Transition from inspiral to plunge in precessing binaries of spinning black holes

We investigate the nonadiabatic dynamics of spinning black-hole binaries by using an analytical Hamiltonian completed with a radiation-reaction force, containing spin couplings, which matches the known rates of energy and angular-momentum losses on quasicircular orbits. We consider both a straightforward post-Newtonian-expanded Hamiltonian (including spin-dependent terms), and a version of the resummed post-Newtonian Hamiltonian defined by the effective one-body approach. We focus on the influence spin terms have on the dynamics and waveforms. We evaluate the energy and angular momentum released during the final stage of inspiral and plunge. For an equal-mass binary the energy released between 40 Hz and the frequency beyond which our analytical treatment becomes unreliable is found to be, when using the effective one-body dynamics, $0.6\%M$ for antialigned maximally spinning black holes, $5\%M$ for aligned maximally spinning black holes, and $1.8\%M$ for nonspinning configurations. In confirmation of previous results, we find that, for all binaries considered, the dimensionless rotation parameter $J/E^2$ is always smaller than unity at the end of the inspiral, so that a Kerr black hole can form right after the inspiral phase. By matching a quasinormal mode ring down to the last reliable stages of the plunge, we construct complete waveforms approximately describing the gravitational-wave signal emitted by the entire process of coalescence of precessing binaries of spinning black holes.

Figure 1

The energy for circular orbits as a function of the frequency evaluated using the PN-expanded Hamiltonian (left panel) and the PN expansion of the analytically computed function given by Eq. (2.32) (right panel) at various PN orders for maximal spins and equal-mass binaries.

Figure 2

The energy for circular orbits as a function of the frequency evaluated from the EOB Hamiltonian at various PN orders for maximal spins and equal-mass binaries.

Figure 3

The energy (left panel) and the frequency (right panel) at the LSSO as a function of ${\chi }_L\equiv S_{\mathrm{eff}}·\widehat{\mathit{L}}/M^2$ in the equal-mass case for the EOB Hamiltonian and PN-expanded, analytically computed function $E(\Omega )$ (see right panel of Fig. 1). The horizontal dashed line in the right panel marks the highest LSSO angular frequency for BBHs with total mass in the range $10–40M_{\odot }$, assuming the LIGO frequency band $40\le f_{\mathrm{GW}}\le 240\mathrm{Hz}$.

Figure 4

Signal-to-noise ratio versus binary total mass at 100 Mpc for equal-mass binaries with LSSO determined by the 3PN-EOB Hamiltonian.

Figure 5

Newton-normalized flux in the equal-mass case with both BH spins aligned (and maximal $\chi ={\chi }_1={\chi }_2$) with orbital angular momentum when $T$ approximants (left panel) and (lower-diagonal) $P$ approximants (right panel) are used.

Figure 6

Newton-normalized flux in the equal-mass case with both BH spins antialigned (and maximal $\chi ={\chi }_1={\chi }_2$) with orbital angular momentum when $T$ approximants (left panel) and (lower-diagonal) $P$ approximants (right panel) are used.

Figure 7

Comparison between the $T$- and (lower-diagonal) $P$-approximant Newton-normalized flux in the equal-mass, nonspinning case at 3PN and 3.5PN order.

Figure 8

Oscillations in $\dot{r}$ when spin-spin interactions are present, in $(10+10)M_{\odot }$ (left panel) and $(15+5)M_{\odot }$ (right panel) binaries. Dark curves show $\dot{r}/(r\omega )$ as functions of $f_{\mathrm{GW}}$ when both spin-orbit and spin-spin interactions are take into account, while light curves show the same quantity when only spin-orbit interactions are included.

Figure 9

Direct comparison of orbital phases measured with respect to the frame determined by ${\mathbf{S}}_{\mathrm{tot}}\times {\widehat{\mathbf{L}}}_N$, when different initial orbital phases are used. For maximally spinning $(15+15)M_{\odot }$ binaries, we start evolution at 40 Hz, with $({\theta }_{S_1},{\varphi }_{S_1};{\theta }_{S_2},{\varphi }_{S_2})=(60°,90°;60°,0°)$, and initial orbital phases of ${\varphi }_{\mathrm{orb}}(0)=0$, and ${\varphi }_{\mathrm{orb}}^{\prime }(0)=\pi /4$, $\pi /2$, and $3\pi /4$. We use the evolution with ${\varphi }_{\mathrm{orb}}(0)=0$ as a reference, and plot (in degrees) the direct differences $\Delta {\varphi }_{\mathrm{orb}}(t)\equiv {\varphi }_{\mathrm{orb}}^{\prime }(t)-{\varphi }_{\mathrm{orb}}(t)$ in orbital-phase evolution [cf. Eq. (5.3)].

Figure 10

Comparison of orbital phases measured with respect to the frame determined by ${\mathbf{S}}_{\mathrm{tot}}\times {\widehat{\mathbf{L}}}_N$, when different initial orbital phases are used—with a relative time shift applied to reduce secular growth in the differences. We take the same binary and initial conditions as in Fig. 9, but apply a time shift $\delta t$ on evolutions with ${\varphi }_{\mathrm{orb}}^{\prime }(0)=\pi /4$, $\pi /2$, and $3\pi /4$, before comparing them to the evolution with ${\varphi }_{\mathrm{orb}}(0)=0$. We plot (in radians) the resulting ${\Delta }^{\prime }{\varphi }_{\mathrm{orb}}$ [cf. (5.4)], when $\delta t$ is optimized in the reduction of secular growths in phase differences. The $\delta t/M$ used are 69, 139, and 71, respectively. (The initial orbital period is 338 M.)

Figure 11

Comparison between the three different prescriptions, $\delta E_I$ (dashed curves), $\delta E_h$ (dotted curves), and $\delta E_H$ (solid curves), for calculating energy losses. We use Newtonian-order radiation reaction in the left panel, and Padé at 3.5PN order in the right panel. We use the EOB Hamiltonian at 3PN order.

Figure 12

Accumulative energy release (upper panels) and instantaneous values of $|\mathbf{J}|/E^2$ (lower panels) of $(15+15)M_{\odot }$ (left panels) and $(15+5)M_{\odot }$ (right panels) binaries. LSSO frequencies for the antialigned, generic-down, and nonspinning configurations are shown in vertical grid lines, while LSSOs of generic-up and aligned configurations are above the ranges of our plots. (Spin-spin terms are not included in these evolutions.) We use the SEHPF(3,3.5) model.

Figure 13

Inspiral waveforms (which end at $t=0$ in our plot) matched to ring-down waveforms for nonspinning (light solid curve) and half-maximally spinning $(15+15)M_{\odot }$ binaries (left panel) and $(15+5)M_{\odot }$ binaries (right panel) in the generic-up (dark solid curve) and generic-down (dark dashed curve) configurations. We start our evolutions at 40 Hz, and use $(F_{+},F_{\times };\Theta ,\phi )=(1,0;\pi /4,0)$. In the plot, we mark the position of the LSSO with solid curves. The waveforms have been shifted in time such that the end of the inspiral occurs at $t/M=0$.