Measuring supermassive black hole properties via gravitational radiation from eccentrically orbiting stellar mass black hole binaries

There may exist stellar-mass binary black holes (BBH) which merge while orbiting nearby a supermassive black hole (SMBH). In such a triple system, the SMBH will modulate the gravitational waveform of the BBH through orbital Doppler shift and de Sitter precession of the angular momentum. Future space-based gravitational wave (GW) observatories focused on the milli- and decihertz band will be uniquely poised to observe these waveform modulations, as the GW frequency from stellar-mass BBHs varies slowly in this band while modulation effects accumulate. In this work, we apply the Fisher information matrix formalism to estimate how well space-borne GW detectors can measure properties of $\mathrm{BBH}+\mathrm{SMBH}$ hierarchical triples using the GW from orbiting BBH. We extend previous work by considering the more realistic case of an eccentric orbit around the SMBH, and notably include the effects of orbital pericenter precession. We find that for detector concepts such as LISA, B-DECIGO, and TianGO, we can extract the SMBH mass and semimajor axis of the orbit with a fractional uncertainty below the 0.1% level over a wide range of triple system parameters. Furthermore, we find that the effects of pericenter precession and orbital eccentricity significantly improve our ability to measure this system. We also find that while LISA could measure these systems, the decihertz detector concepts B-DECIGO and TianGO would enable better sensitivity to the triple's parameters.