Universal relation between the variances of distortions of gravitational waves owing to gravitational lensing

Gravitational waves from the distant sources are gravitationally lensed during their propagation through the intervening matter inhomogeneities before arriving at detectors. It has been proposed in the literature that the variance of the lensed waveform can be used to extract information of the matter power spectrum at very small scales and of low-mass dark halos. In this paper, we show that the variance of the amplitude fluctuation and that of the phase fluctuation of the lensed waveform obey a simple relation irrespective of the shape of the matter power spectrum. We study conditions under which this relation can be violated and discuss some potential applications of the relation. This relation may be used to confirm the robustness of claimed observations of gravitational lensing of gravitational waves and the subsequent reconstruction of the matter power spectrum.