LIGO/Virgo BBHs, which typically have $m_1\sim10\,M_\odot$ and $f\sim10^2$ Hz, the gas effect is negligible unless the gas density $n$ is orders of magnitude higher than $10^{16}\,{\rm cm^{-3}}$. This is the reason that for LIGO/Virgo sources, significant gas effect is expected only in stellar cores, where $n$ could be as high as $(10^{28}-10^{31})\,{\rm cm^{-3}}$ [e.g. @fedrow17]. For LISA sources with $f\sim3$ mHz, however, gas effect is already important when $n\sim10^{16}\,{\rm cm^{-3}}$ according to Equation (\[eq:Gamm\]) or $n\sim10^{12}\,{\rm cm^{-3}}$ according to Equation (\[eq:Gamm2\]). These two characteristic densities can be found, respectively, in common envelopes and AGN accretion disks [e.g. @antoni19]. Therefore, the gaseous environments common for BBHs would affect the LISA signals more than the LIGO/Virgo ones.

Faking a chirp signal {#sec:signal}
=====================

Having understood the effect and the relative importance of gas, we now compute the chirp signal of a BBH embedded in a gaseous environment. In the following, we assume $z=0$ for simplicity ($f_o=f$). When there is no gas, we calculate the time derivative of the GW frequency ($\dot{f}_{\rm gw}$) using a 3.5 post-Newtonian (PN) approximation presented in @sathyaprakash09. When gas is present, we have $\dot{f}_o=\dot{f}_{\rm gw}+\dot{f}_{\rm gas}$, where $\dot{f}_{\rm gas}$ is calculated with $\dot{f}_{\rm gas}=-3f/(2\tau_{\rm
gas})$ according to Equation 