Dynamical interactions and the black-hole merger rate of the Universe

Binary black holes can form efficiently in dense young stellar clusters, such as the progenitors of globular clusters, via a combination of gravitational segregation and cluster evaporation. We use simple analytic arguments supported by detailed $N$-body simulations to determine how frequently black holes born in a single stellar cluster should form binaries, be ejected from the cluster, and merge through the emission of gravitational radiation. We then convolve this “transfer function” relating cluster formation to black-hole mergers with (i) the distribution of observed cluster masses and (ii) the star formation history of the Universe, assuming that a significant fraction $g_{\mathrm{cl}}$ of star formation occurs in clusters and that a significant fraction $g_{\mathrm{evap}}$ of clusters undergo this segregation and evaporation process. We predict future ground-based gravitational wave detectors could observe $\sim 500(g_{\mathrm{cl}}/0.5)(g_{\mathrm{evap}}/0.1)$ double black-hole mergers per year, and the presently operating LIGO interferometer would have a chance (50%) at detecting a merger during its first full year of science data. More realistically, advanced LIGO and similar next-generation gravitational wave observatories provide unique opportunities to constrain otherwise inaccessible properties of clusters formed in the early Universe.

Figure 1

The distribution of GW merger time scales for circular (solid step function) and eccentric (solid smooth function, based on ${10}^4$ samples) $14M_{\odot }+14M_{\odot }$ binaries, assuming the binding energy is distributed logarithmically from ${10}^3{k}_{b}T$ to ${10}^5{k}_{b}T$ for $k_{b}T$ the characteristic cluster energy. This figure assumes $k_{b}T=0.5M_{\odot }(10\mathrm{km}{\mathrm{s}}^{-1}{)}^2/3$.

Figure 2

The distribution of binary birth times for those binary black holes merging at present, given three different assumptions for the star formation history in the Universe: Nagamine et al. [37] (solid line), Porciani and Madau’s [49] Eqs. (5) (dotted line) and (6) (dashed line). In all three cases, clusters formed more than 5 Gyr ago produce more BH-BH mergers than more recent cluster formation.