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 black hole (SMBH). It is suggested that the gravitational wave (GW) emitted by a BBH has a high probability to be lensed by the SMBH if the BBH's orbit around the SMBH (i.e., the outer orbit) has a period of less than a year and is less than the duration of observation of the BBH by a space-borne GW observatory. For such a "BBH +SMBH " triple system, the de Sitter precession of the BBH's orbital plane is also significant. In this work, we thus study GW waveforms emitted by the BBH and then modulated by the SMBH due to effects including Doppler shift, de Sitter precession, and gravitational lensing. We show specifically that for an outer orbital period of 0.1 yr and an SMBH mass of 107 M⊙, there is a 3-10% chance for the standard, strong lensing signatures to be detectable by space-borne GW detectors such as LISA and/or TianGO. For more massive lenses (≳108 M⊙) and more compact outer orbits with periods ≲0.1 yr , retrolensing of the SMBH (which is closely related to the glory scattering) might also have a 1%-level chance of detection. Furthermore, by combining the lensing effects and the dynamics of the outer orbit, we find that the mass of the central SMBH can be accurately determined with a fraction error of ∼10-4. This is much better than the case of static lensing because the degeneracy between the lens' mass and the source's angular position is lifted by the outer orbital motion. Including lensing effects also allows the de Sitter precession to be detectable at a precession period 3 times longer than the case without lensing. Finally, we demonstrate that one can check the consistency between the SMBH's mass determined from the orbital dynamics and the one inferred from gravitational lensing, which serves as a test on theories behind both phenomena. The statistical error on the deviation between two masses can be constrained to a 1% level.