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

We present the first attempt at calibrating the effective-one-body (EOB) model to accurate numerical relativity simulations of spinning, nonprecessing black-hole binaries. Aligning the EOB and numerical waveforms at low frequency over a time interval of $1000M$, we first estimate the phase and amplitude errors in the numerical waveforms and then minimize the difference between numerical and EOB waveforms by calibrating a handful of EOB-adjustable parameters. In the equal-mass, spin aligned case, we find that phase and fractional amplitude differences between the numerical and EOB $(2,2)$ mode can be reduced to 0.01 radian and 1%, respectively, over the entire inspiral waveforms. In the equal-mass, spin antialigned case, these differences can be reduced to 0.13 radian and 1% during inspiral and plunge, and to 0.4 radian and 10% during merger and ringdown. The waveform agreement is within numerical errors in the spin aligned case while slightly over numerical errors in the spin antialigned case. Using Enhanced LIGO and Advanced LIGO noise curves, we find that the overlap between the EOB and the numerical $(2,2)$ mode, maximized over the initial phase and time of arrival, is larger than 0.999 for binaries with total mass $30M_{\odot }–200M_{\odot }$. In addition to the leading $(2,2)$ mode, we compare four subleading modes. We find good amplitude and frequency agreements between the EOB and numerical modes for both spin configurations considered, except for the $(3,2)$ mode in the spin antialigned case. We believe that the larger difference in the $(3,2)$ mode is due to the lack of knowledge of post-Newtonian spin effects in the higher modes.

Figure 1

Numerical error estimates for the UU configuration. We show the phase difference between several numerical ${\Psi }_4^{22}$ waveforms aligned using the procedure defined by Eq. (35).

Figure 2

Numerical error estimates for the DD configuration. We show the phase difference between several numerical ${\Psi }_4^{22}$ waveforms aligned using the procedure defined by Eq. (35). The dashed vertical line marks the peak amplitude time of the reference numerical strain waveform $h_{22}$ (N6, $n=3$).

Figure 3

Phase and relative amplitude difference between the $(l,m)=(2,2)$ modes of the RWZ waveform $h_{\mathrm{RWZ}}$ and NP scalar ${\Psi }_4$ for the UU case.

Figure 4

Phase and relative amplitude difference between the $(l,m)=(2,2)$ modes of the RWZ waveform $h_{\mathrm{RWZ}}$ and NP scalar ${\Psi }_4$ for the DD case. 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% of the maximal value. (The solid blue lines are smoothed; the grey data in the background represent the unsmoothed data.)

Figure 5

Comparison between the numerical and EOB waveform for the UU configuration using $b(\nu )=-1.65$ and $a_{\mathrm{SS}}^{3\mathrm{PN}}=1.5$. The top panels show the real part of the numerical and EOB $h_{22}$, and the bottom panels show the amplitude and phase differences between them. The left panels show times $t=0$ to $2950M$, whereas the right panels present an enlargement of the later portion of the waveform. The EOB waveforms in the top panels are not quite visible since they are covered by the very similar NR waveforms.

Figure 6

Comparison between the numerical and EOB waveform for the DD configuration using $b(\nu )=-1.65$ and $a_{\mathrm{SS}}^{3\mathrm{PN}}=1.5$. The top panels show the real part of the numerical and EOB $h_{22}$, and the bottom panels show the amplitude and phase differences between them. The left panels show times $t=0$ to $2300M$, and the right panels show times $t=2300M$ to $t=2480M$ on a different vertical scale.

Figure 7

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 ${\Psi }_4^{22}$ for the DD configuration. The vertical line labeled $t_{\mathrm{peak}}$ marks the peak of the amplitude of the numerical waveform. The EOB light ring is $0.6M$ before the peak and is too close to be shown in the figure.

Figure 8

Comparison of the numerical (solid lines), EOB (dashed lines), and Taylor-expanded (dotted lines) amplitudes of the dominant and leading subdominant $(l,m)$ modes for the UU (left panel) and DD (right panel) configurations. The inset shows the amplitudes for the dominant $(2,2)$ mode during the late inspiral and plunge in the DD configuration, without the addition of EOB-NQC and EOB-waveform adjustable parameters.