Multi-frequency gravitational wave (GW) observations are useful probes of the formation processes of coalescing stellar-mass binary black holes (BBHs). We discuss the phase drift in the GW inspiral waveform of the merging BBH caused by its center-of-mass acceleration. The acceleration strongly depends on the location where a BBH forms within a galaxy, allowing observations of the early inspiral phase of LIGO-like BBH mergers by the Laser Interferometer Space Antenna (LISA) to test the formation mechanism. In particular, BBHs formed in dense nuclear star clusters or via compact accretion disks around a nuclear supermassive black hole in active galactic nuclei would suffer strong acceleration, and produce large phase drifts measurable by LISA. The host galaxies of the coalescing BBHs in these scenarios can also be uniquely identified in the LISA error volume, without electromagnetic counterparts. A non-detection of phase drifts would rule out or constrain the contribution of the nuclear formation channels to the stellar-mass BBH population.
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