Binary black hole coalescence in the extreme-mass-ratio limit: testing and improving the effective-one-body multipolar waveform

TL;DR

The paper addresses the accuracy of analytical gravitational-wave waveforms for extreme-mass-ratio coalescences by comparing the effective-one-body (EOB) multipolar waveform to Regge-Wheeler-Zerilli (RWZ) numerical waveforms extracted at ${\cal I}^+$ using a hyperboloidal layer. It demonstrates that the analytically resummed EOB waveform preserves phase coherence with RWZ during a long inspiral (about 37 orbits) with dephasing around $-2.5\times 10^{-3}$ rad, and it improves the merger and ringdown by introducing and calibrating next-to-quasi-circular (NQC) corrections for multiple multipoles, achieving phase differences as small as $\pm 0.015$ rad and a fractional amplitude agreement of about $2.5\%$ near merger. The authors fix four NQC parameters per multipole by enforcing compatibility at the light ring $t_m=t_{\rm LR}$ and explore shifted QNM matching times to further refine the ringdown, showing robust improvements across multipoles up to $\ell=4$. The work supports the use of EOB waveforms for LISA EMRI studies and provides a systematic method for NQC calibration that is applicable to NR benchmarks for comparable-mass binaries.

Abstract

We discuss the properties of the effective-one-body (EOB) multipolar gravitational waveform emitted by nonspinning black-hole binaries of masses $μ$ and $M$ in the extreme-mass-ratio limit, $μ/M=ν\ll 1$. We focus on the transition from quasicircular inspiral to plunge, merger and ringdown.We compare the EOB waveform to a Regge-Wheeler-Zerilli (RWZ) waveform computed using the hyperboloidal layer method and extracted at null infinity. Because the EOB waveform keeps track analytically of most phase differences in the early inspiral, we do not allow for any arbitrary time or phase shift between the waveforms. The dynamics of the particle, common to both wave-generation formalisms, is driven by leading-order ${\cal O}(ν)$ analytically--resummed radiation reaction. The EOB and the RWZ waveforms have an initial dephasing of about $5\times 10^{-4}$ rad and maintain then a remarkably accurate phase coherence during the long inspiral ($\sim 33$ orbits), accumulating only about $-2\times 10^{-3}$ rad until the last stable orbit, i.e. $Δφ/φ\sim -5.95\times 10^{-6}$. We obtain such accuracy without calibrating the analytically-resummed EOB waveform to numerical data, which indicates the aptitude of the EOB waveform for LISA-oriented studies. We then improve the behavior of the EOB waveform around merger by introducing and tuning next-to-quasi-circular corrections both in the gravitational wave amplitude and phase. For each multipole we tune only four next-to-quasi-circular parameters by requiring compatibility between EOB and RWZ waveforms at the light-ring. The resulting phase difference around merger time is as small as $\pm 0.015$ rad, with a fractional amplitude agreement of 2.5%. This suggest that next-to-quasi-circular corrections to the phase can be a useful ingredient in comparisons between EOB and numerical relativity waveforms.

Figures (10)

• Figure 1: Multipolar "convergence" of the ${\cal R} h_+/(M\nu)$ polarization of the Regge-Wheeler-Zerilli multipolar waveform. Top panel: The complete wave train ($\sim 37$ cycles). Bottom panel: Impact of subdominant multipoles around the merger time. The vertical dashed line indicates the light-ring crossing.
• Figure 2: Testing the waveform resummation for $\ell=2$ at the beginning of the inspiral. Top panel: Insplunge and RWZ waveforms extracted at ${\cal I}^+$. Bottom panel: The phase differences $\Delta\phi^{\rm EOBRWZ}_{\ell m}=\phi^{\rm EOB}_{\ell m}-\phi^{\rm RWZ}_{\ell m}$, for RWZ waveforms measured at ${\cal I}^+$, are contrasted with the corresponding ones for RWZ waveforms measured at a finite extraction radius $r_*^{\rm obs}=1000M$.
• Figure 3: Addition of NQC corrections and matching to QNMs; $\ell=2$ multipoles. The (light) dashed lines refer to the bare insplunge waveform, without the addition of NQC corrections (dash-dotted line, blue online) nor of QNM ringdown (dashed line, red online). The vertical (light) dashed line indicates the location of the maximum of $M\Omega$.
• Figure 4: Time evolution of the phase difference $\Delta\phi^{\rm EOBRWZ}_{\ell m}=\phi^{\rm EOB}_{\ell m}-\phi^{\rm RWZ}_{\ell m}$ between the full EOB and RWZ multipolar waveforms. The dash-dotted vertical line locates the light ring.
• Figure 5: Time evolution of the relative amplitude difference $(\Delta A/A)_{\ell m} = (A_{\ell m}^{\rm EOB}- A_{\ell m}^{\rm RWZ})/A_{\ell m}^{\rm RWZ}$ between the full EOB and RWZ multipolar waveforms. The dash-dotted vertical line locates the light ring.