Order in the chaos: eccentric black hole binary mergers in triples formed via strong binary-binary scatterings

Black hole (BH) triples represent one of the astrophysical pathways for BH mergers in the Universe detectable by LIGO and VIRGO. We study the formation of BH triples via binary-binary encounters in dense clusters, showing that one-third of triples formed through this channel are in a non-hierarchical, unstable configuration. We build a database of $32,000$ $N$-body simulations to investigate the evolution of these BH triples. Varying the mutual orbital inclination, the three BH masses and the inner and outer eccentricities, we show that retrograde, nearly planar configurations lead to a significant shrinkage of the inner binary. We find an universal trend of triple systems, namely that they tend to evolve toward prograde configurations and that the orbital flip, driven by the torque exerted on the inner BH binary (BHB) by the outer BH, leads in general to tighter inner orbits that, in some cases, merge within a Hubble time. We find that the mergers' mass distribution maps the original BH binary spectrum. A large fraction of merging BHBs with initial separation $1$ AU enter the $10^{-3}-10^{-1}$ Hz frequency band with large eccentricities, thus representing potential eccentric LISA sources. Mergers originating from initially tighter BHB ($a\sim 0.01$ AU), instead, often have eccentricities above 0.7 in the $1$ Hz band. This might imply that eccentric sources detected in high-frequency detectors are most likely connected with a high-velocity dispersion stellar environment, whereas eccentric sources detected in low-frequency detectors are likely to develop in low-density clusters. (Abridged)