Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism

TL;DR

The paper presents an improved nonspinning effective-one-body (EOB) model calibrated to numerical-relativity (NR) simulations for mass ratios $q=1$–$6$, incorporating multiple subleading gravitational-wave modes. It combines a Padé-resummed potential and adjustable coefficients with a factorized, resummed mode framework, aligning NR and EOB waveforms at low frequency and refining peak properties via NR-informed non-quasi-circular corrections and QNM ringdown matching. The calibrated model yields subdominant-mode performance close to NR, with phase differences under 0.1 rad for the dominant mode and mismatches significantly reduced when including higher-order modes, thereby improving both detection (effectualness) and measurement (parameter biases) for Advanced LIGO. The study highlights the practical impact of higher-order modes on parameter estimation and sets the stage for further refinements in ringdown modeling and longer NR datasets for highly asymmetric binaries.}

Abstract

We calibrate an effective-one-body (EOB) model to numerical-relativity simulations of mass ratios 1, 2, 3, 4, and 6, by maximizing phase and amplitude agreement of the leading (2,2) mode and of the subleading modes (2,1), (3,3), (4,4) and (5,5). Aligning the calibrated EOB waveforms and the numerical waveforms at low frequency, the phase difference of the (2,2) mode between model and numerical simulation remains below 0.1 rad throughout the evolution for all mass ratios considered. The fractional amplitude difference at peak amplitude of the (2,2) mode is 2% and grows to 12% during the ringdown. Using the Advanced LIGO noise curve we study the effectualness and measurement accuracy of the EOB model, and stress the relevance of modeling the higher-order modes for parameter estimation. We find that the effectualness, measured by the mismatch, between the EOB and numerical-relativity polarizations which include only the (2,2) mode is smaller than 0.2% for binaries with total mass 20-200 Msun and mass ratios 1, 2, 3, 4, and 6. When numerical-relativity polarizations contain the strongest seven modes, and stellar-mass black holes with masses less than 50Msun are considered, the mismatch for mass ratio 6 (1) can be as high as 5% (0.2%) when only the EOB (2,2) mode is included, and an upper bound of the mismatch is 0.5% (0.07%) when all the four subleading EOB modes calibrated in this paper are taken into account. For binaries with intermediate-mass black holes with masses greater than 50Msun the mismatches are larger. We also determine for which signal-to-noise ratios the EOB model developed here can be used to measure binary parameters with systematic biases smaller than statistical errors due to detector noise.

Figures (17)

• Figure 1: Amplitude of extrapolated numerical-relativity waveforms for the dominant modes. The two panels from top to bottom are for mass ratios $q=1$ and $q=6$, respectively. Each curve is labeled with its respective $(l,m)$ mode, and the time-delay $\Delta t_\mathrm{ peak}^{\ell m}$ between the extrema of $|h_{22}|$ and $|h_{lm}|$. The horizontal axis measures the time-difference to the peak of $|h_{22}|$.
• Figure 2: We compare the numerical-relativity and EOB $h_{22}$ amplitudes with and without the NQC corrections $N_{\ell m}$ given in Eq. \ref{['Nlm']}. We also plot the numerical and EOB gravitational frequency of the $(2, 2)$ mode and twice the EOB orbital frequency. The left panel refers to $q = 1$ and the right panel to $q =6$. The horizontal axis is the retarded time in the numerical-relativity simulation. The vertical lines mark the peaks of the numerical-relativity $h_{22}$ amplitudes.
• Figure 3: Amplitude (in units of $M/{\cal R}$) and frequency (in units of $1/M$) comparison between the full "NR" waveform and the "NR$+$QNM" waveform generated by attaching QNMs to the inspiral-plunge numerical waveform. We show also the relative amplitude and phase differences. In the left panel, we compare $h_{22}$. In the right panel, we compare the numerical $h_{44}$ mode with two "NR$+$QNM" mode. One of them is generated by attaching the physical QNMs, the other is generated by attaching both the physical QNMs and the pseudo QNM. The former is very different from the numerical-relativity mode and we do not show their amplitude and phase differences. All $h_{44}$ amplitudes have been multiplied by a factor of 20, so that they are more visible. The horizontal axis is the retarded time in the numerical-relativity simulation.
• Figure 4: We show the amplitude of the numerical-relativity $h_{22}$ for mass ratios $q = 1, 2, 3, 4, 6$. We have time shifted the modes so that their peaks are aligned. We have also rescaled them by $\nu$. The horizontal axis is the retarded time in the numerical-relativity simulation. The inset shows an enlargement of the merger region.
• Figure 5: We calibrate adjustable parameters of the EOB dynamics. For mass ratios $q=1, 2, 3, 4$ and $6$, the shaded regions correspond to $(a_5, a_6)$ values for which the EOB and numerical-relativity $h_{22}$ agree within $0.2$ rad at merger, i.e., at the peak of the numerical $h_{22}$. In the inserted subplot, for $a_6(\nu)/\nu = 184$, we show $a_5(\nu)/\nu$ values constrained by the shaded regions and by the test-particle ISCO-shift result BarackSago09, and also the quadratic fit (red curve) given by Eq. \ref{['a5cal']}.