Initially, the GW frequency is $f=3$ mHz, and the corresponding GW radiation timescale is about $10^4$ years. In this particular example the observational period is set to $T_{\rm obs}=1.25$ years, but later we will show the FF for different $T_{\rm
obs}$. We match the signal using the templates developed for vacuum BBHs, i.e., the 3.5 PN approximation described above. We use a simulated annealing algorithm to search for the highest FF in the parameter space of $({\cal
M}_c,\,q,\,\phi)$. The best FF is found at a chirp mass of ${\cal
M}_c\simeq37\,M_\odot$. It is offset from the real chirp masses by a factor of about $4.2$, which is consistent with our Equation (\[eq:calM\_o\]). This result confirms our prediction that ignoring the gas effect could result in a significant overestimation of the mass of a LISA BBH.

![Dependence of the FF on the chirp mass ${\cal M}_c$ and initial phase $\phi$ of the template $h_2$. The dependence on $q$ is relatively weak and is not shown here. The signal $h_1$ is generated using the parameters ${\cal M}_c=8.7\,M_\odot$, $q=0.7$, and $\tau_{\rm gas}=10^3$ years. The plus symbol marks the location of the maximum FF, which is offset from the real chirp mass ($8.7\,M_\odot$).[]{data-label="fig:heatmap"}](heat_map.pdf){width="50.00000%"}

We note 