Non-Hermitian Gravitational Wave Scattering

Burak Pekduran; Ekrem Ayd{\i}ner; Mustafa Sar{\i}saman

REVIEW 4 major objections 4 minor 115 references

Non-Hermitian Gravitational Wave Scattering

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A non-Hermitian transfer-matrix model of gravitational wave scattering predicts spectral-singularity frequencies that match the Hubble constant.

desk verdict Load-bearing algebraic slip (dropped 1/i in Eq. 16), unjustified asymptotic limit, and circular H0 input sink the paper's claim of perfect agreement. read the letter →

arxiv 2506.08567 v1 pith:VY2URK3Q submitted 2025-06-10 gr-qc math-phmath.MPquant-ph

Burak Pekduran , Mustafa Sar{i}saman , Ekrem Ayd{i}ner This is my paper

classification gr-qcmath-phmath.MPquant-ph PACS 02.40.Hw03.65.-w03.65.Nk03.65.Pm03.75.-b04.20.-q04.25.Nx04.30.-w

keywords Non-HermitianPhysicsGravitationalWaveScatteringTheoryTransferMatrixSpectralSingularityHubbleConstantFLRWuniverseSphericalBesselfunctions

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

Gravitational waves are treated as a non-Hmitian scattering problem: a wave from a compact binary merger propagates through an expanding FLRW universe and scatters off a spherical cosmic object. The paper constructs the transfer matrix for the Bessel-mode expansion of the metric perturbation and identifies spectral singularities—states with real energy that produce purely outgoing waves—as the real zeros of a particular transfer-matrix component. At these singularities the model yields a discrete set of gravitational wave frequencies paired with Hubble constant values; for Bessel order $\ell = 12564$, the frequency $\omega = 10^{-16}\,\mathrm{Hz}$ is paired with $H_0 = 69.98\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$, which the authors describe as a perfect agreement with measured values. If correct, the non-Hermitian spectral-singularity condition would encode the cosmic expansion rate, offering a new way to read the Hubble constant from gravitational wave data and potentially informing the Hubble tension.

What carries the argument

The central object is the transfer matrix $M = V^{-1}U$ that connects the coefficients of the gravitational wave mode expansion inside and outside a spherical scattering region of radius $R$. A spectral singularity—a state with real energy and purely outgoing waves, corresponding to a zero-width resonance in non-Hermitian scattering—appears when the $(2,2)$ component of $M$ vanishes, which is equivalent to the Wronskian condition $W[j_\ell, h_\ell^{(2)}](z_R) = 0$, where $z_R = aR\sqrt{\omega^2 + 3i\omega H}$ and $j_\ell$, $h_\ell^{(2)}$ are spherical Bessel and Hankel functions. Using the large-argument asymptotic expansions (11), this condition reduces to the complex transcendental equation $\tan(z_R - \pi\ell/2) = \zeta_\ell(z_R)$, whose real and imaginary parts give the two real equations (17) and (18) for $z_{R,r}$ and $z_{R,i}$. The paper defines functions $F$ and $G$ (Eqs. 23–24) whose real zeros locate the spectral singularities in the $\ell$–$\omega$ and $\omega$–$H_0$ planes; the hybrid scale factor of Eq. (19), anchored to the matter- and dark-energy-dominated eras, fixes the Hubble parameter appearing in $z_R$.

