gas these binaries have a typical evolutionary timescale of $$\begin{aligned}
    \tau_{\rm gw}&=\frac{5}{64}\frac{c^5a^4}{G^3m_1m_2m_{12}}\\
&\simeq\frac{9.1\times10^3}{q(1+q)^{-1/3}}
\left(\frac{m_1}{10\,M_\odot}\right)^{-5/3}
\left(\frac{f}{3\,{\rm mHz}}\right)^{-8/3}\,{\rm years},\end{aligned}$$ where $q$ denotes the mass ratio $m_2/m_1$ of the two BHs (we assume $m_1\ge m_2$).

As for $\tau_{\rm gas}$, we first use the hydrodynamic drag derived in @ostriker99 [@sanchez99] to estimate its value. The drag force on the secondary BH ($m_2$) is calculated with $F\sim4\pi\rho(Gm_2/v)^2$, where $\rho$ is the mass density of the background gas and $v$ is the Kepler velocity of the secondary. For circular orbits, we have $v\propto a^{-1/2}$, so that $a/\dot{a}_{\rm gas}=-v/(2\dot{v})$. Moreover, since $|\dot{v}|=F/m_2$, we derive that $$\tau_{\rm gas}=\frac{m_2v}{2F}\simeq\frac{1.1\times10^4}{q(1+q)^{2}}
\left(\frac{n}{10^{16}\,{\rm cm^{-3}}}\right)^{-1}
    \left(\frac{f}{3\,{\rm mHz}}\right) \,{\rm yrs},$$ where $n$ is the number density of hydrogen atoms in the gas background. It follows that $$\begin{aligned}
\Gamma\simeq4.3\,\left(\frac{1+q}{2}\right)^{7/3}
\left(\frac{n}{10^{16}\,{\rm cm^{-3}}}\right)
\left(\frac{m_1}{10\,M_\odot}\right)^{-5/3}
    \left(\frac{f}{3\,{\rm mHz}}\right)^{-11/3}.\label{eq:Gamm}\end{aligned}$$

In the above derivation of $\tau_{\rm gas}$, it is assumed that the gas background is homogeneous and the small body is moving in a straight line. However, for the BHs in binaries, which move along Keplerian orbits, it has been shown that the formula for the drag force will be modified, because the shape of the density wake is different [@sanchez01; @escala04; @kim07; @kim08]. More recently, @antoni19 showed that when $a$ 