Weak lensing of gravitational waves in wave optics: Beyond the Born approximation

Abstract: The Universe's matter inhomogeneity gravitationally affects the propagation of gravitational waves (GWs), causing the lensing effect. Particularly, the weak lensing of GWs has been studied within the range of the Born approximation to constrain the small-scale power spectrum. In this work, the validity of the Born approximation is investigated by accounting for the higher-order terms in the gravitational potential $\Phi$. To do so, we formulate the post-Born approximation and derive the magnification $K$ and the phase modulation $S$ up to third order in $\Phi$. We find that the average of $S$ and $K$ is non-zero and that the average of $S$ depends on the size of the point mass. Due to this size dependency, the signal is enhanced, and the number of GW events required for detecting the average of $S$ decreases. We find that this number can become comparable to or even smaller than the number required for detecting the variance of $S$ in certain scenarios. In addition, it is verified that, for lensing by dark low-mass halos, the post-Born corrections are a few orders of magnitude smaller than the Born approximation at $f\geq0.01$~Hz. However, in the presence of the point mass, there is a condition under which the Born approximation fails. We derive the correction terms to the Born approximation and identify the condition under which the Born approximation no longer holds. For the magnification, the Born approximation is valid as long as the wavelength of GWs is larger than the Schwarzschild radius of lenses, while for the phase modulation, this condition is modified due to the physical size of the point mass.