Faithful Effective-One-Body waveforms of small-mass-ratio coalescing black-hole binaries

TL;DR

This paper tackles the challenge of producing high-fidelity analytic gravitational wave templates for the full coalescence of black-hole binaries in the small-mass-ratio regime. By refining the Effective-One-Body formalism and calibrating against Regge-Wheeler-Zerilli waveforms, the authors implement a resummed 3PN inspiral-plus-plunge waveform with non-quasi-circular corrections, a refined ring-down model with multiple quasi-normal modes, and a novel comb-based matching strategy. The resulting analytic waveforms achieve phase accuracy within about 1% of a cycle and accurate amplitude through inspiral to ring-down, illustrating that purely analytic templates can be both faithful and potentially useful for detectors. The methodology promises extension to comparable-mass binaries, broadening the applicability of EOB-based templates for ground-based GW observations.

Abstract

We address the problem of constructing high-accuracy, faithful analytic waveforms describing the gravitational wave signal emitted by inspiralling and coalescing binary black holes. We work within the Effective-One-Body (EOB) framework and propose a methodology for improving the current (waveform)implementations of this framework based on understanding, element by element, the physics behind each feature of the waveform, and on systematically comparing various EOB-based waveforms with ``exact'' waveforms obtained by numerical relativity approaches. The present paper focuses on small-mass-ratio non-spinning binary systems, which can be conveniently studied by Regge-Wheeler-Zerilli-type methods. Our results include: (i) a resummed, 3PN-accurate description of the inspiral waveform, (ii) a better description of radiation reaction during the plunge, (iii) a refined analytic expression for the plunge waveform, (iv) an improved treatment of the matching between the plunge and ring-down waveforms. This improved implementation of the EOB approach allows us to construct complete analytic waveforms which exhibit a remarkable agreement with the ``exact'' ones in modulus, frequency and phase. In particular, the analytic and numerical waveforms stay in phase, during the whole process, within $\pm 1.1 %$ of a cycle. We expect that the extension of our methodology to the comparable-mass case will be able to generate comparably accurate analytic waveforms of direct use for the ground-based network of interferometric detectors of gravitational waves.

Figures (6)

• Figure 1: Relative dynamics for $\nu =\mu/M=0.01$ and initial separation $r=7$. The plot shows the (nearly indistinguishable) trajectories with $\hat{\cal F}_{\varphi}^K$ (black line) and with $\hat{\cal F}_{\varphi}$ (red line). The dashed circle indicates the LSO (which is crossed at the dynamical time $t/2 \simeq 240$). The light ring (at $r=3$), near which the plunge waveform will be matched to a ring-down one, is indicated (dotted circle). It is crossed at the dynamical time $t/2 \simeq 300$Nagar:2006xv.
• Figure 2: Comparison between various angular momentum losses: GW fluxes at infinity (solid lines; computed à la Regge-Wheeler-Zerilli including up to $\ell=4$ radiation multipoles, using two types of radiation reaction in the driving dynamics: 'K' (black) or 'not K' (red)); or mechanical angular momentum losses: $(dJ/dt)^{K}=-\hat{\cal F}^K_{\varphi}$ ( hyphenated black lines), $dJ/dt = -\hat{\cal F}_{\varphi}$ (hyphenated red lines). The dotted line refers to the mechanical losses with non-quasi-circular corrections. See text for discussion.
• Figure 3: Left panel: Quadrupole waveforms ($\ell=m=2$, even parity): exact Zerilli-type (black line) and "matched" improved EOB-type (red line). Right panel: exact and matched-EOB modulus and instantaneous gravitational wave frequency. The blue line displays twice the orbital frequency $\Omega$. The two vertical dash-dot lines mark the ends (at 297.8 and 301.4) of our matching interval (which is centered, at 299.6 on the maximum of the modulus, and which includes the light ring, at 300.9).
• Figure 4: Illustration of some of the important physical features of the excitation of quasi-normal modes by a small-mass-ratio coalescing binary system. The upper panel shows that, during all the plunge, one remains in a "quasi-stationary" near-zone regime (which does not excite QNMs). The crucial feature which can excite the QNMs is the non-adiabaticity of the evolution of the exciting orbital frequency near its maximum (similarly to Ref. Damour:2006tr). The lower panel illustrates why the (rather short) non-adiabatic behavior of $2 \Omega(t)$ near its maximum preferentially excites the positive-frequency QNMs.
• Figure 5: Analytic and numerical gravitational wave phases versus (shifted) retarded time. As they are very nearly superposed, the inset shows the difference between the two phases (computed, as a check, by means of two independent methods).