Scattering perspective on gravitational lensing

Abstract

Gravitational waves propagating across gravitational potentials undergo lensing effects that, in the wave-optics regime, manifest as frequency-dependent amplitude and phase modulations. In this work, we revisit the diffraction integral formalism of gravitational lensing and demonstrate that it admits a natural and transparent interpretation within the framework of scattering theory. We establish a direct correspondence between the lensing amplification factor and the scattering amplitude of waves propagating in curved spacetime, clarifying how familiar lensing limits map onto distinct scattering regimes. In particular, we show that the diffraction integral matches exactly the eikonal limit of the scattering amplitude at lowest post-Minkowskian order, after a change in coordinates and the inclusion of finite-distance effects. We further extend the standard formalism by including subleading corrections to the post-Minkowskian and eikonal approximations. Our results provide a unified theoretical framework for the interpretation of lensed gravitational-wave signals and open the way to more accurate waveform modeling for future lensed observations.

Figures (7)

• Figure 1: Schematic illustration of the lensing/scattering event, showing an incoming scalar wave $\phi^0_{\omega, {\rm L}}$ interacting with a gravitational lens (placed at the center of the reference frame). The interaction generates an outgoing wave $\phi_{\omega}$ that eventually reaches an observer located far away.
• Figure 2: Behavior of the modulus of the diffraction integral in terms of the dimensionless frequency $\nu$, including the PM corrections at order $\mathcal{O}(G^2)$. Different colors show distinct values of the observer location $\varphi$ and the Einstein ring $\tilde{\theta}_{\rm E}$. The bottom panel is a close-up of the top panel for $\nu \in (40 , 50)$.
• Figure 3: Percentage relative variation of the absolute value of the diffraction integral, accounting for the leading $\mathcal{O}(G^2)$ PM correction to the phase shift. Left panel: Behavior in terms of the dimensionless impact parameter $\varphi$, changing the strength of the PM correction ${\cal C}|_{G^2} (\tilde{\theta}_{\rm E})$. Right panel: Behavior in terms of the wave coefficient $\nu = 4 G M \omega$, for different values of the PM and impact parameters.
• Figure 4: Same as right panel of Fig. \ref{['fig:beyond-PM']}, but for larger values of the wave coefficient $\nu$. The left, central and right panels correspond to the choices of the impact parameter $\varphi = \{ 0.5, 1.5, 2.5 \}$, respectively. Red and blue curves stand for different choices of the Einstein angle, namely $\tilde{\theta}_{\rm E} = \{ 0.01, 0.001 \}$ in order.
• Figure 5: Percentage variation of the amplitude of the modulus of the amplification factor including only PM corrections (red lines) or only beyond-eikonal (BE) corrections (blue lines), as a function of $\nu$, for fixed $\varphi = 1.5$ and $\tilde{\theta}_E = \{ 0.03, 0.05\}$.