Characterization of numerical relativity waveforms of eccentric binary black hole mergers

We introduce a method to quantify the initial eccentricity, gravitational wave frequency, and mean anomaly of numerical relativity simulations that describe nonspinning black holes on moderately eccentric orbits. We demonstrate that this method provides a robust characterization of eccentric binary black hole mergers with mass ratios $q\le 10$ and eccentricities $e_0\le 0.2$ fifteen cycles before merger. We quantify the circularization rate of a variety of eccentric numerical relativity waveforms introduced in E. A. Huerta et al. [arXiv:1901.07038] by computing overlaps with their quasicircular counterparts, finding that $50M$ before merger they attain overlaps $\mathcal{O}\ge 0.99$, furnishing evidence for the circularization of moderately eccentric binary black hole mergers with mass ratios $q\le 10$. We also quantify the importance of including higher-order waveform modes for the characterization of eccentric binary black hole mergers. Using two types of numerical waveforms, one that includes $(\ell ,|m|)=\{(2,2),(2,1),(3,3),(3,2),(3,1),(4,4),(4,3),(4,2),(4,1)\}$ and one that only includes the $\ell =|m|=2$ mode, we find that the overlap between these two classes of waveforms is as low as $\mathcal{O}=0.89$ for $q=10$ eccentric binary black hole mergers, underscoring the need to include higher-order waveform modes for the description of these gravitational wave sources. We discuss the implications of these findings for future source modeling and gravitational wave detection efforts.

Figure 1

A schematic of the algorithm used to characterize NR waveforms. First, $f_0$ and $e_0$ are roughly estimated using a grid search determining the optimal coordinate pair $(e_0^{*},f_0^{*})$. An iterative search is then performed on $f_0^{*}$ and $e_0^{*}$ to increase precision. With these refined ${\widehat{f}}_0$ and ${\widehat{e}}_0$, a second grid search is used to find the optimal ${\widehat{l}}_0$. The output is the optimal triple $({\widehat{e}}_0,{\widehat{f}}_0,{\widehat{l}}_0)$.

Figure 2

Removal of high-frequency noise from $M\omega$ using a Savitsky-Golay filter. Notice that this filtering scheme does not remove the true eccentricity features from the time-series data.

Figure 3

Comparison between the dimensionless orbital frequency, $M\omega$, of numerical relativity simulations and their corresponding post-Newtonian counterparts. Each post-Newtonian evolution was constructed using the ENIGMA model and optimal values for the triplet $({\widehat{e}}_0,{\widehat{f}}_0,{\widehat{l}}_0)$ provided by the algorithm described in Sec. 2. The extrema used in search cost functions are marked.

Figure 4

Comparison between NR waveforms and their ENIGMA counterparts. The ENIGMA waveforms were constructed using the optimal values for the triplet $({\widehat{e}}_0,{\widehat{f}}_0,{\widehat{l}}_0)$ as determined by the algorithm introduced in Sec. 2.

Figure 5

Overlap between eccentric numerical relativity waveforms and their quasicircular counterparts. We show the overlap, computed between a fiducial time, $t_0$, and merger, $t=0M$. These results show that moderately eccentric systems circularize prior to merger. Note also that the decrease in overlap as we go backwards in time is not monotonically decreasing. This is expected, since the overlap between the quasicircular and eccentric systems will be different if we start the comparison when the eccentric system is close to apoapse or periapse. This distinct property between eccentric and quasicircular orbits is not washed away by maximizing over phase and time of coalescence when computing the overlap.

Figure 6

Overlap between numerical relativity waveforms that include either the modes $(\ell ,|m|)=\{(2,2),(2,1),(3,3),(3,2),(3,1),(4,4),(4,3),(4,2),(4,1)\}$ (labeled as all modes in the panels), or just the $(\ell =|m|=2)$ mode. Higher-order waveform models become significant for asymmetric mass-ratio binary black hole systems.