tokens: taking its place. It is only when m 3 ∼ m 2 that genuinely chaotic dynamics can take place, in some cases leading to coalescence. Finally, Fig. 5 shows the merger fraction as a function of the initial eccentricity of the inner and outer binary. We note that the merger fraction increases with increasing e in , while it decreases with increasing e out . The dependence upon e in is readily understood, since highly eccentric inner binaries are closer to the efficient GW-emission stage and can easily be driven to coalescence by a relatively mild perturbation from a third body. The dependence upon e out is likely due to the fact that quasi-circular outer binaries form a stable hierarchical triplet for a comparatively longer time during the inspiral of m 3 , hence leaving more room to the development of K–L resonances, which are efficient at driving the inner binary to coalescence. Conversely, in very eccentric outer binaries, m 3 soon interacts with the inner binary at pericentre, entering the chaotic phase. Chaotic interactions are more likely to result in ejections rather than mergers, hence suppressing the overall merger fraction. Figure 5. Merger fraction (colour coded) as a function of the initial inner ( e in ) and outer ( e out ) binary eccentricities. 3.1.1 Importance of PN corrections As pointed out in the previous section, the K–L mechanism can be suppressed by general relativistic effects. In particular, relativistic precession tends to destroy the coherent pile-up of the perturbation that the third body induces on the inner binary, hence effectively damping the K–L resonances. 5 In order to quantify the impact of the relativistic precession on the merger fraction and to compare our results with previous work that neglected this effect (e.g. Iwasawa, Funato Makino 2006 ; Hoffman Loeb 2007 ), in Table 3 and in Fig. 6 we compare, only for the case m 1 = 10 9 M ⊙ , the merger fraction obtained with and without 1PN corrections. Overall, the merger fraction is substantially higher in the case without 1PN terms (right-hand panel of Fig. 6 ). As can be seen, the largest differences compared to the full case occur for q in ≪ 1, where, because of octupole-order terms, the K–L mechanism is maximally effective. On the contrary, for large q in the merger fractions with or without 1PN corrections are similar, because, as previously discussed, coalescences are mainly due to chaotic strong encounters, rather than to K–L oscillations. Our results highlight the importance of K–L resonances in inducing MBH mergers in triple systems, and the need to account at least for 1PN corrections. Figure 6. Comparison between the merger fraction in the case m 1 = 10 9 M ⊙ , when the dynamics is evolved with (left-hand panel) and without (right-hand panel) 1PN term. The suppression of the merger fraction for low q in , due to 1PN precession, is clearly visible (see discussion in the main text). Table 3. Comparison of the merger fraction from the simulations with m 1 =
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