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REVIEW 3 major objections 7 minor 14 references

Cosmic expansion bends gravitational wave signals into a measurable angle

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2026-07-09 11:55 UTC pith:DQ5OKKE4

load-bearing objection Letter on arXiv:2607.07398 the 3 major comments →

arxiv 2607.07398 v1 pith:DQ5OKKE4 submitted 2026-07-08 gr-qc astro-ph.COhep-th

Impact of H₀ on gravitational wave propagation: prospects for the LISA mission

Nil Blanco , Domenec Espriu This is my paper
classification gr-qc astro-ph.COhep-th
keywords lisagravitationalpropagationtimingwaveaboveanalyzeangle
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.

The paper argues that the Hubble parameter H0 modifies gravitational wave propagation in a way that goes beyond the standard redshift of frequency: it shifts the effective wavenumber, producing an angular modulation in the timing residual measured along a detector arm. For LISA's 2.5-million-kilometer arms, this modulation peaks at a specific incidence angle alpha_optim = 2 arcsin(sqrt(H0 * ZA / 2)), where ZA is the comoving distance to the source. The key claim is that this optimal angle depends only on the product H0 * ZA and is independent of gravitational wave frequency above 100 mHz. By measuring the angle at which the timing residual is maximized, one could read off H0 * ZA directly. If the source distance ZA is known independently from the source's chirp mass and frequency evolution, this yields a measurement of H0 that does not rely on standard sirens, electromagnetic counterparts, or any distance ladder. The authors estimate LISA could determine H0 to within a few percent by this method. The central mechanism is a stationary-phase condition in the integral of the metric perturbation along the detector arm: the phase of the accumulated signal becomes stationary at a particular angle, concentrating the signal there, and that angle encodes H0. The paper supports this with both the analytical stationary-phase derivation and numerical integration of the timing residual for single-arm and two-arm LISA configurations, showing that a +/-10% change in H0 shifts the angular enhancement by 2-7 degrees depending on source distance.

Core claim

The optimal incidence angle for gravitational wave signals arriving at a LISA arm, at which the timing residual is maximized, is given by alpha_optim = 2 arcsin(sqrt(H0 * ZA / 2)). This angle is determined solely by the product of the Hubble parameter and the source's comoving distance, is independent of gravitational wave frequency for frequencies above 100 mHz, and provides a route to measuring H0 that is geometrically and methodologically independent of the standard siren approach.

What carries the argument

The effective wavenumber k_eff = omega(1 - R*H0/2), which differs from the naive redshift-only wavenumber omega_eff = omega(1 - R*H0). This distinction means the gravitational wave phase varies spatially along the detector arm in a way that depends on H0, producing an angular enhancement in the timing residual whose peak position encodes H0 * ZA.

Load-bearing premise

The method requires observable signals above 100 mHz, but LISA's arm-length transfer function suppresses response in exactly this band, and the astrophysical sources radiating there are predominantly nearby stellar-origin black hole binaries at low redshifts where the H0-dependent angular shift is smallest. The paper does not perform a signal-to-noise calculation to confirm that the effect survives this tension.

What would settle it

If a signal-to-noise calculation incorporating LISA's full transfer function, TDI noise budget, and realistic source populations above 100 mHz shows that the angular enhancement is not detectable above instrument noise, the method would be impractical regardless of the correctness of the underlying physics.

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

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 / 7 minor

Summary. The manuscript extends a formalism for cosmological corrections to gravitational wave propagation—previously developed for Pulsar Timing Arrays—to the LISA mission geometry. The authors derive the timing residual for a single LISA arm, identify an optimal incidence angle α_optim = 2 arcsin(√(H₀Z_A/2)) via a stationary phase condition, and confirm this formula numerically. They further propose a two-arm correlation method and argue that LISA could determine H₀ to within a few percent. The analytical derivation is internally consistent and the numerical results confirm the predicted angular scaling. However, the central feasibility claim is not supported by a signal-to-noise analysis, and the regime where the effect is visible coincides with reduced LISA sensitivity and unfavorable source populations.

