Prototype effective-one-body model for nonprecessing spinning inspiral-merger-ringdown waveforms

This paper presents a tunable effective-one-body (EOB) model for black-hole (BH) binaries of arbitrary mass ratio and aligned spins. This new EOB model incorporates recent results of small-mass-ratio simulations based on Teukolsky’s perturbative formalism. The free parameters of the model are calibrated to numerical-relativity simulations of nonspinning BH-BH systems of five different mass ratios and to equal-mass nonprecessing BH-BH systems with dimensionless BH spins ${\chi }_i\simeq \pm 0.44$. The present analysis focuses on the orbital dynamics of the resulting EOB model, and on the dominant $(\ell ,m)=(2,2)$ gravitational-wave mode. The calibrated EOB model can generate inspiral-merger-ringdown waveforms for nonprecessing, spinning BH binaries with any mass ratio and with individual BH spins $-1\le {\chi }_i\lesssim 0.7$. Extremizing only over time and phase shifts, the calibrated EOB model has overlaps larger than 0.997 with each of the seven numerical-relativity waveforms for total masses between $20M_{\odot }$ and $200M_{\odot }$, using the Advanced LIGO noise curve. We compare the calibrated EOB model with two additional equal-mass highly spinning (${\chi }_i\simeq -0.95,+0.97$) numerical-relativity waveforms, which were not used during calibration. We find that the calibrated model has an overlap larger than 0.995 with the simulation with nearly extremal antialigned spins. Extension of this model to black holes with aligned spins ${\chi }_i\gtrsim 0.7$ requires improvements of our modeling of the plunge dynamics and inclusion of higher-order PN spin terms in the gravitational-wave modes and radiation-reaction force.

Figure 1

We show in the spacetime diagram $(\widehat{t},r^{*})$ the trajectory of the effective particle in the EOB description (black solid line in the left part of the diagram) and the EOB (2, 2) gravitational mode (red solid oscillating line) for an equal-mass nonspinning black-hole binary. Although we only need to evolve the EOB trajectory until the orbital frequency reaches its maximum (“light ring”), the model’s dynamics allows the trajectory to continue to negative $r^{*}$ (short-dashed black line in the left part of the diagram). The blue dashed lines represent $\widehat{t}\pm r^{*}=\mathrm{const}$ surfaces and ingoing/outgoing null rays. The EOB (2,2) mode is a function of the retarded time $\widehat{t}-r^{*}$, plotted here orthogonal to $\widehat{t}-r^{*}=\mathrm{const}$ surfaces, at a finite $\widehat{t}+r^{*}$ distance. The two outgoing null rays are drawn at the $\widehat{t}-r^{*}$ retarded times when the EOB particle crosses the EOB ISCO and light-ring radii, respectively. The shaded green area is a rough sketch of the potential barrier around the newborn black hole.

Figure 2

We show the quadratic fit in $\nu$ for the adjustable parameter $K$. This parameter is calibrated using the five nonspinning NR waveforms, assuming ${\rho }_{22}^{(4)}(\nu )$ in Eq. (36). The error bars are determined by the intersection of the contours of $\Delta {\varphi }_{\mathrm{global}}=0.1\mathrm{rads}$ with ${\rho }_{22}^{(4)}(\nu )$ for each mass ratio considered.

Figure 3

Comparison of the NR and EOB (2, 2) mode for $q=1$, ${\chi }_1={\chi }_2=0$. In the upper panels we show the comparison between the real part of the two waveforms, zooming into the merger region in the upper right plot. In the lower panels we show the dephasing and relative amplitude difference over the same time ranges as the upper panels. A vertical dashed line marks the position of the NR amplitude peak. The dotted curves are the NR errors.

Figure 4

Same as in Fig. 3 but for $q=1/6$, ${\chi }_1={\chi }_2=0$.

Figure 5

Same as in Fig. 3 but for $q=1$, ${\chi }_1={\chi }_2=-0.43655$.

Figure 6

Same as in Fig. 3 but for $q=1$, ${\chi }_1={\chi }_2=+0.43756$.

Figure 7

Same as in Fig. 3 but for $q=1$, ${\chi }_1={\chi }_2=-0.94905$. This NR waveform was not used to calibrate the adjustable parameters $d_{\mathrm{SO}}$ and $d_{\mathrm{SS}}$. Alignment between the NR and EOB waveforms was performed using Eq. (26), with $t_1=860M$ and $t_2=2470M$.

Figure 8

Same as in Fig. 3 but for $q=1$, ${\chi }_1={\chi }_2=+0.9695$ and only the inspiral portion. This NR waveform was not used to calibrate the adjustable parameters $d_{\mathrm{SO}}$ and $d_{\mathrm{SS}}$. Also, in the aligned case our prototype EOB model only covers ${\chi }_{1,2}\lesssim 0.7$. Note that in this plot we do not include spinning NQC corrections in our EOB waveform. Alignment between the NR and EOB waveforms was performed using Eq. (26), with $t_1=1170M$ and $t_2=2790M$.