Testing an exact diffraction formula with gravitational wave source lensed by a supermassive black hole in binary systems

Abstract

When it comes to long-wavelength gravitational waves (GWs), diffraction effect becomes significant when these waves are lensed by celestial bodies. Typically, the traditional diffraction integral formula neglects large-angle diffraction, which is often adequate for most of cases. Nonetheless, there are specific scenarios, such as when a GW source is lensed by a supermassive black hole in a binary system, where the lens and source are in close proximity, where large-angle diffraction can play a crucial role. In our prior research, we have introduced an exact, general diffraction integral formula that accounts for large-angle diffraction as well. This paper explores the disparities between this exact diffraction formula and the traditional, approximate one under various special conditions. Our findings indicate that, under specific parameters — such as a lens-source distance of D LS = 0.1 AU and a lens mass of M L = 4 × 106 M ⊙ — the amplification factor for the exact diffraction formula is notably smaller than that of the approximate formula, differing by a factor of approximately rF ≃ 0.806. This difference is substantial enough to be detectable. Furthermore, our study reveals that the proportionality factor rF gradually increases from 0.5 to 1 as D LS increases, and decreases as M L increases. Significant differences between the exact and approximate formulas are observable when D LS ≲ 0.2 AU (assuming M L = 4 × 106 M ⊙) or when M L ≳ 2 × 106 M ⊙ (assuming D LS = 0.1 AU). These findings suggest that there is potential to validate our general diffraction formula through future GW detections.