Comparison of numerical and post-Newtonian waveforms for generic precessing black-hole binaries

We compare waveforms and orbital dynamics from the first long-term, fully nonlinear, numerical simulations of a generic black-hole binary configuration with post-Newtonian (PN) predictions. The binary has mass ratio $q\sim 0.8$ with arbitrarily oriented spins of magnitude $S_1/m_1^2\sim 0.6$ and $S_2/m_2^2\sim 0.4$ and orbits 9 times prior to merger. The numerical simulation starts with an initial separation of $r\approx 11M$ and orbital parameters determined by 2.5 PN and 3.5 PN evolutions of a quasi-circular binary starting from $r=50M$. The resulting binaries have very little eccentricity according to the 2.5 PN and 3.5 PN systems, but show eccentricities of $e\sim 0.01–0.02$ and $e\sim 0.002–0.005$ in the respective numerical simulations, thus demonstrating that 3.5 PN significantly reduces the eccentricity of the binary compared to 2.5 PN. We perform three numerical evolutions from $r\approx 11M$ with maximum resolutions of $h=M/48$, $M/53.3$, $M/59.3$, to verify numerical convergence. We observe a reasonably good agreement between the PN and numerical waveforms, with an overlap of nearly 99% for the first six cycles of the ($\ell =2$, $m=\pm 2$) modes, 91% for the ($\ell =2$, $m=\pm 1$) modes, and nearly 91% for the ($\ell =3$, $m=\pm 3$) modes. The phase differences between numerical and post-Newtonian approximations appear to be independent of the $(\ell ,m)$ modes considered and relatively small for the first 3–4 orbits. An advantage of the 3.5 PN model over the 2.5 PN one seems to be observed, which indicates that still higher PN order (perhaps even 4.0 PN) may yield significantly better waveforms. In addition, we identify features in the waveforms likely related to precession and precession-induced eccentricity.

Figure 1

The ($\ell =2$, $m=2$) component of ${\psi }_4$ for the G2.5 configuration for the three resolutions. Note the excellent phase agreement until about $t=1400M$.

Figure 2

Convergence of the amplitude of the G2.5 ($\ell =2$, $m=2$) component of ${\psi }_4$. The amplitude shows between third- and fourth-order convergence (as demonstrated by multiplying the deviations in the amplitude by 1.358 and 1.5098, respectively).

Figure 3

Convergence of the G2.5 phase of the ($\ell =2$, $m=2$) component of ${\psi }_4$. The phase shows eighth-order convergence up to $t=1200M$, decreasing to between third- and fourth-order convergence during the plunge (as demonstrated by multiplying the phase deviations by 2.305, 1.5098, and 1.358, respectively).

Figure 4

The amplitude of the G2.5 ($\ell =2$, $m=2$) component of ${\psi }_4$ versus the phase. Note that the phase becomes more negative as $t$ increases.

Figure 5

A comparison of $\ddot{h}$ and ${\psi }_4$ for the ($\ell =2$, $m=1$) mode for the G2.5 configuration. The plot demonstrates that the windowing procedure apparently does not contaminate the waveform to a significant degree.

Figure 6

The trajectory difference ${\overrightarrow{x}}_1–{\overrightarrow{x}}_2$ for the G2.5 configuration. Note the orbital plane precession and the very good agreement between trajectories at the different resolutions (the tracks from the different resolutions are not distinguishable on this scale).

Figure 7

An $xy$ projection of the trajectory difference ${\overrightarrow{x}}_1–{\overrightarrow{x}}_2$ for the G2.5 configuration. Note the very good agreement between trajectories at the different resolutions. The initial orbital plane is inclined with respect to the $xy$ plane, making the orbit appear more eccentric.

Figure 8

An $xy$ projection of the trajectory difference ${\overrightarrow{x}}_1–{\overrightarrow{x}}_2$ for the G3.5 configuration. The initial orbital plane is inclined with respect to the $xy$ plane, making the orbit appear more eccentric.

Figure 9

The coordinate distance $r=|{\overrightarrow{x}}_1–{\overrightarrow{x}}_2|$ between punctures versus time for the G2.5 and G3.5 configurations. Note that the large eccentricity in the orbit (apparent in the oscillation in $r$) is reduced by using the 3.5 PN equations to generate the initial data. Unlike in Figs. 6, 7, here the differences between resolution becomes apparent during the plunge. These differences drive the phase error. Also note that the G3.5 configuration merges more slowly.

