Comparison of Numerical and Post-Newtonian Waveforms for Generic Precessing Black-Hole Binaries

TL;DR

The paper benchmarks long-term numerical-relativity simulations of a generic precessing black-hole binary against post-Newtonian predictions, focusing on a mass ratio $q\sim0.8$ with misaligned spins. It demonstrates that a 3.5PN EOM more accurately captures the inspiral dynamics and waveform phases than 2.5PN, and that NR-PN overlaps for key modes exceed 0.9 in early cycles, with higher overlaps for subleading modes in the presence of precession. The analysis highlights two sources of eccentricity— residual PN-driven eccentricity and precession-induced eccentricity that grows at small separations— and shows that precession imprint appears prominently in the $\ell=2,m=1$ and $\ell=3,m=3$ modes. The results underscore the need for even higher-order PN corrections, especially spin-dependent terms, to produce accurate waveform templates for generic precessing binaries and motivate iterative eccentricity-reduction approaches for better initial data. Overall, the work provides a detailed NR-PN calibration framework for complex black-hole binaries and supports continued development of higher-order PN models and hybrid waveform construction.

Abstract

We compare waveforms and orbital dynamics from the first long-term, fully non-linear, numerical simulations of a generic black-hole binary configuration with post-Newtonian predictions. The binary has mass ratio q~0.8 with arbitrarily oriented spins of magnitude S_1/m_1^2~0.6 and S_2/m_2^2~0.4 and orbits 9 times prior to merger. The numerical simulation starts with an initial separation of r~11M, with orbital parameters determined by initial 2.5PN and 3.5PN post-Newtonian evolutions of a quasi-circular binary with an initial separation of r=50M. The resulting binaries have very little eccentricity according to the 2.5PN and 3.5PN systems, but show significant eccentricities of e~0.01-0.02 and e~0.002-0.005 in the respective numerical simulations, thus demonstrating that 3.5PN significantly reduces the eccentricity of the binary compared to 2.5PN. We perform three numerical evolutions from r~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 (l=2,m=+-2) modes, 91% for the (l=2,m=+-1) modes, and nearly 91% for the (l=3,m=+-3) modes. The phase differences between numerical and post-Newtonian approximations appear to be independent of the (l,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 4.0PN) may yield significantly better waveforms. In addition, we identify features in the waveforms likely related to precession and precession-induced eccentricity.

Figures (39)

• 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 3rd- and 4th-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 8th-order convergence up to $t=1200M$, decreasing to between 3rd- and 4th-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 $(l=2,m=1)$ mode for the G2.5 configuration. The plot demonstrates that the windowing procedure apparently does not contaminate the waveform to a significant degree.