Significance. The parameter-free derivation of α_optim (Eq. 16) and its numerical confirmation (Tables II–III) are genuine strengths. The idea of using angular modulation of the GW timing residual as a redshift-independent handle on H₀ is novel for the LISA context and complementary to standard-siren methods. The two-arm correlation proposal (Sec. VI) is an interesting observational strategy. However, the significance is substantially diminished by the absence of any quantitative detectability assessment, which is needed to evaluate whether the effect is observable in practice.

major comments (3)
  1. Abstract and Sec. V: The claim that 'LISA could determine H₀ to within a few percent' is the central feasibility assertion of the paper, but no signal-to-noise ratio calculation is performed anywhere in the manuscript. The amplitude ε is set to 1 throughout (stated in Fig. 3 caption and Sec. V), which means all results show only relative angular modulation, not absolute detectability. For realistic LISA sources at z ~ 0.05–1, the strain is h ~ 10⁻²²–10⁻²⁴, and the timing residual τ ~ h·L is extremely small. An O(1) angular modulation of an undetectably small signal remains undetectable. The paper acknowledges this gap in Sec. VI and the Conclusions ('a rigorous signal-to-noise assessment... is required'), yet the abstract and Sec. V still assert the few-percent precision claim. Either an SNR calculation using the LISA noise curve (ref [15]) should be included, or the feasibility claim in
  2. Sec. IV, Eqs. (11)–(14): The stationary phase approximation is not valid for LISA parameters. The coefficient A = (ωH₀L²/2)(cosα−2)cosα (Eq. 11) is of order 10⁻¹⁶ for LISA (L ≈ 8.3 s, H₀ ≈ 2.3×10⁻¹⁸ s⁻¹, ω ~ 6 s⁻¹), as the authors themselves note ('the constant A is totally negligible because is many orders of magnitude smaller than B'). With A ≈ 0, the phase Θ ≈ Bx + C is effectively linear, and there is no genuine stationary point x₀ = −B/(2A). The maximum at B = 0 corresponds to coherent accumulation when the phase stops oscillating, not to a stationary phase phenomenon. While the numerical results (Tables II–III) confirm that the maximum occurs at the predicted angle, the theoretical justification via the stationary phase approximation (Eqs. 14–15) is misleading for the LISA regime. The authors should clarify that the SPA is not the appropriate framework here and reframe the B = 0条件
  3. Secs. V–VI and Conclusions: There is a fundamental tension that is acknowledged but not quantified. The angular enhancement is visible only for f > 100 mHz (Sec. V, Fig. 3), but LISA's arm-length transfer function T(f) ∝ sin(πfL)/(πfL) suppresses the response precisely in this band. Additionally, the sources radiating above 100 mHz are predominantly stellar-origin black hole binaries at z ≲ 0.1 (Table I), where the H₀-dependent correction is smallest. The paper states that 'the geometric enhancement may partially compensate for this suppression' (Conclusions) but provides no calculation to support this. A quantitative comparison of the angular enhancement factor against the transfer-function suppression and the LISA noise curve is needed to assess whether the effect survives.
minor comments (7)
  1. Eq. (3): The notation 'wef f' and 'w' should use ω (omega) consistently; the subscript formatting is also inconsistent ('wef f' vs. 'k ef f').
  2. Sec. IV, Eq. (9): The factor sin²(α) in the denominator of the integrand diverges as α → 0. The paper should clarify how this is handled numerically, particularly since the redshift-only case (Eqs. 17–18) predicts an enhancement near α = 0.
  3. Sec. VI: The assumption of 500 sources per comoving distance bin is acknowledged as potentially unrealistic, but no estimate of the actual expected number is given. The dN/dz distribution in Fig. 6 is described as 'qualitative'—a more quantitative estimate, even order-of-magnitude, would strengthen the discussion.
  4. Figures 7–9: The color scales and selection criteria ('minimum criteria indicated in the graphic') are not clearly defined. The reader cannot easily reproduce the selection cuts or assess their impact on the claimed distinguishability of ±10% H₀ variations.
  5. Sec. II: LISA's strain sensitivity is quoted as h ≈ 10⁻²¹, but this is frequency-dependent. The relevant sensitivity at f ~ 1 Hz should be cited from the LISA sensitivity curve (ref [15]).
  6. Reference [10]: 'J. Stuart, B. Wyithe and A. Loeb' — the author name is likely 'J. S. B. Wyithe' or similar; please verify.
  7. Sec. IV, after Eq. (13): The statement 'the cosinus is anyway extremely close to one in this case' is unclear. Please rephrase to specify which cosine and why it is close to unity.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for a careful and substantive report. The referee raises three major points: (1) the absence of an SNR calculation to support the 'few-percent' feasibility claim, (2) the inapplicability of the stationary phase approximation in the LISA regime, and (3) the tension between the >100 mHz requirement and reduced LISA sensitivity plus unfavorable source populations. We agree with points (1) and (3) in large part and will revise the manuscript accordingly—specifically by softening the feasibility claim in the abstract and Sec. V, and by adding a quantitative comparison of the angular enhancement against the transfer-function suppression. On point (2), we agree the SPA framework as presented is misleading for LISA parameters and will reframe the derivation in terms of coherent phase accumulation, while preserving the final formula and its numerical confirmation. We note that a full SNR calculation using the LISA noise curve and TDI response is beyond the scope of the current theoretical study, and we acknowledge this as a standing limitation.

