⚛️ general relativity

Multi-messenger lensing time delay as a probe of the graviton mass

This paper demonstrates that a single strongly lensed multi-messenger event can constrain the graviton mass to $m < 3 \cdot 10^{-23}$ eV/c $^{2}$ using time-delay measurements, offering a model-independent test that significantly outperforms constraints derived from image magnification.

Original authors: Elena Colangeli, Charles Dalang, Tessa Baker

Published 2026-02-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Elena Colangeli, Charles Dalang, Tessa Baker

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean. Usually, we think of ripples in this ocean (gravitational waves) and flashes of light (electromagnetic waves) as traveling at exactly the same speed, like two identical runners in a race. But what if the gravitational waves were actually slightly heavier than we thought? What if they had a tiny bit of "mass"?

This paper explores a scenario where gravitational waves have a small mass. If they do, they can't travel at the absolute speed limit of the universe (the speed of light). Instead, they would be slightly slower, like a runner carrying a heavy backpack.

Here is how the authors propose we can catch this "heavy runner" in the act, using a cosmic trick called gravitational lensing.

The Cosmic Funhouse Mirror

Imagine a massive galaxy cluster sitting between us and a distant event, like two black holes smashing together. This cluster acts like a giant, warped glass lens. Just as a funhouse mirror bends light to create multiple images of a single object, this galaxy cluster bends the path of the gravitational waves and the light from the same event.

Because the paths are bent, the signals arrive at Earth at different times. Sometimes, you might see two images of the same explosion: one arriving a few days after the other. This gap in time is called a time delay.

The Race with a Twist

The authors realized that if gravitational waves have mass, two things happen that are different from light:

They run slower: Because they have mass, they travel slightly slower than light.
They take a different route: Because they are slower, the "gravity lens" bends their path slightly differently than it bends the path of light.

Usually, scientists have to build complex computer models to guess how the galaxy lens is shaped and exactly where the source is to calculate these delays. It's like trying to guess the winner of a race without knowing the track layout or the runners' starting positions.

The Magic Cancellation

Here is the paper's big discovery: The two differences cancel each other out perfectly.

Think of it this way:

The "heavy" gravitational waves take a slightly longer, more winding path around the galaxy (like a runner taking a scenic detour). This usually makes them arrive later.
However, because they take this wider path, they spend less time passing close to the heavy gravity of the galaxy itself. This makes them arrive earlier.

The authors show that these two effects (the longer path vs. the less time near gravity) perfectly balance each other out. The only thing that remains is the fact that the gravitational waves are simply slower than the light.

The "Golden" Test

This leads to a incredibly simple test. If we catch a "Golden Event"—a moment where we see both the light and the gravitational waves from a lensed explosion—we can compare their arrival times.

The Light: Arrives at time $T_{light}$ .
The Gravity: Arrives at time $T_{gravity}$ .

Because the messy details of the galaxy's shape and the universe's expansion cancel out in the comparison, we don't need to know any of those details. We just look at the difference between the two arrival times. If the gravitational waves are even a tiny bit late compared to the light, we can calculate exactly how much mass the "graviton" (the particle of gravity) must have to cause that delay.

The Results

The paper calculates that if we observe just one of these rare, lensed events, we could prove that the graviton is incredibly light—lighter than $3 \times 10^{-23}$ electron volts.

They also looked at another method: comparing how much the galaxy "magnifies" (brightens) the light versus the gravity waves. They found this method is much weaker. It's like trying to guess the weight of a backpack by looking at how much a runner's shadow stretches; it's very sensitive to the angle of the sun and the shape of the ground. The time delay method is much more reliable because it doesn't depend on those messy details.

Summary

In short, this paper proposes a clever way to weigh the "particle of gravity" without needing a scale. By watching a cosmic race between light and gravity around a galaxy, and noticing that the gravity runner is just a tiny bit slower, we can prove it has mass. The best part? We don't need to know the details of the track or the weather; the math cancels out all the confusion, leaving us with a clean, direct measurement.

Problem Statement
Gravitational lensing serves as a powerful probe for cosmology and astrophysics. With the anticipated detection of strongly lensed gravitational waves (GWs), there is an emerging opportunity to test fundamental physics, specifically the nature of gravity. While modified gravity models have been explored in the context of unlensed multimessenger signals, fewer studies have addressed GW lensing. This work addresses the specific scenario where the graviton possesses a nonzero mass ( $m$ ). In such a theory, GWs follow timelike geodesics rather than null geodesics, altering their dispersion relation, propagation paths, and interaction with gravitational potentials. The authors aim to derive the standard gravitational lensing equations (scattering angle, time delay, and magnification) under this assumption and determine if a single strongly lensed multimessenger event (a "golden" scenario involving both GWs and electromagnetic (EM) counterparts) can constrain the graviton mass.

