Lensing-Driven Early Warning

• Lensing-driven early-warning is a method that uses strong gravitational lensing time delays to predict subsequent gravitational-wave images with advance notice.
• The approach relies on detailed host-galaxy imaging and lens modeling to convert Fermat potential differences into precise delay forecasts.
• This technique offers lead times from hours to days, providing targeted optical localization and improved prioritization for sub-threshold gravitational-wave searches.

Lensing-driven early-warning is the use of strong-gravitational-lens time delays to forecast the later arrival of additional images of a transient after an earlier image has already been observed. In the current arXiv literature, the concept is developed primarily for strongly lensed gravitational-wave events, especially binary neutron star and neutron star–black-hole mergers with electromagnetic counterparts, and for the subsequent recovery of additional lensed gravitational-wave images that may be sub-threshold in detector data. The central idea is that lensing converts an observed event into a predictable “action replay”: once the lens geometry is constrained, the remaining inter-image delay can provide advance warning on timescales of $\mathcal{O}({\rm hours})$ to $\mathcal{O}({\rm days})$, rather than the $\sim 10$ s to $\sim 30$ s scale associated with standard inspiral-only early-warning in O4 and O5, with the additional advantage that localization can become effectively optical rather than GW-only (Magare et al., 2023, Magare et al., 9 Sep 2025).

1. Scope and operational meaning

Two distinct but related meanings of lensing-driven early-warning appear in the literature. The first is genuinely pre-merger for a later lensed image: after one lensed GW image and its electromagnetic counterpart are observed, the host galaxy can be identified as a strongly lensed optical system, modeled, and used to predict when the next GW image will merge. This is the route emphasized for BNS and NSBH multimessenger astronomy, where the intended payoff is advance scheduling of observations of pre-merger, merger, and immediate post-merger emission (Magare et al., 2023, Magare et al., 9 Sep 2025).

The second meaning is post-detection forecasting within GW data analysis. Here the merger has already occurred in the source frame, and one or more super-threshold lensed GW images have already been detected. The objective is then to predict when an additional image should appear in the detector data, or whether it may have appeared earlier, so that targeted sub-threshold searches can be narrowed, re-ranked, or partially vetoed. This route is not a replacement for inspiral-based warning; it is a lensing-conditioned alert and follow-up framework for subsequent image arrivals (Ng et al., 2024).

A foundational distinction follows from this split. Lensing-driven early-warning does not derive its lead time from the detector’s low-frequency sensitivity to an incoming chirp. Its lead time is the remaining astrophysical lensing time delay between images. This is why the literature repeatedly contrasts it with standard GW early-warning, which yields only about $\sim 10$ s in O4, $\sim 30$ s in O5, and $\sim 1$ min in Voyager for BNSs, often with coarse sky localization, whereas the lensing route targets $\mathcal{O}(\mathrm{hours})$ to $\mathcal{O}(\mathrm{days})$ and, once a host is identified, sub-arcsecond pointing (Magare et al., 2023, Magare et al., 9 Sep 2025).

2. Strong-lensing basis and timing formalism

The physical basis is the geometric-optics limit of strong lensing. A compact-binary signal can be split into multiple images with identical intrinsic frequency evolution but different arrival times, magnifications, and lensing-induced phase shifts. One formulation used for the lensed GW waveform is

$$\tilde{h}_L^j (f; \vec{\theta}, \vec{\Lambda}_j) = \sqrt{\mu_j}\,e^{(2\pi i f t_j - i \pi n_j \,\mathrm{sign}(f))}\, \tilde{h}_U (f; \vec{\theta})\,,$$

where $\mathcal{O}({\rm days})$0 is the magnification, $\mathcal{O}({\rm days})$1 the excess time delay, and $\mathcal{O}({\rm days})$2 the Morse factor of image $\mathcal{O}({\rm days})$3 (Ng et al., 2024). A related convention writes image types I, II, and III in terms of Morse indices $\mathcal{O}({\rm days})$4 rather than $\mathcal{O}({\rm days})$5; the underlying content is the same, but the phase-index normalization differs across papers (Lo et al., 2021).

