Large Gravitational Wave Phase Shifts from Strong 3-body Interactions in Dense Stellar Clusters

Abstract

The phase evolution of gravitational waves (GWs) can be modulated by the astrophysical environment surrounding the source, which provides a probe for the origin of individual binary black holes (BBHs) using GWs alone. We here study the evolving phase of the GW waveform derived from a large set of simulations of BBH mergers forming in dense stellar clusters through binary-single interactions. We uncover that a well-defined fraction of the assembled eccentric GW sources will have a notable GW phase shift induced by the remaining third object. The magnitude of the GW phase shift often exceeds conservative analytical estimates due to strong 3-body interactions, which occasionally results in GW sources with clearly shifted and perturbed GW waveforms. This opens up promising opportunities for current and future GW detectors, as observing such a phase shift can identify the formation environment of a BBH, as well as help to characterise the local properties of its surrounding environment.

Figures (6)

• Figure 1: Illustration of 3-body interaction resulting in a BBH merger with an observable GW Phase Shift.Top left: A stellar cluster with highlighted BH interactions, each of which is able to produce BBH mergers. Top right: Zoom-in on a binary-single interaction resulting in a BBH merger with the third object still bound (3-body merger). Turquoise- and pink lines show the trajectory of the merging BHs (true path), where the orange lines illustrate the path the BBH would have taken without the third object (reference path). The trajectory of the third BH is depicted in red. The white lines show lines-of-sight for an observer located in the lower right corner, where the green lines illustrate the spacial distance between the true- and the reference paths along the sight-lines, respectively. Bottom: GW strain as a function of time. The turquoise curve shows the GW signal for the observed BBH inspiral (true path), where the orange shows what the isolated BBH merger signal would look like (reference path). The two signals are shifted by the light crossing time between the true and the reference BBH paths, i.e. the time it takes the GWs to travel along the green lines. This gives rise to a unique observable GW phase shift that can be directly mapped to the BBH formation and environment.
• Figure 2: Distribution of maximum GW phase shifts from 3-body BBH mergers. Results from ${\cal PN}$ simulations between a BBH and a single incoming BH that concludes with two of the three BHs merging while the third BH is still bound (see Fig. \ref{['fig:GWPH_ill_1']} and Fig. \ref{['fig:orbit_ex']}). Each dot shows the maximum observable GW phase shift (y-axis) for each of these mergers, as a function of the corresponding GW peak frequency (bottom x-axis) or merger time (top x-axis), derived using the methods outlined in Sec. \ref{['sec:The Origin of GW Phase shifts']}. Each colour refers to a different SMA of the initial BBH before interaction: $\sim 1$ AU ( orange), $\sim 0.1$ AU ( red), and $\sim 0.01$ AU ( blue). The dashed coloured lines illustrate the asymptotic limit $\Delta{\phi} \propto f^{-13/3}$, where the horizontal dotted line indicates the analytical maximum value for $\Delta{\phi}$ assuming the distance between the BBH and the single BH, $R$, is similar to the initial SMA, $a$ (see Methods). The three highlighted examples, (A,B,C), are shown and studied further in Fig. \ref{['fig:orbit_ex']}.
• Figure 3: Examples of 3-body BBH mergers resulting in significant GW phase shifts. The shown interactions involve BHs with equal mass $m = 20M_{\odot}$, and initial SMA $a = 0.1$ AU for case (A) and (B), where case (C) has $a = 0.01$ AU. The filled-circles in the left figures indicate the initial positions of the BHs, where the filled-circles in the right figures show the end positions at merger. The orange lines illustrate the trajectory the merging BBHs would take if the third BH was not there (see Fig. \ref{['fig:GWPH_ill_1']}). Common for the cases that give rise to significant GW phase shifts is that the chaotic nature of the 3-body problem brings the BBH close to the remaining bound single BH near merger. This is clearly seen in these examples. GW phase shifts and corresponding peak frequencies for (A,B,C) are shown in Fig. \ref{['fig:dist_dphifp']}.
• Figure 4: Evolution of GW phase shift with GW peak frequency. The figure shows evolutionary tracks of the GW phase shift as a function of the GW peak frequency, $f$, from its initial value at formation, $f_0$, derived using Eq. \ref{['eq:AP_Dphi_f']}. The data is the same as the one shown in Fig. \ref{['fig:dist_dphifp']}, but only including the GW sources with a maximum GW phase shift of $>10^{-2}$. The filled-circles are identical to the ones shown in Fig. \ref{['fig:dist_dphifp']}, and illustrate correctly (within our analytical approximations) the maximum value for the GW phase shift along each track.
• Figure 5: Characteristics of phase-shifted 3-body BBH mergers.Top: the GW phase shift as a function of the peak GW frequency (see Fig. \ref{['fig:dist_dphifp']}) for the population with $a_0=0.1 AU$. The $+$-marked systems are those binaries whose time to merger at assembly ($t_m$) exceeds the orbital period of the initial binary ($T_0$). The black dashed line is the asymptotic limit $f^{-13/3}$ and the dotted line is the expected analytical upper threshold according to our analytical model. Middle: the distance $R$ at merger between the binary and third body, relative to the semi-major axis of the initial binary, $a_0$. Bottom: the maximum GW phase shift as a function of the merger time at the maximum (where $e=e_m$), relative to the outer orbital period of the tertiary in the 3-body merger ($T_\mathrm{out}$).