Intermediate-mass-ratio black hole binaries: Intertwining numerical and perturbative techniques

We describe in detail full numerical and perturbative techniques to compute the gravitational radiation from intermediate-mass-ratio black-hole-binary inspirals and mergers. We perform a series of full numerical simulations of nonspinning black holes with mass ratios $q=1/10$ and $q=1/15$ from different initial separations and for different finite-difference resolutions. In order to perform those full numerical runs, we adapt the gauge of the moving punctures approach with a variable damping term for the shift. We also derive an extrapolation (to infinite radius) formula for the waveform extracted at finite radius. For the perturbative evolutions we use the full numerical tracks, transformed into the Schwarzschild gauge, in the source terms of the Regge-Wheller-Zerilli Schwarzschild perturbations formalism. We then extend this perturbative formalism to take into account small intrinsic spins of the large black hole, and validate it by computing the quasinormal mode frequencies, where we find good agreement for spins $|a/M|<0.3$. Including the final spins improves the overlap functions when comparing full numerical and perturbative waveforms, reaching 99.5% for the leading $(\ell ,m)=(2,2)$ and (3, 3) modes, and 98.3% for the nonleading (2, 1) mode in the $q=1/10$ case, which includes 8 orbits before merger. For the $q=1/15$ case, we obtain overlaps near 99.7% for all three modes. We discuss the modeling of the full inspiral and merger based on a combined matching of post-Newtonian, full numerical, and geodesic trajectories.

Figure 1

The puncture separation as a function of time for three $q=1/10$ simulations. The solid curve shows a high-eccentricity simulation obtained from PN quasicircular parameters (with particle limit corrections); the dotted curve shows the results for a binary with the same initial separation after a few iterations to reduce eccentricity; the dot-dashed curve shows an even further separated binary with still smaller eccentricity. Note that the initial jump in the orbit does not appear to be a strong function of the eccentricity or initial separation.

Figure 2

The magnitude of the puncture separation ($|{\overrightarrow{x}}_1-{\overrightarrow{x}}_2|$) as a function of time for a $q=1/10$ and $q=1/15$ binary at similar initial separations. Note that the initial jump in the orbit appears to be independent of $q$. Also note that the $q=1/15$ run inspirals more slowly.

Figure 3

An (xy) projection of the puncture separation (${\overrightarrow{x}}_1-{\overrightarrow{x}}_2$) for a $q=1/10$ and $q=1/15$ binary at similar initial separations. The trajectories have been rotated so that they overlap during the plunge and merger. Note the “universal” plunge trajectory.

Figure 4

The real part of the ($\ell =2$, $m=2$) mode of ${\psi }_4$ for a $q=1/10$ and $q=1/15$ binaries starting at similar separations. The waveform from the $q=1/15$ binary was rescaled by a factor of 1.5 ($15/10$).

Figure 5

The convergence of the phase and amplitude of the ($\ell =2$, $m=2$) mode of ${\psi }_4$ for the $q10r7.3\mathrm{PN}$ configuration. Note that here the three resolutions consist of a low resolution with grid spacing 1.2 times larger than the low resolution runs for $q10r7.3$, $q10r8.4$, $q15r7.3$ configurations. Eighth-order convergence implies ${\psi }_4(1.2h_0)-{\psi }_4(h_0)=4.29982({\psi }_4(h_0)-{\psi }_4(h_0/1.2))$, while fourth-order convergence implies ${\psi }_4(1.2h_0)-{\psi }_4(h_0)=2.0736({\psi }_4(h_0)-{\psi }_4(h_0/1.2))$. Initially, the error in ${\psi }_4$ is very small and dominated by grid noise. Eighth-order convergence in the amplitude is apparent beginning at $t=320M$, while eighth-order convergence in the phase becomes apparent at $t=420M$. The dashed vertical line shows the time when the wave frequency is $M\omega =0.2$. The phase error at this frequency is $\delta \varphi \le 0.2$ rad.

Figure 6

(Top) The phase of ($\ell =2$, $m=2$) mode of ${\psi }_4$ for a $q=1/15$ BHB for three resolutions. Note that the phase error only converges to fourth order and that the highest resolution is refined by a factor of ${1.2}^2$ rather than 1.2 with respect to the medium resolution. (Bottom) A convergence plot showing the initial (better than) fourth-order convergence of the waveform. Note here that the differences ${\psi }_4(1.2h_0)-{\psi }_4(h_0)=1.39895({\psi }_4(h_0)-{\psi }_4(h_0/{1.2}^2))$ if the waveform is fourth-order convergent.

Figure 7

The real part of the ($\ell =2$, $m=2$) mode of $dh/dt$ for the $q=1/10$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively.

Figure 8

The real part of the ($\ell =2$, $m=1$) mode of $dh/dt$ for the $q=1/10$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively.

Figure 9

The real part of the ($\ell =3$, $m=3$) mode of $dh/dt$ for the $q=1/10$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively.

Figure 10

The real part of the ($\ell =2$, $m=2$) mode of $dh/dt$ for the $q=1/15$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively.

Figure 11

The real part of the ($\ell =2$, $m=1$) mode of $dh/dt$ for the $q=1/15$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively.

Figure 12

The real part of the ($\ell =3$, $m=3$) mode of $dh/dt$ for the $q=1/15$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively.

Figure 13

The phase evolution of the ($\ell =2$, $m=2$) wave for the $q=1/10$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively. The inset shows the zoom-in for the quasinormal region.

Figure 14

The phase evolution of the ($\ell =2$, $m=1$) wave for the $q=1/10$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively. The inset shows the zoom-in for the quasinormal region.

