Survival Analysis of Intermediate-Mass Black Holes in Dense Star Clusters

Abstract

Recently, an intermediate-mass black hole (IMBH) candidate was announced in the Galactic globular cluster Omega Centauri. IMBHs at the lower end of the traditional mass range have also been detected through gravitational-wave transients, though their formation and subsequent growth linking the two mass scales remains a mystery. One way IMBHs may be produced is through the collapse of very massive stars produced by stellar collisions in dense stellar environments. However, IMBHs may be ejected from such environments by either dynamical recoil from binary-single scattering or gravitational-wave recoil following the merger of two black holes. We conduct Newtonian and post-Newtonian binary-single scattering experiments to study dynamical ejection in greater detail. We obtain fits to the probabilities for dynamical ejection, gravitational wave capture, and per-encounter hardening as a function of the binary mass ratio and hardness with respect to its environment. We borrow techniques from survival analysis (commonly used in studies of medicine, epidemiology, engineering, etc.) to develop a model to calculate the probability of IMBH binary ejection vs in-cluster merger. We confirm that the dynamical ejection probability strongly depends on both the mass ratio of the IMBH compared to other BHs in its environment and on the environment's velocity dispersion. We estimate that for a typical Milky Way globular cluster, IMBHs with mass $\lesssim10^3\,\mathrm{M}_\odot$ are unlikely to be retained until the present. Our results also suggest that IMBH mergers with $q\lesssim0.2$ may be detectable at higher redshifts with future gravitational wave instruments such as the Einstein Telescope and Cosmic Explorer.

Figures (11)

• Figure 1: (Left) Cumulative distributions of binary $v_\mathrm{kick}/v_\mathrm{orb}$ for different values of binary hardness $v_\mathrm{orb}/v_\mathrm{inf}$. The vertical dashed lines shows the escape speed of the background cluster in the corresponding normalized units for the CDF of the same color. Binaries with kick velocities to the right of this line are ejected. (Right) Normalized distributions of the fractional energy change of the binary $\Delta$ as a result of the interaction for the same selected values of $v_\mathrm{orb}/v_\mathrm{inf}$. Both sets of distributions show results from $q=0.6$ scattering experiments. In physical units, these selected values of $v_\mathrm{orb}/v_\mathrm{inf}$ correspond to $100\,\mathrm{M}_\odot$--$60\,\mathrm{M}_\odot$ binaries with semimajor axis $a_0=1.6\,\mathrm{au}$ (solid green) and $a_0=0.06\,\mathrm{au}$ (dashed purple) for $v_\mathrm{inf}=20\,\mathrm{km/s}$. In the limit $v_\mathrm{crit}\gg v_\mathrm{inf}$, the distribution of $\Delta$ converges to a single distribution. Thus, in the normalized units used here, making the binary harder or softer corresponds to changing the boundary for escape. For relatively soft binaries (green), the possibility of escape lies only in the tail of the recoil kick distribution, whereas for harder binaries (purple), escape may be possible in the bulk of the distribution.
• Figure 2: Branching ratios for ejection of the most massive BH in our $q=0.6$ interactions as a function of $v_\mathrm{orb}/v_\mathrm{inf}$. In most cases, the BH will be ejected as part of a binary (orange points), though in a minority of cases it may be ejected as a single object (blue points). We show the fit from Eq. (\ref{['eq:hillfit']}) in black. The fit parameters for the fit to the binary ejection probability are provided in Tab. \ref{['tab:hillfit']}.
• Figure 3: Branching ratios for ejection of the massive object in a binary for all simulated mass ratios (diamonds) with fits (dashed lines). Fit parameters are provided in Tab. \ref{['tab:hillfit']}.
• Figure 4: Distributions of $T_\mathrm{final}/T_\mathrm{int}$ (left) and normalized $v_\mathrm{kick}$ (right) for all interactions (solid blue) and subsets of the interactions (dashed orange) with $q=0.2$ and $v_\mathrm{orb}/v_\mathrm{inf} \approx73$. In the left panel, the subset includes only interactions for which the change in energy $\Delta\leq-0.1$, whereas in the right panel, it includes only interactions for which $T_\mathrm{final}\geq10\,T_\mathrm{min}$.
• Figure 5: Branching ratios for mass ratios $q = 0.6$ (top) and $q = 0.1$ (bottom) with different assumptions of the background cluster potential as a function of binary hardness. The Plummer model is shown in blue. The King models are shown in green, orange, red and purple with their associated $W_0=3,\,6,\,9,\,12$ respectively. The range of $W_0$ values chosen correspond to a representative range of measured concentration parameters $c=\log(r_t/r_0)$ reported by the Harris catalog of Milky Way GCs HarrisCatalog.