Our Galaxy and the nearby Andromeda galaxy (M31) are the most massive members of the Local Group, and they seem to be a bound pair, despite the uncertainties on the relative motion of the two galaxies. A number of studies have shown that the two galaxies will likely undergo a close approach in the next 4$-$5 Gyr. We used direct $N$-body simulations to model this interaction to shed light on the future of the Milky Way - Andromeda system and for the first time explore the fate of the two supermassive black holes (SMBHs) that are located at their centers. We investigated how the uncertainties on the relative motion of the two galaxies, linked with the initial velocities and the density of the diffuse environment in which they move, affect the estimate of the time they need to merge and form ``Milkomeda. After the galaxy merger, we follow the evolution of their two SMBHs up to their close pairing and fusion. Upon the fiducial set of parameters, we find that Milky Way and Andromeda will have their closest approach in the next 4.3 Gyr and merge over a span of 10 Gyr. Although the time of the first encounter is consistent with other predictions, we find that the merger occurs later than previously estimated. We also show that the two SMBHs will spiral in the inner region of Milkomeda and coalesce in less than 16.6 Myr after the merger of the two galaxies. Finally, we evaluate the gravitational-wave emission caused by the inspiral of the SMBHs, and we discuss the detectability of similar SMBH mergers in the nearby Universe ($zleq 2$) through next-generation gravitational-wave detectors.
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