New Consistency Relations between Averages and Variances of Weakly Lensed Signals of Gravitational Waves

Weak lensing of gravitational waves (GWs) occurs when GWs experience fluctuations in the local gravitational potential. Due to the long wavelength of GWs compared to light, wave effects become relevant, causing a shift in the phase (phase modulation) as well as (de)magnification of the amplitude. The lensing effect is sensitive to matter fluctuation scales corresponding to the Fresenl scale of GWs and this feature can be used to probe the small-scale matter power spectrum. In addition, a previous study showed that the variances of the magnification and phase modulation satisfy a nontrivial consistency relation. In this presentation, I will show that there are two additional consistency relations between the averages and variances of the weakly lensed GW signals in wave optics. These consistency relations are derived as the weak lensing limit of the full-order relations for the averages of the amplification factor and its absolute square. I will demonstrate that these full-order relations originate from the energy conservation of GWs and the Shapiro time delay, and they hold irrespective of the shape of the matter power spectrum.