Gravitational-Wave Lensing Tests

Gravitational-wave lensing can repeat a compact-binary signal, impose a Morse phase, or produce frequency-dependent distortions. A confirmed common lens could compare tensor-wave and electromagnetic propagation through the same potential. The difficult first step is association: similar binary parameters are common, detector duty cycles hide images, and a catalog search tests a population as well as a lens model. This frontier status is current through 2026-08-19.

Required background. Geodesics and Jacobi Fields supplies geometric-optics propagation and lens delays; Probability and Statistical Inference supplies association odds, selection effects, and hierarchical likelihoods.

Helpful background. Weak-Field Gravity supplies the lens potential; Relativistic Cosmology supplies distance, source-rate, and lens-population models.

The common-source waveform relation

Adopt $h(t)=\int_0^\infty \tilde h(f)e^{-2\pi i f t}df+\mathrm{c.c.}$, consistent with the site time convention. In geometric optics, image $j$ of one source has

$$\tilde h_j(f)= \lvert\mu_j\rvert^{1/2} \exp\!\left(2\pi i f\Delta t_j-i\pi n_j\right) \tilde h_0(f),$$

where $\mu_j$ is signed magnification in the lens map, $\Delta t_j$ is the delay, and $n_j=0,\tfrac12,1$ denotes a minimum, saddle, or maximum image. The detector response and Earth rotation still differ between arrival times. Intrinsic detector-frame masses and spins, sky location, and unlensed waveform must agree; luminosity distance, coalescence phase, and amplitude are altered or degenerate with lensing.

A multimessenger comparison is especially clean when the same source and multiple images are identified in both channels. The differential quantity

$$\delta(\Delta t)_{AB} =\Delta t^{\rm GW}_{AB}-\Delta t^{\rm EM}_{AB}$$

cancels the unknown absolute emission time and a common intrinsic GW–EM emission lag. Explicitly, write $t_A^X=t_{\rm em}^X+T_A^X$ and $t_B^X=t_{\rm em}^X+T_B^X$ for messenger $X$. Then $\Delta t_{AB}^X=T_A^X-T_B^X$, so the source emission times drop out before the two messengers are compared. For example, two inter-image delays of $10.000$ and $10.002$ days differ by $172.8\,\mathrm{s}$ regardless of an unknown common one-hour source lag. These are illustrative values, not an observed system.

This cancellation requires correct image and source-feature association. Spatially separated emission regions, image-dependent electromagnetic reprocessing or microlensing delays, plasma dispersion, and frequency-dependent GW diffraction can leave residual delays. A residual therefore tests a declared propagation-and-lens model rather than directly giving a universal speed difference. With only one image in each channel, intrinsic emission lag is inseparable from propagation without additional assumptions. Polarization tests also require the detector network response and possible electromagnetic polarization transport; an amplitude ratio alone cannot identify a tensor polarization or a distance law.

Association likelihood and release boundary

For two candidates $d_a,d_b$, lensing odds compare a common-source hypothesis $L$ with unrelated events $U$:

$$\mathcal O_L^U= \frac{p(L)}{p(U)} \frac{\int p(d_a,d_b\mid\lambda,\ell,L) p(\lambda,\ell\mid L)\,d\lambda\,d\ell} {p(d_a\mid U)p(d_b\mid U)}.$$

The source parameters $\lambda$ and lens variables $\ell$ include delay, magnification, Morse type, lens population, and sky position. Prior odds depend on merger rate, optical depth, magnification bias, detector duty cycle, and the search threshold. Catalog-trigger selection and subthreshold-counterpart selection must be normalized; otherwise a high waveform-overlap Bayes factor is mistaken for association probability.

The peer-reviewed full-O3 analysis used calibrated strain, GWTC-3 posteriors, targeted subthreshold searches, pair overlap, joint parameter estimation, Morse-phase tests, and point-lens distortions. It found no compelling lensing evidence within those searches LIGO–Virgo–KAGRA Collaboration 2024, §§II–V and Appendices A–B, with machine-readable products in its 2023 official data release.

