Post-Newtonian dynamics in dense star clusters: Formation, masses, and merger rates of highly-eccentric black hole binaries

Using state-of-the-art dynamical simulations of globular clusters, including radiation reaction during black hole encounters and a cosmological model of star cluster formation, we create a realistic population of dynamically formed binary black hole mergers across cosmic space and time. We show that in the local universe, 10% of these binaries form as the result of gravitational-wave emission between unbound black holes during chaotic resonant encounters, with roughly half of those events having eccentricities detectable by current ground-based gravitational-wave detectors. The mergers that occur inside clusters typically have lower masses than binaries that were ejected from the cluster many Gyrs ago. Gravitational-wave captures from globular clusters contribute $1–2{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ to the binary merger rate in the local universe, increasing to $\gtrsim 10{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ at $z\sim 3$. Finally, we discuss some of the technical difficulties associated with post-Newtonian scattering encounters, and how care must be taken when measuring the binary parameters during a dynamical capture.

Figure 1

Formation rate of GCs as a function of cosmic time (where $t=0$ at $z=10$). In the top panel, we show the rate of GC formation over all galaxy halo masses, based on the model of [68], and the three metallicity bins that we use to assign individual ages to our GC models. On the bottom panel, we show the overall merger rate of BBHs from our GC models, convolved with the GC formation rate. The numbers are normalized to the total number of BBH mergers from clusters of all metallicities.

Figure 2

A pN integration of an isolated binary with close separation and high eccentricity, typical of GW capture BBH mergers with high eccentricities. In the top panel, we show the energy of the binary over two orbits when only the conservative terms are included. Because the pN expansion explicitly does not conserve energy beyond the highest order in $v/c$, the binary introduces a significant energy increase during the high-velocity pericenter passage. This error does not occur in the purely Newtonian case. In the bottom panel, we show the variation in the Newtonian eccentricity near merger. The integration of Eqs. (3) and (4) from initial conditions extracted at instantaneous separations of $10M$ and $100M$ are shown in dashed orange and dotted red, respectively.

Figure 3

The eccentricities at a GW frequency of 10 Hz for all BBHs from GCs. We show separately the eccentricities for the primordial binaries (BBHs from preexisting stellar binaries, in dotted green), the BBHs that merge after being ejected from the cluster (in dot-dashed blue), the BBHs that merge in the cluster as isolated binaries (in dashed orange), and the binaries which merge due to GW emission during resonant three- and four-body encounters between BHs (in solid red). Each curve is normalized to the total number of BBH mergers (in solid gray) and weighted using the cosmological model described in Sec. 2a. The top panel shows all mergers, while the bottom panel is restricted to mergers in the local universe ($z<1$). The insert in the bottom panel shows the cumulative distribution of eccentricities for all BBH mergers from GCs at different redshifts. In each plot, we show the minimum measurable eccentricity of BBH mergers in Advanced LIGO/Virgo [$e\sim 0.05$, from 106].

Figure 4

The semimajor axes and eccentricities of the ejected and in-cluster BBH mergers immediately following their last dynamical interaction in the cluster. The ejected mergers typically leave the cluster with smaller semimajor axes (dot-dashed blue, in the top panel) than the in-cluster mergers (dashed orange), and follow a roughly thermal eccentricity distribution (middle panel). The in-cluster mergers typically occur when a BBH leaves a resonant encounter with a very high eccentricity, prompting a rapid merger before another BH or stellar encounter can occur. In the bottom panel, we show jointly the semimajor axis and eccentricity for 1000 BBHs drawn from our weighted population of BBH mergers, along with the merger times for $20M_{\odot }+20M_{\odot }$ binaries at those semimajor axes and eccentricities.

Figure 5

The fraction of energy lost to GWs during a resonant three- or four-body encounter, normalized to the total initial energy (kinetic and potential) of the encounter. The ejected and in-cluster mergers (dot-dashed blue and dashed orange) typically lose only a small fraction of their energy to GWs, while the GW capture events (in solid red) lose significant amounts of energy to GWs during the Fewbody encounter.

Figure 6

The masses of BBH mergers from different formation pathways for all mergers that occur in the local universe ($z<1$). In the top panel, we show the weighted total masses of the BBHs, normalized by the total number of mergers. Although there is a trend for BBHs which merge in the cluster (both isolated binaries and GW captures) to have lower masses, this trend becomes less clear past $M\sim 80M_{\odot }$ because of the contribution from second-generation mergers (those with at least one component formed in a previous BBH merger). In the bottom panel, we show the fraction of mergers of each type that contribute to the total number of mergers at a given mass.

Figure 7

Similar to Fig. 6, but excluding any mergers whose components were formed in a previous BBH merger. This enforces a much stronger cutoff in total mass, with only a few objects having total masses greater than $81M_{\odot }$ (the result of mass transfer onto BHs during previous star-BH encounters). Here the trend towards ejected mergers having larger masses than in-cluster mergers and GW captures is significantly more pronounced.

Figure 8

The formation of the highly eccentric GW capture events. In the top plot, we show the measured eccentricities at a separation of $500M$ (solid gray), as well as the eccentricities once the binaries reach a peak GW frequency of 10 Hz and 40 Hz (in solid black and dotted red, respectively). All of the binaries whose peak GW frequency is already in the LIGO band ($f>10\mathrm{Hz}$), which we record as having eccentricities of 0.99, evolve to lower, measurable values by 40 Hz. In the bottom panel, we show the pericenter separation for all the binaries at $500M$. Those highly eccentric captures with peak frequencies $\gtrsim 10\mathrm{Hz}$ all form with very low pericenters distances, typically less than $30M$. In both cases, the peak of systems with $e>1$ are recorded as direct collisions. In reality, these systems involved close pericenter passages of less than $10M$, which we cannot resolve using our pN scattering code.

Figure 9

The cumulative distribution of merger times for the ejected mergers, in-cluster mergers, and GW captures. As expected, the ejected BBHs typically merge later due to the large delay times between ejection and merger that many systems can experience. The in-cluster mergers and GW captures typically merge earlier, with nearly identical distributions. Note that these distributions are for the merger times after GC formation, and have not been convolved with the cluster formation rate from Fig. 1.

Figure 10

The merger rate of GW captures across cosmic time. Here, we use the merger rate from the in-cluster mergers of [69], normalized to the fraction of mergers that occur as highly eccentric captures. We show the merger rate of all GW captures in red, and show how the merger rate depends on the initial virial radii of the clusters. In black, we show the fraction of these events which merge with eccentricities greater than 0.05 and 0.1 in dashed and solid black, respectively.