Lensing of gravitational waves: universal signatures in the beating pattern

Abstract

When gravitational waves propagate near massive objects, their paths curve resulting in gravitational lensing, which is expected to be a promising new instrument in astrophysics. If the time delay between different paths is comparable with the wave period, lensing may induce beating patterns in the waveform, and it is very close to caustics that these effects are likely to be observable. Near the caustic, however, the short-wave asymptotics associated with the geometrical optics approximation breaks down. In order to describe properly the crossover from wave optics to geometrical optics regimes, along with the Fresnel number, which is the ratio between the Schwarzschild diameter of the lens and the wavelength, one has to include another parameter — namely, the angular position of the source with respect to the caustic. By considering the point mass lens model, we show that in the two-dimensional parameter space, the nodal and antinodal lines for the transmission factor closely follow hyperbolas in a wide range of values near the caustic. This allows us to suggest a simple formula for the onset of geometrical-optics oscillations which relates the Fresnel number with the angular position of the source in units of the Einstein angle. We find that the mass of the lens can be inferred from the analysis of the interference fringes of a specific lensed waveform.