SNR is derived in the scenario of LISA observations. It does not apply to LIGO/Virgo because in the latter case the assumption $T_{\rm obs}\ll |f/\dot{f}|$ is invalid. In fact, gas will reduce the SNR for LIGO/Virgo sources by suppressing $|h_c|$. Nevertheless, the corresponding change of SNR is small because, as has been explained in §\[sec:compare\], the acceleration factor $\Gamma$ is small when a BBH enters the LIGO/Virgo band.

Therefore, we can use Equation (\[eq:SNR\]) to estimate the detection horizon and, based on it, discuss the detectability of the fake massive BBHs embedded in gaseous environment. For ${\cal M}_c=10\,M_\odot$, the detection horizon corresponding to a SNR of $10$ is approximately $13$ Mpc, assuming $5$ years of observation. If ${\cal M}_c=30\,M_\odot$, as the LIGO/Virgo observations tend to suggest [@ligo18a], the detection horizon elongates to about $80$ Mpc.

To estimate the number of fake massive BBHs within the detection horizon, we start from the event rate in the LIGO/Virgo band, which is estimated to be ${\cal O}(10^2)\,{\rm Gpc^{-3}\,yr^{-1}}$ [@ligo18a]. According to this rate, the number of BBHs in the last year before their coalescence is about $N(\tau=1\,{\rm yr})=100$ per ${\rm Gpc}^{3}$, where $\tau=|a/\dot{a}|$ denotes the orbital evolutionary timescale. For the other BBHs at 