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

TL;DR

This paper investigates how a nonzero graviton mass modifies the propagation of gravitational waves in strong lensing, focusing on a golden multimessenger event. Starting from a massive dispersion relation, it derives how geodesics, scattering angles, and time delays between images change, and shows a remarkable cancellation between geometric and Shapiro delays, leaving the time-delay shift controlled by the GW group velocity. The key result is a model-independent bound on the graviton mass from comparing electromagnetic and gravitational-wave time delays: Delta t_g = (1 + m^2/(2 omega^2)) Delta t_gamma, which for LISA-scale masses and cluster lenses yields m < around 3e-23 eV/c^2. Magnification-based constraints, while computable via Kirchhoff diffraction, are weaker and depend on lens and cosmology, making time delays the most robust probe; future detectors could enable such tests with golden lensed events.

Abstract

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.

Figures (4)

• Figure 1: Schematic of the lensing configuration: a signal traveling from a source and encountering a lens travels along a deflected path $d_{ls}+d_{l}$ before reaching the observer. Angular positions of the source $\boldsymbol{\beta}$ and image $\boldsymbol{\theta}$ are shown along with the scattering angle $\boldsymbol{\hat{\alpha}}$. Vectors $\boldsymbol{\eta}$ and $\boldsymbol{\xi}$ represent the physical positions on the source and lens plane respectively, where we highlight the lens plane $E$. The distance $d_s$ is the undeflected path the signal would take in the absence of the lens.
• Figure 2: We show schematically the paths followed by the massive GW (in blue) which differs from the path followed by the massless photon ($\gamma$) in red. This results in different apparent images forming at different angles $\boldsymbol{\theta}_{\rm g}$ and $\boldsymbol{\theta}_\gamma$ for these signals, as found in Eq. \ref{['eq:images_GW_vs_EM']}. The fact that the path for massive gravitons is a little bit longer geometrically turns out to be canceled by the fact that it also experiences less Shapiro time delay. This lucky cancellation results in the time delay between different images to differ only by the different group velocity between photons and massive gravitons as in Eq. \ref{['eq:model-independent delay']}. We also show the source angle $\boldsymbol{\beta}$ which is the angle between the source and the optical axis, which connects the observer with a reference point in the lens.
• Figure 3: Upper bound on the graviton mass as a function of gravitational wave frequency. The solid line corresponds to an optimistic scenario with an error on EM time-delay measurements of $0.1$s, while the dashed line corresponds to an error of $10^5$s. The gray areas above the lines correspond to the values of $m$ excluded at $95\%$ confidence. The area shaded in blue corresponds to the LISA frequency band, while the green area corresponds to ground-based detector bands. Constraints shown are for a galactic cluster lens.
• Figure 4: The Kirchhoff theorem setup. We consider a volume $V$ in blue bounded by a surface $S$ to evaluate the wave amplitude at point $\boldsymbol{x}_o$, which corresponds to the observer. The source is located outside the volume $V$. We also depict a sphere of small radius $\epsilon$ around the observer (the excluded region in our calculation) and the vector $\boldsymbol{n}$, which is normal to the surface $S$. Additionally, though the Kirchhoff theorem is general, we show the positions of a source, observer and lens together with the paths followed by the geometric optics images which form at impact parameters $\boldsymbol{b}_{\rm G}(\boldsymbol{\theta}^{\rm{I}}_{\rm g})$ and $\boldsymbol{b}_{\rm G}(\boldsymbol{\theta}^{\rm{J}}_{\rm g})$ in the lens plane to facilitate the connection between our derivation of the theorem and its use in the context of lensing.