disks, $10^{16}-10^{19}\,{\rm cm^{-3}}$ in common envelops, and even higher in stellar cores [see @antoni19 for a summary].

Several earlier works studied the impact of gas on the GW signal of merging binaries, including BBHs in the LIGO/Virgo band [@fedrow17; @ginat19; @cardoso19] and extreme-mass-ratio inspirals (another type of binary composed of a stellar-mass BH orbiting a SMBH) in the LISA band [@yunes11; @kocsis11; @barausse14]. They focused on the final evolutionary stage when the semi-major axes ($a$) of the binaries are only $10-10^2$ times the Schwarzschild radius ($r_S$) of the bigger BHs. At this stage, GW radiation predominates and gas plays a minor role. Nevertheless, these works showed that gas could induce a small phase shift to the GW signal. The phase shift does not significantly affect the mass measurement but can be used to identify the mergers happening in gas.

The BBHs in the LISA band are very different from those considered in the earlier works. They have much greater semi-major axes [$a\sim10^3-10^4\,r_S$, e.g., @chen17] since GW frequency is proportional to $a^{-3/2}$ . Such wide binaries have much weaker GW radiation because the GW power scales with $a^{-5}$ [@peters64]. Consequently, gas could play a dominant role in the evolution of these 