At the cutoff, the submitted GWTC-4.0 lensing analysis is the dedicated O4a extension and reports no strongly lensed signal while retaining a point-lens outlier for further study LIGO–Virgo–KAGRA Collaboration 2025, official DCC release P2500419-v10. The draft also discloses a late-discovered likelihood-normalization error in parts of the underlying parameter-estimation work, documents analysis-specific reweighting or reruns, and anticipates further updates LIGO–Virgo–KAGRA Collaboration 2025, §3. Its null and outlier statements must therefore remain tied to version 10 and the stated mitigation. The current catalog boundary is GWTC-5.0, whose official documentation supplies event strain and stable parameter-estimation release 9 GWOSC 2026, official GWTC-5.0 data release documentation. That catalog supersedes earlier event bookkeeping, but it is not itself a GWTC-5 lensing result. Any new search must pin strain and calibration, posterior version, waveform family, power spectrum, event and subthreshold cuts, sampler, lens prior, and injection set.

Results and limitations through 2026-08-19

Evidence or position	Result supported	Limitation or scope
Peer-reviewed full-O3 multi-method search	No compelling strong- or microlensing evidence in its tested catalog and models	O3 sensitivity, waveform systematics, lens populations, and search efficiency bound the null
Submitted O4a GWTC-4.0 lensing search	Extends pair, Morse-phase, subthreshold, and point-lens tests to O4a	Submitted status, disclosed likelihood-normalization correction, and superseded catalog bookkeeping require explicit labeling
GWTC-5.0 strain and stable posterior release 9	Defines the current event-data input for a new catalog analysis	A data catalog alone neither identifies lensed pairs nor tests propagation

The conservative position is that no common-lens association has yet earned a propagation test in these official analyses. A competing interpretation may emphasize a high-overlap pair or distortion outlier, but population coincidence, glitches, waveform mismatch, microlensing, and trials provide evidence against promotion without a lens-consistent counterpart. A null result constrains only the declared event selection, lens mass and redshift distribution, and source population.

The adversarial check injects both true common-lens pairs and unrelated look-alikes through the same detector duty cycle and catalog pipeline. Association odds must be calibrated, and recovered pairs must agree in sky position, intrinsic parameters, delay, Morse phase, magnification, and an independently reconstructed lens. Only then should GW–EM delay or distance residuals be interpreted. Until such an association exists, the propagation conclusion remains unresolved and model-dependent.

What could change the conclusion. The interpretation could change when LVK publishes a superseding dedicated lensing search, revises GWTC-5 strain or posterior products, or demonstrates injection-calibrated common-lens odds with an electromagnetic lens model that materially changes the propagation boundary.

Exercise

Two putative images have magnifications $\mu_A=4$ and $\mu_B=1$ and otherwise share a geometric-optics waveform. What is their strain-amplitude ratio, and why is agreement insufficient to establish lensing?

Solution

Geometric optics multiplies strain by $\lvert\mu\rvert^{1/2}$, so the ratio is $\sqrt{4/1}=2$. However, luminosity distance, inclination, detector antenna response, and calibration also change amplitude. Unrelated binaries can have similar intrinsic parameters, and a true repeated image must additionally satisfy sky, time-delay, phase, lens-population, and selection predictions. Therefore the amplitude relation is a consistency check inside a common-source likelihood, not standalone evidence for lensing.

References

• Gravitational Wave Open Science Center (2026), “GWTC-5.0 Data Release Documentation,” stable candidate, strain, posterior, and sky-localization products. Official release.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2023), “Search for Gravitational-Lensing Signatures in the Full Third Observing Run of the LIGO–Virgo Network,” data release, version 1.0.0, 17 April 2023. Zenodo. DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2024), “Search for Gravitational-Lensing Signatures in the Full Third Observing Run of the LIGO–Virgo Network,” Astrophysical Journal 970, 191. DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025), “GWTC-4.0: Searches for Gravitational-Wave Lensing Signatures,” submitted manuscript P2500419-v10. Official DCC record.