What would settle it

Recompute $z_R = aR\sqrt{\omega^2 + 3i\omega H_0}$ using the paper's central parameters ($\omega = 10^{-16}\,\mathrm{Hz}$, $H_0 = 69.98\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$, $R \approx 2.5 \times 10^4\,\mathrm{km}$) and the scale factor from Eq. (19) at the present epoch. If $|z_R|$ is not much greater than 1, the asymptotic expansion (11) is invalid and the spectral-singularity condition must be solved with exact Bessel functions; solving the exact Wronskian condition $W[j_\ell, h_\ell^{(2)}](z_R) = 0$ at these parameters and checking whether any real frequency satisfies it would settle the claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the spectral singularity condition for gravitational wave scattering in a non-Hermitian FLRW setup is satisfied at gravitational wave frequencies that reproduce the Hubble constant. For a configuration with mode numbers $m = 25128$, $\ell = 12564$, and radius $R \approx 2.5 \times 10^4\,\mathrm{km}$, the real and imaginary parts of the spectral-singularity condition—Eqs. (17) and (18), derived from the asymptotic reduction of the Wronskian condition $W[j_\ell, h_\ell^{(2)}](z_R) = 0$—have a real zero at $\omega = 10^{-16}\,\mathrm{Hz}$ and $H_0 = 69.98\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$. The same condition produces additional spectral singularity points for $\ell = 20$ and $\ell = 3740$, with frequencies from $10^{-23}\,\mathrm{Hz}$ to $10^{-12}\,\mathrm{Hz}$ and Hubble constant values around $70\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ or below $1\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$. The authors take this agreement as evidence that gravitational waves exhibit non-Hermitian characteristics and that the transfer-matrix spectral-singularity framework can reproduce known cosmological parameters from the frequency of gravitational waves measured on Earth.

Load-bearing premise

The derivation of Eq. (13) from Eq. (9) assumes the large-argument asymptotic expansions of the spherical Bessel and Hankel functions, valid only for $z_R \gg 1$; with the paper's stated $\omega = 10^{-16}\,\mathrm{Hz}$, $R \approx 2.5 \times 10^4\,\mathrm{km}$, and a scale factor of order one, the argument $z_R$ is about $10^{-8}$, so the asymptotics and the spectral-singularity equations (17)–(18) break down unless the scale factor takes an enormous, unspecified value.

Editorial extensions

If this is right

If the spectral-singularity condition is correct, the frequency of a gravitational wave measured at a singularity point on Earth would determine the Hubble constant at the scattering epoch, turning gravitational wave observatories into cosmological probes.
The discrete Bessel orders $\ell = 20$, $3740$, and $12564$ select a small set of allowed gravitational wave frequencies; a broadband observatory scanning the $10^{-16}\,\mathrm{Hz}$ to $10^4\,\mathrm{Hz}$ band could test whether these particular frequencies are special.
The model also predicts spectral singularity points with very small Hubble constant values (around $0.7\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$) at low frequencies, which would correspond to a different scattering geometry or cosmological epoch and could be checked against future data.
Because the predicted $H_0 \approx 69.98\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ lies between the CMB-based and distance-ladder measurements, the formalism offers a potential intermediate route for addressing the Hubble tension if the scattering geometry is identified with a real astrophysical object.

Reading between the lines

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

The paper never states the numerical value of the scale factor $a(t)$ used to compute $z_R$; since every step from Eq. (9) to Eqs. (17)–(18) depends on $z_R$ through the asymptotic expansions, the claimed agreement is only as robust as that unstated choice. A testable extension is to fix $a(t)$ from Eq. (19) at the present epoch and recompute the spectral singularities.
The scattering radius $R \approx 2.5 \times 10^4\,\mathrm{km}$ is close to the Earth's radius, but the paper does not explicitly identify the scatterer. If the scatterer is the Earth, one could search existing gravitational wave data for spectral-singularity enhancements at the listed frequencies.
The low Hubble constant values near $0.7\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ in Table I are far below any observed expansion rate; determining whether those points are physical solutions or artifacts of the parameter choice would clarify which entries of the table are meaningful.

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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 one thing to know: the central numerical claim—that spectral singularities in a non-Hermitian GW scattering model reproduce the Hubble constant—does not survive contact with the paper's own equations. The transfer-matrix machinery is standard, but the derivation contains a load-bearing algebraic slip, and the parameter choices break the asymptotic approximation the derivation depends on.

