{"id":"d9e8337c-a772-4003-8dea-58a881590a5e","arxiv_id":"2608.18310","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A Fourier-based, quasi-model-independent reconstruction of the GW/EM luminosity-distance ratio finds GR consistency in GWTC-5 and projects 3-sigma sensitivity to 5% deviations with Einstein Telescope-like data.","lead":"This paper introduces a flexible 'weakly modelled' method that reconstructs how gravitational wave distances deviate from light-based distances as a function of cosmic time, without assuming a particular formula. Applied to 231 black hole mergers from GWTC-5, it finds no deviation from general relativity, and forecasts that next-generation detectors could spot 5% deviations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"As written, the likelihood in Eqs. (6)-(7) and (9) drops the per-event measurement errors that the mock pipeline injects; the reported 3-sigma ET detection and 5% threshold are therefore not supported.","rationale":"I read the paper as a proof-of-concept for a weakly-modelled Fourier reconstruction of r(z) = d_GW/d_EM. For that claim to hold, the inference must correctly handle the noise that the mock generator inserts; otherwise the 3-sigma contours and detectability thresholds are not trustworthy. The reader's chosen weakest assumption—the fixed power-spectrum hyperparameters A, l, a in Eq. (11)—is a genuine sensitivity question, but it concerns the calibration of the reported thresholds rather than the internal validity of the inference. The likelihood issue is more load-bearing: the text's generative model (Gamma-distributed observed values and posterior samples) is not reflected in the written inference model (delta functions), so the results cannot be reproduced as described. If the code in fact marginalizes over the Gamma posterior samples, the paper needs a corrected likelihood and a rerun; if it does not, the apparent ET detections may be an artifact of treating point estimates as exact. The abstract's '5% at 3-sigma' claim is additionally in tension with the reported average 3-sigma width of 9% and with the 8-9% amplitude of the injected deviations, so the headline sensitivity claim is not demonstrated even conditional on the method working. I keep the manuscript in conditional territory because the issue is concrete and testable with the public code, and the Fourier-basis idea is not inherently invalidated by a reporting error; but the condition must be a noise-aware likelihood and a re-derived detectability threshold, not merely a variation of the power-spectrum priors. My agreement with the reader is partial: the reader's rationale lists the likelihood issue as one of three concerns, but the formal weakest_assumption selects the hyperparameters, which I regard as secondary.","tokens_in":18185,"tokens_out":12003,"duration_ms":126798,"concrete_test":"Open the CosmoPyro repository (pin a specific commit) and locate the likelihood used for the ET bright-siren analysis and the O5/GWTC-5 dark-siren analyses. If Eq. (9) is implemented literally, re-run the ET four-injection analysis with the delta replaced by the same Gamma likelihood used to generate the data, e.g. L(d_GW_i | z_i, r) = Gamma(shape = alpha+1, rate = hat d_GW(z_i, r)) with alpha set to the stated 8.3% distance error (and analogously for mass ratio and chirp mass in the dark-siren case), then recompute the 3-sigma contours for the four injected r(z). If the injected curves are no longer excluded at 3-sigma, or if the average 3-sigma width grows beyond the injected amplitudes, the central detectability claim fails. The same check also verifies whether the '5% at 3-sigma' statement is reproducible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—ET bright sirens can recover injected deviations at 3-sigma and detect about 5% deviations—requires a likelihood that correctly propagates measurement uncertainties. The displayed likelihood does not. Eq. (6) replaces the per-event measurement distribution with delta functions in detector-frame mass and GW distance; Eq. (7) then evaluates the population and redshift terms at the single point (hat m_s, hat z) fixed by those deltas. Eq. (9) does the same for bright sirens, placing a delta on d_GW. Yet the mock generation in Section 4.4 explicitly draws an observed value from Gamma(alpha, x_true) and posterior samples from Gamma(alpha+1, x_obs), injecting the asymmetric distance, mass-ratio and chirp-mass errors quoted in the text (20%, 14%, 3% for O5; 8.3%, 2.8%, 1% for ET). Nowhere is this per-event posterior averaged over or convolved into the likelihood. If the equations are what the code implements, every quoted width—O5 24%/10%, ET 9%/4%, and the abstract's '5% at 3-sigma'—is conditional on treating noisy point estimates as exact, which