Wave propagation in a weak gravitational field and the validity of the thin lens approximation

Wave effects can be important for the gravitational lensing of gravitational waves. In such a case, wave optics must be used instead of geometric optics. We consider a plane wave entering a lens object and solve numerically the wave equation for three lens models: the uniform density sphere, the singular isothermal sphere, and the Hernquist model. By comparing our numerical solutions with the analytical solutions under the thin lens approximation, we evaluate the error of this approximation. The results show that the relative error of the thin lens approximation is small if the geometrical thickness of the lens is much smaller than the distance between the lens and the observer.

Figure 1

Lensing configuration. The lens is the origin of coordinate axes, while the observer position is ($r$, $\overrightarrow{\theta }$) with $\theta \ll 1$. The incident wave is a plane wave propagating in the $\mathrm{z}$-direction.

Figure 2

${\delta }_{\ell }$ as a function of $\ell$ for the uniform density sphere.

Figure 3

Error of the amplification factor for the uniform density sphere. The horizontal axis is the radius $\omega R$ (top left), the distance $\omega r$ (top right), the mass $2GM\omega (=p)$ (bottom left), and the angle $\theta$ (bottom right). The dashed lines denote the ratio of radius to distance $R/r$.

Figure 4

Same as Fig. 3, but for the SIS model. The dashed lines denote the ratio of cutoff radius to distance $r_c/r$.

Figure 5

Same as Fig. 3, but for the Hernquist model. The dashed lines denote the ratio of cutoff radius to distance $r_c/r$.