For host-galaxy modeling, the arrival-time structure is written in the standard Fermat-potential form. With lens potential $\mathcal{O}({\rm days})$6, source position $\mathcal{O}({\rm days})$7, and image position $\mathcal{O}({\rm days})$8,

$\mathcal{O}({\rm days})$9

These expressions encode the geometric and Shapiro contributions to the delay and are the direct bridge between optical lens modeling and forecasted arrival times of later GW images (Magare et al., 9 Sep 2025).

The literature on galaxy-lensed GW images further exploits image ordering. For galaxy lenses, the expected observable ordering is approximately I–I–II–II, with a central type-III image usually too demagnified to matter. Typical galaxy-lens delays are days to months. This ordering information, together with measured relative delays and phase information, becomes an important source of predictive power once multiple images have been observed (Ng et al., 2024).

3. Host-galaxy route to predictive replay

The host-galaxy route begins after a first GW image has been detected and an electromagnetic counterpart has identified the host galaxy. Because the compact binary sits in that host, the host itself should also appear as a strongly lensed optical system. The forecasting workflow is then: identify multiple host images, measure their angular positions, obtain the source and lens redshifts, infer the lens mass distribution and source position under a parametric lens model, and convert the inferred Fermat-potential differences into a predicted time delay for the next GW image (Magare et al., 2023).

The initial formulation used a singular isothermal ellipsoid lens model and emphasized two practical scenarios. In the first, one lensed CBC image with electromagnetic counterpart is observed and at least two images of the lensed host galaxy are identified; their positions, plus $\sim 10$0 and $\sim 10$1, are used under the SIE model to infer the lens parameters and source position, then predict the time delays to future images. In the second, if two CBC/EM images have already been observed in a quad and at least three host images are known, the measured delay between the first two CBC images calibrates the time-delay distance directly, so future delays can be forecast without requiring source/lens redshifts or an assumed cosmology (Magare et al., 2023).

A later implementation turned this idea into an explicit imaging-and-modeling pipeline. It assumes a galaxy-scale lens modeled as a Singular Isothermal Ellipsoid plus external shear, with the foreground lens light following a de Vaucouleurs profile and the source/host galaxy light following a Sérsic profile with $\sim 10$2. Sixteen mock galaxy-scale lens systems were constructed: 4 doubles and 12 quads, with the quads divided into fold, cusp, and cross configurations, four examples each. Each system was rendered twice, once as an HST-like image and once as an HSC/Subaru-like image, using glafic for the image simulation (Magare et al., 9 Sep 2025).

The instrumental assumptions in that study are explicit. The HST mocks use a pixel scale of $\sim 10$3 per pixel and a Gaussian PSF with standard deviation $\sim 10$4, whereas the HSC mocks use $\sim 10$5 per pixel and a Gaussian PSF with standard deviation $\sim 10$6. Background noise is Gaussian with standard deviation 0.05 for HST and 0.12 for HSC, Poisson noise is added assuming an exposure time of 2100 s, and the bright lens galaxy is subtracted with imcascade, which represents the galaxy light profile as a Multi-Gaussian Expansion / Gaussian Mixture Model. After subtraction, the residual arcs are fit with an SIE + external shear mass model and a Sérsic source, using Differential Evolution rather than MCMC for speed; the DE algorithm is run 600 times with different seeds for each lens, and the ensemble of best fits is used to estimate medians and $\sim 10$7 confidence intervals on parameters and derived time delays (Magare et al., 9 Sep 2025).

This host-based route is what gives lensing-driven early-warning its most distinctive observational feature. Once a host image is identified, the location of the replay is no longer a tens-to-hundreds of square degrees GW skymap but effectively the known position of another host image, i.e. sub-arcsecond. The literature therefore frames the method not merely as a timing forecast but as a high-precision, pre-pointed replay forecast for multimessenger campaigns (Magare et al., 2023).