What is actually new: applying the spectral-singularity condition of a transfer matrix to gravitational-wave scattering in an FLRW background, and looking for a connection to H0, is a fresh combination. The wave-equation derivation in the TT gauge and the Wronskian-based transfer matrix construction are clean and clearly presented. Credit where due.

The soft spots are serious, and they are internal. First, Eq. (16) misquotes the complex arctangent identity: the factor 1/i is dropped, so the real and imaginary parts in Eqs. (17)-(18) are wrong. The functions F and G in Eqs. (23)-(24), the plots, and every row of Table I depend on those wrong equations. This is not a nitpick; it is an internal contradiction with the identity stated just above.

Second, Eq. (13) comes from asymptotic expansions valid for z >> 1. For the flagship solution, ω = 10^-16 Hz and R = 2.5e4 km, z_R is around 10^-8 if the scale factor is O(1). The paper never gives the numerical value of a(t) used, so the table rests on a hidden, unjustified assumption.

Third, the comparison is partly circular: H0 = 70.88 is inserted as an input via Eq. (20), then recovered in Table I. Most rows actually give H0 around 0.7 km/s/Mpc; only the 69.98 row is advertised as "perfect agreement." The text also asserts without a reference that a 10^-16 Hz gravitational wave has been measured.

In sum: the idea is worth a footnote, but the paper as written is not a reliable result. It deserves a referee in the limited sense that a careful review would catch these errors quickly; I wouldn't bring it to reading group or cite it until the algebra and the asymptotics are actually fixed.

Referee Report

4 major / 4 minor

Summary. The manuscript considers non-Hermitian scattering of gravitational waves in an FLRW background, constructs a transfer matrix from spherical Bessel/Hankel boundary conditions, and identifies spectral singularities at real zeros of a transfer-matrix component. It then derives, via asymptotic expansions and an arctangent identity, a pair of equations that relate the gravitational-wave frequency and the Hubble constant at spectral singularities. Using a complex scale factor and fixing H0 from the cosmological constant, the authors report gravitational-wave frequencies and Hubble-constant values that they claim agree perfectly, with Table I presenting several spectral-singularity entries.

Significance. If the central claim were correct, the paper would offer a striking new connection between non-Hermitian scattering physics, gravitational-wave frequencies, and the cosmological Hubble constant, potentially relevant to the Hubble-tension discussion. The manuscript presents a full derivation, explicit transfer-matrix expressions, and a numerical table, which makes the logic checkable. However, the derivation contains an algebraic sign/factor error that invalidates the later equations, and the numerical results depend on an unjustified asymptotic limit and on H0 being inserted before solving. Because the claimed agreement is therefore not an independent prediction, the paper's central contribution is not established.

major comments (4)

