REVIEW
Simulations of black-hole binaries with unequal masses or non-precessing spins: accuracy, physical properties, and comparison with post-Newtonian results
General Relativity and Quantum Cosmology · Submitted 27 Jul 2010 · 122 references
Pith's verdict: A solid, useful simulation-and-calibration paper whose parameter coverage and error accounting are real; the one thing to fix is an internal inconsistency in the mismatch numbers.
T0 · AI review
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Referee report
The paper
The paper claims
Simulated black-hole mergers now come with a stated error budget, and among the standard analytic approximants TaylorT1 tracks the phase of spinning binaries most reliably, while TaylorT4 wins for nonspinning ones.
The reading
What carries the argument
Two pieces of machinery carry the argument. The first is a pipeline for low-eccentricity puncture initial data: integrate the PN equations of motion inward from a wide separation using a 3PN Hamiltonian and 3.5PN flux, now including next-to-leading-order spin terms, read off the momenta at the separation where the full simulation begins, and then refine them by asking how much a momentum change would have to be to reproduce the residual eccentricity seen in the numerical run, scaling both momenta by a common factor (0.15% in the worked example). The second is the comparison protocol itself: express both numerical and PN signals as phase functions of gravitational-wave frequency, align them where $Momega = 0.1$ (about 1.5 orbits before merger), and report both the accumulated phase difference and its square-averaged version over a fixed number of cycles. Because the absolute phase is unobservable, this alignment choice is what makes the comparison meaningful, and the authors show that the same freedom changes numerical error estimates by an order of magnitude.
Extended reading notes
Core claim
The central claim is a map of where each standard PN approximant can be trusted in the last few orbits before merger. Working with the $(2,2)$ harmonic, the authors align PN and numerical phases at $Momega = 0.1$ and measure the disagreement accumulated over the preceding 8-10 cycles. For nonspinning binaries, including the unequal-mass cases up to $q=4$, the 3.5PN TaylorT4 phase remains the closest to the numerical result, with a disagreement of roughly $0.2$ rad at $q=4$. For equal-mass binaries with equal aligned or anti-aligned spins, TaylorT1, which carries spin terms only to 2.5PN order, is the most robust across the whole spin range, staying between about 1 and 2 rad of accumulated disagreement; the truncated TaylorT4 does poorly at large spin, especially large anti-aligned spin, while the full TaylorT4 is the best choice for $S_i/M_i^2 > -0.75$. On amplitude, the authors find that once the highest available corrections are included (3PN nonspinning, 2PN spinning) the PN amplitude of the $(2,2)$ mode exceeds the numerical one by only 2-4%, whereas the plain quadrupole formula overestimates by up to 13% at $S_i/M_i^2 = +0.85$. They also report the physical outputs of the runs: final mass and spin of the remnant, ringdown frequencies, the distribution of radiated energy among harmonics, and recoils of $140$-$155$ km/s in the unequal-mass cases.
Load-bearing premise
The low-eccentricity claim rests on assuming that a momentum tweak which removes a given eccentricity in a post-Newtonian integration removes the same eccentricity in the full numerical simulation, even though the eccentricity itself is read off the coordinate motion of the punctures, which the authors acknowledge is a gauge artifact.
Editorial extensions
If this is right
- Template builders gain a concrete rule for hybrid waveform construction: attach TaylorT4 to nonspinning numerical waveforms, TaylorT1 to spinning ones, with TaylorT4 acceptable for spins above -0.75 if its higher-order spin terms are retained.
- The simulated waveforms can be used as calibration targets for phenomenological and effective-one-body models without further error correction, since the mismatch from finite resolution is below $10^{-6}$ and the dominant error, from finite extraction radius, is at most a few times $10^{-4}$.
- The 2-4% residual in the corrected PN amplitude sets a floor on how accurately amplitude-based parameter estimation can be calibrated against these hybrids, and warns that the largest-amplitude, highest-spin aligned cases are also the ones where the uncorrected quadrupole formula errs most.
- Final mass, final spin, ringdown frequency and recoil values tabulated here give independent checks on analytic remnant formulas, with agreement at the 1% level for aligned spins and disagreement around 10% for strongly anti-aligned spins in one published fit.
- Residual eccentricity of a few times $10^{-3}$ is shown to be achievable across the spin and mass-ratio range studied, which is the level needed before phase comparisons at the 0.1 rad scale are meaningful.
Reading between the lines
- The paper's accuracy claims are stated for binaries heavy enough that the whole numerical waveform sits in the detector band (masses above roughly $120 M_\odot$); the same waveforms say nothing about the low-mass regime, where the answer depends on how well PN extends the signal, not on how well the simulation is resolved.
