Measuring eccentricity in binary black hole inspirals with gravitational waves

When binary black holes form in the field, it is expected that their orbits typically circularize before coalescence. In galactic nuclei and globular clusters, binary black holes can form dynamically. Recent results suggest that $\approx 5\%$ of mergers in globular clusters result from three-body interactions. These three-body interactions are expected to induce significant orbital eccentricity $\gtrsim 0.1$ when they enter the Advanced LIGO and Virgo band at a gravitational-wave frequency of 10 Hz. Measurements of binary black hole eccentricity therefore provide a means for determining whether or not dynamic formation is the primary channel for producing binary black hole mergers. We present a framework for performing Bayesian parameter estimation on gravitational-wave observations of eccentric black hole inspirals. Using this framework, and employing the nonspinning, inspiral-only EccentricFD waveform approximant, we determine the minimum detectable eccentricity for an event with masses and distance similar to GW150914. At design sensitivity, we find that the current generation of advanced observatories will be sensitive to orbital eccentricities of $\gtrsim 0.05$ at a gravitational-wave frequency of 10 Hz, demonstrating that existing detectors can use eccentricity to distinguish between circular field binaries and globular cluster triples. We compare this result to eccentricity distributions predicted to result from three black hole binary formation channels, showing that measurements of eccentricity could be used to infer the population properties of binary black holes.

Figure 1

Posterior distributions of eccentricity at 10 Hz for two simulated GW150914-like events with eccentricities of $e=0.1$ (blue) and $e=0$ (red). The dotted vertical lines correspond to the 95% credible intervals.

Figure 2

Posterior distributions for the primary black hole mass ($m_1$), secondary black hole mass ($m_2$), eccentricity ($e$), distance ($d_L$), and inclination ($\iota$). The waveform is an eccentric event ($e=0.1$) with GW150914-like black hole masses and distance, with the true values given by the orange lines. Contours in the two-dimensional posteriors represent the 68%, 95%, and 99% confidence intervals. The mean recovered posteriors and 95% confidence intervals are displayed at the top of each one-dimensional posterior distribution.

Figure 3

Overlap (red) for Advanced LIGO and Virgo (dash-dotted), CE (dashed) and ET (dotted), and the Bayes factor (blue points fit by nonlinear least squares, shaded region is the five-sigma fit error) as a function of the initial eccentricity, defined at $f_{\mathrm{GW}}=10\mathrm{Hz}$ for events with GW150914-like parameters. The horizontal lines correspond to thresholds of $\ln (\mathcal{B})=8$ and $1-\mathcal{O}=1/{\rho }_0^2$. The inset shows a zoom-in on the points at which the waveform overlap results for the CE and ET detectors cross their respective thresholds (dashed and dotted horizontal lines).

Figure 4

Eccentricity distributions at $f_{\mathrm{GW}}=10\mathrm{Hz}$ for different eccentric BBH with GW150914-like masses. The green distribution shows binaries formed in globular clusters [43]. From left to right: the first peak is from ejected binaries, the second is from two-body mergers in the binary, and the third peak is from three-body mergers [44]. Field triples are shown in purple [85]. Direct capture within galactic nuclei is shown in orange [80]. Vertical lines correspond to the different minimum distinguishable eccentricities calculated from the overlap (red) and Bayes factor (blue) methods for the eccentric GW150914-like events analyzed in Table 2.

Figure 5

Posterior distributions for the select waveform parameters not shown in Fig. 2. Contours in the two-dimensional posteriors represent the 68%, 95%, and 99% confidence intervals and the true values are indicated by orange lines. Note the coalescence time ($t_c$) is in units of ms either side of the true value of 1180002601 s.

Figure 6

Growth of Bayes factor for events with increasing combined Hanford and Livingston S/N. The dashed horizontal line corresponds to the detection threshold at $\ln (\mathcal{B})=8$. The black curve corresponds to the line of best fit.

Figure 7

Comparison between the Bayes factor for events with total masses of $M_{\mathrm{tot}}=30{\mathrm{M}}_{\odot }$, $60{\mathrm{M}}_{\odot }$ or $90{\mathrm{M}}_{\odot }$. Each of the events being compared has the same S/N and mass ratio. The black curve corresponds to the line of best fit. The dashed horizontal line corresponds to the detection threshold of $\ln (\mathcal{B})=8$.