Eccentric binary black hole inspiral-merger-ringdown gravitational waveform model from numerical relativity and post-Newtonian theory

We present a prescription for computing gravitational waveforms for the inspiral, merger and ringdown of nonspinning moderately eccentric binary black hole systems. The inspiral waveform is computed using the post-Newtonian expansion and the merger waveform is computed by interpolating a small number of quasicircular NR waveforms. The use of circular merger waveforms is possible because binaries with moderate eccentricity circularize in the last few cycles before the merger, which we demonstrate up to mass ratio $q=m_1/m_2=3$. The complete model is calibrated to 23 numerical relativity (NR) simulations starting $\approx 20$ cycles before the merger with eccentricities $e_{\mathrm{ref}}\le 0.1$ and mass ratios $q\le 3$, where $e_{\mathrm{ref}}$ is the eccentricity $\approx 7$ cycles before the merger. The NR waveforms are long enough that they start below 30 Hz (10 Hz) for BBH systems with total mass $M\ge 80M_{\odot }$ ($230M_{\odot }$). We find that, for the sensitivity of advanced LIGO at the time of its first observing run, the eccentric model has a faithfulness with NR of over 97% for systems with total mass $M\ge 85M_{\odot }$ across the parameter space ($e_{\mathrm{ref}}\le 0.1$, $q\le 3$). For systems with total mass $M\ge 70M_{\odot }$, the faithfulness is over 97% for $e_{\mathrm{ref}}\lesssim 0.05$ and $q\le 3$. The NR waveforms and the Mathematica code for the model are publicly available.

Figure 1

The NR configurations plotted as a function of eccentricity $e_{\mathrm{comm}}$ and mean anomaly $l_{\mathrm{comm}}$ at the common frequency $x_{\mathrm{comm}}=0.075$ for different mass ratios $q$.

Figure 2

An NR waveform with eccentricity $e_{\mathrm{comm}}=0.189$ (Case 9). The top panel shows the real part and amplitude of the dominant $\ell =2$, $m=2$ spherical harmonic mode of the strain, and the bottom panel shows the frequency of this mode, computed as ${\omega }_{22}=\frac{d}{dt}\arg h_{22}$. The oscillations in $|h_{22}|$ and ${\omega }_{22}$ are characteristic features of eccentricity.

Figure 3

Circularization of $q=3$ nonspinning binary black hole waveforms. Shown in the upper panels are the amplitude, $A$, and frequency, ${\omega }_{22}$, of the $l=2$, $m=\pm 2$ mode of the gravitational wave strain. The lower panels show the fractional deviations from the noneccentric results $A_{22,\mathrm{circ}}$ and ${\omega }_{22,\mathrm{circ}}$. The gray horizontal lines in the lower panels indicate $\pm 4\%$, and the vertical lines indicate $t=t_{\mathrm{peak}}-30M$.

Figure 4

PN fit to NR frequency to measure eccentric parameters at $x=0.11$ shortly before merger. The vertical lines indicate the interval in which the fit was performed.

Figure 5

Tests of the circular merger model (CMM) for NR waveforms which were not used to construct it. Plotted in (a) and (b) are the GW frequency and amplitude from NR simulations (solid curves) and the CMM (dashed curves). We see good agreement. In (c) and (d) are plotted the phase and relative amplitude differences between NR and the CMM. (e) shows the real part of the $\ell =2$, $m=2$ mode of the gravitational wave strain, and (f) shows the unfaithfulness between NR and the CMM using the aLIGO O1 detector configuration (see Sec. 7a).

Figure 6

Agreement of the time-to-merger model with NR. The time between the reference point and the peak of $|h_{22}|$ is shown as a circle for each NR simulation, and a cross for the fitted model from Eq. (9).

Figure 7

Combining the different ingredients to produce the IMR waveform. The top panel shows the transition function $\alpha$ which is used to blend the amplitude, $A$, and frequency, $\omega$, between the eccentric PN and circular NR waveforms. The middle and bottom panels show $A$ and $\omega$ from the NR simulation, the PN model fitted to it at the reference point $t_{\mathrm{ref}}$, and the circular merger model (CMM) with peak $t_{\mathrm{peak}}$ determined from the time-to-merger fit $\mathrm{\Delta }T$. The IMR model is constructed by blending the PN and CMM quantities using $\alpha$. $t_2$ indicates the end of the fitting window.

Figure 8

Fourier transforms of circular and eccentric waveforms. The eccentric waveform has been truncated in the time domain to two different lengths. This allows us to assess the effect of time-domain truncation on the Fourier transform.

Figure 9

Gravitational wave frequency comparison between IMR model and NR.

Figure 10

Gravitational wave strain comparison between IMR model and NR. Top: quasicircular case with $q=3$; some dephasing is visible. Middle: an eccentric equal-mass case, which shows excellent agreement. Bottom: $q=3$ with the highest eccentricity configuration, which shows the worst agreement of all the cases.

Figure 11

Phase difference between IMR model and NR.

Figure 12

Amplitude difference between IMR model and NR.

Figure 13

The unfaithfulness, $1-F$, for the aLIGO O1 detector configuration, between the eccentric IMR model and the NR simulations, as a function of the total binary mass. Masses for which the NR waveform starts at a frequency higher than the detector’s $f_{\min }$ are omitted from the plot. The horizontal line shows the 3% unfaithfulness target, and the vertical line shows $70M_{\odot }$, roughly corresponding to the mass of GW150914.

Figure 14

The unfaithfulness, $1-F$, for the aLIGO design detector configuration, between the eccentric IMR model and the NR simulations, as a function of the total binary mass. Masses for which the NR waveform starts at a frequency higher than the detector’s $f_{\min }$ are omitted from the plot. The horizontal line shows the 3% unfaithfulness target, and the vertical line shows $70M_{\odot }$, roughly corresponding to the mass of GW150914.