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

TL;DR

This paper develops an analytical framework to model the non-adiabatic transition from inspiral to plunge in precessing spinning black-hole binaries using two Hamiltonian descriptions: a post-Newtonian (PN) expanded Hamiltonian and a resummed Effective One Body (EOB) Hamiltonian with spin couplings. It derives a spin-dependent radiation-reaction force consistent with energy and angular-momentum fluxes, compares PN and EOB predictions, and demonstrates that PN-expanded dynamics fail to capture the last stable spherical orbit while the EOB approach remains robust and predictive. By combining a 3PN-accurate EOB Hamiltonian with Padé-resummed fluxes, the study estimates the energy and angular momentum radiated during the final stages, confirms that the dimensionless ratio $|f J|/E^2$ stays below unity at the end of inspiral, and constructs complete waveforms by matching to Kerr quasi-normal modes for ringdown. These results yield physically plausible, analytically tractable templates for the gravitational waves from spinning, precessing binaries and provide a foundation for more accurate waveform modeling in gravitational-wave data analysis. The work highlights the limitations of purely Taylor-expanded PN approaches near the transition to plunge and argues for the continued use of the EOB framework, augmented with spin effects and Padé-resummed radiation reaction, to capture the late-inspiral and plunge dynamics relevant to ground-based detectors.

Abstract

We investigate the non-adiabatic 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 quasi-circular 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 of spin terms onto 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 40Hz and the frequency beyond which our analytical treatment becomes unreliable is found to be, when using the more reliable Effective One-Body dynamics: 0.6% M for anti-aligned maximally spinning black holes, 5% M for aligned maximally spinning black hole, and 1.8% M for non-spinning 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 quasi-normal mode ringdown 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.

Figures (13)

• Figure 1: The energy for circular orbits as function of the frequency evaluated using the PN-expanded Hamiltonian (left panel) and the PN-expansion of the analytically computed function given by Eq. (\ref{['s1']}) (right panel) at various PN orders for maximal spins and equal mass binaries.
• Figure 2: The energy for circular orbits as 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 function of $\chi_L/M^2 \equiv S_{\rm eff} \cdot \hbox{\boldmath${\hat{L}}$}/M^2$ in the equal mass case for EOB Hamiltonian and PN-expanded analytically computed function ${E}(\Omega)$ [see right panel of Fig. \ref{['fig:eomega']}]. The horizontal dashed line in the right panel marks the highest LSSO angular frequency for BBHs with total mass in the range $10\hbox{--}40 M_\odot$, assuming the LIGO frequency band $40 \leq f_{\rm GW} \leq 240 Hz$.
• 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 (upper-diagonal) P-approximants (right panel) are used.