Arrival Time Differences of Lensed Massive Gravitational Waves

It is of fundamental importance to know the mass of gravitons. A simple method for constraining the graviton mass is to compare the arrival time of light and that of gravitational waves provided that both waves are simultaneously emitted from the same source. To date, from observations of gravitational waves by the LIGO, the upper bound on the graviton mass $m_g$ is given by $m_g\lesssim 5.0 \times 10^{-23}$eV. However, when we compare the arrival time of light and gravitational waves, lensing effects could be important for some cases. Moreover, in many cases, the wavelength of gravitational waves is comparable with the gravitational radius of a lens object. Hence, we calculate arrival time differences between electromagnetic waves and massive gravitational waves by taking into account the effect of the gravitational wave optics. Here we take two lens models, a point mass lens and a singular isothermal sphere lens. We find that the lensing changes the arrival time difference of two waves by more than a second for the massive gravitational waves detectable by the LISA.