Detecting intermediate-mass black holes in Milky Way globular clusters and the Local Volume with LISA and other gravitational wave detectors
The next generation of gravitational wave (GW) observatories would enable the detection of intermediate mass ratio inspirals (IMRIs), tight binaries composed of an intermediate-mass black hole (IMBH) and a compact stellar object. Here, we study the formation of IMRIs triggered by the interactions between two stellar BHs and an IMBH inhabiting the centre of a globular cluster via direct $N$-body models. Varying the IMBH mass, the stellar BH mass spectrum, and the star cluster properties, we find that IMRIs' formation probability depends on the initial conditions adopted, attaining overall values $\sim5-50\%$. Merging IMRIs tend to map the stellar BH mass spectrum, thus suggesting that their detections can provide clues about the co-evolution of IMBHs and stellar BHs. Given the post-merger GW recoil, typical globular clusters containing an IMBH with mass $M_{\rm IMBH}\gtrsim 10^3$ M$_\odot$ have a retention probability $>50\%$ only if the companion has a mass $M_{\rm BH}\leq 30$ M$_\odot$. Lower IMBHs are ejected from the cluster if $M_{\rm BH}>10$ M$_\odot$. At masses $M_{\rm IMBH}=10^2-10^3$ M$_\odot$, the remnant spin is strongly affected by the spin of the companion, whereas for $M_{\rm IMBH}>10^4$ M$_\odot$ it is preserved upon multiple mergers. In a 4 yr long mission, we suggest that LISA can unravel IMBHs in Milky Way globular clusters with a signal-to-noise ratio SNR$=10-100$, or in the Large Magellanic Cloud (SNR$=8-40$). Our inferred merger rate suggests that LIGO could detect $0.9-2.5$ yr$^{-1}$ IMRIs with $M_{\rm IMBH}\sim100$ M$_\odot$, whereas for LISA we find $6-60$ yr$^{-1}$ detections ($M_{\rm IMBH} \sim 4\times 10^4$ M$_\odot$). Future detectors like Einstein Telescope can deliver up to $10^3$ yr$^{-1}$ IMRIs, whereas DECIGO might allow $400-3800$ yr$^{-1}$ detections for $M_{\rm IMBH}<5\times10^4$ M$_\odot$.