read point-by-point responses

  1. Referee: Abstract and Sec. V: The claim that 'LISA could determine H0 to within a few percent' is the central feasibility assertion of the paper, but no signal-to-noise ratio calculation is performed anywhere in the manuscript. The amplitude epsilon is set to 1 throughout, which means all results show only relative angular modulation, not absolute detectability. For realistic LISA sources at z ~ 0.05-1, the strain is h ~ 10^-22-10^-24, and the timing residual tau ~ h*L is extremely small. An O(1) angular modulation of an undetectably small signal remains undetectable. The paper acknowledges this gap in Sec. VI and the Conclusions, yet the abstract and Sec. V still assert the few-percent precision claim. Either an SNR calculation using the LISA noise curve (ref [15]) should be included, or the feasibility claim should be removed.

    Authors: The referee is correct that the manuscript does not contain an SNR calculation and that the 'few-percent' claim in the abstract and Sec. V is not supported by a quantitative detectability assessment. We acknowledge this gap. We will revise the abstract and Sec. V to replace the assertion of percent-level precision with a more measured statement: that the angular modulation is of order unity and that, if detectable, it would provide a direct handle on H0*ZA. We will explicitly state that a rigorous SNR calculation incorporating the LISA noise curve (ref [15]) and the full TDI response is required before any precision claim can be made, and that this is left for future work. We agree that setting epsilon = 1 throughout means all results are relative and do not establish absolute detectability; we will make this caveat more prominent. We note that a full SNR assessment requires modeling the complete interferometric response (not just a single arm), realistic source populations, and the TDI noise budget, which is a substantial study beyond the scope of the present proof-of-principle. revision: yes

  2. Referee: Sec. IV, Eqs. (11)-(14): The stationary phase approximation is not valid for LISA parameters. The coefficient A is of order 10^-16 for LISA, as the authors themselves note. With A approximately 0, the phase is effectively linear, and there is no genuine stationary point x0 = -B/(2A). The maximum at B = 0 corresponds to coherent accumulation when the phase stops oscillating, not to a stationary phase phenomenon. While the numerical results confirm that the maximum occurs at the predicted angle, the theoretical justification via the stationary phase approximation is misleading for the LISA regime. The authors should clarify that the SPA is not the appropriate framework here and reframe the B = 0 condition.

