On the Assembly Rate of Highly Eccentric Binary Black Hole Mergers

The most likely outcome of a binary–single interaction is a binary with an unbound single (Heggie 1975). In this study we generally refer to the final binary from that outcome as the post-interaction binary. Such post-interaction binaries are generally formed with an eccentricity distribution that to leading order follows a thermal distribution, and binding energies that are slightly higher than that of the initial target binary (Heggie 1975). As a result, the GW merger time of post-interaction binaries can be significantly shorter than the GW merger time of the initial target binary (Peters 1964). For this reason, the binary–single channel is considered to be an effective pathway for transforming binaries with long GW inspiral times into binaries that could merge within a Hubble time.

The rate of BBH mergers forming in this way has recently been studied by Rodriguez et al. (2015, 2016a) using detailed dynamical simulations of GCs. The rate of BBH mergers forming in GCs is estimated by these authors to be somewhere between 2 and 50 Gpc⁻³ yr⁻¹, with all of their resulting mergers having circularized long before entering the LIGO band. The population of post-interaction BBHs is thus not expected to contribute to the rate of highly eccentric BBH mergers.

High eccentricity GW inspirals have recently been shown to be produced during resonant binary–single interactions when PN terms are included in the three-body EOM (Gültekin et al. 2006; Samsing et al. 2014, 2016). As described in Samsing et al. (2014, 2016), a resonant interaction can be envisioned as a series of intermediate states that are each composed of a binary with a bound single. Between each of these states, the three objects undergo a strong interaction that results in a semi-random redistribution of orbital energy and angular momentum. Each intermediate state binary therefore has a finite probability for being formed with a high eccentricity, even if the initial target binary is circular. If the eccentricity is high enough, the intermediate state binary will undergo a GW inspiral and merge while still being bound to the single object. On the other hand, if the eccentricity is too low, the corresponding GW inspiral time is too long, and the single will simply return to the intermediate state binary after which a new binary–single state will form. As a result, GW inspirals will generally form at high eccentricity.

Although such GW inspirals are generally assembled with an initial high eccentricity, they do not necessarily come into the LIGO frequency band with the same high eccentricity. In fact, as described in Samsing et al. (2014), a finite fraction of the GW inspirals will circularize before entering the LIGO band. This population originates from the fraction of interactions, where the bound single in the intermediate state is sent out on an orbit with close to zero orbital energy. For this reason, it is important to not only identify and count the number of GW inspirals, but also to evolve them until they reach the frequency band observable by LIGO. We thus now turn our attention to the eccentricity distribution of BBH mergers forming through the binary–single channel at the time that their GW frequency is 10 Hz. We refer to this frequency threshold as coming into the LIGO band. As inferred from our previous arguments, the eccentricity distribution is expected to have two peaks: one arising from the post-interaction binary GW mergers, and another one from the GW inspirals forming during the three-body interaction.

For this study, we perform $2.5\times {10}^{5}$ numerical binary-single scatterings with 2.5 PN corrections included in the EOM (for details on our N-body code, the reader is referred to Samsing et al. 2016). The scatterings are between [BH($20{M}_{\odot }$), BH($20{M}_{\odot }$)] binaries with an initial semimajor axis (SMA) distribution that is uniform in $\mathrm{log}({a}_{0})$, where a₀ denotes the initial SMA between 0.1–0.2 au, and an incoming single BH($20{M}_{\odot }$). The relative velocity at infinity is set to ${v}_{\infty }=10$ km s⁻¹; however, our presented relative rate estimates do not depend on this value as long as we are in the HB limit (Samsing et al. 2014). These initial conditions (ICs) are inspired by the derived distributions of BBH mergers in our local universe from Rodriguez et al. (2016a). For each BBH that forms and eventually merges, we follow its SMA and eccentricity using the GW quadrupole equations given in Peters (1964), together with its GW peak frequency derived using the fitting formulae by Wen (2003). We only include BBHs that merge within a Hubble time; for the post-interaction binaries, we further require that the dynamical kick velocity in the three-body center of mass is >50 km s⁻¹. These two requirements are included for consistency, but do not significantly affect our results.