Figure 15

The phase evolution of the ($\ell =3$, $m=3$) wave for the $q=1/10$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively. The inset shows the zoom-in for the quasinormal region.

Figure 16

The phase evolution of the ($\ell =2$, $m=2$) wave for the $q=1/15$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively. The inset shows the zoom-in for the quasinormal region.

Figure 17

The phase evolution of the ($\ell =2$, $m=1$) wave for the $q=1/15$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively. The inset shows the zoom-in for the quasinormal region.

Figure 18

The phase evolution of the ($\ell =3$, $m=3$) wave for the $q=1/15$ case. The (black) solid, (red) dotted, and (blue) dashed curves show the NR, spin-off, and spin-on calculations, respectively. The inset shows the zoom-in for the quasinormal region.

Figure 19

The amplitude of the ($\ell =2$, $m=2$) mode of the NR $dh/dt$ for the $q=1/10$ and $1/15$ cases. The (black) thick solid and (red) solid curves show the $q=1/10$ and $1/15$ amplitudes, respectively. The (red) dashed curve denotes $\eta (q=1/10)/\eta (q=1/15)\sim 1.41$ times the $q=1/15$ amplitude.

Figure 20

The amplitude difference in the ($\ell =2$, $m=2$) mode between the NR and perturbative $dh/dt$ for the spin-off cases. The (black) thick solid curve shows the $q=1/10$ case. The (red) solid, dotted, and dashed curves show the amplitude differences for the $q=1/15$ case rescaled by factors of 1, 1.41, and ${1.41}^2$, respectively.

Figure 21

The amplitude difference in the ($\ell =2$, $m=2$) mode between the NR and perturbative $dh/dt$ for the spin-on cases. The (black) thick solid curve shows the $q=1/10$ case. The (red) solid, dotted, and dashed curves show the amplitude differences for the $q=1/15$ case rescaled by factors of 1, 1.41, and ${1.41}^2$, respectively.

Figure 22

The radial trajectory $R(t)$ obtained from the PN and NR evolutions for the $q=1/10$ case in the Schwarzschild coordinates. The (black) solid, (red) dashed, and (blue) dotted curves show the NR, resummed, and Taylor PN ones, respectively. From the lower-right inset, we can choose the matching radius between the NR and resummed PN evolutions as $R(t)/M=9.35123$. The upper-left inset is the zoom-in around the matching time.

Figure 23

The radial trajectory $R(t)$ obtained from the PN and NR evolutions for the $q=1/15$ case in the Schwarzschild coordinates. The (black) solid, (red) dashed, and (blue) dotted curves show the NR, resummed, and Taylor PN ones, respectively. From the lower-right inset, we can choose the matching radius between the NR and resummed PN evolutions as $R(t)/M=8.28796$. The upper-left inset is the zoom-in around the matching time.

Figure 24

The orbit with imaginary eccentricities discussed in 64. The thick and thin curves show the $q=1/10$ and $q=1/15$ cases, respectively. Here we show the orbits with various eccentricities.

Figure 25

The quasinormal frequencies, $\omega$ around $\chi =0$. We have used the same expression as 67. The (red) circles show our result, and the $+$ marks denote the values given in Table II of 68.

Figure 26

The real part of the quasinormal frequencies, $\omega$ around $\chi =0$. The horizontal axis denotes the nondimensional spin parameter, $\chi =S/M^2$. The (red) circles show our result, and the $+$ marks denote the values given in Table II of 68.

Figure 27

The (minus) imaginary part of the quasinormal frequencies, $\omega$ around $\chi =0$. The horizontal axis denotes the nondimensional spin parameter, $\chi =S/M^2$. The (red) circles show our result, and the $+$ marks denote the values given in Table II of 68.

Figure 28

The quasinormal frequencies, $\omega$ for $-0.9\le \chi \le 0.9$. We have used the same expression as 67. The (red) circles show our result, and the $+$ marks denote the values given in Table II of 68.

Figure 29

The real part of the quasinormal frequencies, $\omega$ for $-0.9\le \chi \le 0.9$. The horizontal axis denotes the nondimensional spin parameter, $\chi =S/M^2$. The (red) circles show our result, and the $+$ marks denote the values given in Table II of [68].

Figure 30

The (minus) imaginary part of the quasinormal frequencies, $\omega$ for $-0.9\le \chi \le 0.9$. The horizontal axis denotes the nondimensional spin parameter, $\chi =S/M^2$. The (red) circles show our result, and the $+$ marks denote the values given in Table II of [68].

Figure 31

The error in the real part of the quasinormal frequencies, i.e., $-\mathrm{Er}{\mathrm{r}}_{\mathfrak{F}}$. The horizontal axis denotes the nondimensional spin parameter, $\chi =S/M^2$.

Figure 32

The error in the (minus) imaginary part of the quasinormal frequencies, i.e., $\mathrm{Er}{\mathrm{r}}_{\mathfrak{R}}$. The horizontal axis denotes the nondimensional spin parameter, $\chi =S/M^2$.

Figure 33

The absolute value of the relative errors in quasinormal frequencies in the region $-0.5\le \chi \le 0.5$. The (red) circles and (blue) boxes show those of the real and imaginary parts of the frequencies, i.e., $|\mathrm{Er}{\mathrm{r}}_{\mathfrak{F}}|$ and $|\mathrm{Er}{\mathrm{r}}_{\mathfrak{R}}|$, respectively. The horizontal axis denotes the nondimensional spin parameter, $\chi =S/M^2$.