we will show that this method cannot distinguish light BBHs embedded in certain gaseous environments from those massive ones in vacuum.

Matched filtering
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Given two waveforms, $h_1$ and $h_2$, their similarity is quantified by a “fitting factor” (FF), which is defined as $${\rm FF}=\frac{\left<h_1|h_2\right>}{\sqrt{\left<h_1|h_1\right>\left<h_2|h_2\right>}}.$$ The term $\left<h_1|h_2\right>$ means an inner product of $$\left<h_1|h_2\right>=2\int_0^\infty\frac{\tilde{h}_1(f)\tilde{h}_2^*(f)+\tilde{h}_1^*(f)\tilde{h}_2}{S_n(f)}df,$$ where the tilde symbols stand for the Fourier transformation, the stars stand for the complex conjugation, and $S_n(f)$ is the spectral noise density of LISA [@klein16]. Identical waveforms have ${\rm FF}=1$.

In our problem, $h_1(t)$ is the chirp signal of a BBH embedded in a gaseous environment, and $h_2(t)$ is the waveform of a inspiraling BBH in a vacuum. By tuning the parameters of $h_2$, we want to maximize the FF. We follow @cutler94 and compute the waveforms using $$h(t)=\frac{Q(\theta, \varphi, \psi, \iota) \mu M}{d_L\, a(t)} \cos \left(\int 2 \pi f d t\right),$$ where $Q(\theta, \varphi, \psi, \iota)$ is a function depending on the sky location and orientation of the BBH. In the integrand, the frequency $f$ is a function of $a$. It is computed using the 3.5 PN approximation for $h_2$ [@sathyaprakash09] and using the gas model described in §\[sec:signal\] for $h_1$.

Because the two 