circularity Check

2 steps flagged

Maximum-mass 'limits' are imposed, not derived: α, β and B_g are tuned until compactness reaches the photon-sphere threshold, and Tables III–IV report that same threshold as the new mass limit.

specific steps
  1. fitted input called prediction [Sec. III, bullet (iii)–(iv); Tables I–IV]
    "Suitable combinations of the non-minimal matter coupling parameter, β and the quadratic curvature corrections, α are considered based on the physically acceptable TOV solutions. ... we find that to investigate GW echoes while preserving the presence of photon sphere within the allowed parameter space, the lower bound of Bg must be increased to 61 MeV/fm3 for β=−0.5 and α=10."

    The paper first states that echoes require compactness in [1/3, 4/9−a/6], then selects α, β, and B_g so that the TOV solutions attain compactness ≥1/3. The 'new M-R limits' in Tables III–IV are the points where this imposed threshold is met. The central output—configurations that can produce echoes and their revised maximum mass—is therefore the selection criterion itself, not an independent prediction.

  2. self definitional [Sec. III, paragraph after Tables I–II; Tables III–IV]
    "The intersection of the photon sphere with M-R curves reveal a very interesting result: the nodes of intersection are the brinks of GW echoes. Below these points, photon sphere is absent and echo is not possible."

    Here the 'result' is defined as the intersection of the M-R curve with the prescribed compactness cut. The maximum mass that satisfies the cut is then relabeled as a 'new maximum mass-radius limit.' Since the free parameters were already chosen so that some solutions cross the M/R ≥ 1/3 line, the output is the input constraint applied back to the tuned solutions, rather than a genuinely derived bound.

full rationale

The central derivation—action to TOV equations to M-R curves—is self-contained and not circular: the modified field equations are solved with the MIT bag EoS, and the M-R curves in Figs. 1–2 are genuine outputs of those equations. What is circular is the use made of those curves. The paper defines the echo-allowed band by the compactness condition 1/3 ≤ M/R ≤ 4/9 − a/6, then reports as its headline 'new maximum mass-radius limits' the points of the M-R curves that satisfy M/R ≥ 1/3. Since α, β are not independently constrained but scanned until 'physically acceptable TOV solutions' with compactness ≥1/3 are obtained, and B_g's lower bound is raised from 57.55 to 61 MeV/fm^3 precisely to preserve the photon sphere, Tables III–IV do not predict a mass limit; they re-state the imposed threshold at the chosen parameter values. The echo times/frequencies (Eqs. 12–13) are then computed from these already-selected configurations. Eq. (12) is also dimensionally inconsistent (length² vs length) and is not the standard round-trip echo-delay integral; this is a correctness/validity problem rather than a definitional circularity, but it compounds the issue. There is one self-citation ([37]) introducing the f(R+αR^2,T) model, but the equations are re-derived here and the modified-Buchdahl/photon-sphere constraints come from independent references, so the self-citation is not load-bearing for circularity. Overall, the mass-limit claim is partially circular: the output is the selection criterion with tuned parameters; the echo frequencies are not independently predicted.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The calculation depends on three free parameters (α, β, and the effective lower bound of B_g) that are tuned to produce the photon-sphere condition, plus several domain assumptions about the EoS, exterior solution, and Buchdahl bound. No new particles or fields are introduced beyond the modified-gravity parameters.

free parameters (3)
  • α (quadratic curvature coupling) = 10, 8.1, 6.4, 5.0 (units unspecified)
    Chosen so that the TOV solutions yield compactness ≥1/3; only α>0 is required for ghost-freedom, but no independent constraint or physical unit is given.
  • β (non-minimal matter-curvature coupling) = -0.5, -0.98, -0.83, -0.65
    Selected so that TOV solutions are 'physically acceptable' and produce M/R > 1/3; the paper notes that only β<0 works, so the sign and magnitude are effectively tuned to the target effect.
  • Lower bound of bag constant B_g = raised to 61 MeV/fm³ (from 57.55 MeV/fm³) for one parameter set
    The stable SQM range 57.55–95.11 MeV/fm³ is shortened to 61–95.11 MeV/fm³ when needed 'to investigate GW echoes while preserving the presence of photon sphere'—selection to retain the phenomenon.
axioms (4)
  • domain assumption MIT bag model EoS p = (ρ − 4B_g)/3 describes strange quark matter
    Adopted in Sec. III bullet 1 as the interior matter model; this is a phenomenological EoS with no direct QCD derivation.
  • domain assumption The exterior spacetime in f(R+αR²,T) gravity is Schwarzschild, so the photon-sphere radius is R_ps = 3M
    Stated in Sec. III: 'in the vacuum and asymptotic limit, the exterior solution resembles the Schwarzschild one'. Junction conditions and possible surface layers in f(R,T) theory are not discussed.
  • domain assumption The modified Buchdahl bound M/R ≤ 4/9 − a/6 (Burikham et al. [66]) applies to these configurations
    Used in Sec. III to define the upper compactness bound; the paper does not verify that the conditions of Ref. [66] hold for the stellar solutions obtained.
  • ad hoc to paper The echo time is given by Eq. (12), τ_echo = ∫_0^{3M} dr r e^{2ν}/(1 − 2M(r)/r)
    This formula is asserted as the 'typical light time from the stellar centre to the photon sphere' with no derivation, and it does not match the standard echo-cavity integrals in Refs. [67,68].

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read the original abstract

Gravitational wave astronomy provides an exemplary avenue to study exotic compact stars with utmost precision. Recent analyses of GW170817 have reported possible post-merger gravitational wave echoes with a significance of $4.2\sigma$ and a dominant frequency near $72$ Hz. Such echoes may originate from ultracompact remnants possessing photon spheres that partially trap gravitational perturbations. In general relativity, photon-sphere formation requires the stellar compactness to lie within one-third and four-ninths, which is challenging even for a realistic equations of state. Here, we explore this possibility in quadratic curvature gravity with non-minimal matter coupling, considering strange stars described by the MIT bag model equation of state. By solving the modified Tolman-Oppenheimer-Volkoff equations, we obtain the mass-radius relations and identify configurations capable of supporting photon spheres and GW echoes. In the proposed framework, the modified Buchdahl limit allows more compact stellar solutions, while photon-sphere constraints restrict the viable parameter space. We find that increasing the bag constant, decreases the maximum mass and echo time, shifting the echo frequency toward the kHz regime. The echo constraints yield more stringent maximum mass-radius limits than hydrostatic equilibrium, suggesting a revised maximum mass bounds for strange stars. These results highlight the potential of post-merger strange stars as GW echo sources and demonstrate the role of echoes as probes of modified gravity and high-frequency gravitational waves.

Figures

Figures reproduced from arXiv: 2607.19298 by Debadri Bhattacharjee, Kazuharu Bamba, Pradip Kumar Chattopadhyay.

Figure 2
Figure 2. Figure 2: FIG. 2. M-R plot for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Variation of compactness with maximum [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Variation of compactness with maxi [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Variation of echo time ( [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Variation of maximum mass (M [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

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