Merging stellar and intermediate-mass black holes in dense clusters: implications for LIGO, LISA, and the next generation of gravitational wave detectors

Context. The next generation of gravitational wave (GW) observatories would enable the detection of intermediate-mass black holes (IMBHs), an elusive type of BH expected to reside in the centres of massive clusters, dwarf galaxies, and possibly the accretion discs of active galactic nuclei. Intermediate-mass ratio inspirals (IMRIs), which are composed of an IMBH and a compact stellar object, constitute one promising source of GWs detectable by this new generation of instruments. Aims: We study the formation and evolution of IMRIs triggered by interactions between two stellar BHs and an IMBH inhabiting the centre of a dense star cluster, with the aim of placing constraints on the formation rate and detectability of IMRIs. Methods: We exploit direct N-body models varying the IMBH mass, the stellar BH mass spectrum, and the star cluster properties. Our simulations take into account the host cluster gravitational field and general relativistic effects via post-Newtonian terms up to order 2.5. These simulations are coupled with a semi-analytic procedure to characterise the evolution of the remnant IMBH after the IMRI phase. Results: Generally, the IMRI formation probability attains values of ∼5−50%, with larger values corresponding to larger IMBH masses. Merging IMRIs tend to map out the stellar BH mass spectrum, suggesting that IMRIs could be used to unravel the role of dynamics in shaping BH populations in star clusters harbouring an IMBH. After the IMRI phase, an initially almost maximal(almost non-rotating) IMBH tends to significantly decrease(increase) its spin. Under the assumption that IMBHs grow mostly via repeated IMRIs, we show that only sufficiently massive (Mseed > 300 M⊙) IMBH seeds can grow up to MIMBH > 103 M⊙ in dense globular clusters (GCs). Assuming that these seeds form at a redshift of z ∼ 2−6, we find that around 1−5% of them would reach typical masses of ∼500−1500 M⊙ at redshift z = 0 and would exhibit low spins, generally SIMBH < 0.2. Measuring the mass and spin of IMBHs involved in IMRIs could help to unravel their formation mechanism. We show that LISA can detect IMBHs in Milky Way GCs with a signal-to-noise ratio S/N = 10−100, or in the Large Magellanic Cloud, for which we get a S/N = 8−40. More generally, we provide the IMRI merger rate for different detectors, namely LIGO (ΓLIGO = 0.003−1.6 yr−1), LISA (ΓLISA = 0.02−60 yr−1), ET (ΓET = 1−600 yr−1), and DECIGO (ΓDECIGO = 6−3000 yr−1). Conclusions: Our simulations explore one possible channel for IMBH growth, namely via merging with stellar BHs in dense clusters. We find that the mass and spin of the IMRI components and the merger remnant encode crucial information about the mechanisms that regulate IMBH formation. Our analysis suggests that the future synergy among GW detectors will enable us to fully unravel IMBH formation and evolution....