Gravitational lensing beyond the eikonal approximation

TL;DR

This work develops a systematic BE (beyond eikonal) expansion for wave propagation in curved spacetime by solving the massless Klein-Gordon equation with the Newman-Penrose formalism, yielding corrections to amplitude and phase beyond geometric optics. It shows a dichotomy between Weyl (vacuum) lensing, which has no BE correction at linear order in $G$ and first BE effects at $\mathcal{O}(G^2)$, and Ricci (matter) lensing, which generates BE corrections at $\mathcal{O}(G)$, including a phase shift immediately after a thin lens that decays with distance. The authors derive a complete BE differential system, provide weak-field analytic results, and verify them via numerical simulations around a Schwarzschild black hole and through a toy Tolman–Oppenheimer–Volkoff star, obtaining compact expressions for the leading BE terms. The findings refine the understanding of wave-optics effects in gravitational lensing and establish criteria for the validity of geometric optics in regimes relevant to astrophysical and gravitational-wave observations, such as LISA.

Abstract

Waves propagating through a gravitational potential exhibit wave-optics effects when their wavelength is not significantly smaller than the lensing scales. We study the propagation of a scalar wave, governed by the Klein-Gordon equation in curved spacetime, to focus on effects on amplitude and phase, while leaving aside the issue of wave polarization which affects electromagnetic and gravitational waves. Using the Newman-Penrose formalism, we obtain the first corrections beyond the geometric optics in the expansion in the inverse frequency. In vacuum, that is for Weyl tensor lensing, there is no wave effect at first order in $G$ and wave effects start at order $G^2$. Conversely, if the wave travels through a non-vanishing matter density, the first corrections start at order $G$. We check these analytic results by solving numerically the equations dictating the evolution of the corrections either in the vicinity of a Schwarzschild black hole or through a transparent star.

Figures (5)

• Figure 1: The main system of coordinates is a spherical coordinates system centered on the lens in which we express the metric of the lenses [Eqs. (\ref{['BHmetric']}) and (\ref{['TOVmetric']})]. If the geodesic was not deviated, the closest radial distance when $s =s_{\rm L}$ would be the impact parameter $b$. When placing initial conditions close to the source, we assume the region is very well described by a flat space time and we use a system of spherical coordinates centered on the source, as detailed in \ref{['App:Flat']}.
• Figure 2: Numerical integration (continuous lines) and analytic approximation (dashed lines) for the first NP scalars (parameters given in main text).
• Figure 3: The r.h.s. of (\ref{['DA1Final']}) is decomposed into its main contributions (parameters given in main text). The vertical bar is located at the position of the black hole lens.
• Figure 4: The intermediate quantities, needed to estimate the source of $A_1$, are determined numerically (continuous lines) or analytically approximated (dashed lines) for $r_{\rm i}/R_{\rm S}=2.5\times 10^5$ with expressions of section (\ref{['SecFirstG2']}).
• Figure 5: Intermediate quantities and $\Im(A_1)/A_0$ are solved either numerically (continuous lines) or analytically approximated (dashed lines) with expressions reported in section \ref{['SecOG1']} for $R/R_{\rm S}=10^4$ and $r_{\rm i}/R_{\rm S}=2.5\times10^5$. The vertical bands denote the star region with $\rho_{\rm mat} \neq 0$.