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Probing the theory of gravity with gravitational lensing of gravitational waves and galaxy surveys

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that weak gravitational lensing of gravitational waves can be measured by cross-correlating LIGO/Virgo sources with galaxy surveys, forecasting detection at z<0.5 within a decade.

desk verdict A plausible forecast paper for a genuinely new observable—GW lensing cross-correlations—whose central unbiasedness assumption is asserted, not proven, and whose 'testing gravity' claim is overstated. read the letter →

arxiv 1908.08951 v2 pith:HJHKW4SS submitted 2019-08-23 astro-ph.CO astro-ph.GAgr-qc

Suvodip Mukherjee , Benjamin D. Wandelt , Joseph Silk This is my paper · ORCID
classification astro-ph.COastro-ph.GAgr-qc
keywords gravitationallensingwavesweakgalaxysurveyscross-correlationluminositydistancemodifiedgravityLISA
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

The paper proposes that weak gravitational lensing of gravitational waves can be measured statistically by cross-correlating gravitational-wave sources with galaxy catalogs. In general relativity, gravitational waves and light follow the same perturbed spacetime geodesics, so the strain of a gravitational-wave signal should be magnified or demagnified by foreground matter in the same way as electromagnetic radiation. Using an estimator that compares the luminosity distance inferred from the gravitational-wave signal with the distance expected from an electromagnetic redshift, the authors forecast that Advanced LIGO and Virgo combined with planned galaxy surveys should detect this lensing at redshifts below $z<0.5$ within about ten years. Their most specific prediction is that black-hole--neutron-star mergers with electromagnetic counterparts, cross-correlated with galaxy overdensity, will reach a greater-than-$3\sigma$ detection at $z>0.35$ within roughly five years. A successful measurement would test how gravitational waves propagate through spacetime and probe the gravitational imprint of dark matter.

What carries the argument

The load-bearing object is the estimator $\hat{D}_L(\hat{n})$, the fractional deviation of the gravitational-wave-inferred luminosity distance from the electromagnetic-based expected distance, along with its cross-correlation against galaxy overdensity $\delta_g$ and galaxy lensing convergence $\kappa_g$. The lensing kernel $W_{\kappa_{\rm gw}}(\chi(z)) = (1/H(z))\int_z^\infty dz'\, (dn_{\rm gw}/dz')(\chi(z')-\chi(z))/\chi(z')$ projects the matter power spectrum along the line of sight, so the theory spectra $C^{\kappa_{\rm gw}\kappa_g}_\ell$ and $C^{\kappa_{\rm gw}\delta}_\ell$ follow from the same Limber-approximation machinery used in galaxy weak lensing. The estimator converts a per-event multiplicative distortion $(1+\kappa_{\rm gw})$ into an additive field whose mean can be extracted: detector noise, redshift error, and sky-localization uncertainty enter as additive noise terms in the covariance, and averaging over many sources suppresses them. This same comparison of gravitational-wave lensing with light lensing is what tests gravity, because modified theories alter either the gravitational-wave propagation equation or the relation between metric potentials and matter density.

What would settle it

Run the proposed estimator on a forward simulation that assigns photometric redshifts to gravitational-wave sources with realistic environmental correlations and source--lens clustering: if $\langle\epsilon_s\delta\rangle$ or $\langle\epsilon_s\kappa\rangle$ does not vanish, the recovered $\hat{C}^{\kappa_{\rm gw}\delta}_\ell$ will be biased at the level of the signal. On real data, split the galaxy catalog by redshift-error properties and check that the measured cross-correlation is stable between subsamples.

