Relative amplitudes of strongly lensed LISA binaries as probes of galaxy-lens potentials

Strongly lensed gravitational waves (GWs) provide precise time-delay measurements. Inferring the Hubble constant, H_0, also requires the corresponding Fermat-potential differences. We investigated whether the relative amplitudes of lensed massive black hole binaries observed by the Laser Interferometer Space Antenna (LISA) improve this lens reconstruction. We analysed the images of each binary jointly. The binary parameters were common to all images, while their arrival times and effective luminosity distances were allowed to differ. The time-dependent LISA response was evaluated at every arrival time, allowing the effective-distance ratios to be interpreted as lensing magnification ratios. We used one four-year realisation of the HS-nod-SN-high-accr heavy-seed population. Of 51 lensed binaries, 43 have at least two detectable non-overlapping images; the median 1σ uncertainty of their magnification ratios is 4.76%. For identified source and lens galaxies with measured redshifts, a singular isothermal ellipsoid with external shear in the simulation and analysis, a fractional time-delay uncertainty of 10^-4, 10 mas source-plane astrometry, and σ(κ_ ̊m ext )=0.02, the seven usable systems with source redshift z_ ̊m s <10 give fractional 1σ uncertainties in H_0 of 2.217% without the ratios and 2.038% with them. For all 43 GW-selected systems, the corresponding uncertainties are 0.747% and 0.685%, depending on redshift measurements for every source and lens. Neither sample was selected by electromagnetic detectability. Resampling the full lens sample gives a 95% interval of 6.00--10.75% for the relative reduction. The relative reduction ranged from 3.5% to 15.0% when the lens-prior widths were varied and from 6.5% to 10.0% when σ(κ_ ̊m ext ) was varied from 0.03 to 0.01. The magnification ratios are independent of the unlensed luminosity distance and remained unchanged under a mass-sheet transformation. They therefore constrained the angular structure of the lens potential without determining its common mass-sheet scaling. This is an idealised best-case statistical forecast for the adopted population, lens model, and waveform.