Accurate effective-one-body waveforms of inspiralling and coalescing black-hole binaries

The effective-one-body (EOB) formalism contains several flexibility parameters, notably $a_5$, $v_{\mathrm{pole}}$, and ${\overline{a}}_{\mathrm{RR}}$. We show here how to jointly constrain the values of these parameters by simultaneously best-fitting the EOB waveform to two, independent, numerical relativity (NR) simulations of inspiralling and/or coalescing binary black-hole systems: published Caltech-Cornell inspiral data (considered for gravitational wave frequencies $M\omega \le 0.1$) on one side, and newly computed coalescence data on the other side. The resulting, approximately unique, “best-fit” EOB waveform is then shown to exhibit excellent agreement with NR coalescence data for several mass ratios. The dephasing between this best-fit EOB waveform and published Caltech-Cornell inspiral data is found to vary between $-0.0014$ and $+0.0008$ radians over a time span of $\sim 2464M$ up to gravitational wave frequency $M\omega =0.1$, and between $+0.0013$ and $-0.0185$ over a time span of $96M$ after $M\omega =0.1$ up to $M\omega =0.1565$. The dephasings between EOB and the new coalescence data are found to be smaller than: (i) $\pm 0.025$ radians over a time span of $730M$ (11 cycles) up to merger, in the equal-mass case, and (ii) $\pm 0.05$ radians over a time span of about $950M$ (17 cycles) up to merger in the $2:1$ mass-ratio case. These new results corroborate the aptitude of the EOB formalism to provide accurate representations of general relativistic waveforms, which are needed by currently operating gravitational wave detectors.

Figure 1

Differences between the phase extracted at $R_{\mathrm{ex}}=90M$, and the phase extrapolated to infinity based on two choices of the retarded time and on two choices of the extrapolating polynomial, as described in the text. The choice of retarded time makes little difference to the result.

Figure 2

Comparison between Caltech-Cornell and Jena actual numerical data: the phase difference $\Delta {\varphi }_{22}^{\mathrm{CCJena}}={\varphi }^{\mathrm{CC}}-{\varphi }_{90M}^{\mathrm{Jena}}$ is shown versus Caltech-Cornell GW frequency ${\omega }_{\mathrm{CC}}$.

Figure 3

Functional relationships linking $v_{\mathrm{pole}}$ and ${\overline{a}}_{\mathrm{RR}}$ to $a_5$ obtained by imposing the two constraints (16, 17) based on published Caltech-Cornell inspiral waveform data.

Figure 4

Top panel: near-perfect agreement between T4-EOB and T4-NR phase differences when $a_5=25$, ${\overline{a}}_{\mathrm{RR}}=27.9197$, and $v_{\mathrm{pole}}=0.51563$. Here NR refers to the published results of the Caltech-Cornell inspiral simulation. The corresponding EOB-NR phase difference (bottom panel) is of the order of ${10}^{-3}$ radians over the 30 GW cycles of the Caltech-Cornell inspiral simulation.

Figure 5

$L_{\infty }$ norm of the EOB-NR late-inspiral ($[t_{\mathrm{L}},t_{\mathrm{R}}]$) phase difference, as a function of $a_5$ for $\nu =0.25$ ($1:1$ mass ratio) and $\nu \simeq 0.2222$ ($2:1$ mass ratio). NR refers to results of Jena coalescence simulations reported here.

Figure 6

Comparison between NR and EOB waveforms for $\nu =0.25$ (top), $\nu =0.2222$ (bottom). The left panels depict the EOB-NR phase difference; the right panels show the real part of the metric waveforms. Here, NR refers to the full results of Jena coalescence simulations, from early inspiral to ring-down, by contrast to the $L_{\infty }$ norm of Fig. 5 which concerned a late-inspiral stage $[t_{\mathrm{L}},t_{\mathrm{R}}]$. This interval is indicated on the figures. The pinching times $(t_1,t_2)$ of Table  are also shown. The dash-dot and the dash vertical lines at the extreme right of the figures mark the location of the EOB adiabatic LSO and the “EOB light-ring,” respectively.

Figure 7

Comparison between NR and EOB waveforms for $\nu =0.16$. The left panel depicts the EOB-NR phase difference; the right panels show the real part of the metric waveforms. The pinching times $(t_1,t_2)$ of Table  are also shown. The dash-dot and the dash vertical lines at the extreme right of the figures mark the location of the EOB adiabatic LSO and the “EOB light-ring,” respectively.

Figure 8

Comparison between EOB and NR instantaneous gravitational wave frequencies (and moduli) for the equal-mass case, $\nu =0.25$. Here, as in Fig. 9, NR refers to the Jena coalescence simulation. The dash-dot vertical line indicates the EOB adiabatic LSO, while the dash one indicates the “EOB light-ring.” The pinching frequencies $({\omega }_1^{22},{\omega }_2^{22})$ of Table  are also indicated.

Figure 9

Fractional EOB-NR differences in the gravtitational wave (metric) amplitudes, $A\equiv |{\Psi }_{22}|$, versus NR time for different mass ratios.

Figure 10

Computation of the Zerilli-normalized metric waveform ${\Psi }_{22}^{(\mathrm{e})}$ from $r{\psi }_4^{22}$ via two backward time integrations starting with zero integration constants at the final time. The data refer to the $1:1$ mass ratio ($\nu =0.25$) numerical simulation. Unphysical oscillations in the modulus are quite visible at early times.

Figure 11

Computation of the metric waveform ${\Psi }_{22}^{(\mathrm{e})}$ from $r{\psi }_4^{22}$ via two time integrations starting at $t=0$. The upper panel has zero integration constants and exhibits clear linear drifts $\alpha t+\beta$. The bottom panel shows the result of subtracting the linear drift of the waveform obtained by fitting the upper panel over the entire time interval starting at $t=0$.

Figure 12

Same as Fig. 11 except that the integration and the linear fit have been done starting at time $t\sim 150$, i.e. at the beginning of the inspiral signal. Note the oscillations in the modulus of the bottom panel, and the fact that the linear drifts (visible in the upper panel) are much larger than in Fig. 11.

Figure 13

Numerical relativity (Jena) and TaylorT4: comparison between the instantaneous gravitational wave frequencies for the equal-mass case ($\nu =0.25$).

Figure 14

Comparison between phase-acceleration curves $a_{\omega }$ of EOB (solid line) and TaylorT4 (dash-dot line) for $\nu =0.25$ (left panel) and $\nu =0.16$ (right panel). The rightmost dashed vertical line indicates the location of the EOB adiabatic LSO. The leftmost vertical line on the left panel indicates the EOB GW frequency corresponding the instant when T4 blows up.

Figure 15

This figure (done for $\nu \ll 1$) illustrates, in particular, the fact that the domain of validity of TaylorT4 when $\nu \lesssim 0.16$ reduces to the “normal” 3.5 PN one, namely $\omega \lesssim 0.06$.