Galaxy lens reconstruction based on strongly lensed gravitational waves: the mass-sheet and similarity transformation degeneracy

Einstein's general relativity posits that gravitational waves, like light, can be gravitationally lensed by intervening mass distributions. If lensing by galaxies is detected in the future, these gravitational waves allow us to reconstruct the strong lenses that produced them, potentially allowing for several new science applications. However, such gravitational-wave-based strong lensing reconstructions suffer from the well-known mass sheet degeneracy and the less well-known similarity transformation degeneracy. We review these two degeneracies and discuss their implications for gravitational wave-based lens reconstructions and two notable gravitational wave lensing science cases: the Hubble constant measurement and tests for modified gravitational wave propagation. Furthermore, we review methods to break these degeneracies: although binary black holes are standard sirens, they cannot break the mass-sheet degeneracy with electromagnetic lens reconstructions and photometric redshift measurements of the lens and source galaxy in cosmological applications. Indeed, any science application measuring the Hubble constant will be affected by the mass-sheet degeneracy unless complementary measurements, such as the lens galaxy's velocity dispersion, are available. Fortunately, modified gravitational-wave propagation tests are unaffected by the mass-sheet degeneracy. However, both applications suffer from the similarity transformation degeneracy, which can only be broken with complementary electromagnetic observation, which provides redshifts and Einstein radius of the lens system. Breaking these degeneracies is critical to unlocking the full potential of gravitational wave-based lens reconstructions for future scientific applications.