Comparison between numerical-relativity and post-Newtonian waveforms from spinning binaries: the orbital hang-up case

TL;DR

This study assesses how well post-Newtonian (PN) waveforms describe spinning equal-mass black-hole binaries in the orbital hangup regime by comparing NR simulations with PN templates TaylorT1, TaylorT4, and TaylorEt up to 2.5PN spin effects and, where available, 3.5PN non-spinning terms. Using long NR runs with spins aligned to the orbital angular momentum, the authors analyze phase and amplitude during the ten cycles preceding $M \omega = 0.1$, and they examine inspiral–merger overlaps to gauge spin-estimation potential. They find that NR–PN phase differences remain modest (a few radians) over this interval, with TaylorT1 showing spin-independent behavior and TaylorT4/Et providing improvements in many spin regimes, though 3.5PN results are mixed due to incomplete terms; NR–PN amplitude disagreements rise with spin (roughly 6% nonspinning to 11–12% at high spin) indicating the need for higher-order spin amplitude corrections. The results imply that merger waveforms play a crucial role in accurately estimating spins and that NR–PN hybrids remain viable tools for spinning binaries, provided the parameter space is adequately explored and higher-order spin corrections are incorporated.

Abstract

We compare results from numerical simulations of spinning binaries in the "orbital hangup" case, where the binary completes at least nine orbits before merger, with post-Newtonian results using the approximants TaylorT1, T4 and Et. We find that, over the ten cycles before the gravitational-wave frequency reaches $Mω= 0.1$, the accumulated phase disagreement between NR and 2.5PN results is less than three radians, and is less than 2.5 radians when using 3.5PN results. The amplitude disagreement between NR and restricted PN results increases with the black holes' spin, from about 6% in the equal-mass case to 12% when the black holes' spins are $S_i/M_i^2 = 0.85$. Finally, our results suggest that the merger waveform will play an important role in estimating the spin from such inspiral waveforms.

Figures (6)

• Figure 1: The accumulated phase difference between nonspinning and spinning binaries for the ten cycles before $M\omega = 0.1$, aligned such that the phase disagreement is zero when $M\omega = 0.1$. The black-hole spins are, in order of increasing magnitude of the phase disagreement, $S_i/M_i^2 = 0.25,0.5,0.75,0.85$. Note that the differences are in some cases comparable to the phase disagreements with PN results that we will present in Section \ref{['sec:pncomparison']}.
• Figure 2: An example of a phase comparison between NR and PN results. The case shown here is $S_i/M_i^2 = 0.5$, and the PN results were obtained with the TaylorT1 approximant at 2.5PN order. The accumulated phase error over the ten GW cycles before $M\omega = 0.1$ is $\Delta \phi = -2.69$ radians.
• Figure 3: The accumulated phase disagreement between NR and PN results over the ten cycles before $M\omega = 0.1$. The five different spin values are $S_i/M_i^2 = 0,0.25,0.5,0.75,0.85$, and we compare with the three approximants TaylorT1, TaylorT4 and TaylorEt. All approximants are calculated at 2.5PN order.
• Figure 4: The same comparison as in Figure \ref{['fig:PhaseComparison25']}, but this time the PN approximants are evaluated at 3.5PN order in those terms where this is possible (see text).
• Figure 5: The disagreement between restricted NR and PN amplitudes of $r\Psi_{4,22}$ as a function of GW frequency $M\omega$. The lines below correspond, from bottom to top, to the cases $S_i/M_i^2 = 0,0.25,0.5,0.75$. (Every second line is dashed, to make them easier to distinguish.) In the nonspinning case the disagreement is roughly 6% over this frequency range. The disagreement increases as spin is added, and is about 11% for $S_i/M_i^2 = 0.75$. The large oscillations in this last case are due to the relatively high eccentricity of that system, $e \sim 0.006$.