Measuring the polarization content of gravitational waves with strongly lensed binary black hole mergers

Alternative modified theories of gravity predict up to six distinct polarization modes for gravitational-wave (GW) sources. In order to measure the relative amplitudes for each mode, we require at least six linearly-independent GW detectors, as they measure the projection of the GW signal onto their geometry. This projection is encoded in the antenna pattern functions of the instruments, which modulate the detectability of each mode as a function of time due to Earth's rotation. Strong gravitational lensing of gravitational waves allows us to probe the polarization content of these signals by effectively increasing the number of possible observations from the same astrophysical source. Given that the lensed images will arrive at different times, each measures a different projection of the GW waveform originating from the same astrophysical system, effectively doubling the number of detectors that observe the same event (for a pair of lensed events) and allowing us to measure the relative amplitudes of additional polarization modes. To measure these amplitudes, we jointly fit the lensed image observations to a single GW signal model, taking into account the image magnifications, time delays, and polarization mode amplitudes. We show that for specific GW signals from binary black hole mergers, we can make a measurement of the relative mode amplitudes for strongly lensed events with at least two detectable images.