[Section II, Eq. (16)] The step from Eq. (15) to Eq. (16) drops the factor 1/i from the arctangent identity. The paper correctly writes tan^{-1}(z) = πm + (1/(2i)) ln((1+iz)/(1-iz)), but Eq. (16) instead has (1/2) ln((1+iζ)/(1-iζ)) with no 1/i. The correct substitution yields z_R - πℓ/2 = πm + (1/(2i)) ln w, whose real and imaginary parts are (1/2) Arg w and -(1/2) ln|w|. Equations (17) and (18) therefore have the logarithmic and arctangent contributions interchanged and sign-flipped, and the definitions of F and G in Eqs. (23)-(24), all subsequent figures, and every row of Table I inherit this algebraic error. The numerical agreement claimed for ω = 10^{-16} Hz and H0 = 69.98 km s^{-1} Mpc^{-1} is unsupported.
[Section II, Eqs. (11)-(13)] Equation (13) is derived from Eq. (9) using the large-argument asymptotic expansions (11), which are valid for z >> 1. For the central solution quoted in the abstract and Table I (ω = 10^{-16} Hz, R = 2.5 × 10^4 km), and using H0 ≈ 70.88 km s^{-1} Mpc^{-1} ≈ 2.3 × 10^{-18} s^{-1}, the argument is z_R = a R sqrt(ω² + 3iωH). With ω = 2π × 10^{-16} s^{-1}, this gives z_R ≈ 1.5 × 10^{-8} a. Thus z >> 1 fails for any order-unity scale factor, and the manuscript never specifies the numerical value of a(t) used to compute z_R. The central derivation therefore rests on an unstated and physically implausible scale-factor value.
[Section III, Eqs. (20), (23)-(24), and Table I] The agreement claimed in the abstract is partly circular: the value H0 ≈ 70.88 km s^{-1} Mpc^{-1} is inserted from Eq. (20) as an input before solving the zero conditions for F and G. Since z_R = a R sqrt(ω² + 3iωH) contains both ω and H, equations (17)-(18) define a relation between these quantities, so recovering a value near the input is not an independent test. To support the claim, the authors need to specify which parameters are free, show that the solution is not selected by the input, and demonstrate that the recovered H0 is robust to variations in the unstated scale factor.
[Table I and Section III] The table does not support the stated 'perfect agreement.' Of eight listed spectral-singularity solutions, only rows 3, 6, and 8 have H0 near 70 km s^{-1} Mpc^{-1}; the other five rows give H0 ≈ 0.7 km s^{-1} Mpc^{-1}, a factor of 100 smaller. Moreover, for the same angular order ℓ = 12564 and the same frequency ω = 10^{-16} Hz, the table lists two different H0 values (69.98 and 0.698 km s^{-1} Mpc^{-1}), indicating that the spectral-singularity condition does not uniquely determine H0 for a given frequency. The selective emphasis on one row is not justified.

minor comments (4)

[Throughout] The acronym FLRW is consistently typeset as 'FLR W', and several author affiliation strings contain nonstandard characters such as 'T¨ urkiye'; these should be corrected.
[Section II, Eq. (5)] The solution (5) writes spherical Bessel functions for all real ℓ, but spherical Bessel functions of negative integer order are related to positive orders; the later claim that no spectral singularities occur for ℓ = -3, -2, -1 is confusing because ℓ is conventionally a nonnegative integer in partial-wave expansions.
[Section III, Eq. (25)] The parameters m = 25128 and ℓ = 12564 are introduced without any derivation or physical justification; the text states that they 'correspond to gravitational waves generated in the early times of the universe,' but no quantitative argument links these particular integers to a cosmological source or detector.
[Section II, Eq. (4)] The dimensionful status of a(t) should be clarified: if a is the dimensionless FLRW scale factor, then the term a² ω(ω + 3iH) in Eq. (4) has units of inverse time squared while ∂² has units of inverse length squared, so the equality requires an explicit convention for the speed of light and coordinate units.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in the derivation chain; the H0 agreement is a consistency check with an externally supplied cosmological parameter, though the paper has unrelated algebraic and evidential flaws.

full rationale

The derivation is not circular. The Hubble constant enters from independent cosmology via Eq. (20) with Lambda approximately 2 x 10^-35 s^-2, giving H0 approximately 70.88 km/s/Mpc, and the spectral-singularity condition is derived from the transfer-matrix Wronskian condition (Eq. 9) and the large-argument asymptotic expansions (Eq. 11), not by assuming the entries of Table I. Solving Eqs. (23)-(24) for real zeros of F and G is a genuine two-variable consistency problem: one finds pairs (omega, H0) for which the spectral condition holds, and the occurrence of a zero near H0 = 69.98 km/s/Mpc at omega = 10^-16 Hz is a nontrivial check against an externally measured value. The self-citations ([64], [77], [114]) concern standard transfer-matrix methods or a hybrid scale factor that the paper explicitly does not use in the numerical analysis, so they are not load-bearing. The serious problems with the paper are correctness and evidence problems, not circularity: Eq. (16) drops the factor 1/i from the arctangent identity, which invalidates Eqs. (17)-(18) and the F/G definitions; the z >> 1 asymptotic regime is not justified for the claimed solution without a specified enormous scale factor; and Table I lists four solutions for ell = 12564, only one of which is near the measured H0, indicating selection rather than a robust prediction. None of these issues makes the output equivalent to the input by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The model rests on hand-picked free parameters and a nonstandard complex scale factor rather than on new physical entities. No new particles, forces, or conserved quantities are introduced.

