to claim a detection, the corresponding criterion of distinguishing two different waveforms becomes ${\rm FF}<0.995$.

According to this criterion, the gas and vacuum waveforms used in Figure \[fig:FFsingle\] are indistinguishable during the first $1-2$ years of observation. Only in the third year could one start to tell the difference and prove that the signal is not produced by a massive BBH of ${\cal
M}_c\simeq37\,M_\odot$ residing in a vacuum environment.

For completeness, we show in Figure \[fig:FFmultiple\] the FF derived assuming different values for $\tau_{\rm gas}$. The other model parameters are the same as in Figure \[fig:heatmap\]. As $\tau_{\rm gas}$ increases, the FF gets better at later times because the gas effect becomes weaker. We note that when $\tau_{\rm gas}\ga5,000$ years, the FF is better than $0.995$ for almost five years, which is equivalent to the canonical mission duration of LISA. As a result, LISA may identify our BBHs of a chirp mass of $8.7\,M_\odot$ as more massive binaries. The measured chirp mass is approximately $18\,M_\odot$ when $\tau_{\rm gas}=5,000$ years and $14\,M_\odot$ when $\tau_{\rm gas}=10^4$ years. In both cases, the overestimation of the mass, and hence the distance (see Eq. (\[eq:bigdo\])), is substantial.

![Variation of the maxim FF with time assuming 