Evolution of the semi-major axis and eccentricity of the three different binaries formed with the IMBH. Shortly after the beginning of the simulation, the IMBH builds a binary with the SBH with the 11th most massive mass, m•, 11. This corresponds to the first part of the curve (dashed blue curve). Later, there is an interaction which leads to a companion exchange for the binary, the SBH with the 18th most massive mass, m•, 18. This binary lives for about 40 Myr. We can see that the first two binaries have phases of very high eccentricity, ebin ~ 1, but not high enough to lead to a coalescence. The jumps in ebin indicate that the IMBH-SBH is still in a regime in which interactions with other stars play an important role. The system continues to shrink further and further until there is a three-body interaction. The binary is unbound and for a short period of time the IMBH has no companion, as indicated in the zoom-in subplots embedded in both the upper and lower panels. Then the final binary forms, with the second most massive SBH. This binary is very hard and quickly loses energy via GWs radiation, which very efficiently leads to circularization and the final merger.
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