  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.