narrows contours and can manufacture 3-sigma exclusions. Independently of the noise issue, the abstract's 5%-at-3-sigma detectability statement is in tension with the reported average 3-sigma error of 9% in Section 5.2, since the injected deviations are 8-9%, not 5%. The public code can settle whether the likelihood actually marginalizes over posterior samples; if it does, the equations must be corrected, and if it does not, the quantitative claims are invalid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quasi-model-independent method to reconstruct the ratio r(z)=d_GW/d_EM as a function of redshift, using a Fourier basis with a fixed power-spectrum prior, jointly inferred with GW source-population parameters via spectral sirens. The method is validated on 500 mock LVK O5 dark-siren events and 1000 mock Einstein Telescope bright-siren events for four injected non-GR cosmologies, and it is applied to 231 GWTC-5 BBH events, finding consistency with GR. The headline quantitative claims are that O5-like data cannot resolve deviations below about 30%, while third-generation detectors can detect deviations of order 5% at 3-sigma significance.","tokens_in":18520,"tokens_out":5168,"duration_ms":57904,"significance":"If the quantitative claims survive scrutiny, this is a useful and timely contribution: it widens the space of modified-gravity distance-ratio behaviours beyond the usual one- or two-parameter ansatzes, and it provides a public pipeline (CosmoPyro) with mock-injection validation over monotonic, oscillatory, and localised-feature models. The demonstration that the mass-distribution posteriors are not biased by the weakly modelled MG sector, and the application to the external GWTC-5 dataset with results consistent with GR, are concrete strengths. However, the central detectability numbers are not supported by the likelihood as written, because the displayed likelihood collapses per-event measurement errors to delta functions. The conceptual framework is sound and the required fixes are local, so the paper is a candidate for major revision rather than rejection, provided the likelihood issue is resolved and the claimed detection thresholds are recomputed or rephrased.","major_comments":[{"comment":"The single-event likelihood in Eqs. (6)–(7) is written with delta functions in detector-frame mass and GW distance, so the integral in Eq. (5) collapses to a single point evaluation. Section 4.4, however, describes injecting Gamma-distributed observed values and posterior samples with O5 relative errors of about 20% in d_GW, 14% in q, and 3% in chirp mass. If the code implements the displayed likelihood, the injected measurement noise is ignored and all quoted widths and significances are computed from noisy point estimates treated as exact; this can artificially narrow contours and manufacture 3-sigma exclusions. If the code instead averages the population terms over the Gamma-distributed posterior samples, then Eqs. (6)–(7) are not what the analysis does and must be rewritten. Either way, the displayed likelihood must be fixed before the paper's quantitative claims can be accepted.","section":"§3, Eqs. (6)–(7)"},{"comment":"The bright-siren likelihood in Eq. (9) places a delta function on d_GW, yet Section 4.5 states that the ET mock events are generated with Gamma-distributed distance errors corresponding to sigma_dGW ~ 8.3%. The reported ET results—3-sigma recovery of all four injected models and the abstract's '5% at 3-sigma' threshold—therefore inherit the same issue as the dark-siren likelihood: the per-event distance posterior is not marginalised over. The authors should include the per-event distance likelihood in Eq. (9) (and in the reported inference) and show how the reconstructed contours change when the distance errors are propagated.","section":"§4.5 and §5.2, Eq. (9)"},{"comment":"The power-spectrum hyperparameters A=2.3e-4, l=2.4e-2, and a=4 in Eq. (11) are fixed by hand, and Section 6 states that varying them has not been explored. Because the prior band directly controls the allowed amplitude and correlation length of r(z), and because the posteriors in Figs. 4 and 6 relax to the prior where data are absent, the detectability thresholds ('O5 cannot resolve below 30%', 'ET can detect 5% at 3-sigma') are conditional on this specific prior choice. A sensitivity test over these hyperparameters, or a rephrasing of the thresholds as explicitly prior-conditional, is needed; without it the robustness of the headline numbers is not established.","section":"§4.3 and §6"},{"comment":"The abstract claims that third-generation detectors can 'confidently detect deviations of order 5% to 3-sigma significance', but Section 5.2 reports an average 3-sigma error of 9% on the reconstructed ratio, and the