Our scattering results are shown in Figure 1. We first note that our derived post-interaction binary eccentricity distribution at 10 Hz is very similar to that obtained by Rodriguez et al. (2016a), using a more sophisticated GC modeling approach. While this is expected due to our choice of ICs, it gives credence to the validity of the formalism used here. We note that the main shape of the distribution depicted in Figure 1 originates from the thermal distribution of eccentricities (Heggie 1975), where both the BBH mass and initial SMA determine the location of the peak. The high-eccentricity component of the distribution, originating from our inclusion of GW emission in the EOM through the 2.5 PN term, is clearly evident in Figure 1 and contains $\gtrsim 1 \%$ of all BBH mergers. If 2.5 PN corrections are not included, this high-eccentricity component within the LIGO frequency band is absent from the distribution.

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Having numerically derived the relative fraction of highly eccentric mergers, it is important to explore how this fraction changes when the initial SMA a₀ and BH mass ${m}_{\mathrm{BH}}$ are altered. This is done in the following, assuming the equal-mass case. We start by noting that low-eccentricity mergers are dominated by post-interaction binaries that merge within a Hubble time, ${t}_{{\rm{H}}}$, while high-eccentricity ones are dominated by GW inspirals. The relative rate of highly eccentric BBH mergers can thus be approximated by the ratio between the GW inspiral cross-section, ${\sigma }_{{\rm{I}}}$, and the cross-section for post-interaction binaries to merge within time ${t}_{{\rm{H}}}$, denoted here as ${\sigma }_{{\rm{M}}}$. The cross-section ${\sigma }_{{\rm{I}}}$ was shown in Samsing et al. (2014) to scale as ${\sigma }_{{\rm{I}}}\propto {a}_{0}^{2/7}{m}_{\mathrm{BH}}^{12/7}$. For deriving ${\sigma }_{{\rm{M}}}$, we first express it as the cross-section for a close binary–single interaction, which scales $\propto \,{a}_{0}{m}_{\mathrm{BH}}$ (e.g., Samsing et al. 2014), times the probability for a post-interaction binary to merge within time ${t}_{{\rm{H}}}$, $P(\lt {t}_{{\rm{H}}})$. In the limit ${\tau }_{0}\lt {t}_{{\rm{H}}}$, where ${\tau }_{0}$ is the average merger time of post-interaction binaries with $e\approx 0$, the cross-section scales as ${\sigma }_{{\rm{M}}}\propto {a}_{0}{m}_{\mathrm{BH}}$, since $P(\lt {t}_{{\rm{H}}})\approx 1$. In the limit ${\tau }_{0}\gt {t}_{{\rm{H}}}$, the post-interaction binary eccentricity must instead be above a certain value, ${e}_{{\rm{M}}}$, for the corresponding merger time to be $\lt {t}_{{\rm{H}}}$. Assuming a thermal distribution for the eccentricities (Heggie 1975), one finds $P(\lt {t}_{{\rm{H}}})\approx 1-{e}_{{\rm{M}}}^{2}$. From relating this probability to the binary merger time (Peters 1964), we find ${\sigma }_{{\rm{M}}}\propto {a}_{0}^{-1/7}{m}_{\mathrm{BH}}^{13/7}$. As a result, the fraction of BBHs that merge with notable eccentricity, ${f}_{e\gt 0}$, can then be written as

$$\begin{eqnarray}&&{f}_{e\gt 0}\propto {a}_{0}^{-5/7}{m}_{\mathrm{BH}}^{5/7},{\tau }_{0}\lt {t}_{{\rm{H}}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{f}_{e\gt 0}\propto {a}_{0}^{3/7}{m}_{\mathrm{BH}}^{-1/7},{\tau }_{0}\gt {t}_{{\rm{H}}}.\end{eqnarray} \tag{ 2 }$$

We note here that ${\tau }_{0}$ in the equal-mass case is approximately given by the merger time of the initial target binary. If we keep ${m}_{\mathrm{BH}}$ fixed, we then see that ${f}_{e\gt 0}$ will always increase with a₀, provided that ${\tau }_{0}\approx {t}_{{\rm{H}}}$. As ${\tau }_{0}\approx {t}_{{\rm{H}}}$ for our ICs, we can safely conclude that an increase in a₀ will actually lead to a larger ${f}_{e\gt 0}$ than the one reported in this Letter. The scaling solutions further illustrate that ${f}_{e\gt 0}$ is not expected to change much within the variations reported by Rodriguez et al. (2016a). The fraction of high-eccentricity BBH mergers derived here is likely to be an accurate estimate of the fraction for a broad range of GC properties.