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

Gravitational wave (GW) galaxy lens reconstruction is a crucial step for many GW lensing science applications. However, dark siren GW lensing (i.e. lensed GW without observed electromagnetic (EM) counterpart) suffers from similarity transformation degeneracy and mass-sheet degeneracy. We review these two degeneracies and discuss their implications on GW-based lens reconstruction and two well-known GW lensing science cases: the Hubble constant measurement and test for modified GW propagation. Building upon previous works, our conclusions are:1) GWs can only infer the scale-free lens mass model parameters, the dimensionless source position, the GW luminosity distance and the time delay scaling (a combination of Einstein radius, lens redshift, and cosmology).2) Lens reconstruction (of singular isothermal ellipsoid lens) with only two GW signals is unlikely to yield a complete lens model, while four (three) signals can measure all the above parameters accurately (with large uncertainties).3) The similarity transformation degeneracy causes the lens redshift/Einstein radius/cosmology to be degenerate in dark siren measurements. Breaking the degeneracy can be achieved by supplementing the GWs with EM observation of lens redshifts/Einstein radius (source redshift is not required).4) The mass-sheet degeneracy causes the GW luminosity distance to be entirely degenerate with a constant mass sheet.5) Contrary to expectation, the Hubble constant is degenerate with the mass-sheet even when supplemented with lens reconstruction/redshift/Einstein radius and can only be lifted with lens galaxy velocity dispersion measurement, while modified GW propagation test discussed in prior literature is unaffected by the degeneracy. These properties highlight the need for GW observations to be supplemented by EM observations, which could become accessible through a lens archival search or a rapid EM follow-up.