Post-Newtonian evolution of massive black hole triplets in galactic nuclei -- III. A robust lower limit to the nHz stochastic background of gravitational waves

Inspiraling massive black-hole binaries (MBHBs) forming in the aftermath of galaxy mergers are expected to be the loudest gravitational-wave (GW) sources relevant for pulsar-timing arrays (PTAs) at nHz frequencies. The incoherent overlap of signals from a cosmic population of MBHBs gives rise to a stochastic GW background (GWB) with characteristic strain around $h_c\sim10^{-15}$ at a reference frequency of 1 yr$^{-1}$, although uncertainties around this value are large. Current PTAs are piercing into the GW amplitude range predicted by MBHB-population models, but no detection has been reported so far. To assess the future success prospects of PTA experiments, it is therefore important to estimate the minimum GWB level consistent with our current understanding of the formation and evolution of galaxies and massive black holes (MBHs). To this purpose, we couple a semianalytic model of galaxy evolution and an extensive study of the statistical outcome of triple MBH interactions. We show that even in the most pessimistic scenario where all MBHBs stall before entering the GW-dominated regime, triple interactions resulting from subsequent galaxy mergers inevitably drive a considerable fraction of the MBHB population to coalescence. At frequencies relevant for PTA, the resulting GWB is only a factor of 2-to-3 suppressed compared to a fiducial model where binaries are allowed to merge over Gyr timescales. Coupled with current estimates of the expected GWB amplitude range, our findings suggest that the minimum GWB from cosmic MBHBs is unlikely to be lower than $h_c\sim 10^{-16}$ (at $f = 1$ yr$^{-1}$), well within the expected sensitivity of projected PTAs based on future observations with FAST, MeerKAT and SKA.