Eccentric black hole mergers forming in globular clusters

Johan Samsing

Phys. Rev. D 97, 103014 – Published 21 May 2018

Abstract

We derive the probability for a newly formed binary black hole (BBH) to undergo an eccentric gravitational wave (GW) merger during binary-single interactions inside a stellar cluster. By integrating over the hardening interactions such a BBH must undergo before ejection, we find that the observable rate of BBH mergers with eccentricity $> 0.1$ at 10 Hz relative to the rate of circular mergers can be as high as $\sim 5 %$ for a typical globular cluster (GC). This further suggests that BBH mergers forming through GW captures in binary-single interactions, eccentric or not, are likely to constitute $\sim 10 %$ of the total BBH merger rate from GCs. Such GW capture mergers can only be probed with an $N$ -body code that includes general relativistic corrections, which explains why recent Newtonian cluster studies have not been able to resolve this population. Finally, we show that the relative rate of eccentric BBH mergers depends on the compactness of their host cluster, suggesting that an observed eccentricity distribution can be used to probe the origin of BBH mergers.

Received 28 November 2017

DOI:https://doi.org/10.1103/PhysRevD.97.103014

Physics Subject Headings (PhySH)

General relativity Gravitation Gravitational wave sources Gravitational waves

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Johan Samsing^*

Department of Astrophysical Sciences, Princeton University, Peyton Hall, 4 Ivy Lane, Princeton, New Jersey 08544, USA

^*jsamsing@gmail.com

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Vol. 97, Iss. 10 — 15 May 2018

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Figure 1
Formation of an eccentric BBH GW merger during a resonating binary-single interaction between three equal mass BHs. The location of the eccentric GW capture merger is denoted by “GW capture,” where the initial paths of the incoming BBH and single BH, are denoted by “Binary” and “Single,” respectively. The GW capture forms as a result of GW emission during a close encounter between two of the three BHs while they temporarily form a bound three-body state. Such GW capture mergers often appear highly eccentric at 10 Hz.
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Figure 2
Illustration of a BBH undergoing hardening binary-single interactions in a stellar cluster. Initially the BBH (labeled by “initial“) forms with a SMA $a_{ej} < a_{in} < a_{HB}$ , either dynamically or primordially, after which it sinks to the core due to dynamical friction. The BBH here undergoes a HB binary-single interaction, which classically concludes with the BBH receiving a kick velocity $v_{B}$ that unbinds it from the single and sends it back into the cluster. It then sinks back to the core, after which the process repeats. Each of these HB binary-single interactions gradually decreases the SMA of the BBH, which correspondingly leads to increasing dynamical kicks. When the SMA of the BBH reaches $a \approx a_{ej}$ , i.e., when the dynamical kick velocity is about the escape velocity of the cluster, then the following binary-single interaction will eject the BBH out of the cluster (labeled by “ejected”), after which it merges in isolation. However, if GW emission is included in the $N$ -body solver, then the BBH can also undergo a GW capture merger inside the cluster core during one of its hardening binary-single interactions, as illustrated in Fig. 1. The grey inset circle shows a zoom-in on the core region. As described, the BBH here undergoes binary-single interactions that either will lead to hardening (the SMA changes from $a$ to $δ a$ , labeled “hardening”), or a GW merger during the interaction if GR effects are included (labeled “GW merger”).
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Figure 3
The black solid contours show the ratio $P_{EM} (a_{in}, a_{ej}) / P_{CM}^{< t_{H}} (a_{ej})$ , evaluated for the relevant LIGO values $e_{f} = 0.1$ and $f = 10 Hz$ , as a function of the escape velocity of the host cluster $v_{esc}$ (x-axis), and the BH mass $m$ (y-axis). As described in Sec. 4, this ratio approximately equals the ratio between the present rate of eccentric GW capture mergers and ejected circular mergers, $R_{E / C} = Γ_{EM} / Γ_{CM}$ , which observationally can be used to constrain the fraction of all BBH mergers forming in clusters. As seen, the relative rate of BBH mergers with $e \geq 0.1$ at 10 Hz is $\sim 5 %$ for a typical GC, which interestingly suggests that eccentric LIGO sources assembled in clusters will be relatively frequent. The blue dotted contours show the integrated probability for a given BBH to undergo an isolated merger between encounters during hardening from SMA $a_{in}$ to $a_{ej}$ , $P_{IM} (a_{in}, a_{ej})$ , derived in Sec. 3d2. For this estimate we have assumed that $n_{s} = 10^{6} {pc}^{- 3}$ and that $v_{disp} \approx v_{esc}$ . The green region indicates where our estimate of $R_{E / C}$ is valid [ $P_{IM} (a_{in}, a_{ej}) ≪ 0.1$ , i.e., merger before ejection is unlikely], the red region where it breaks down (all BBHs will merge before ejection), and the grey region the transition. As seen, our estimate is valid for classical GC systems, but corrections are needed for describing dense nuclear star clusters.
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