- Because the spin terms in the phase approximants stop at 2.5PN while the nonspinning terms reach 3.5PN, the apparent superiority of TaylorT1 for spinning cases may be a statement about which truncation happens to err in a compensating direction, rather than about the underlying physics; a future 3.5PN spin completion could reshuffle the ranking entirely.
- The eccentricity-reduction step assumes the numerical and PN responses to a momentum perturbation agree in sign and size. If a cheaper, phase-aware iteration were adopted, the quoted residual eccentricities would be the first thing to shift, and with them the sub-0.1-rad phase agreements.
- The recoil numbers come from only two usable extraction radii, so the extrapolation to infinite radius is doing more work than the data strictly supports; independent simulations with better-resolved extraction spheres would be the natural cross-check.
Original abstract
We present gravitational waveforms for the last orbits and merger of black-hole-binary (BBH) systems along two branches of the BBH parameter space: equal-mass binaries with equal non-precessing spins, and nonspinning unequal-mass binaries. The waveforms are calculated from numerical solutions of Einstein's equations for black-hole binaries that complete between six and ten orbits before merger. Along the equal-mass spinning branch, the spin parameter of each BH is $\chi_i = S_i/M_i^2 \in [-0.85,0.85]$, and along the unequal-mass branch the mass ratio is $q =M_2/M_1 \in [1,4]$. We discuss the construction of low-eccentricity puncture initial data for these cases, the properties of the final merged BH, and compare the last 8-10 GW cycles up to $M\omega = 0.1$ with the phase and amplitude predicted by standard post-Newtonian (PN) approximants. As in previous studies, we find that the phase from the 3.5PN TaylorT4 approximant is most accurate for nonspinning binaries. For equal-mass spinning binaries the 3.5PN TaylorT1 approximant (including spin terms up to only 2.5PN order) gives the most robust performance, but it is possible to treat TaylorT4 in such a way that it gives the best accuracy for spins $\chi_i > -0.75$. When high-order amplitude corrections are included, the PN amplitude of the $(\ell=2,m=\pm2)$ modes is larger than the NR amplitude by between 2-4%.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We present gravitational waveforms for the last orbits and merger of black-hole-binary (BBH) systems along two branches of the BBH parameter space: equal-mass binaries with equal non-precessing spins, and nonspinning unequal-mass binaries. The waveforms are calculated from numerical solutions of Einstein's equations for black-hole binaries that complete between six and ten orbits before merger. Along the equal-mass spinning branch, the spin parameter of each BH is $\chi_i = S_i/M_i^2 \in [-0.85,0.85]$, and along the unequal-mass branch the mass ratio is $q =M_2/M_1 \in [1,4]$. We discuss the construction of low-eccentricity puncture initial data for these cases, the properties of the final merged BH, and compare the last 8-10 GW cycles up to $M\omega = 0.1$ with the phase and amplitude predicted by standard post-Newtonian (PN) approximants. As in previous studies, we find that the phase from the 3.5PN TaylorT4 approximant is most accurate for nonspinning binaries. For equal-mass spinning binaries the 3.5PN TaylorT1 approximant (including spin terms up to only 2.5PN order) gives the most robust performance, but it is possible to treat TaylorT4 in such a way that it gives the best accuracy for spins $\chi_i > -0.75$. When high-order amplitude corrections are included, the PN amplitude of the $(\ell=2,m=\pm2)$ modes is larger than the NR amplitude by between 2-4%.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present gravitational waveforms for the last orbits and merger of black-hole-binary (BBH) systems along two branches of the BBH parameter space: equal-mass binaries with equal non-precessing spins, and nonspinning unequal-mass binaries. | contribution 1.00 | 0.95 | 0.08 | asserted 1.00 |
| s1 | The waveforms are calculated from numerical solutions of Einstein's equations for black-hole binaries that complete between six and ten orbits before merger. | method 0.98 | 0.92 | 0.03 | asserted 1.00 |
| s2 | Along the equal-mass spinning branch, the spin parameter of each BH is $\chi_i = S_i/M_i^2 \in [-0.85,0.85]$, and along the unequal-mass branch the mass ratio is $q =M_2/M_1 \in [1,4]$. | method 0.67 | 0.61 | 0.03 | asserted 0.99 |
| s3 | We discuss the construction of low-eccentricity puncture initial data for these cases, the properties of the final merged BH, and compare the last 8-10 GW cycles up to $M\omega = 0.1$ with the phase and amplitude predicted by standard post-Newtonian (PN) approximants. | contribution 0.68 | 0.93 | 0.08 | asserted 0.88 |
| s4 | As in previous studies, we find that the phase from the 3.5PN TaylorT4 approximant is most accurate for nonspinning binaries. | result 1.00 | 0.85 | 0.15 | asserted 1.00 |
| s5 | For equal-mass spinning binaries the 3.5PN TaylorT1 approximant (including spin terms up to only 2.5PN order) gives the most robust performance, but it is possible to treat TaylorT4 in such a way that it gives the best accuracy for spins $\chi_i > -0.75$. | result 1.00 | 0.87 | 0.19 | asserted 0.61 |
| s6 | When high-order amplitude corrections are included, the PN amplitude of the $(\ell=2,m=\pm2)$ modes is larger than the NR amplitude by between 2-4%. | result 1.00 | 0.84 | 0.09 | asserted 1.00 |
What was examined, and what was not
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- P abstract
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Figures
Figures reproduced from arXiv:1007.4789 by the authors.