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Extended reading notes

Core claim

The paper's central claim is that the weak-lensing convergence of gravitational waves, $\kappa_{\rm gw}$, is observable as a statistical signal rather than only as a source of noise. The observed strain is written as $\tilde{h} = h(f_z)[1 + \kappa_{\rm gw}(\hat{n})]$, which makes the apparent luminosity distance $D_L = d_L/(1 + \kappa_{\rm gw}) + \epsilon_{\rm gw}$. For sources with electromagnetic counterparts, the authors build $\hat{D}_L \equiv 1 - D_L/d_L^{\rm es}$, with $d_L^{\rm es}$ the luminosity distance computed from the electromagnetic redshift and best-fit cosmological parameters, and show that to first order $\hat{D}_L \approx \kappa_{\rm gw} - \epsilon_{\rm gw}/d_L + \epsilon_s - \epsilon_s \kappa_{\rm gw}$. Cross-correlating $\hat{D}_L$ with the galaxy convergence $\hat{C}^{\kappa_{\rm gw}\kappa_g}_\ell$ and with the galaxy overdensity $\hat{C}^{\kappa_{\rm gw}\delta}_\ell$ recovers the lensing signal, provided the redshift-error term $\epsilon_s$ averages to zero. The forecasts place the strongest LIGO-era signal in the black-hole--neutron-star channel: $\hat{C}^{\kappa_{\rm gw}\delta}_\ell$ from these mergers exceeds $3\sigma$ at $z>0.35$ within five years, making it the most promising avenue for Advanced LIGO, and the multi-detector network plus galaxy surveys detects gravitational-wave lensing at $z<0.5$ within a decade. For LISA, the same estimator yields high-signal-to-noise measurements for supermassive black hole binaries of $10^4$--$10^7\,M_\odot$, extending the probe to high redshift.

Load-bearing premise

The entire measurement rests on one assumption: the fractional error in the luminosity distance calculated from an electromagnetic redshift and the assumed cosmology is uncorrelated with the galaxy density or lensing field, so that it averages to zero. If redshift errors are tied to environment, the measured cross-correlation is biased at the level of the lensing signal itself.

Editorial extensions

If this is right

  • Within about ten years, Advanced LIGO, Virgo, and planned galaxy surveys should detect weak lensing of gravitational waves at $z<0.5$, giving a new multi-messenger view of cosmic structure.
  • Black-hole--neutron-star mergers with electromagnetic counterparts are the most promising Advanced-LIGO source: their cross-correlation with galaxy overdensity should exceed $3\sigma$ at $z>0.35$ after roughly five years.
  • For LISA, the same cross-correlation estimator gives high signal-to-noise measurements across supermassive black hole masses $10^4$--$10^7\,M_\odot$, extending gravity tests to large redshift.
  • A measured $\kappa_{\rm gw}$ would probe the growth of the gravitational potential and the gravitational distribution of dark matter, and comparing the gravitational-wave and galaxy lensing signals tests whether gravitational waves propagate exactly as general relativity predicts.

Reading between the lines

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

  • The paper does not quantify what happens if photometric redshift errors correlate with environment; a simulation with realistic magnification and source--lens clustering could test whether the claimed unbiased estimator survives, since the lensing signal is only about $10^{-3}$ while per-source redshift error is an order of magnitude larger.
  • If modified gravity changes the distance--redshift relation, the 'true' distance used in $\hat{D}_L$ is itself theory-dependent; recovering $\kappa_{\rm gw}$ and testing gravity would then require fitting cosmological parameters jointly with the lensing signal rather than fixing them to the standard model.
  • The same cross-correlation approach can be pointed at CMB lensing or 21-cm intensity maps at $z>3$, where galaxy surveys lose sensitivity; the paper gestures toward CMB lensing, and this would be a direct extension of the forecast.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a new observational probe: measuring the weak lensing convergence of gravitational waves (GWs) by cross-correlating GW-derived luminosity distances with galaxy density and galaxy-lensing fields. The authors derive analytic estimators for the cross-spectra C^{κgwκg}_l and C^{κgwδ}_l, including separate treatments for GW sources with and without electromagnetic counterparts, and forecast signal-to-noise ratios for Advanced LIGO/Virgo, LISA, and various galaxy surveys. Their central forecast is that black-hole--neutron-star mergers with EM counterparts, cross-correlated with galaxy overdensity, can yield a >3σ detection at z>0.35 within about five years, and that the method tests general relativity and modified-gravity theories by probing GW propagation and the relation between metric perturbations and matter.

