$o$ in $\dot{f}_o$ for simplicity. (ii) In this approximation, the characteristic amplitude defined as $h_c(f):=2f\tilde{h}(f)$ becomes proportional to $A\dot{f}^{-1/2}$, where $A$ is a function of ${\cal M}_c$, $q$, $f$, and $d_L$, as well as the sky location and orientation of the binary. Noticing that $\dot{f}=\dot{f}_{\rm gw}+\dot{f}_{\rm gas}>\dot{f}_{\rm gw}$, we find that gas in general reduces the characteristic amplitude at any frequency. The reduction is due to a faster drift of the signal in the frequency domain. (iii) Using $h_c$, we can rewrite the SNR defined in Equation (\[eq:SNR2\]) as $${\rm SNR}^2=\int_{f_1}^{f_2}\frac{|h_c(f)|^2}{f^2S_n(f)}df,$$ where $f_1$ and $f_2$ denote the minimum and maximum frequencies during the observational period. Because $\Delta f=f_2-f_1\ll f$, the integration becomes proportional to $|h_c|^2\Delta f/(f^2S_n)$. (iv) In our problem we have $T_{\rm
obs}\ll |f/\dot{f}|$. Therefore, we can write $\Delta f\simeq \dot{f}T_{\rm
obs}$. Again, by noticing that $\dot{f}>\dot{f}_{\rm gw}$, we find that $\Delta f$ is broader when gas is present. (v) Finally, the $\dot{f}^{-1}$ from the term $|h_c|^2$ cancels the $\dot{f}$ from the term $\Delta f$, so that the SNR does not depend on $\dot{f}$. Physically, this means the reduction of the characteristic amplitude is compensated by the larger frequency drift.

The above conclusion that gas does not affect the 