Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism

We calibrate an effective-one-body (EOB) model to numerical-relativity simulations of mass ratios 1, 2, 3, 4, and 6, by maximizing phase and amplitude agreement of the leading (2, 2) mode and of the subleading modes (2, 1), (3, 3), (4, 4) and (5, 5). Aligning the calibrated EOB waveforms and the numerical waveforms at low frequency, the phase difference of the (2, 2) mode between model and numerical simulation remains below $\sim 0.1$ rad throughout the evolution for all mass ratios considered. The fractional amplitude difference at peak amplitude of the (2, 2) mode is 2% and grows to 12% during the ringdown. Using the Advanced LIGO noise curve we study the effectualness and measurement accuracy of the EOB model, and stress the relevance of modeling the higher-order modes for parameter estimation. We find that the effectualness, measured by the mismatch between the EOB and numerical-relativity polarizations which include only the (2, 2) mode, is smaller than 0.2% for binaries with total mass $20–200M_{\odot }$ and mass ratios 1, 2, 3, 4, and 6. When numerical-relativity polarizations contain the strongest seven modes, and stellar-mass black holes with masses less than $50M_{\odot }$ are considered, the mismatch for mass ratio 6 (1) can be as high as 7% (0.2%) when only the EOB (2, 2) mode is included, and an upper bound of the mismatch is 0.5% (0.07%) when all the four subleading EOB modes calibrated in this paper are taken into account. For binaries with intermediate-mass black holes with masses greater than $50M_{\odot }$ the mismatches are larger. We also determine for which signal-to-noise ratios the EOB model developed here can be used to measure binary parameters with systematic biases smaller than statistical errors due to detector noise.

Figure 1

Amplitude of extrapolated numerical-relativity waveforms for the dominant modes. The two panels from top to bottom are for mass ratios $q=1$ and $q=6$, respectively. Each curve is labeled with its respective ($l$, $m$) mode, and the time-delay $\Delta t_{\mathrm{peak}}^{\ell m}$ between the extrema of $|h_{22}|$ and $|h_{lm}|$. The horizontal axis measures the time-difference to the peak of $|h_{22}|$.

Figure 2

We compare the numerical-relativity and EOB $h_{22}$ amplitudes with and without the NQC corrections $N_{\ell m}$ given in Eq. (22). We also plot the numerical and EOB gravitational frequency of the (2, 2) mode and twice the EOB orbital frequency. The left panel refers to $q=1$ and the right panel to $q=6$. The horizontal axis is the retarded time in the numerical-relativity simulation. The vertical lines mark the peaks of the numerical-relativity $h_{22}$ amplitudes.

Figure 3

Amplitude (in units of $M/\mathcal{R}$) and frequency (in units of $1/M$) comparison between the full “NR” waveform and the “$\mathrm{NR}+\mathrm{QNM}$” waveform generated by attaching QNMs to the inspiral-plunge numerical waveform. We show also the relative amplitude and phase differences. In the left panel, we compare $h_{22}$. In the right panel, we compare the numerical $h_{44}$ mode with two “$\mathrm{NR}+\mathrm{QNM}$” modes. One of them is generated by attaching the physical QNMs, the other is generated by attaching both the physical QNMs and the pseudo QNM. The former is very different from the numerical-relativity mode and we do not show their amplitude and phase differences. All $h_{44}$ amplitudes have been multiplied by a factor of 20, so that they are more visible. The horizontal axis is the retarded time in the numerical-relativity simulation.

Figure 4

We show the amplitude of the numerical-relativity $h_{22}$ for mass ratios $q=1$, 2, 3, 4, 6. We have time shifted the modes so that their peaks are aligned. We have also rescaled them by $\nu$. The horizontal axis is the retarded time in the numerical-relativity simulation. The inset shows an enlargement of the merger region.

Figure 5

We calibrate adjustable parameters of the EOB dynamics. For mass ratios $q=1$, 2, 3, 4 and 6, the shaded regions correspond to ($a_5$, $a_6$) values for which the EOB and numerical-relativity $h_{22}$ agree within 0.2 rad at merger, i.e., at the peak of the numerical $h_{22}$. In the inserted subplot, for $a_6(\nu )/\nu =184$, we show $a_5(\nu )/\nu$ values constrained by the shaded regions and by the test-particle ISCO-shift result [61], and also the quadratic fit (red curve) given by Eq. (36a).

Figure 6

For the equal-mass case, we compare the numerical-relativity and calibrated EOB (2, 2) mode. The top panels show the real part of numerical and EOB $h_{22}$, the bottom panels show amplitude and phase differences between them. The horizontal axis is the retarded time in the numerical-relativity simulation. The left panels show retarded times $t-r_{*}=0$ to $3850M$, and the right panels show retarded times $t-r_{*}=3850M$ to $t-r_{*}=4070M$ on a different vertical scale. The dotted curves are the numerical-relativity errors.

