  The cross-correlation of gravitational wave strain with upcoming galaxy
surveys probe theories of gravity in a new way. This method enables testing the
theory of gravity by combining the effects from both gravitational lensing of
gravitational waves and the propagation of gravitational waves in spacetime. We
find that within 10 years, the combination of the Advanced-LIGO and VIRGO
detector networks with planned galaxy surveys should detect weak gravitational
lensing of gravitational waves in the low redshift Universe ($z<0.5$). With the
next generation gravitational wave experiments such as Voyager, LISA,
Cosmic-Explorer and Einstein Telescope, we can extend this test of the theory
of gravity to larger redshifts by exploiting the synergies between
electromagnetic wave and gravitational wave probes.
