different values of $\tau_{\rm gas}$.[]{data-label="fig:FFmultiple"}](FF-T.pdf){width="50.00000%"}

Discussions {#sec:Dis}
===========

We have seen that those BBHs with $\Gamma=\tau_{\rm gw}/\tau_{\rm gas}\ga1$ could be misidentified by LISA as more massive binaries residing in vacuum environments. To what distance could LISA detect such fake massive binaries? The standard way of addressing this question is to derive the maximum distance at which the SNR drops to a threshold, say ${\rm SNR}=10$. This distance is known as the “detection horizon”. In the case without gas, the detection horizon has been derived in several works assuming different LISA configurations. For example, @kyutoku17 showed that $$\begin{aligned}
    d_L\simeq&13\,\left(\frac{{\cal M}_c}{10\,M_\odot}\right)^{5/3}
    \left(\frac{T_{\rm obs}}{5\,{\rm yr}}\right)^{1/2}
    \left(\frac{{\rm SNR}}{10}\right)^{-1}\nonumber \\
&\times
\left(\frac{S_n(f)}{10^{-40}\,{\rm Hz^{-1}}}\right)^{-1/2}
    \left(\frac{f}{3\,{\rm mHz}}\right)^{2/3}\,{\rm Mpc}\label{eq:SNR}\end{aligned}$$ for the N2A5 configuration of LISA. Here ${\cal M}_c$ and $d_L$ refer to the real chirp mass and real distance of the source.

In fact, the last equation is valid also in the case with gas. This is so because of the following reasons. (i) The stationary phase approximation [@thorne87] in which the last equation is derived remains valid, since the evolutionary timescale $f/\dot{f}$ for the frequency is much longer than the GW period $1/f$. We note that $\dot{f}$ here stands for the observed frequency. We dropped the subscript 