injected deviations in Eqs. (16) have amplitudes of 8–9%, not 5%. A 5% deviation would be below the average 3-sigma error, so the claim as stated is not supported by the reported numbers. Either an explicit injection-recovery calculation at 5% amplitude is required, or the abstract and conclusions should be revised to state what the simulations actually demonstrate.","section":"§5.2 and the abstract"}],"minor_comments":[{"comment":"The phrase 'deviations of order 5% to 3-sigma significance' is ambiguous: it could mean 5% deviations detected at 3-sigma, or a 5-sigma detection of a 3% deviation. Please rephrase for clarity.","section":"Abstract / §5.2"},{"comment":"There is a typo: 'sceneario' should be 'scenario'.","section":"§4.5"},{"comment":"The sentence beginning 'The sum over modes runs up to n' uses 'infer the phases and amplitudes of the Fourier modes, encoded in F(z_i)' but F(z_i) is the inverse-Fourier-transformed field; the wording conflates the sampled Fourier coefficients with their real-space realisation.","section":"§4.2, after Eq. (12)"},{"comment":"The text contains a duplicated lead-in: 'We can simplify the expression using the relation between detector and source frame masses We additionally rewrite' is missing a period or conjunction.","section":"§2.2"},{"comment":"There is an incomplete sentence fragment, 'The rest of the uncertainti', which should be completed or removed.","section":"Figure 2 caption / surrounding text"},{"comment":"The reference 'Zazzera et al. 2026' is given as 'MNRAS, 547, stag307'; 'stag307' appears to be a placeholder or formatting error and should be corrected.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The most important issue is the mismatch between the displayed likelihood (delta functions) and the mock-injection description (Gamma-distributed errors and posterior samples). Since CosmoPyro is public, the editor can ask the authors to verify whether the code marginalises over posterior samples. If it does, a concise textual correction may suffice; if it does not, the numerical results and the abstract's detectability statements will need to be recomputed. I recommend requiring this verification explicitly in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the novel piece is the Fourier/power-spectrum prior on r(z)=dGW/dEM, with a low-z suppression and a positivity constraint, applied through spectral sirens. The ET mock run is a solid proof of concept—it recovers monotonic, bump, and damped oscillatory injections, and the GWTC-5 result is a clean null. That deserves a referee's time.\n\nThe real problem is the statistical presentation. Eqs. (6)–(7) and (9) write the per-event likelihood as delta functions in detector-frame mass and GW distance, meaning exact measurements. The mock generation in Sec. 4.4, however, explicitly draws observed values and posterior samples from Gamma distributions. If the code implements the delta-function likelihood, the quoted 3-sigma ET detection and the 5% threshold are not supported—the contours would be too narrow because they ignore the injected noise. If the code instead marginalizes over posterior samples, the equations are misleading and must be corrected. Either way, the paper cannot be checked without digging into CosmoPyro, and the public repo has no version or commit hash. This needs fixing before the numbers can be taken at face value.\n\nIndependently, the abstract's '5% at 3-sigma' is in tension with Sec. 5.2, where the average 3-sigma error is 9% and the injected deviations are 8–9%. A 5% deviation would sit inside the stated error. That claim should be revised or substantiated.\n\nThe 'without biasing population parameters' claim is also a bit strong. Table A1 shows gamma and alpha shifted about 1–2 sigma from truth in most mock runs. Mild, and plausibly a selection effect, but not 'no bias'. And the detectability thresholds depend on hand-fixed power-spectrum hyperparameters (A=2.3e-4, l=2.4e-2, a=4), which the authors explicitly do not vary. A wider prior would widen the contours. They acknowledge this, but it means the quoted thresholds are conditional on untested choices.