4. Forecast accuracy, morphology dependence, and imaging requirements

Forecast quality depends strongly on image multiplicity, morphology, and angular resolution. In the initial forecasting study, the headline result was that with angular resolution of $\sim 10$8, one can predict time delays of $\sim 10$9 day with $\sim 30$0 error-bar of $\sim 30$1(hours) at best. For one representative O5-detectable lensed BNS system with $\sim 30$2, $\sim 30$3, $\sim 30$4, and $\sim 30$5, absolute errors on predicted delays were typically within a day, relative errors for most source positions spanned roughly $\sim 30$6 to $\sim 30$7, doubles typically had larger delays than quads, and some doubles reached delays up to 36 hr. A separate scan over redshift showed that for sources out to $\sim 30$8 and lenses out to $\sim 30$9, the $\sim 10$0 confidence-interval width on predicted delays can exceed 2 weeks even with 50 mas resolution (Magare et al., 2023).

The later optical-imaging study sharpened these conclusions. Its central quantitative claim is that quads tend to provide accurate time-delay predictions with typical relative error $\sim 10$1 in HST-like imaging. HST quads are generally more accurate and less biased than HSC-like quads because HST better recovers the lens velocity dispersion and source position, which enter directly into the Fermat potential and hence $\sim 10$2. By contrast, HSC quads can be significantly biased, sometimes with the true delay outside the inferred $\sim 10$3 interval for some image pairs (Magare et al., 9 Sep 2025).

Morphology matters even within quads. Cross configurations show the largest bias in predicted time delays because the images lie closer to the lens center, are less well resolved, and are more contaminated by lens light. Cusp systems are the most favorable for ground-based data: they have smaller relative errors when using HSC images and can be a better candidate for getting accurate time delays when using ground-based telescopes. Fold systems are intermediate. Doubles behave differently again: their relative errors are comparable between HST and HSC, and no bias is noted in either, because their images are usually farther from the lens center and better separated from lens light. That said, the main $\sim 10$4 relative-error result applies to quads with HST, not to doubles in general (Magare et al., 9 Sep 2025).

The real-data test underscores the operational stakes. For the lens HSCJ230335+003703, observed by both HST and HSC, the maximum difference between HST-based and HSC-based inferred time delays is $\sim 10$5 days. The authors attribute this primarily to the poorer resolution of HSC relative to HST, while also noting that imperfect lens-light subtraction and configuration details likely contribute. A plausible implication is that seeing-limited imaging alone may be inadequate when the next GW image is expected on a similar timescale (Magare et al., 9 Sep 2025).

The computational side is nontrivial but not presented as prohibitive. A single DE run on one CPU core takes a wall-clock time of $\sim 10$6–$\sim 10$7 hours for the 12-parameter model, whereas MCMC convergence would be of order $\sim 10$8 hours. The same work nonetheless concludes: “We are able to predict time delays in a few hours if optical imaging of the lensed host galaxy is available.” This suggests that the intended operational regime is rapid post-identification imaging plus immediate modeling, rather than fully general real-time lens inference from arbitrary survey data (Magare et al., 9 Sep 2025).

5. Forecasting missing GW images and reprioritizing sub-threshold searches

A distinct line of work addresses the case in which two or three lensed GW images are already detected at super-threshold significance and additional images may exist below threshold. Here the objective is not optical lens reconstruction but event-conditioned prediction of the missing image’s arrival time, so that targeted searches can be narrowed and reprioritized. The key quantity is the predictive distribution for the time delay of a sub-threshold counterpart relative to the first detected super-threshold image:

$\sim 10$9

with $\sim 30$0 denoting relative magnification, relative delay, and relative Morse factor between images. In practice, the analysis approximates the super-threshold arrival times as known exactly and fixes relative magnifications to their median, reducing the forecasting problem to $\sim 30$1 (Ng et al., 2024).

