{"id":"4650ae4e-e6bd-40a3-a792-922a5767edc5","arxiv_id":"2607.07398","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"The Hubble parameter produces a frequency-independent angular enhancement in LISA timing residuals above 100 mHz, potentially enabling a percent-level H0 measurement without standard sirens.","lead":"The paper proposes that LISA could measure the Hubble constant by detecting an angular enhancement in gravitational wave timing residuals caused by cosmological expansion modifying the effective wavenumber. A smart generalist might read it because it sketches a new, redshift-independent route to H0 using a future space mission.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The analytical formula for α_optim is sound and numerically confirmed, but the claim that LISA could measure H0 to within a few percent is unsupported: no SNR calculation is performed, and the O(1) angular modulation occurs in a frequency band where LISA's transfer function suppresses the response, ","rationale":"The reader's weakest_assumption correctly identifies the load-bearing concern: the absence of an SNR analysis and the tension between the frequency band where the effect is visible (f>100 mHz) and LISA's degrading sensitivity in that band. I verified the algebra of Eq. 16 and it is correct. I also verified numerically that the stationary phase approximation is not strictly valid for LISA parameters (A∼10⁻¹⁶), but the maximization condition B=0 still correctly identifies the peak location, and the numerical results confirm this. The formula itself is not the problem; the problem is the feasibility claim. The paper is honest about its limitations (Sec. VI, Conclusions) but the abstract's claim of 'a few percent' precision is unsupported. The CONDITIONAL verdict is appropriate: the analytical result is a valid proof-of-principle, but the feasibility claim requires an SNR analysis incorporating the LISA sensitivity curve, TDI response, transfer function, and realistic source populations. The concrete test I propose would settle whether the O(1) angular modulation is detectable above noise for any realistic source configuration. If it fails even for optimistic assumptions, the headline claim should be retracted to a theoretical prediction awaiting feasibility assessment.","tokens_in":11443,"tokens_out":6299,"duration_ms":382789,"concrete_test":"Compute the SNR for a representative source using the full pipeline: take a SOBHB at z=0.05 (ZA≈200 Mpc) with f=1 Hz and strain h=10⁻²², evaluate the timing residual from Eq. 9 with the physical strain amplitude (not ε=1), apply the arm-length transfer function T(f)=sin(πfL)/(πfL) with L=8.3 s, and compare the angular modulation amplitude |τ(α_optim)−τ(α_other)| to the LISA noise spectral density from Babak et al. 2021 (ref [15]) over a 1000 s integration. If the modulation is below the noise floor by more than a factor of ~3 even for this optimistic case, the single-source feasibility claim does not hold. For the two-arm stacking method, repeat with N=500 sources but using a realistic redshift distribution for f>100 mHz sources (predominantly z<0.1) rather than the assumed 500 sources at each of 100 Mpc, 500 Mpc, and 1 Gpc.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the load-bearing concern. The derivation of α_optim = 2 arcsin(√(H0·ZA/2)) (Eq. 16) is algebraically correct: setting B=0 in Eq. 12 with TA≈ZA yields cos(α)=1−H0·ZA, which is equivalent to Eq. 16 via the double-angle identity. The numerical results in Tables II–III confirm this formula. However, the stationary phase approximation that motivates the B=0 condition is not strictly 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), meaning the quadratic phase term Ax² is negligible over the integration domain. The phase is effectively linear (Θ≈Bx+C), so there is no genuine stationary point. The maximum at B=0 still occurs (coherent accumulation when the phase stops oscillating), but the enhancement is O(1), not the dramatic peak seen in PTA where A∼O(0.1). The paper acknowledges this ('much less marked than for PTA') but then claims the effect is 'of order one and therefore measurable' — an O(1) modulation of an undetectably small signal is still undetectable. The paper sets ε=1 throughout, masking the absolute magnitude. For a realistic source (e.g., SOBHB at z∼0.05, h∼10⁻²²), the timing residual is τ∼h·L∼10⁻²¹ s, and the angular modulation is O(1) relative to this. Meanwhile, LISA's transfer function T(f)=sin(πfL)/(πfL) suppresses the response by a factor of ~30 at f=1 Hz and ~250 at f=10 Hz. No comparison to the LISA noise curve (ref [15]) is made. The paper itself acknowledges this tension in Sec. VI and the Conclusions but does not resolve it.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","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.","tokens_in":11763,"tokens_out":2556,"duration_ms":180769,"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":[{"comment":"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","section":null},{"comment":"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","section":null},{"comment":"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.","section":null}],"minor_comments":[{"comment":"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').","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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]).","section":null},{"comment":"Reference [10]: 'J. Stuart, B. Wyithe and A. Loeb' — the author name is likely 'J. S. B. Wyithe' or similar; please verify.","section":null},{"comment":"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.","section":null}],"recommendation":"major_revision","confidential_remarks":"The analytical core of this paper (the α_optim formula and its numerical confirmation) is sound and represents a legitimate extension of the PTA formalism. However, the gap between what is proven (an angular scaling relation) and what is claimed (few-percent H₀ measurement) is large. The authors themselves acknowledge most of the limitations in Sec. VI and the Conclusions, which suggests good faith, but the abstract still overstates the result. If the authors either (a) perform a basic SNR estimate or (b) substantially soften the feasibility claim to match what is actually demonstrated, the paper could be acceptable. The stationary phase approximation issue (Major Comment 2) is a conceptual mischaracterization that should be corrected regardless. I would not recommend rejection because the analytical result is correct and novel, but the feasibility framing needs substantial revision."