References
How the literature uses it
32 arXiv papers cite this in 52 sentences
Papers citing it · 24 sentences without a confident label
Contests it 1 paper
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Our findings disagree with the results of Hannam [ref] , despite following the identical comparison protocol.
Suitability of hybrid gravitational waveforms for unequal-mass binaries · Phase errors between PN and NR · contest 0.77
Uses its method or tool 6 papers
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In this work, we will average over different realizations of $n$ by setting $n=0$ and use synthetic numerical relativity (NR) waveforms [ref] both as a model of the input GW signal $h_inj$ (or injection) and as templates $h(λ⃗)$.
Tracking black hole kicks from gravitational wave observations · Introduction · 0.91
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Following Hannam et al. [ref] , we quantify this by aligning the two waveforms in phase at a certain frequency and calculating the phase difference accumulated over a number of GW cycles before this frequency.
Phenomenological gravitational waveform model of binary black holes incorporating horizon fluxes · Phase comparison of inspiral and NR waveforms · 1.00
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We estimate initial momenta for low-eccentricity inspiral using the post-Newtonian methods outlined in [ref] .
Black-hole hair loss: learning about binary progenitors from ringdown signals · Numerical simulations of merger and ringdown signals · 1.00
Uses its data or values 4 papers
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We use results from simulations of non-spinning binaries with mass ratios $q = {1,2,3,4}$ that were previously presented in Refs. [ref] and an additional simulation of a $q=11$ binary that was carried out as a part of this study.
Black-hole hair loss: learning about binary progenitors from ringdown signals · Numerical simulations of merger and ringdown signals · 0.94
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The data-set #1 covers the parameter space of non-spinning systems for several mass ratios during at least the last 5 orbits before merger (length $∼1100-1450 M$, where $M$ is the total ADM mass of the spacetime) [ref] .
Matching post-Newtonian and numerical relativity waveforms: systematic errors and a new phenomenological model for non-precessing black hole binaries · NR waveforms and codes · 1.00
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Seven sets of simulations were used in this paper: (1) Equal-mass binaries with equal, non-precessing spins $χ_i = ±{0.25,0.5,0.75,0.85}$, described in [ref] .
Inspiral-merger-ringdown waveforms for black-hole binaries with non-precessing spins · Introduction · 0.91
Compares with it 2 papers
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Figure 8 of [ref] shows a roughly constant phase difference between TaylorT1 and the numerical simulation of $∼0.5$ radians, for all considered mass ratios ($q=1,2,3,4$), whereas we find a steadily declining phase difference reaching zero near $q∼5$.
Suitability of hybrid gravitational waveforms for unequal-mass binaries · Phase errors between PN and NR · 0.81
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Our results are also qualitatively consistent with the results described in [ref] as we find that for equal mass binaries, the Taylor T4 approximant performs better than the Taylor T1 approximant (both for aligned and anti-aligned spins).
Comparing Post-Newtonian and Numerical-Relativity Precession Dynamics · Orbital Phase Comparisons · 0.88
Cites it as support 3 papers
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This is motivated by PN-NR comparisons of equal-mass spinning binaries, in which the accuracy of the TaylorT1 approximant was found to be the most robust [ref] .
Inspiral-merger-ringdown waveforms for black-hole binaries with non-precessing spins · Introduction · 1.00
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Nonprecessing binaries also emit a significant fraction of their total (band-limited) signal $ρ^2$ in higher harmonics, at high mass [ref] or mass ratio [ref] .
Is J enough? Comparison of gravitational waves emitted along the total angular momentum direction with other preferred orientations · Failure of antipodal reconstruction · 0.97
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These subdominant modes become more important with higher mass ratio (see [ref] and Fig. [ref]), and we argue that this increases the need for reflection minimizing boundary conditions, as those applied here.