    Authors: We agree with the referee's analysis. The coefficient A is indeed negligibly small for LISA parameters, and the maximum of the timing residual at B = 0 arises because the phase becomes constant (linear with zero slope) along the integration path, allowing coherent accumulation rather than the oscillatory cancellation that occurs when B is nonzero. This is physically distinct from the stationary phase mechanism that operates in the PTA regime, where A is non-negligible. We will reframe Sec. IV to clarify that the condition B = 0 corresponds to coherent phase accumulation—where the integrand ceases to oscillate and contributions add constructively—rather than to a stationary phase point in the traditional sense. The formula alpha_optim = 2 arcsin(sqrt(H0*ZA/2)) remains valid as the condition B = 0, and its numerical confirmation in Tables II-III is unaffected. We will remove or substantially qualify the references to the stationary phase approximation in the LISA context and present the derivation in terms of the coherence condition directly. revision: yes

  3. Referee: Secs. V-VI and Conclusions: There is a fundamental tension that is acknowledged but not quantified. The angular enhancement is visible only for f > 100 mHz, but LISA's arm-length transfer function T(f) suppresses the response precisely in this band. Additionally, the sources radiating above 100 mHz are predominantly stellar-origin black hole binaries at z < 0.1, where the H0-dependent correction is smallest. The paper states that the geometric enhancement may partially compensate for this suppression but provides no calculation to support this. A quantitative comparison of the angular enhancement factor against the transfer-function suppression and the LISA noise curve is needed to assess whether the effect survives.

    Authors: The referee correctly identifies the central tension of the paper. We will add a quantitative comparison in the revised manuscript: we will compute the angular enhancement factor (the ratio of the timing residual at alpha_optim to the residual at generic angles, as shown in Figs. 3-5) and compare it against the transfer-function suppression T(f) = sin(pi*f*L)/(pi*f*L) at the relevant frequencies (100 mHz to 10 Hz). This will allow a direct assessment of whether the O(1) angular modulation survives the geometric suppression. We expect that at f ~ 1 Hz, where T(f) is already significantly suppressed, the enhancement factor (which can be several-fold for distant sources) may partially compensate but is unlikely to fully overcome the suppression for the faintest sources. We will present this comparison explicitly and discuss its implications. Regarding the source population issue, we agree that SOBHBs at z < 0.1 are the dominant population above 100 mHz and that the H0 correction is smallest there. We will strengthen the discussion of this point and note that the method is most promising for intermediate-redshift sources (z ~ 0.3-1) if any can be detected at frequencies above 100 mHz, or for stacking analyses. We acknowledge that without a full SNR calculation we cannot definitively establish that the effect survives in practice; this is a genuine limitation of the current work. revision: partial

standing simulated objections not resolved
  • A full SNR calculation using the LISA noise curve, TDI response, and realistic source populations is beyond the scope of the present theoretical study. While we will soften the feasibility claims and add a quantitative comparison of the enhancement factor against the transfer-function suppression, we cannot provide a complete detectability assessment without substantially extending the analysis to include the full interferometric response. We acknowledge this as an open limitation that requires further work.

Circularity Check

0 steps flagged

No significant circularity found; derivation is algebraically self-contained with verifiable self-citations