Significance. If the estimator is unbiased as claimed, this is a genuinely novel and timely multi-messenger cosmological probe. The analytic signal equations are standard and internally consistent, and the forecasts are concrete, falsifiable predictions rather than post-hoc fits; the BH-NS result is a specific, testable claim for the LIGO/Virgo era. The paper also usefully lays out the combined covariance of GW lensing, galaxy clustering, and galaxy lensing. However, the central gravity-testing framing is not backed by quantitative forecasts for any modified-gravity model, and the unbiasedness of the estimator rests on an unverified correlation assumption that is load-bearing because the signal is orders of magnitude smaller than per-source distance errors. The paper's value as a forecast is solid, but its stronger claim to 'probe the theory of gravity' is not yet established.

major comments (3)
  1. [Sec. 4.2.1, Eq. (17)] The unbiasedness of the EM-counterpart estimator is asserted rather than demonstrated. The estimator is unbiased only if the correlators ⟨ϵ_s κ_g⟩ and ⟨ϵ_s δ⟩ vanish, where ϵ_s is the fractional error in the 'true' luminosity distance computed from the EM redshift and best-fit cosmological parameters. This is load-bearing because the lensing signal is at the few×10^-3 level at z~0.5, while ϵ_s from a 3% photometric redshift is an order of magnitude larger per source and is suppressed only by 1/√N_gw. Realistic photo-z errors can correlate with environment through magnification bias or source-lens clustering, and modified-gravity deviations in the distance--redshift relation would appear in ϵ_s. The authors provide no derivation, simulation, or bound on residual ⟨ϵ_s δ⟩ and ⟨ϵ_s κ_g⟩; without such a bound, the claimed unbiasedness and the resulting SNR forecasts are not fully supported.
  2. [Sec. 3 and Eq. (16)] The paper's central claim to 'probe the theory of gravity' is not operationalized. The 'true' luminosity distance in Eq. (16) is defined using the fiducial ΛCDM distance–redshift relation, so any modified-gravity effect on d_L(z) is absorbed by construction into ϵ_s and removed from the estimator. The cross-correlation signal is therefore sensitive only to modifications in the lensing kernels and the Poisson/GR relation, not to the distance-redshift modifications listed in Sec. 3. The forecasts in Sec. 5 are computed entirely from GR + ΛCDM inputs with no modified-gravity model, no predicted signal difference, and no distinguishability criterion. The title and abstract oversell what is currently demonstrated; the paper needs either a quantitative modified-gravity forecast or a careful statement of which gravity modifications the proposed estimator can actually constrain.
  3. [Sec. 5, Eq. (21)] The noise model in Eq. (21) treats σ_b, the error from 'uncertain values of cosmological parameters,' as an additional variance term, but a common shift in cosmological parameters induces a correlated error in d_L^est across all sources. Such a correlated error enters ϵ_s and will not average down like independent noise; depending on how the source redshift distribution overlaps the galaxy survey, it can bias C^{κgwδ}_l coherently rather than merely inflating its variance. The same issue affects the no-EM-counterpart estimator in Eq. (18), where the multiplicative factor ⟨1/(1+ϵ_s)⟩ is assumed removable once the GW source redshift distribution N_gw(z) is known from clustering, but clustering redshifts do not remove a cosmology-dependent or environment-correlated ϵ_s. The paper should quantify this correlated-bias contribution to the SNR or explicitly justify why it is subdominant.
minor comments (5)
  1. [Abstract] The sentence 'The cross-correlation ... probe theories of gravity' has a subject-verb agreement issue; also, the abstract says 'within 10 years' while the BH-NS forecast in Sec. 5 is quoted for five years, so the time-line wording should be made consistent.
  2. [Sec. 4.2.1, Eq. (17)] The notation ⟨·⟩ in Eq. (17) is used for an average over source–galaxy pairs, but the distinction between ensemble average and survey average is not defined; please specify how the estimator is computed in practice over a finite sky area and finite N_gw.
  3. [Fig. 4 caption] The caption says the detection threshold for NS-NS at z>0.2 'is going to be less than 10−σ'; this phrase appears garbled and should be rewritten to state whether the events are below the detection threshold or the SNR is below 10.
  4. [Eq. (21)] The parameter θ_min is described as the sky localization area, but it appears in the exponential as an angle; please clarify whether θ_min is an angular radius or a solid angle and give its units consistently.
  5. [Sec. 5] The text states C=0 for spectroscopic redshifts for some EM-counterpart cases but later uses photometric redshifts for LISA sources; please state explicitly which source classes are assigned C=0 and which use C=0.03, and discuss the implication for the claimed unbiasedness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a forward-model SNR forecast computed from GR+ΛCDM inputs, and the self-citations are auxiliary and non-load-bearing.