Figure 10

The eccentricity $e_D(t)$ of the G3.5 and G2.5 configurations, as calculated using the techniques of 92.

Figure 11

The eccentricity $e_r(t)$ of the G3.5 and G2.5 configurations and the 3.5 PN prediction for the G2.5 configuration (the 3.5 PN prediction for G3.5 is a factor of 10 smaller than the NR prediction). The inset shows the “eccentricity” at early times when gauge effects dominate the trajectories. Note that the eccentricity of G2.5 decays throughout the evolution while the smaller eccentricity for G3.5 remains roughly constant beyond $t\sim 630M$. At later times the eccentricities of G2.5 and G3.5 begin to agree.

Figure 12

The coordinate displacement $\overrightarrow{r}={\overrightarrow{x}}_1–{\overrightarrow{x}}_2$ between punctures versus time for the G2.5 configuration after performing a constant rotation that maps the initial orbital plane onto the $xy$ plane. Precession is responsible for driving the amplitude of $r^z$.

Figure 13

The coordinate displacement $\overrightarrow{r}={\overrightarrow{x}}_1–{\overrightarrow{x}}_2$ between punctures versus time for the G3.5 configuration after performing a constant rotation that maps the initial orbital plane onto the $xy$ plane. Precession is responsible for driving the amplitude of $r^z$.

Figure 14

The components of the spin for the more massive black hole in configuration G3.5 as a function of time. The precession of the spin drive the orbital plane precession. Here the precession time scale is of order $1000M$.

Figure 15

$e_r$ versus radius for a binary with spins aligned with the angular momentum. Here the eccentricity decreases with $r$ for all radii.

Figure 16

$e_r$ versus radius for the G3.5 configuration and a very similar binary, with parameters chosen to reduce the (PN) initial eccentricity. Note that the eccentricity at $r<10$ is constant and roughly the same for both configurations.

Figure 17

The evolution of the orbital radius for the G2.5 configuration from the numerical, 2.5 PN, and 3.5 PN simulations. The residual eccentricity in the 2.5 PN evolution is due to our using a different 2.5 PN radiation-reaction term from that used in the original evolution beginning at $r=50M$. Note that both 3.5 PN and the numerical simulation indicate that this configuration has relatively large eccentricity.

Figure 18

The evolutions of the orbital radius for the G3.5 configuration from the numerical and 3.5 PN simulations. Here the numerical simulations shows that the eccentricity was reduced, but is still relatively large, while the 3.5 PN evolution indicates that the binary is noneccentric.

Figure 19

The real part of the ($\ell =2$, $m=2$) mode of $h$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. Note that the 3.5 PN prediction is closer to the numerical waveform and that 3.5 PN predicts an early merger while 2.5 PN predicts a late merger (as is evident by the amplitude of the mode versus time).

Figure 20

The real part of the ($\ell =2$, $m=2$) mode of $h$ for the G3.5 configuration from the numerical and 3.5 PN simulations. Here too, 3.5 PN predicts an early merger (as is evident by the amplitude of the mode versus time).

Figure 21

The real part of the ($\ell =2$, $m=1$) mode of $h$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. Note the precession-induced modulation in the amplitude of the oscillations.

Figure 22

The real part of the ($\ell =2$, $m=1$) mode of $h$ for the G3.5 configuration from the numerical and 3.5 PN simulations. Note the precession-induced modulation in the amplitude of the oscillations.

Figure 23

The real part of the ($\ell =3$, $m=3$) mode of $h$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. Note the relatively high-frequency oscillations in the amplitude (roughly corresponding to the orbital period).

Figure 24

The real part of the ($\ell =3$, $m=3$) mode of $h$ for the G3.5 configuration from the numerical and 3.5 PN simulations. Note the relatively high-frequency oscillations in the amplitude (roughly corresponding to the orbital period).

Figure 25

The amplitude of the ($\ell =2$, $m=2$) mode of $h$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. The oscillations in the amplitude are much more pronounced in the numerical and 3.5 PN simulations, indicating that these oscillations are likely due to eccentricity.

Figure 26

The amplitude of the ($\ell =2$, $m=2$) mode of $h$ for the G3.5 configuration from the numerical and 3.5 PN simulations. The amplitude oscillations in the numerical waveform are much larger than those in the 3.5 PN waveform, indicating that they are likely due to eccentricity.

