Invariance transformations in wave-optics lensing: implications for gravitational-wave astrophysics and cosmology

Gravitational lensing offers unique opportunities to learn about the astrophysical origin of distant sources, the abundance of intervening objects acting as lenses, and gravity and cosmology in general. However, all this information can only be retrieved as long as one can disentangle each effect from the finite number of observables. In the geometric optics regime, typical of electromagnetic radiation, when the wavelength of the lensed signal is small compared to the size of the lens, there are invariance transformations that change the mass of the lens and the source-lens configuration but leave the observables unchanged. Neglecting this ``mass-sheet degeneracy'' can lead to biased lens parameters or unrealistic low uncertainties, which could then transfer to an incorrect cosmography study. This might be different for gravitational waves as their long wavelengths can be comparable to the lens size and lensing enters into the wave-optics limit. We explore the existence of invariance transformations in the wave-optics regime of gravitational-wave lensing, extending previous work and examining the implications for astrophysical and cosmological studies. We study these invariance transformations using three different methods of increasing level of complexity: template mismatch, Fisher Matrix, and Bayesian parameter estimation. We find that, for a sufficiently loud signal, the degeneracy is partially broken and the lens and cosmological parameters, e.g. $H_0$, can be retrieved independently and unbiased. In current ground-based detectors, though, considering also population studies, a strong constraint on these parameters seems quite remote and the prevailing degeneracy implies a larger uncertainty in the lens model reconstruction. However, with better sensitivity of the third-generation ground-based detectors, a meaningful constraint on $H_0$ is possible to obtain.