Operationally, the pipeline starts from joint parameter estimation on the super-threshold set, using GOLUM to infer relative arrival times, relative magnifications, and Morse-factor differences. It then enumerates allowed image-order sub-cases consistent with those Morse differences, filters a large simulated lens catalog generated by LeR, constructs a Gaussian KDE for the missing image’s time delay under each allowed sub-case, and combines the sub-case KDEs into a final predictive distribution under the lensed hypothesis. The ranking ratio is verbally defined as the lensed-hypothesis timing probability divided by the unlensed-hypothesis timing probability, and this ratio is multiplied by an initial chirp-mass posterior-overlap statistic $\sim 30$2 to produce the final ranking used for follow-up (Ng et al., 2024).

The simulation assumptions are specific: the catalog contains $\sim 30$3 lensed systems, uses an elliptical power-law lens with shear, a PowerLaw+Peak BBH source population from O3, source redshift distribution from Oguri (2018), and an HLV network at expected O4 sensitivity. Super-threshold images satisfy network SNR $\sim 30$4 and sub-threshold images satisfy $\sim 30$5. A central qualitative result is that conditioning on already observed images can compress the forecast for the missing image from a broad year-scale window to about a week-scale window in an example figure (Ng et al., 2024).

The reported gains are substantial but differ sharply between doubles and triples. For doubles, the average trigger-list reduction is only 2 triggers, but the average genuine counterpart rank improves from 66 to 5, i.e. from top 48.5% to top 3.7%. For triples, the average genuine counterpart rank improves from 82 to 9, while the average trigger list shrinks by a factor 2.6, from 133 to 52 triggers, i.e. from top 61.6% to top 17.3%. Overall, the average counterpart rank improves from 75 out of 133 to 7 out of 122, and the average trigger-list reduction factor is 1.2 overall, with stronger than halving when three super-threshold images are available (Ng et al., 2024).

This route is best understood as forecast-driven follow-up rather than pre-coalescence warning in the ordinary sense. It does not warn from inspiral buildup before the first observed merger image. Instead, it forecasts the next or missing copy of an already observed event and uses that forecast to focus searches and human attention. In that narrower but operationally important sense, it is a direct implementation of lensing-driven early-warning (Ng et al., 2024).

6. Bayesian identification backbone and localization gains

A more general Bayesian framework formalizes the hypothesis test underlying all such systems: whether a set of observed GW events is better explained as multiple strongly lensed images of the same source or as independent mergers. The framework compares $\sim 30$6, the shared-source lensed hypothesis, with $\sim 30$7, the independent-source hypothesis, through posterior odds

$\sim 30$8

and decomposes the Bayes factor into a coherence term and selection effects. The common parameters include detector-frame intrinsic source parameters and sky location, while image-specific parameters include arrival times, apparent luminosity distances, and image type. This is the statistical structure needed to decide whether a newly observed event is another image of a previously seen source (Lo et al., 2021).

Selection effects are central in this formulation. Under the unlensed hypothesis the detectable fraction is $\sim 30$9, while under the lensed hypothesis the fraction of sources that produce $\sim 1$0 detectable lensed images is $\sim 1$1. The framework emphasizes that the normalized Bayes factor must include these terms, and that selection effects typically penalize the lensed hypothesis by roughly a factor $\sim 1$2. This prevents accidental coherence from being overinterpreted as lensing evidence and is especially relevant when one contemplates low-threshold searches over large candidate pools (Lo et al., 2021).

The same work proposes a two-step hierarchical analysis that isolates the source redshift as a one-dimensional hyperparameter, exploiting the degeneracy

$\sim 1$3

This separation permits more efficient joint inference of source parameters free from bias introduced by lensing. The framework is explicitly described as foundational for any operational early-warning system, but it does not itself provide a full prospective forecasting module that, after the first image alone, returns a calibrated prediction for the arrival time and SNR of future images (Lo et al., 2021).