Figure 27

The amplitude of the ($\ell =2$, $m=2$) mode of ${\psi }_4$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. Note the very good agreement between the 3.5 PN and numerical waveforms.

Figure 28

The amplitude of the ($\ell =2$, $m=2$) mode of ${\psi }_4$ for the G3.5 configuration from the numerical and 3.5 PN simulations.

Figure 29

The amplitude of the ($\ell =2$, $m=1$) mode of $h$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. The secular oscillation in the numerical amplitude occurs at roughly the precessional frequency. Here the shorter time scale oscillations apparent in the PN waveforms are much smaller in the numerical waveform.

Figure 30

The amplitude of the ($\ell =2$, $m=1$) mode of $h$ for the G3.5 configuration from the numerical and 3.5 PN simulations. The secular oscillation in the numerical amplitude occurs at roughly the precessional frequency. Here the shorter time scale oscillations (corresponding roughly to the orbital period) are present in both waveforms with very similar amplitudes.

Figure 31

The amplitude of the ($\ell =2$, $m=1$) mode of ${\psi }_4$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. Note the very good agreement between the 3.5 PN and numerical waveforms.

Figure 32

The amplitude of the ($\ell =2$, $m=1$) mode of ${\psi }_4$ for the G3.5 configuration from the numerical and 3.5 PN simulations.

Figure 33

The amplitude of the ($\ell =3$, $m=3$) mode of $h$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. Note the very good agreement between the 3.5 PN and numerical waveforms. Also note that the short time scale oscillations (orbital period) grow with time at later times, indicating that, at least at later times, they are due mainly to precession.

Figure 34

The amplitude of the ($\ell =3$, $m=3$) mode of $h$ for the G3.5 configuration from the numerical and 3.5 PN simulations. Note that the short time scale oscillation at later times grow with time, indicating that these later-time oscillations are due to precession. The early-time oscillations in the numerical waveform (at the same frequency) are likely due to eccentricity.

Figure 35

The amplitude of the ($\ell =3$, $m=3$) mode of ${\psi }_4$ for the G2.5 configuration from the numerical, truncated 2.5 PN, and 3.5 PN simulations. Note the very good agreement between the 3.5 PN and numerical waveforms.

Figure 36

The amplitude of the ($\ell =3$, $m=3$) mode of ${\psi }_4$ for the G3.5 configuration from the numerical and 3.5 PN simulations.

Figure 37

The phase differences in $h$ between the numerical and 3.5 PN simulations for the G2.5 configuration in the ($\ell =2$, $m=1$), ($\ell =2$, $m=2$), and ($\ell =3$, $m=3$) modes. We multiplied the phase differences in the modes by a factor of $1/(\ell \pi )$. We divide by $\ell \pi$, rather than $m\pi$, because the ($\ell =2$, $m=1$) mode is dominated by mode mixing from the ($\ell =2$, $m=\pm 2$) modes (see text for more details). The vertical lines shows the times when the ($\ell =2$, $m=2$) frequency is $M\omega =0.05$ ($t\sim 323M$) and $M\omega =0.075$ ($t\sim 1075M$).

Figure 38

The phase differences in $h$ between the numerical and 3.5 PN simulations for the G3.5 configuration in the ($\ell =2$, $m=1$), ($\ell =2$, $m=2$), and ($\ell =3$, $m=3$) modes. We multiplied the phase differences in the modes by a factor of $1/(\ell \pi )$. Note that the normalized phase differences are qualitatively independent of the mode and arise from the orbital phase error in the PN approximation. We divide by $\ell \pi$, rather than $m\pi$, because the ($\ell =2$, $m=1$) mode is dominated by mode mixing from the ($\ell =2$, $m=\pm 2$) modes (see text for more details). The vertical lines show the times when the ($\ell =2$, $m=2$) frequency is $M\omega =0.05$ ($t\sim 360M$), $M\omega =0.075$ ($t\sim 1252M$), and $M\omega =0.1$ ($t\sim 1493M$).

Figure 39

The phase difference in the ($\ell =2$, $m-2$) mode of $h$ between the NR and PN waveforms for the G2.5 configuration. The vertical axis denotes the number of orbital rotations derived from GW cycle. Note that the normalized phase differences are qualitatively independent of the mode and arise from the orbital phase error in the PN approximation.