a $10M_\odot-10M_\odot$ binary in a vacuum (red dashed curve), because GW radiation predominates during the merger.

![Evolution of the GW frequency as a function of the time before coalescence. The blue solid curve shows a $10M_\odot-10M_\odot$ circular BBH embedded in a gaseous environment. The model parameters are chosen such that ${\Gamma}=10$ when the GW frequency is $f=3$ mHz (see Eq.(\[eq:Gamm2\])). The two vertical lines mark the typical LISA observational window of five years. The red dashed curve shows the chirp signal of the same BBH but placed in a vacuum. The coalescence times of the previous two BBHs are aligned for easier comparison. The dot-dashed curve shows another BBH merging in vacuum but with higher masses. It is offset in time so that the GW frequency in the LISA observational window is also $3$ mHz. []{data-label="fig:long"}](longscale.pdf){width="50.00000%"}

To see more clearly the chirp signal in the observation window of LISA, we show in Figure \[fig:short\] the evolution of $f$ during a period of $1-2$ years, around the moment when $f$ is approximately $3$ mHz. Now the chirp signals look like straight lines because the observational period is orders of magnitude shorter than the evolutionary timescales of the BBHs. Although the variation 