free parameters (6)

Angular mode order ℓ = 12564
Chosen in Section III (Eq. 25) to make the spectral singularity condition have zeros near the desired frequency and H0; no physical justification is given.
Mode index m = 25128
Same ad hoc choice as ℓ, introduced in Eq. (25) without derivation.
Scattering radius R = 2.5e4 km
Assumed radius of the spherical scatterer; does not equal Earth's radius (6371 km) and is not justified.
Hubble constant input H0 = 70.88 km/s/Mpc
Taken from Eq. (20) using the cosmological constant; used as an input to solve the spectral conditions, then recovered as output.
Scale factor a(t) = not stated
The numerical value of the scale factor used to compute z_R is never given; the results depend on it.
Gravitational wave frequency ω = 1e-16 Hz
Presented as a measured value, but no detector has measured 1e-16 Hz; the value is chosen as a round power of ten.

assumptions (5)

domain assumption The wave equation (4) reduces gravitational wave propagation to a scalar Helmholtz equation for tensor perturbations in a perfect-fluid FLRW background.
Derived in Appendix B with questionable steps, including Eq. (44) which sets (ρ+p) h_munu u^mu u^nu = 0; the tensor structure is suppressed to a single scalar mode.
domain assumption The transfer-matrix spectral singularity condition is M22 = 0, as in 1D scattering.
Borrowed from Mostafazadeh's 1D and optical scattering literature; the 3D partial-wave extension is assumed without proof.
ad hoc to paper The asymptotic expansions in Eq. (11) are valid for the parameters used.
The paper states these hold for z >> 1 but applies them to ω = 10^-16 Hz where z_R is tiny unless an unstated huge scale factor is used.
ad hoc to paper The scale factor can be complex and takes the form in Eq. (21), with a^2 = a0^2 + t^2.
Eq. (22) is dimensionally inconsistent (a0 is set to the Planck length while t is in seconds) and is not used consistently in the numerical work.
domain assumption Spectral singularity points correspond to stable states and require real k_r.
Stated in Section IV, but the equations then allow complex z_R, so the requirement is not enforced in the numerical solution.

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Cite this review

Pith. "Pith review of Non-Hermitian Gravitational Wave Scattering." pith.science (2026). https://pith.science/paper/VY2URK3Q

@misc{pith2026250608567,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian Gravitational Wave Scattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VY2URK3Q}},
  note         = {Machine review of arXiv:2506.08567}
}

read the original abstract

In this study, the non-Hermitian scattering of gravitational waves is examined, and their behavior at spectral singularities is discussed. We investigate the non-Hermitian properties of gravitational waves through the construction of a transfer matrix. By examining spectral singularity points obtained from the transfer matrix, we explore the behavior of gravitational waves at these spectral singularity points and compare the theoretical results with observed measurements. Our findings demonstrate that the frequency values of gravitational waves measured on Earth at spectral singularity points exhibit perfect agreement with the corresponding Hubble constant values. This alignment underscores the significance of the non-Hermitian characteristics of gravitational waves. We anticipate that this research will contribute to a deeper understanding of the role of non-Hermitian phenomena in gravitational physics and provide a foundation for future studies in the field.

Figures

Figures reproduced from arXiv: 2506.08567 by the authors.

**Figure 1.** FIG. 1. (Color online) Pictorial demonstration of Gravitational Wave production (from a binary system) and its propagation [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** FIG. 2. (Color online) Figure denotes the scattering configuration of the gravitational wave. Notice that the scatterer has [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color online) The figures illustrate the locations of the spectral singularity points, derived from the real zeros of the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. (Color online) The figures display the graphs of the functions [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. (Color online) The figure shows the spectral singularity points in the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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