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Coalescing massive black hole binaries (MBHBs) of $10^{4-7} \rm M_{\odot}$,forming in the aftermath of galaxy mergers, are primary targets of the spacemission LISA, the {\it Laser Interferometer Space Antenna}. An assessment ofLISA detection prospects requires an estimate of the abundance and propertiesof MBHBs that form and evolve during the assembly of cosmic structures. To thisaim, we employ a semi-analytic model to follow the co-evolution of MBHBs withintheir host galaxies. We identify three major evolutionary channels driving thebinaries to coalescence: two standard paths along which the binary evolution isdriven by interactions with the stellar and/or gaseous environment, and a novelchannel where MBHB coalescence occurs during the interaction with a third blackhole. For each channel, we follow the orbital evolution of MBHBs withphysically motivated models that include a self-consistent treatment of theorbital eccentricity. We find that LISA will detect between $\approx 25$ and$\approx 75$ events per year depending on the seed model. We show thattriple-induced coalescences can range from a few detected events up to $\sim30\%$ of the total detected mergers. Moreover, even if the standardgas/stars-driven evolutionary channels should fail and MBHBs were to stall,triple interactions would still occur as a result of the hierarchical nature ofgalaxy formation, resulting in about $\approx 10$ to $\approx 20$ LISAdetections per year. Remarkably, triple interactions among the black holes canproduce coalescing binaries with large eccentricities ($\gtrsim 0.9$) uponentrance into the LISA band. This eccentricity will remain significant ($\sim0.1$) also at merger, requiring suitable templates for parameter estimation.

Post-Newtonian Evolution of Massive Black Hole Triplets in Galactic Nuclei: IV. Implications for LISA