full rationale

The paper is a forward-modeling forecast. The theoretical signals C^{κgwκg} and C^{κgwδ} in Eq. (10) are computed from the standard GR lensing kernels, a linear galaxy bias bg=1.6, and the CLASS nonlinear matter power spectrum; no parameter is fitted to the quantity being predicted. The SNR estimates in Figs. 4–7 use detector noise curves, assumed event rates, and assumed redshift errors, with sensitivity reported as a function of these inputs. The estimator in Eqs. (15)–(17) defines a statistic from the GW distance and an EM-redshift-based 'true' distance, but the predicted C_l is not recovered from data in this paper, so the forecast is not forced by construction. The main caveat is methodological rather than circular: Sec. 4.2.1 states that ϵ_s is uncorrelated with δ and κ_gw, but this is asserted, not derived; if redshift or cosmological-parameter errors correlate with environment, or if modified gravity changes d_L(z) relative to the fiducial ΛCDM relation, the proposed estimator could be biased at the signal level. That is a validity risk for the advertised 'test of gravity,' not a circular derivation. The self-citations (Mukherjee & Wandelt 2018; Mukherjee & Silk 2019; Mukherjee et al. 2019) are auxiliary: clustering redshifts are invoked for EM-counterpart-free sources, but the main BH-NS forecast explicitly uses EM counterparts, and the BH-BH forecast treats redshift as unknown. No load-bearing self-citation chain or imported uniqueness theorem is present.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or fields. Its forecast rests on standard cosmological inputs plus a set of user-chosen event rates, localization errors, and survey parameters. The most fragile input is not a number but the statistical assertion that redshift and cosmological-distance errors are uncorrelated with the density field; this is treated as an assumption rather than a derived or simulated property.

free parameters (7)
  • GW event rate, NS-NS = range; mean and upper/lower in Figs. 4-5
    Number of NS-NS mergers per year is a free parameter; SNR scales roughly with N_gw^1/2. The detection claim depends on assumed rates.
  • GW event rate, BH-NS = range; mean and upper/lower in Figs. 4-5
    Most promising source for detection; the central C^{κgwδ}_l forecast with z>0.35 uses the assumed BH-NS rate.
  • GW event rate, BH-BH = range; mean and upper/lower in Figs. 4-5
    Stellar-mass BH-BH rate affects the marginal detection forecasts for sources without EM counterparts.
  • LISA event rate per unit redshift = 50, 100, and 300 per unit redshift
    Three scenarios for supermassive BH-BH mergers; SNR forecasts depend on these choices.
  • Sky localization error θ_min = 10 sq deg (ground, no EM), <1 arcsec (with EM), 0.2-1 deg (LISA)
    Enter N^DD_l via exp(l^2 θ_min^2/(8 ln 2)); determines the effective l_max and dominates the BH-BH forecasts.
  • Galaxy bias b_g = 1.6
    Adopted from previous LSS analyses; C^{κgwδ}_l scales linearly with b_g and no marginalization is performed.
  • Redshift error coefficient C = 0.03 (photometric) or 0 (spectroscopic)
    σ_z = C(1+z) enters σ_s and N^DD_l; for BH-NS with spectroscopic counterparts, C=0 is used.
assumptions (5)
  • domain assumption Gravitational waves propagate on the perturbed FLRW metric in the geometric optics limit, with lensing convergence given by Eq. 6.
    The forecast signal assumes GR and LCDM; the same equations are what the method aims to test, so systematic deviations are not included in the null forecast.
  • domain assumption Galaxy density is a linear, scale-independent, biased tracer of dark matter: δ_g(k) = b_g δ_DM(k) with b_g = 1.6.
    Used in Eq. 10 for C^{κgwδ}_l; non-linear bias or scale dependence would change the signal amplitude.
  • ad hoc to paper The error ϵ_s in the estimated true luminosity distance is zero-mean and uncorrelated with galaxy density and lensing fields.
    Asserted in Sec. 4.2.1, Eq. 17; required for the unbiased estimator and not demonstrated by simulation.
  • standard math The Limber approximation k = (l+1/2)/χ(z) is valid for these cross-spectra.
    Used in Eq. 10; standard for broad redshift kernels but approximate at low l.
  • domain assumption For sources without EM counterparts, the redshift distribution N_gw(z) can be recovered from clustering with galaxies.
    Invoked in Sec. 4.2.2 and in the conclusions; depends on large samples and on bias modeling of GW sources.

