BBH, as well as the distance. To see this effect, it is useful to first define an acceleration factor $$\Gamma:=\dot{a}_{\rm gas}/\dot{a}_{\rm gw}.$$ From this definition, it follows that $\dot{f}_o=(1+\Gamma)(1+z)^{-2}\dot{f}$, i.e., the chirp signal evolves faster by a factor of $1+\Gamma$. Finally, by revisiting Equations (\[eq:M\_o\]) and (\[eq:d\_o\]), we find that the apparent mass and distance become $$\begin{aligned}
    {\cal M}_o&=(1+\Gamma)^{3/5}{\cal M}(1+z),\label{eq:calM_o}\\
    d_o&=(1+\Gamma)\,d_C(1+z)=(1+\Gamma)\,d_L.\label{eq:bigdo}\end{aligned}$$

The factor $\Gamma$ in general is a function of $a$, because both $\dot{a}_{\rm
gw}$ and $\dot{a}_{\rm gas}$ depend on $a$. Such a dependence has two consequences. (i) If the semi-major axis changes substantially during the observational period, one would see a significant variation of ${\cal M}_o$ and $d_o$ with time. This result is inconsistent with the dynamical evolution of a BBH in a vacuum, and hence can be used to prove the presence of a environmental factor, such as gas. (ii) Otherwise, if $a$ evolves very slowly, i.e., the corresponding evolutionary timescale $a/|\dot{a}_{\rm gw}+\dot{a}_{\rm gas}|$ is much longer than the observational period $T_{\rm obs}$, the acceleration factor $\Gamma$ would be more or less a constant. In this case, the measurement of ${\cal M}_o$ and $d_o$ would be relatively consistent during the observational period, and both values would be greater 