Effective-one-body waveforms calibrated to numerical relativity simulations: Coalescence of nonspinning, equal-mass black holes

We calibrate the effective-one-body (EOB) model to an accurate numerical simulation of an equal-mass, nonspinning binary black-hole coalescence produced by the Caltech-Cornell Collaboration. Aligning the EOB and numerical waveforms at low frequency over a time interval of $\sim 1000M$, and taking into account the uncertainties in the numerical simulation, we investigate the significance and degeneracy of the EOB-adjustable parameters during inspiral, plunge, and merger, and determine the minimum number of EOB-adjustable parameters that achieves phase and amplitude agreements on the order of the numerical error. We find that phase and fractional amplitude differences between the numerical and EOB values of the dominant gravitational-wave mode $h_{22}$ can be reduced to 0.02 radians and 2%, respectively, until a time $20M$ before merger, and to 0.04 radians and 7%, respectively, at a time $20M$ after merger (during ringdown). Using LIGO, Enhanced LIGO, and Advanced LIGO noise curves, we find that the overlap between the EOB and the numerical $h_{22}$, maximized only over the initial phase and time of arrival, is larger than 0.999 for equal-mass binary black holes with total mass $30–150M_{\odot }$. In addition to the leading gravitational mode (2, 2), we compare the dominant subleading modes (4, 4) and (3, 2) for the inspiral and find phase and amplitude differences on the order of the numerical error. We also determine the mass-ratio dependence of one of the EOB-adjustable parameters by calibrating to numerical inspiral waveforms for black-hole binaries with mass ratios $2:1$ and $3:1$. The results presented in this paper improve and extend recent successful attempts aimed at providing gravitational-wave data analysts the best analytical EOB model capable of interpolating accurate numerical simulations.

Figure 1

Numerical error estimates. Phase difference between numerical ${\Psi }_4^{22}$ waveforms, when aligned using the same procedure as employed for the EOB-NR alignment [see Eq. (23)]. “N6” and “N5” denote the highest- and next-to-highest numerical resolution, $n$ denotes the order of extrapolation to infinite extraction radius, and ”$r=225M$” denotes waves extracted at finite radius $r=225M$. The data are smoothed with a rectangular window of width $10M$; the light grey dots represent the unsmoothed data for the N5–N6 comparison at $r_{\mathrm{ex}}=225M$.

Figure 2

In the parameter space of the EOB-dynamics adjustable parameters $a_5(1/4)$ and $v_{\mathrm{pole}}(1/4)$ we show the contours of the time $t_{\mathrm{ref}}$ at which the phase difference between the numerical “$30\mathrm{c}1/\mathrm{N}6$, $n=3$” and EOB ${\Psi }_4^{22}$ becomes larger than 0.02 radians. Note that the innermost red contours cover two disjoint regions. The inset shows the effect of numerical uncertainty: The filled contours are the $t_{\mathrm{ref}}=3850M$ and $3900M$ contours from the main panel. The open contours are identical, except that they are computed using the “$30\mathrm{c}1/\mathrm{N}6$, $n=2$” numerical ${\Psi }_4^{22}$. The reference model is shown as a black dot.

Figure 3

For the case $a_5(1/4)=6.344$ and $v_{\mathrm{pole}}(1/4)=0.85$ ($A_8=0$, $a_{\mathrm{RR}}^{{\mathcal{F}}_{\Phi }}=0$ and $a_{\mathrm{RR}}^{{\mathcal{F}}_r}=0$), we show the phase difference between the numerical and EOB mode $h_{22}$ versus the numerical GW frequency $M{\omega }_{22}$ for EOB models in which the GW energy flux is shut down at several EOB orbital frequencies. The vertical line marks the maximum EOB orbital frequency.

Figure 4

Effect of $a_{\mathrm{RR}}^{{\mathcal{F}}_r}$ on contours of acceptable EOB parameters. The solid contours are the $t_{\mathrm{ref}}=3850M$ and $3900M$ contours from Fig. 2. The open contours shifted to the lower-right are the same, but computed with $a_{\mathrm{RR}}^{{\mathcal{F}}_r}=0.5$ instead of $a_{\mathrm{RR}}^{{\mathcal{F}}_{\Phi }}=0$. The reference model is shown as a black dot.

Figure 5

We compare the numerical and EOB $h_{22}$ amplitudes when the EOB model with reference values $a_5(1/4)=6.344$ and $v_{\mathrm{pole}}(1/4)=0.85$ are used. We show the EOB amplitudes without the NQC corrections and the EOB amplitude with the NQC terms suggested in Ref. 31, where the NQC parameters take the values $a=0.75$ and $ϵ=0.09$. When the NQC corrections are not included, we show the EOB amplitude of Eq. (16a), which uses the resummation procedure of Ref. 17, and also the EOB amplitudes of Eq. (16a) when the Padé resummations $P_4^1$ and $P_3^2$ of ${\rho }_{22}$ suggested in Ref. 17 are applied. Note that in this plot, the EOB amplitudes do not contain the merger-ringdown contribution.

