(\[eq:fdottot\]). In this way, the gas effect is included in the model through a parameter $\tau_{\rm gas}$.

Figure \[fig:long\] compares the long-term evolution of the chirp signal in the cases with and without the gas drag. The blue solid curve corresponds to a $10M_\odot-10M_\odot$ BBH (${\cal M}\simeq8.7\,M_\odot$) embedded in a gaseous medium. The two BHs coalesce at the time $t=t_c$. The hydrodynamical timescale $\tau_{\rm gas}$ is computed according to Equation (\[eq:Tgas\]) and the model parameters are chosen such that $\tau_{\rm gw}/\tau_{\rm gas}=10$ when $f=3$ mHz. This chirp signal around $f=3$ mHz, according to Equation (\[eq:calM\_o\]), should resemble a more massive binary with a chirp mass of $(1+\Gamma)^{3/5}{\cal M}\simeq37\,M_\odot$ in a vacuum. The chirp signal of the latter more massive BBH is shown as the dot-dashed curve, and we can see that, indeed, at $f=3$ mHz it matches the signal of the smaller binary embedded in gas. Eventually, the two signals diverge, since $\Gamma$ is decreasing, but the divergence appears more than one hundred years later. If we focus on the LISA observational window of five years (marked by the two vertical lines), the two signals are almost identical. Finally, during the coalescence, the blue curve recovers the chirp signal of 