\n\nNone of this kills the method. The Fourier reconstruction with a derivative-based positivity prior is genuinely new here, the code is public, and the GWTC-5 mass-population consistency is a good sanity check. The paper is a promising addition to the non-parametric literature, but the likelihood presentation and the headline numbers need a major revision. I'd send it to referees, expecting them to require the likelihood to be written correctly and the detectability claims re-derived or retracted.","headline":"The Fourier weakly-modelled reconstruction is a genuinely new tool for GW cosmology, but the likelihood as written drops per-event measurement errors, so the headline detectability claims need a correction before the numbers can be trusted.","tokens_in":19092,"tokens_out":4613,"would_cite":false,"duration_ms":44202,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces a quasi-model-independent Fourier reconstruction of the gravitational-wave-to-electromagnetic luminosity distance ratio and shows that third-generation bright-siren data can detect deviations from general relativity…","keywords":["gravitational wave cosmology","modified gravity","luminosity distance","spectral sirens","dark sirens","Fourier reconstruction","hierarchical Bayesian inference","third-generation detectors"],"falsifier":"Run the identical reconstruction on the 1000-event bright-siren mocks with the amplitude hyperparameter $A$ doubled and then halved, keeping everything else fixed; if the injected 5% deviations no longer reach $3\\sigma$ in either case, the claimed detectability is an artifact of the prior width rather than of the data.","tokens_in":17908,"feed_emoji":"🌌","tokens_out":9509,"duration_ms":77015,"temperature":0.7,"pith_summary":"Cosmological tests of general relativity with gravitational waves usually fix a one- or two-parameter curve for how the gravitational-wave luminosity distance differs from the electromagnetic one. This paper replaces that parametric ansatz with a Fourier reconstruction of the ratio $r(z)=d_{\\rm GW}/d_{\\rm EM}$, letting the data shape any monotonic, irregular, or oscillatory deviation. Testing the method on mock data, the authors show that current-generation (O5-like) dark sirens cannot resolve deviations below roughly 30%, while a third-generation detector with 1000 bright sirens recovers injected deviations of order 5% at $3\\sigma$ significance. The same analysis applied to 231 black-hole mergers from the latest catalog release finds no deviation from general relativity and leaves the inferred black-hole mass distribution unbiased. If the method works as claimed, it widens modified-gravity tests from a few pre-chosen curves to essentially any function of redshift.","feed_headline":"GW sirens flag 5% GR deviations with no fixed formula","feed_subtitle":"Shape-free Fourier reconstruction recovers monotonic, irregular, and oscillatory departures, and current data see none.","key_machinery":"The central object is a weighted Fourier reconstruction of $r(z)$: random Gaussian modes are reweighted by a power spectrum $P(k)=A/[\\ell(1+(\\ell k)^a)]$ with fixed hyperparameters $A=2.3\\times10^{-4}$, $\\ell=2.4\\times10^{-2}$, $a=4$, then combined with a suppression factor $1-\\exp[-(z/z_*)^2]$ that forces $r\\to1$ as $z\\to0$, and a positivity-preserving construction of $d_{\\rm GW}$ as a cumulative sum of the positive EM-distance derivative. The completeness of the Fourier basis is what carries the claim: as the number of modes grows, any functional form of the ratio can in principle be reproduced. The data inform the amplitudes and phases of the modes, so the method needs no theory-specific ansatz; the suppression scale $z_*$ is the only extra parameter beyond the mode coefficients. This machinery is embedded in a hierarchical Bayesian spectral-sirens analysis that jointly infers the ratio and the black-hole mass-population parameters.","core_discovery":"The paper's central claim is that the ratio $r(z)=d_{\\rm GW}/d_{\\rm EM}$ can be reconstructed without choosing a specific theory-motivated functional form, by expanding it in a Fourier basis whose mode amplitudes and phases are inferred from the data. With 500 simulated dark sirens at O5 sensitivity the reconstruction is too broad to resolve the injected non-GR signals, with average $1\\sigma$ errors around 10% and $3\\sigma$ errors around 24%, but it remains unbiased. With 1000 simulated bright sirens at third-generation sensitivity, the same Fourier reconstruction recovers monotonic, gaussian-bump, damped-oscillatory, and $\\alpha_M$-type ratios and excludes $r(z)=1$ at $3\\sigma$ for all four, with average $3\\sigma$ errors of order 9%. On the 231 black-hole mergers of the fifth release, with $H_0$ fixed to either Planck or local values, the reconstructed ratio is fully consistent with GR and the inferred mass distribution agrees with the reference analysis. The authors therefore claim their method is ready for real deployment and can be extended to future detectors.","pith_inferences":["The reported 5% and 30% detectability thresholds are tied to the hand-fixed prior width $A=2.3\\times10^{-4}$; varying $A$ by even a factor of a few could shift these thresholds, so the method's sensitivity should be treated as prior-dependent until that dependence is mapped.","The same Fourier machinery could be turned on the black-hole mass distribution itself, replacing the parametric