Figure 7

Comparison between the (2, 2) modes of the numerical-relativity simulation and the calibrated EOB model for mass ratios $q=2$, 3, 4, 6. In each subplot, the top panels show the real part of the $h_{22}$ mode and the bottom panels show the phase and amplitude differences between numerical and EOB waveform. The dotted curves are the numerical-relativity errors.

Figure 8

Comparison of the (4, 4) mode for mass ratios $q=1$, 3, 6 between numerical and calibrated EOB model. Plotted data as in Fig. 7.

Figure 9

Comparison of the (2, 1) mode for mass ratios $q=3$ (top panel) and $q=6$ (bottom panel) between numerical and calibrated EOB model. Plotted data as in Fig. 7.

Figure 10

Comparison of the (3, 3) mode for mass ratios $q=3$ (top panel) and $q=6$ (bottom panel) between numerical and calibrated EOB model. Plotted data as in Fig. 7.

Figure 11

Amplitude of the Fourier transform of the (2, 2) mode of the numerical-relativity waveforms, scaled to a total mass $M=20M_{\odot }$. We also plot the noise spectral density of the Advanced LIGO detector. The two vertical lines mark the initial gravitational-wave frequency for the numerical waveforms with $q=1$ and $q=4$ (lowest and highest start frequency of all considered waveforms).

Figure 12

The polarization waveform $h_{+}(\theta ,\varphi ;t)$ as emitted into sky direction $\theta =\varphi =\pi /3$. Top panel: mass ratio $q=1$; bottom panel: mass ratio $q=6$. The solid blue curve represents the numerical data, the red dashed curve the EOB model, and only late inspiral, merger and ringdown are shown.

Figure 13

The mismatch versus binary total mass for $q=1$ using Advanced LIGO noise curve. Mismatches with the (2, 2) mode of the EOB waveform are optimized over $\{t_0,{\varphi }_0,\mathbf{\lambda }\}$ whereas mismatches with the 5-mode EOB model are optimized over $\{t_0,{\varphi }_0\}$ only and represent an upper bound. The vertical line represents the maximum total mass for stellar-mass black-hole binaries, assuming a maximum black-hole mass of $50M_{\odot }$. The range of total masses on the right of the vertical line refer to intermediate-mass black-hole binaries.

Figure 14

Mismatch calculation for mass ratio $q=6$. All details as in Fig. 13. This figure and Fig. 13 use the same vertical axis to ease comparisons between them.

Figure 15

The upper bound ${\mathrm{SNR}}^{\mathrm{eff}}\equiv \mathrm{SNR}ϵ$ from the measurement accuracy requirement (46) versus binary total mass for $q=1$ and 6 using Advanced LIGO noise curve. For any ${\mathrm{SNR}}^{\mathrm{eff}}$ below the curves, the EOB polarizations are accurate enough to avoid a systematic bias that is larger than statistical errors when estimating the binary parameters. The horizontal line indicates the single detector SNR for Advanced LIGO which is 8. The two vertical lines represent the maximum total mass for stellar-mass binary black holes with maximum black-hole mass of $50M_{\odot }$ and $q=1$ and $q=6$. The range of total masses on the right of those vertical lines refer to intermediate-mass black-hole binaries. The vertical axis on the right shows $\Vert \delta h\Vert /\Vert h\Vert$.

Figure 16

Investigation of the (3, 2) mode during ringdown for $q=1$ and 6. The top panels show the mode itself, and the lower panels split the mode into amplitude and frequency. The continuous lines represent the numerical-relativity modes. The dotted lines are a fit to eight QNM modes of ($\ell =3$, $m=2$, $n=0$, $1,\dots ,7$). The dashed lines are a fit to eight QNMs of ($\ell =3$, $m=2$, $n=0$, $1,\dots ,4$) and ($\ell =2$, $m=2$, $n=0$, 1, 2). The horizontal axis is the retarded time in the numerical-relativity simulation.

Figure 17

Amplitude and frequency comparisons between full NR (3, 2) mode and “$\mathrm{NR}+\mathrm{QNM}$” (3, 2) mode generated by attaching the QNMs to the inspiral-plunge numerical waveform. The “$\mathrm{NR}+\mathrm{QNM}$” waveform is generated by attaching the following set of eight QNMs ($\ell =3$, $m=2$, $n=0$, $1,\dots ,4$) and ($\ell =2$, $m=2$, $n=0$, 1, 2). Note that the amplitudes have been multiplied by a factor of 20, so that they are more visible. The horizontal axis is the retarded time in the numerical-relativity simulation.