Methodology
The authors adopt a phenomenological approach, assuming a massive graviton leads to a modified dispersion relation $k_\mu k^\mu = -m^2$ (where $k$ is the wave vector). They work within the geometric optics approximation and assume a flat Friedmann–Lemaître–Robertson–Walker (FLRW) background with a weak-field Schwarzschild metric for the lens.

Geodesics and Scattering Angle: Starting from the equation of motion for the metric perturbation, the authors derive the geodesic equation. They calculate the scattering angle for a massive graviton passing a point-mass lens in the weak-field limit. They compare this to the massless case, showing that massive gravitons experience a larger deflection angle due to their subluminal group velocity.
Time Delay Derivation: The total time delay between images is decomposed into geometric and Shapiro (gravitational potential) components. The authors derive corrections to both components arising from the graviton mass. They explicitly calculate the time delay for massive GWs ( $\Delta t_g$ ) and compare it to the standard EM time delay ( $\Delta t_\gamma$ ).
Magnification Derivation: To analyze signal amplification, the authors demonstrate that Kirchhoff's diffraction theorem holds even with a massive dispersion relation. They derive the amplification factor ( $F_g$ ) for massive GWs and compare it to the standard EM amplification factor ( $F_\gamma$ ). They then relate these factors to observable magnifications ( $\mu$ ) for both GW luminosity distances and EM fluxes.

Key Contributions and Results

Scattering Angle: The authors show that massive gravitons are deflected more than massless photons. The image positions for GWs ( $\theta_g$ ) differ from EM images ( $\theta_\gamma$ ) by a factor proportional to $m^2/\omega^2$ .
Time Delay Cancellation: A central result is the derivation of the total time delay difference between GW and EM signals. While the massive graviton travels a longer geometric path (due to the larger scattering angle) and experiences a different Shapiro delay, the authors find that these two effects cancel each other out to first order. Consequently, the difference in time delay between the two signals arises solely from the difference in group velocity ( $v_g \approx 1 - m^2/2\omega^2$ $v_{g} \approx 1 - m^{2} /2 ω^{2}$ ).
- The resulting relation is $\Delta t_g = (1 + m^2/2\omega^2) \Delta t_\gamma$ .
- This leads to a fully model-independent constraint on the graviton mass: $m = \frac{\sqrt{2}\hbar\omega}{c^2} \sqrt{\frac{\Delta t_g}{\Delta t_\gamma} - 1}$ .
Constraints from Time Delay: Applying this formula to a "golden" event (a strongly lensed GW with an EM counterpart), the authors estimate the sensitivity. For a signal in the LISA band ( $f \sim 10^{-4}$ Hz) lensed by a galaxy cluster, a single measurement with optimistic EM time-delay errors (0.1 s) yields a constraint of $m < 3 \cdot 10^{-23}$ eV/c $^2$ . This constraint is independent of the lens model, the waveform model, and cosmological parameters.
Constraints from Magnification: The authors derive the magnification relation $\mu_g = (1 - m^2/\omega^2) \mu_\gamma$ $μ_{g} = (1 - m^{2} / ω^{2}) μ_{γ}$ . However, they demonstrate that using magnification to constrain the graviton mass is significantly less effective than using time delays.
- The magnification method requires precise modeling of the lens (to determine $\mu_\gamma$ ) and a specific cosmological model (to determine the unlensed GW distance).
- Current uncertainties in lens modeling (mass-sheet degeneracy) and GW distance measurements (inclination degeneracy) result in errors that make the magnification-based constraint at least three orders of magnitude weaker than the time-delay constraint.
- Furthermore, for extended sources (like galaxies), the difference in source position between the GW and EM emission could spoil the comparison.

Significance
The paper claims that a single strongly lensed multimessenger event provides a unique, fully model-independent probe of the graviton mass. Unlike previous dispersion relation tests that rely on waveform modeling or comparisons of high- vs. low-frequency components within a single GW signal, this method relies only on the arrival time difference between the GW and its EM counterpart.

The authors highlight that this approach complements existing constraints from Solar System tests, binary pulsars, and weak lensing. They note that while lensed GW events are rare (estimated at 1 in ~1500 detections), the upcoming generation of detectors (LISA, Einstein Telescope, Cosmic Explorer) and wide-field surveys (Euclid, LSST) will likely provide the necessary data to observe such "golden" events. The derived bound of $m < 3 \cdot 10^{-23}$ eV/c $^2$ is comparable to current best limits but offers the distinct advantage of being independent of lens and cosmological models. The paper concludes that while magnification offers a theoretical avenue for testing, the time-delay method is the robust and practical tool for future multimessenger lensing observations.