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Cite this review

Pith. "Pith review of Probing the theory of gravity with gravitational lensing of gravitational waves and galaxy surveys." pith.science (2026). https://pith.science/paper/HJHKW4SS

@misc{pith2026190808951,
  author       = {Pith},
  title        = {Pith review of: Probing the theory of gravity with gravitational lensing of gravitational waves and galaxy surveys},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJHKW4SS}},
  note         = {Machine review of arXiv:1908.08951}
}
abstract

The cross-correlation of gravitational wave strain with upcoming galaxy surveys probe theories of gravity in a new way. This method enables testing the theory of gravity by combining the effects from both gravitational lensing of gravitational waves and the propagation of gravitational waves in spacetime. We find that within 10 years, the combination of the Advanced-LIGO and VIRGO detector networks with planned galaxy surveys should detect weak gravitational lensing of gravitational waves in the low redshift Universe ($z<0.5$). With the next generation gravitational wave experiments such as Voyager, LISA, Cosmic-Explorer and Einstein Telescope, we can extend this test of the theory of gravity to larger redshifts by exploiting the synergies between electromagnetic wave and gravitational wave probes.

Figures

Figures reproduced from arXiv: 1908.08951 by the authors.

Figure 1
Figure 1. We show the correlation for (a) the LIGO sources and (b) the LISA sources between κgw with κg and δ for different tomographic redshift bins indicated by the zbin index given by (zgw, zg/κg ). The redshift range is taken from [0, 1] and [0, 3] for the LIGO and LISA sources with the bin width ∆z = 0.2 and ∆z = 0.5 respectively. For the galaxy survey, we have taken the redshift distribution for (a) Model-I and (b) Mode… view at source ↗
Figure 2
Figure 2. We show the normalized distribution of the galaxy surveys for Model-I and Model-II given in Eq. 12. We have used the case with Model-I and z0 = 0.2 for the cross-correlation with the NS-NS and BH-NS sources which can be observed by LIGO. For the BH-BH sources from LIGO, we have used Model-I with z0 = 0.5. For the LISA gravitational wave sources, we have studied the cross-correlation signal for the galaxy distributio… view at source ↗
Figure 3
Figure 3. We plot the gravitational wave strain noise for Advanced-LIGO and LISA. The Advanced-LIGO noise is from the latest projected noise. The LISA noise is plotted for one year of integration time with the LISA instrument specifications given in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The cumulative SNR for the measurement of the cross-correlation between gravitational wave lensing and galaxy lensing C κgwκg l for an observation time of 5 years with the instrument noise of Advanced-LIGO. (a) For the NS-NS, the detection threshold of the gravitationa…
Figure 5
Figure 5. Figure 5: The cumulative SNR for the measurement of the cross-correlation between gravitational wave lensing and galaxy field C κgwδ l for an observation time of 5 years with the instrument noise of Advanced-LIGO. (a) For the NS-NS, the detection threshold of the gravitational w…
Figure 6
Figure 6. Figure 6: The cumulative SNR for the measurement of the cross-correlation between the gravitational wave lensing and galaxy lensing C κgwκg l from LISA for an observation time of 4 years for equal mass binary black holes with individual masses (a) 105 , (b) 106 and (c) 107 . The…
Figure 7
Figure 7. Figure 7: The cumulative SNR for the measurement of the cross-correlation between the gravitational wave lensing and galaxy field C κgwδ l from LISA for an observation time of 4 years for equal mass binary black holes with individual masses (a) 105 , (b) 106 and (c) 107 . The re…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.