An important practical consequence is sky-localization improvement once multiple images are associated. In one example, the $\sim 1$4 credible region shrinks to $\sim 1$5 for the joint analysis, versus $\sim 1$6 for the brighter image alone and $\sim 1$7 for the fainter image alone. This matters because lensing-driven early-warning is not only about predicting when another image will arrive, but also about reducing the search area for the lens and host once repeated images have been recognized (Lo et al., 2021).

The same framework also clarifies a key limitation. In the $\sim 1$8 case, geometric-optics lensing is weakly informative unless the astrophysical source-population prior is highly constraining or the waveform contains distinctive higher-mode or precessional information that helps identify image type. This suggests that first-image-only warning is intrinsically more fragile than warning based on either a lensed host-galaxy reconstruction or on already identified multiple GW images (Lo et al., 2021).

7. Limits, misconceptions, and regime of applicability

The current literature presents lensing-driven early-warning as physically sound and practically plausible under specific conditions, not as a universal replacement for standard early-warning. The host-galaxy route assumes that the event has already been recognized as lensed to the extent that an electromagnetic counterpart and lensed host galaxy have been identified; it does not solve the separate problem of identifying lensed GW events in real time. Its mass model is relatively idealized—SIE plus external shear with smooth Sérsic/de Vaucouleurs light profiles—and explicitly omits line-of-sight perturbers, mass-sheet-like degeneracies, substructure, and microlensing. Lens-light subtraction is a documented failure mode, especially for cross configurations where the images lie close to the lens center (Magare et al., 9 Sep 2025).

The sub-threshold GW-search route has a different set of restrictions. It treats galaxy lenses only, not clusters; it assumes the galaxy-lens ordering I–I–II–II; it neglects the central type-III image as usually too demagnified to matter; and its demonstration does not use actual sub-threshold pipeline outputs or false alarm rates. Relative magnifications are included but treated as less informative because their measurement errors are larger. The gains it reports are therefore best read as proof of principle for timing-informed reprioritization rather than end-to-end search performance in non-Gaussian detector data (Ng et al., 2024).

The population-level forecasting work adds another practical caveat: the observationally most useful “Scenario 2,” in which multiple detected CBC images directly calibrate the time-delay distance, is expected to be rare in O4 and O5. In near-term networks, detected lensed BNS/NSBH events are expected to be predominantly quads in which only two images are loud enough to be seen. This is why most near-term forecasts rely on Scenario 1, with host-image astrometry and redshift information, rather than on repeated CBC timing alone (Magare et al., 2023).

Several common misconceptions are therefore explicitly contradicted by the literature. Lensing-driven early-warning is not standard inspiral early-warning under another name; the warning time is set by the astrophysical lensing delay, not by low-frequency detector reach. It does not apply to generic GW events, but to the rare subset that are strongly lensed and, for the pre-merger host-based version, also produce detectable electromagnetic counterparts. Nor does the first detected image usually suffice for a robust forecast: in geometric optics, the $\sim 1$9 case is generally weak unless the source-population prior is unusually constraining (Magare et al., 2023, Lo et al., 2021).

Within those limits, the field has converged on a precise operational claim. When the host galaxy is imaged at high resolution comparable to HST or adaptive-optics quality, and the configuration is favorable—most notably in quads rather than central crosses—day-scale lensing delays can be turned into actionable warnings with uncertainty on the scale of hours, while two or three already detected GW images can be used to narrow the search for missing counterparts and reprioritize follow-up substantially. This suggests that lensing-driven early-warning is best understood as a specialized replay-forecasting regime of multimessenger astronomy: rare, technically demanding, and strongly conditioned on imaging quality and lens identification, but uniquely capable of combining hour-to-day lead times with precise pointing (Magare et al., 9 Sep 2025, Ng et al., 2024).