Waveform model for an eccentric binary black hole based on the effective-one-body-numerical-relativity formalism

Binary black hole systems are among the most important sources for gravitational wave detection. They are also good objects for theoretical research for general relativity. A gravitational waveform template is important to data analysis. An effective-one-body-numerical-relativity (EOBNR) model has played an essential role in the LIGO data analysis. For future space-based gravitational wave detection, many binary systems will admit a somewhat orbit eccentricity. At the same time, the eccentric binary is also an interesting topic for theoretical study in general relativity. In this paper, we construct the first eccentric binary waveform model based on an effective-one-body-numerical-relativity framework. Our basic assumption in the model construction is that the involved eccentricity is small. We have compared our eccentric EOBNR model to the circular one used in the LIGO data analysis. We have also tested our eccentric EOBNR model against another recently proposed eccentric binary waveform model; against numerical relativity simulation results; and against perturbation approximation results for extreme mass ratio binary systems. Compared to numerical relativity simulations with an eccentricity as large as about 0.2, the overlap factor for our eccentric EOBNR model is better than 0.98 for all tested cases, including spinless binary and spinning binary, equal mass binary, and unequal mass binary. Hopefully, our eccentric model can be the starting point to develop a faithful template for future space-based gravitational wave detectors.

Figure 1

Test for two identical spinless black holes with an eccentricity $e_0=0$. The four top panels are comparison of the ($\ell =2$, $m=2$) mode of the gravitational waveform for SEOBNRE and SEOBNRv1. The bottom two panels are comparison of the energy flux for the one generated with waveform (60), $-\frac{dE}{dt}$ given in (63) and $-\frac{dE}{dt}{|}_{(C)}$ in (c1). In the plot, the energy flux generated with waveform (60) is marked with “flux based on waveform”. The $-\frac{dE}{dt}$ given in (63) is marked with “with elliptic correction”. The $-\frac{dE}{dt}{|}_{(C)}$ in (c1) is marked with “without elliptic correction”. The SEOBNRE model is used in the energy flux comparison.

Figure 2

Quantitative comparison of waveform $h_{22}$ corresponding to Fig. 1. In the plot, $\mathrm{\Delta }|h_{22}R/M|\equiv \mathrm{abs}(|h_{22,\mathrm{SEOBNRE}}R/M|-|h_{22,\mathrm{SEOBNRv}1}R/M|)$, $\mathrm{\Delta }(\text{phase of}{h}_{22})\equiv \mathrm{abs}(\text{phase of}{h}_{22,\mathrm{SEOBNRE}}-\text{phase of}{h}_{22,\mathrm{SEOBNRv}1})$.

Figure 3

Test for two identical spinless black holes with eccentricity $e_0=0.003$ at $M{f}_0\approx 0.0015$. The comparison of orbital evolution for $e_0=0.003$ and $e_0=0$. The result of $e_0=0$ is from the SEOBNRv1 code, and the $e_0=0.003$ result is from the SEOBNRE model. The inset of the plot is the blowup of the corresponding range.

Figure 4

Similar to Fig. 1 but for two identical spinless black holes with an eccentricity $e_0=0.003$ at $M{f}_0\approx 0.0015$. The result of $e_0=0$ is determined by the SEOBNRv1 code, and the $e_0=0.003$ result is from the SEOBNRE model.

Figure 5

Similar to Fig. 1 but for two identical spinless black holes with an eccentricity $e_0=0.03$ at $M{f}_0\approx 0.0015$. The result of $e_0=0$ is from the SEOBNRv1 code, and the $e_0=0.03$ result is from the SEOBNRE model.

Figure 6

Amplitude and phase comparison for $h_{22}$ corresponding to the case $e_0=0.03$ shown in Fig. 5. The result of $e_0=0$ is from the SEOBNRv1 code, and the $e_0=0.03$ result is from the SEOBNRE model.

Figure 7

Similar to Fig. 1 but for two identical spinless black holes with eccentricity $e_0=0.3$ at $M{f}_0\approx 0.0015$. The result of $e_0=0$ is from the SEOBNRv1 code, and the $e_0=0.3$ result is from the SEOBNRE model.

Figure 8

Amplitude and phase comparison for $h_{22}$ corresponding to the case shown in Fig. 7. The result of $e_0=0$ is from the SEOBNRv1 code, and the $e_0=0.3$ result is from the SEOBNRE model.

Figure 9

Comparison between the ax model and the SEOBNRE model for the $e_0=0$ case.

Figure 10

Comparison between an ax model ($e_0=0.1$, $x_0=0.05$) and the SEOBNRE model ($e_0=0.15$, $M{f}_0\approx 0.001477647$).

Figure 11

Comparison between the ax model ($e_0=0.3$, $x_0=0.05$) and the SEOBNRE model ($e_0=0.4$, $M{f}_0\approx 0.001477647$).

Figure 12

SXS:BBH:0091 one is the spec simulation result for an equal mass spinless binary black hole with $e_0=0.02181$ at an orbital frequency 0.0105565727235. The SEOBNRE one corresponds to $e_0=0.1$ at $M{f}_0\approx 0.001477647$ for an equal mass spinless binary black hole.

Figure 13

SXS:BBH:0106 one is the spec simulation result for two spinless black holes with a mass ratio $1:5$ start orbit with $e_0=0.02181$ at an orbital frequency 0.0105565727235. The SEOBNRE one corresponds to $e_0=0.1$ at $M{f}_0\approx 0.001477647$ for two spinless black holes with a mass ratio $1:5$.

Figure 14

SXS:BBH:0323 one is the spec simulation result for two spinless black holes with an equal mass start orbit with $e_0=0.1935665$ at an orbital frequency 0.0146842176288. SEOBNRE one corresponds to $e_0=0.3$ at $M{f}_0\approx 0.001477647$ for two black holes with a mass ratio $11:9$ and spin ${\chi }_1=0.33$ and ${\chi }_2=-0.44$, respectively, which corresponds to the setting of SXS:BBH:0323.

Figure 15

Comparison between the Teukolsky equation based model (marked with ‘Teukolsky’) and the SEOBNRE model. All cases here admit a mass ratio about $1:1000$, initial eccentricity $e_0=0.3$, and the initial separation parameter $p_0=12$. From top row to bottom row, the corresponding initial orbital angular momentum are $\chi =0.9$, $p_{{\varphi }_0}=3.75$; $\chi =0.5$, $p_{{\varphi }_0}=3.86$; $\chi =-0.5$, $p_{{\varphi }_0}=4.20$, and $\chi =-0.9$, $p_{{\varphi }_0}=4.36$, respectively.

Figure 16

Phase comparison of $h_{+}-i{h}_{\times }$ between the Teukolsky equation based model and the SEOBNRE model. Cases correspond to those of Fig. 15, respectively.