Figure 6

Comparison of numerical waveform to EOB waveform with $a_5(1/4)=6.344$ and $v_{\mathrm{pole}}(1/4)=0.85$, i.e., the same model used in Fig. 5. The top panels show the real part of numerical and EOB $h_{22}$, the bottom panels show amplitude and phase differences between them. The left panels show times $t=0$ to $3900M$, and the right panels show times $t=3900$ to $t=4070M$ on a different vertical scale.

Figure 7

Comparison between EOB ${\ddot{h}}_{22}$ and the numerical ${\Psi }_4^{22}$. The top four panels show the real part of the waveform, on a linear and logarithmic $y$ axis. The bottom two panels show the phase difference (in radians) and the fractional amplitude difference between the two waveforms. The left panels show times $t=0$ to $3900M$, and the right panels show times $t=3900$ to $t=4070M$ with different vertical scales. (The quantities in the lower left panel have been smoothed; the grey data in the background of that panel presents the raw data.) This figure uses the same EOB model as Figs. 5, 6, namely $a_5(1/4)=6.344$ and $v_{\mathrm{pole}}(1/4)=0.85$.

Figure 8

We show the amplitude and frequency of the numerical and EOB mode $h_{22}$, the EOB orbital frequency and the frequency of the numerical mode ${\Psi }_4^{22}$. The vertical line marks the peak of the EOB amplitude and orbital frequency.

Figure 9

Upper panel: Amplitude and phase differences of numerical and EOB mode $h_{32}$ over the inspiral range. Lower panel: Amplitude and phase differences of numerical and EOB mode $h_{44}$ over the inspiral range.

Figure 10

Upper panel: Amplitude of the numerical and EOB modes $h_{32}$ and $h_{44}$. Lower panel: Frequency of the numerical and EOB modes $h_{32}$ and $h_{44}$. The EOB orbital frequency $2M\Omega$ ($4M\Omega$) is indistinguishable from the frequency of the $h_{32}$ ($h_{44}$) mode on the scale of this plot.

Figure 11

Comparison of the numerical data to an EOB model with $v_{\mathrm{pole}}=\infty$. This figure is analogous to Fig. 6, but uses an EOB model that was calibrated with the restriction $v_{\mathrm{pole}}=\infty$ (parameters are given in the main text). Even without $v_{\mathrm{pole}}$, the inspiral can be matched equally well as in Fig. 6; during the ringdown, the phase differences are somewhat larger, but it is possible that refined tuning will reduce them further.

Figure 12

EOB-NR comparison for a BH binary with mass ratio $2:1$. The upper panel shows the numerical and EOB mode ${\Psi }_4^{22}$, and the lower panel shows phase and amplitude differences between EOB and numerical run. The dashed brown line is the estimated phase error of the numerical simulation, obtained as the difference between simulations at high resolution N6 and lower resolution N5.

Figure 13

EOB-NR comparison for a BH binary with mass ratio $3:1$. The upper panel shows the numerical and EOB mode ${\Psi }_4^{22}$, and the lower panel shows phase and amplitude differences between EOB and numerical run. The dashed brown line is the estimated phase error of the numerical simulation, obtained as the difference between simulations at high resolution N6 and lower resolution N5.

Figure 14

The $(l,m)=(2,2)$ mode of the numerical waveform.

Figure 15

Phase and relative amplitude difference between the $(l,m)=(2,2)$ modes of the RWZ waveform $h_{\mathrm{RWZ}}$ and NP scalar ${\Psi }_4$ [see Eqs. (a2, a3, a4, a5)]. The right panel shows an enlargement of merger and ringdown, with the dotted vertical lines indicating time of maximum of $|{\Psi }_4|$, and where $|{\Psi }_4|$ has decayed to 10% and 1% of the maximal value. (The solid blue lines are smoothed; the grey data in the background represents the unsmoothed data.)

Figure 16

The $(l,m)=(4,4)$ mode of the numerical waveform.

Figure 17

Phase and relative amplitude difference between the $(l,m)=(4,4)$ modes of the RWZ waveform $h_{\mathrm{RWZ}}$ and NP scalar ${\Psi }_4$, cf. Eqs (a2, a3, a4, a5). The right panels show an enlargement of merger and ringdown, with the dotted vertical line indicating the position of the maximum of $|{\Psi }_4|$.

Figure 18

The $(l,m)=(3,2)$ mode of the numerical waveform.

Figure 19

Phase difference between the $(l,m)=(3,2)$ modes of the RWZ waveform $h$ and NP scalar ${\Psi }_4$, cf. Eqs (a2, a3, a4, a5). The right panels show an enlargement of merger and ringdown, with the dotted vertical line indicating the position of the maximum of $|{\Psi }_4|$.