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

Gravitational lensing is a powerful probe of cosmology and astrophysics. With the prospect of the first strongly lensed gravitational waves on the horizon, we highlight an opportunity to test fundamental physics. In this work, we assume a nonzero mass for the graviton, which leads to gravitational waves following timelike geodesics instead of null geodesics. We derive standard gravitational lensing equations, such as the scattering angle, the time-delay between different images and the magnification, which normally rely on the assumption of null geodesics. We show that a single strongly lensed multi-messenger event is enough to constrain the graviton mass to $m< 3 \cdot 10^{-23}$ eV/c$^{2}$. Notably this constraint is independent of the lens model, the waveform model, and of cosmology. Additionally, we explore magnification of images and find that they offer at least three orders of magnitude weaker bounds than the time delay, and have a dependence on the correct modeling of the lens and cosmology.

💡 Research Summary

The paper investigates how a non‑zero graviton mass modifies the physics of strongly lensed multimessenger events and shows that a single such “golden” event can place a model‑independent bound on the graviton mass of $m<3\times10^{-23},\mathrm{eV}/c^{2}$. Starting from a massive wave equation $(\Box-m^{2})h_{\mu\nu}=0$, the authors use the geometric‑optics approximation to derive the dispersion relation $k^{\mu}k_{\mu}=-m^{2}$. This implies that massive gravitons follow timelike, not null, geodesics. Decomposing the wave‑vector as $k^{\mu}=\omega u^{\mu}+k d^{\mu}$ yields $\omega^{2}=k^{2}+m^{2}$, where $\omega$ is the observed frequency.

In a weak‑field Schwarzschild lens with potential $U=-R_{s}/(2r)$, the scattering angle becomes \