},"author_rebuttal":{"model":"glm-5.2","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"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."},{"response":"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_made":"yes","referee_comment":"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."},{"response":"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_made":"partial","referee_comment":"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."}],"tokens_in":11550,"tokens_out":1566,"duration_ms":232192,"standing_objections":["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."]},"desk_editor":{"model":"glm-5.2","letter":"The main result here is clean: the authors derive α_optim = 2 arcsin(√(H₀Z_A/2)) for the LISA arm geometry, showing that the incidence angle maximizing the cosmological correction to the GW timing residual depends only on the product H₀Z_A and is frequency-independent above ~100 mHz. This is a genuine adaptation of their PTA formalism to LISA, and the numerical integrations in Tables II–III confirm the analytical formula well. The two-arm correlation method (Sec. VI) is also new and visually intuitive — the cross-shaped pattern in the α₁–α₂ plane shifting with H₀ is a nice idea. Credit is due for the algebra and the honest self-assessment in Sec. VI and the Conclusions, where the authors explicitly flag the central tension without hiding it. That said, the feasibility claim — “LISA could determine H₀ to within a few percent” — is not supported by what the paper actually demonstrates. The stress-test concern lands here: the coefficient A in the quadratic phase is of order 10⁻¹⁶ for LISA parameters, so the stationary phase approximation is not doing what it does for PTA. The phase is effectively linear, and the “enhancement” at B=0 is an O(1) modulation, not a dramatic peak. An O(1) angular modulation of a signal whose absolute magnitude is never compared to the LISA noise floor is not obviously detectable. Setting ε=1 throughout masks this. The paper acknowledges that the effect lives in the frequency band where LISA’s transfer function suppresses response and where sources are predominantly local, but acknowledging a problem is not the same as resolving it. No SNR calculation using the LISA sensitivity curve (ref [15] is cited but never used) is performed. The two-arm statistical method assumes 500 sources per distance bin, which the authors admit may be unrealistic. These are not minor caveats — they are the difference between a proof-of-principle and a feasibility demonstration, and the abstract blurs that line. This paper is for theorists interested in GW propagation effects and LISA mission planners. The analytical result is sound and worth publishing; the feasibility framing needs to be either backed by an SNR analysis or substantially softened. It deserves a serious referee who can push on the detectability question.","headline":"Letter on arXiv:2607.07398","tokens_in":12322,"tokens_out":1173,"would_cite":false,"duration_ms":101952,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Cosmic expansion bends gravitational wave signals into a measurable angle","keywords":[],"falsifier":"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.","tokens_in":11532,"feed_emoji":"📐","tokens_out":739,"duration_ms":192012,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hubble constant from gravitational wave angles, not sirens","feed_subtitle":"LISA could measure cosmic expansion by reading the incidence angle where gravitational wave timing residuals peak","key_machinery":"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.","core_discovery":"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.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["LISA incidence angle as H0 probe, no sirens needed","Gravitational wave arrival angle encodes H0 for LISA","Hubble constant from LISA arm geometry, not sirens","Optimal GW incidence angle traces H0 directly","LISA timing residual angular peak sets H0"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["LISA incidence angle as H0 probe, no sirens needed","Gravitational wave arrival angle encodes H0 for LISA","Hubble constant from LISA arm geometry, not sirens","Optimal GW incidence angle traces H0 directly","LISA timing residual angular peak sets H0","Measure H0 from GW incidence angle at LISA","Angular signature of H0 in LISA timing residuals","H0 from gravitational wave angles, no sirens required"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1123,"prompt_tokens":401,"completion_tokens":722,"prompt_tokens_details":null},"tokens_in":401,"tokens_out":722,"duration_ms":42179,"temperature":1.0,"reasoning_tokens":574,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T11:55:39.944658+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"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.","supporting_citations":[],"review_version":1}