Detecting gravitational lensing in hierarchical triples in galactic nuclei with space-borne gravitational-wave observatories

Stellar-mass binary black holes (BBHs) may merge in the vicinity of a supermassive hole (SMBH). It is suggested that gravitational wave (GW) emitted by BBH has high probability to be lensed SMBH if BBH's orbit around (i.e., outer orbit) period less than year and duration observation space-borne GW observatory. For such ``$\mathrm{BBH}+\mathrm{SMBH}$'' triple system, de Sitter precession orbital plane also significant. In this work, we thus study waveforms then modulated due effects including Doppler shift, precession, lensing. We show specifically for an 0.1 yr mass ${10}^{7}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$, there 3--10% chance standard, strong lensing signatures detectable detectors as LISA and/or TianGO. more massive lenses ($\ensuremath{\gtrsim}{10}^{8}\text{ }{M}_{\ensuremath{\bigodot}}$) compact orbits with periods $\ensuremath{\lesssim}0.1\text{ }\mathrm{yr}$, retrolensing (which closely related glory scattering) might have 1%-level detection. Furthermore, combining dynamics orbit, find central can accurately determined fraction error $\ensuremath{\sim}{10}^{\ensuremath{-}4}$. This much better case static because degeneracy between lens' source's angular position lifted motion. Including allows at 3 times longer without Finally, demonstrate one check consistency SMBH's from inferred lensing, which serves test on theories behind both phenomena. The statistical deviation two masses constrained 1% level.