in this evolutionary phase. Without gas, the semi-major axis $a$ of a BBH decays approximately as $$\dot{a}=\dot{a}_{\rm gw}:=\frac{64}{5}\frac{G^3m_1m_2m_{12}}{c^5a^3}\label{eq:adotgw}$$ [@peters64 assuming near-Keplerian circular orbits], where the dot symbol denotes the time derivative, $G$ is the gravitational constant, $c$ is the speed of light, $m_1$ and $m_2$ are the masses of the two BH members, and $m_{12}=m_1+m_2$ is the total mass of the binary. During the orbital decay, the GW frequency, which can be calculated with $f=\pi^{-1}(Gm_{12}/a^3)^{1/2}$, increases at a rate of $\dot{f}\propto f^{11/3}$. Such a characteristic signature is called the “chirp signal”. From it, one can derive a characteristic mass scale $${\cal
M}=\frac{c^3}{G}\,\left(\frac{5f^{-11/3}\dot{f}}{96\pi^{8/3}}\right)^{3/5}
=\frac{(m_1m_2)^{3/5}}{(m_1+m_2)^{1/5}},\label{eq:M}$$ which is known as the “chirp mass”. It uniquely determines the time evolution of $f$.

From the chirp signal one can also derive the distance $d$ of the BBH [@schutz86]. This is because from $f$ and $\dot{f}$ one can infer the energy-loss rate of the orbit, $\dot{E}\propto f^{-4}\dot{f}^2$, which also equals the GW power. In addition, the frequency $f$ and the GW amplitude $h$, together, determine a flux $S\propto h^2f^2$ which is the GW flux. From the power and the flux, one can derive the distance of the source $$d=\frac{4G}{c^2}\frac{{\cal M}}{h}\left(\frac{G}{c^3}\pi\,f\,{\cal M}\right)^{2/3}.\label{eq:d}$$

If the BBH is at 