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Using few-body simulations, we investigate the evolution of supermassiveblack holes (SMBHs) in galaxies ($M_{\star}=10^{10}-10^{12}{\rm M}_{\odot}$ at$z=0$) at $0<z<4$. Following galaxy merger trees from the Millenniumsimulation, we model BH mergers with two extreme binary decay scenarios for the`hard binary' stage: a full or an empty loss cone. These two models shouldbracket the true evolution, and allow us to separately explore the role ofdynamical friction and that of multi-body BH interactions on BH mergers. Usingthe computed merger rates, we infer the stochastic gravitational wavebackground (GWB). Our dynamical approach is a first attempt to study thedynamical evolution of multiple SMBHs in the host galaxies undergoing mergerswith various mass ratios ($10^{-4} < q_{\star} < 1$). Our main resultdemonstrates that SMBH binaries are able to merge in both scenarios. In theempty loss cone case, we find that BHs merge via multi-body interactions,avoiding the `final parsec' problem, and entering the PTA band with substantialorbital eccentricity. Our full loss cone treatment, albeit more approximate,suggests that the eccentricity becomes even higher when GWs become dominant,leading to rapid coalescences (binary lifetime $\lesssim1 {\rm ~Gyr}$). Despitethe lower merger rates in the empty loss cone case, due to their higher massratios and lower redshifts, the GWB in the full/empty loss cone models arecomparable ($0.70\times10^{-15}$ and $0.53\times10^{-15}$ at a frequency of$1~{\rm yr}^{-1}$, respectively). Finally, we compute the effects of higheccentricities on the GWB spectrum.

Interactions between multiple supermassive black holes in galactic nuclei: a solution to the final parsec problem