tokens: also listed in Table 2 . The decay time-scale is in these cases of the order of or shorter than the decay time of the previous stages, meaning that the coalescence would supposedly happen promptly once the hard binary is formed. None the less, the hardening and inspiral time-scales we inferred in Table 2 are definitely longer than the ≈15 Myr found by Khan et al. ( 2016 ), who addressed the pairing of ∼10 8 M ⊙ MBHs in a massive early-type galaxy at z ≈ 3, which would become a BCG by z ≈ 0. The discrepancy in the time-scales has to be attributed to the large central density in their merging systems, which is nearly 10 5 M ⊙ pc −3 at 10 pc (their fig. 6), i.e. ∼100 times larger than in our case (see also the discussion in Khan et al. 2018 ). 13 Note that the system they explored is not a main-sequence galaxy as the one presented in this paper, but instead it is a more massive and relatively rare galaxy. For this reason, we expect the time-scales we obtain in this paper to be more representative for the bulk of the galaxy population. Note that the hardening and inspiral time-scales extrapolated for our system do not account for the effect of gas (which is expected to assist the orbital decay at scales larger than a pc, but could hinder it at smaller scales; Souza Lima et al. 2020 ), and the possibility that the fragmentation of a non-resolved circumnuclear disc could render even this smaller-scale decay very stochastic (Fiacconi et al. 2013 ; Souza Lima et al. 2017 ). If we neglect these complications, our estimates suggest that the whole pairing process could take as short as a few hundred Myr and probably less than a Gyr for the majority of the studied MBHs; that is, the MBH merger is likely going to occur at z = 5–3.5, except for the MBH that undergoes the bar-induced ejection. Concerning the fate of this latter object, one could wonder how likely this scattering event is. Even if our statistical sample is too limited to make strong statements here, it is worth mentioning that a test simulation with virtually the same initialization as the current one also showed evidence for a bar slingshot, suggesting that the event can be relatively common. 4.5 Caveats and limitations Our simulation has the important limitation that MBH accretion and feedback were not taken into account. However, it is reasonable to expect that feedback would render DF even less efficient by evacuating the gas mass around an MBH (see e.g. Park Bogdanović 2017 ; Souza Lima et al. 2017 ; del Valle Volonteri 2018 for a smaller scale analysis of this aspect). Each inspiralling MBH may also enhance its mass by accreting gas along the decay: in this respect, it is worth remembering that the MBH masses we adopted in the current run are relatively large for z ≳ 6, thus we do not expect this to represent a major issue. We also note that accretion on to MBHs may be used in future high-redshift surveys to sample offset or dual active galactic nuclei (e.g. Capelo et al. 2017 ; De
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O