full rationale

The paper's central result, α_optim = 2 arcsin(√(H₀Z_A/2)) (Eq. 16), is derived through a transparent algebraic chain: the FLRW wave equation (Eq. 2, standard GR) yields the perturbative solution (Eq. 3) with k_eff = ω(1−RH₀/2), which is substituted into the timing residual integral (Eq. 9). The phase Θ(x) = Ax² + Bx + C (Eq. 10) has coefficients A, B, C given explicitly in Eqs. 11–13. Setting B = 0 with T_A ≈ Z_A yields cos(α) = 1 − H₀Z_A, which is equivalent to Eq. 16 via the double-angle identity. This is a parameter-free derivation; no quantity is defined in terms of the result. The numerical results in Tables II–III are independent numerical integrations of Eq. 9, not fits to the analytical formula, so they serve as genuine confirmation rather than circular reproduction. The self-citations [2–5] (co-authored by Espriu) provide the wave equation solution and the PTA formalism, but these are mathematically verifiable results: Eq. 3 can be checked by direct substitution into Eq. 2, and the paper explicitly notes that k_eff = ω_eff does not satisfy the wave equation, which is a checkable mathematical fact rather than an unverified ansatz. No uniqueness theorem is invoked to forbid alternatives. The feasibility claim (few-percent H₀ measurement) lacks SNR support, but that is a correctness risk, not circularity. The only minor concern is that the entire theoretical framework rests on the prior work [2] by the same authors, but since that work is independently verifiable and the present paper's derivation is self-contained from Eq. 3 onward, this does not constitute circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

No new physical entities are postulated. The framework uses standard FLRW cosmology and standard gravitational wave perturbation theory. The effect arises from the known cosmological expansion.

free parameters (3)
  • H0 = 2.26e-18 s^-1
    Used as input throughout; the paper tests sensitivity to +/-10% variations but does not fit it to data.
  • epsilon (GW amplitude) = 1 (arbitrary)
    Set to 1 in all figures; only relative enhancements are studied, so absolute detectability is undetermined.
  • N_sources per distance bin = 500
    Assumed for the two-arm correlation analysis; authors acknowledge this 'may not be realistic.'
axioms (4)
  • domain assumption The FLRW metric perturbation solution at O(H0) is given by Eq. 3 with omega_eff = omega(1-R*H0) and k_eff = omega(1-R*H0/2).
    Invoked in Sec. III; derived in [2]. The statement that k_eff != omega_eff is load-bearing for the entire effect.
  • domain assumption The leading-order approximation where sqrt(Lambda/3) is replaced by H0 is valid for z < 2.
    Stated in Sec. III based on [4]; limits the applicable redshift range.
  • domain assumption ZA >> L (source distance much greater than LISA arm length).
    Used in Sec. IV to simplify R(x) ~ ZA + xL*cos(alpha). Valid for cosmological sources but not for nearby ones.
  • domain assumption The stationary phase approximation accurately captures the integral for f > 100 mHz.
    Stated in Sec. IV.A; verified numerically for PTA in [3] but the LISA regime (k_eff ~ 1/L) is qualitatively different.

pith-pipeline@v1.1.0-glm · 14443 in / 2671 out tokens · 407643 ms · 2026-07-09T11:55:39.944658+00:00 · methodology

0 comments
read the original abstract

It was previously shown that $H_0$ influences gravitational wave propagation beyond simple redshift, by modifying the effective wavenumber. While earlier studies focused on Pulsar Timing Arrays, we analyze the observability of this effect with LISA. Modeling the LISA arm geometry, we derive the timing residual and identify a clear angular enhancement at frequencies above 100 mHz. Using the stationary phase approximation and numerical methods, we show that the optimal incidence angle depends on $Z_A H_0$, providing a direct, redshift-independent measurement of the Hubble parameter. We estimate that LISA could determine $H_0$ to within a few percent using this method, independent of standard siren approaches.

Figures

Figures reproduced from arXiv: 2607.07398 by Domenec Espriu, Nil Blanco.

Figure 1
Figure 1. Figure 1: FIG. 1: PTA results. Left: the statistical significance of the enhancement [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Geometry of the system for LISA. The line AB represents one of LISA’s arms. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Comparison of the cases where [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the cases where the gravitational wave is described in terms of the redshift [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The time average results are not substantially different from those obtained from the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Approximate redshift distribution of GW sources detectable by LISA. The total distribution [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Comparison between the combined signal of two LISA arms for [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The plots depict the sum of the time residuals measured in two arms [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The sum of the timing delays measured by two LISA arms for several values of [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗

discussion (0)

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Reference graph

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