Amplitude and Phase Fluctuations for Gravitational Waves Propagating through Inhomogeneous Mass Distribution in the Universe

When a gravitational wave (GW) from a distant source propagates through the universe, its amplitude and phase change due to gravitational lensing by the inhomogeneous mass distribution. We derive the amplitude and phase fluctuations, and calculate these variances in the limit of a weak gravitational field of density perturbation. If the scale of the perturbation is smaller than the Fresnel scale $\sim 100 {pc} (f/{mHz})^{-1/2}$ ($f$ is the GW frequency), the GW is not magnified due to the diffraction effect. The rms amplitude fluctuation is $1-10 %$ for $f > 10^{-10}$ Hz, but it is reduced less than 5% for a very low frequency of $f < 10^{-12}$ Hz. The rms phase fluctuation in the chirp signal is $\sim 10^{-3}$ radian at LISA frequency band ($10^{-5} - 10^{-1}$ Hz). Measurements of these fluctuations will provide information about the matter power spectrum on the Fresnel scale $\sim 100$ pc.