Multi-Peak model with a non-parametric reconstruction; that would remove the main population-modeling risk the authors identify.","If the recent preference for dynamical dark energy persists, fixing the background to $\\Lambda$CDM could leak evolution of the dark-energy equation of state into reconstructed deviations of $r(z)$; a joint treatment of $w(z)$ and $r(z)$ would be a natural test.","Combining this GW-only reconstruction with non-parametric large-scale-structure probes would provide a fully theory-agnostic consistency test of gravity, since modified gravity generically affects both GW propagation and structure growth."],"forward_implications":["With current O5-like dark-siren data, the method can only exclude deviations from GR larger than about 30%; a null result therefore should not be read as ruling out smaller modifications.","With third-generation bright-siren data, the method can detect deviations of order 5% at $3\\sigma$ across monotonic, bump, and oscillatory shapes, so future detectors would test modified gravity without committing to a parametrisation.","Because the mass-population parameters are recovered without bias in all mock cosmologies, the reconstruction does not compromise standard spectral-sirens cosmology.","Reconstruction errors shrink roughly as $1/\\sqrt{N}$, so reaching $3\\sigma$ errors near a few percent at O5 sensitivity would require about 50,000 events, a number expected from next-generation detectors rather than the current network.","The paper leaves the power-spectrum hyperparameters fixed, so the quoted sensitivity thresholds are conditional on that prior choice; treating them as free parameters is explicitly deferred."],"supporting_citations":[{"why":"Introduces the $\\Xi_0$-$n$ parametrisation and the modified GW friction term that the weakly-modelled approach is designed to supersede.","marker":"Belgacem et al. (2018b)"},{"why":"Supplies the Fourier reconstruction technique that this paper adapts from galaxy-map analyses to the luminosity-distance ratio.","marker":"Leyde et al. (2024)"},{"why":"Establishes the spectral-sirens method used to break the redshift-distance degeneracy for dark sirens.","marker":"Mastrogiovanni et al. (2021)"},{"why":"Provides the hierarchical Bayesian likelihood and selection-bias correction used in the joint inference.","marker":"Mandel et al. (2019)"},{"why":"Sets the selection criteria, mass model, and comparison baseline for the real-data analysis on the fifth catalog release.","marker":"The LIGO Scientific Collaboration et al. (2026c)"},{"why":"Defines the third-generation detector sensitivity and event-rate expectations used to build the bright-siren mock data.","marker":"Abac et al. (2026b)"},{"why":"Supplies the star-formation history used as the redshift prior in generating mock catalogs.","marker":"Madau & Dickinson (2014)"},{"why":"Catalogues modified-gravity theories with non-trivial luminosity-distance relations, motivating the need for a model-independent reconstruction.","marker":"Ezquiaga & Zumalacárregui (2018)"}],"fun_headline_variants":["Shape-free GW tests catch 5% GR breaks at 3σ","No parametrisation needed: GWs map spacetime without bias","Fourier-based siren test: current data clean, future sees 5%","Unbiased GW reconstruction flags no GR deviation yet"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the hand-fixed power-spectrum hyperparameters $A=2.3\\times10^{-4}$, $\\ell=2.4\\times10^{-2}$, and $a=4$, which set the prior size and smoothness of deviations, are representative of the deviations a real modified-gravity theory would produce; if a wider or narrower prior were needed, the reported detection thresholds and the real-data consistency statement could change.","fun_headline_variants_meta":{"raw":{"variants":["Shape-free GW tests catch 5% GR breaks at 3σ","No parametrisation needed: GWs map spacetime without bias","Fourier-based siren test: current data clean, future sees 5%","Unbiased GW reconstruction flags no GR deviation yet"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000807,"raw_usage":{"total_tokens":3557,"prompt_tokens":976,"completion_tokens":2581,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":2515}},"tokens_in":592,"tokens_out":2581,"duration_ms":16109,"temperature":1.0,"reasoning_tokens":2515,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T20:00:28.947436+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the identical reconstruction on the 1000-event bright-siren mocks with the amplitude hyperparameter $A$ doubled and then halved, keeping everything else fixed; if the injected 5% deviations no longer reach $3\\sigma$ in either case, the claimed detectability is an artifact of the prior width rather than of the data.","supporting_citations":[],"review_version":1}