Post-Newtonian Dynamics in Dense Star Clusters: Highly Eccentric, Highly Spinning, and Repeated Binary Black Hole Mergers

We present models of realistic globular clusters with post-Newtonian dynamics for black holes. By modeling the relativistic accelerations and gravitational-wave emission in isolated binaries and during three- and four-body encounters, we find that nearly half of all binary black hole mergers occur inside the cluster, with about 10% of those mergers entering the LIGO/Virgo band with eccentricities greater than 0.1. In-cluster mergers lead to the birth of a second generation of black holes with larger masses and high spins, which, depending on the black hole natal spins, can sometimes be retained in the cluster and merge again. As a result, globular clusters can produce merging binaries with detectable spins regardless of the birth spins of black holes formed from massive stars. These second-generation black holes would also populate any upper mass gap created by pair-instability supernovae.

Figure 1

The eccentricities of BBHs from the 24 GC models that merge at low redshifts. We calculate the eccentricity when the BBH enters the LIGO/Virgo detection band at a (circular) GW frequency of 10 Hz. The distribution is clearly trimodal: the first peak corresponds to BBHs which merger after ejection from the cluster (similar to Ref. [21], Fig. 10). The second peak corresponds to BBH mergers which occur in the cluster. The final peak, at $e>0.1$, corresponds to in-cluster mergers which occur during a strong encounter, when the BBH enters the LIGO/Virgo band during a GW capture. Note that the two distributions are normalized to the total number of mergers (in-cluster and ejected).

Figure 2

The total mass of merging BBHs from all 24 GC models. On the left, we show all mergers as a function of redshift, with the orange diamonds and blue points showing in-cluster and ejected mergers, respectively. The black circles show 2G mergers (both in-cluster and ejected) which have at least one component that was formed from a previous BBH merger. The right panel shows the mass distribution of these mergers at low redshifts ($z<1$). As the spins are increased from ${\chi }_b=0$ to ${\chi }_b=0.4$, the number of 2G mergers decreases significantly, as their progenitors were less likely to be retained in the cluster. See the discussion in Sec. 5. The red-dashed line indicates the maximum mass of first-generation BBHs ($\sim 81M_{\odot }$) with our assumed pair-instability supernova limit. The handful of first-generation BBHs that merge above this are the result of either stable mass transfer or stellar collisions prior to BH formation.

Figure 3

The distributions of ${\chi }_{\mathrm{eff}}$ from BBHs that merge at $z<1$, divided into bins of $15M_{\odot }$. Each bin shows the median (white line), 50th, 90th, and 99th percentiles of ${\chi }_{\mathrm{eff}}$ for all BBH mergers with that mass. For each binary, we average over $N={10}^3$ random spin orientations. For the 2G mergers, we use $N={10}^3$ times the probability of each component having been retained in the cluster following its earlier mergers (see discussion in Sec. 5). As the birth spins (${\chi }_b$) of the BHs are increased, the fraction of 2G BBHs retained in the cluster decreases; however, the overall magnitudes of ${\chi }_{\mathrm{eff}}$ increases, as the first generation of BBHs begin to produce mergers with measurable spins. Note that while the large spin magnitudes for BBHs with total masses above $80M_{\odot }$ does not depend on the birth spins, the number of mergers in that mass range decreases sharply with increasing ${\chi }_b$ (see Fig. 2).