Simulations of non-equal mass black hole binaries with spectral methods · Discussion · 0.97
Cites it as background 9 papers
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On the theoretical side, one has recently obtained a good understanding of all the stages in the dynamics of coalescing BBHs and their emitted gravitational radiation thanks to a combination of theoretical techniques: (i) the early, adiabatic inspiral stage of the evolution can be described in terms of a post-Newtonian (PN) expansion up to 3.5PN order in phase and 3PN in amplitude [ref] ; (ii) the late inspiral, the merger and the ring-down can be computed by numerically solving Einstein's equations [ref] ; and (iii) the effective-one-body (EOB) formalism [ref] has shown its capability of incorporating information coming both from PN theory and numerical relativity (NR) results into an accurate description [ref] of the whole BBHs dynamics and GW radiation, from the early inspiral, right across the last stable orbit, to the “plunge”, the merger and the final ringdown.
Accuracy and effectualness of closed-form, frequency-domain waveforms for non-spinning black hole binaries · Introduction · 0.92
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More recently, there have been extensive studies of waveform models involving high-accuracy numerical relativity (NR) simulations [ref] .
Accuracy of binary black hole waveform models for aligned-spin binaries · Introduction · 0.84
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Both need to be calibrated to numerical relativity (NR) simulations [ref] to faithfully model the late stages of the coalescence.
Detectability of gravitational waves from binary black holes: Impact of precession and higher modes · Introduction · 0.82
What it cites · 129 sentences without a confident label
Builds on 4 papers
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This extends our previous studies of the equal-mass nonspinning binary [ref] and equal-mass binaries in the orbital-hangup configuration [ref] .
Where post-Newtonian and numerical-relativity waveforms meet · Introduction · 1.00
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This extends our previous studies of the equal-mass nonspinning binary [ref] and equal-mass binaries in the orbital-hangup configuration [ref] .
Comparison between numerical relativity and a new class of post-Newtonian gravitational-wave phase evolutions: the non-spinning equal-mass case · Introduction · 1.00
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This extends our previous studies of the equal-mass nonspinning binary [ref] and equal-mass binaries in the orbital-hangup configuration [ref] .
Comparison between numerical-relativity and post-Newtonian waveforms from spinning binaries: the orbital hang-up case · Introduction · 1.00
Uses the method or tool of 19 papers
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For this work we have incorporated recent results [ref] , and include next-to-leading order (NLO) spin terms in the PN equations of motion.
Higher-order spin effects in the dynamics of compact binaries I. Equations of motion · Specification of low-eccentricity initial parameters · 0.83
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For this work we have incorporated recent results [ref] , and include next-to-leading order (NLO) spin terms in the PN equations of motion.
Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling · Specification of low-eccentricity initial parameters · 0.83
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For this work we have incorporated recent results [ref] , and include next-to-leading order (NLO) spin terms in the PN equations of motion.
Higher-order spin effects in the dynamics of compact binaries II. Radiation field · Specification of low-eccentricity initial parameters · 0.83
Uses data or values from 3 papers
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In addition, it was found that the quadrupole amplitude error grew to as much as 12% in high spin cases [ref] .
Comparison between numerical-relativity and post-Newtonian waveforms from spinning binaries: the orbital hang-up case · Late inspiral comparison of NR and PN waveforms · 0.81
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We found in [ref] that in the equal-mass nonspinning case the quadrupole PN amplitude was larger than the full GR amplitude during inspiral by about 7%.
Where post-Newtonian and numerical-relativity waveforms meet · Amplitude comparison · 0.80
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In [ref] the uncertainty due to this effect was estimated at about 3%.
Total recoil: the maximum kick from nonspinning black-hole binary inspiral · Recoil · 0.97
Compares with 14 papers
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We also compare with the PN amplitude that results from using all currently known amplitude corrections (up to 3PN order non-spinning [ref] and up to 2PN order spinning contributions [ref] ).
Higher-order spin effects in the amplitude and phase of gravitational waveforms emitted by inspiraling compact binaries: Ready-to-use gravitational waveforms · Amplitude comparison · 0.80
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We also compare with the PN amplitude that results from using all currently known amplitude corrections (up to 3PN order non-spinning [ref] and up to 2PN order spinning contributions [ref] ).
Gravitational radiation from compact binary systems: gravitational waveforms and energy loss to second post-Newtonian order · Amplitude comparison · 0.80
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We also compare with the PN amplitude that results from using all currently known amplitude corrections (up to 3PN order non-spinning [ref] and up to 2PN order spinning contributions [ref] ).
Using Full Information When Computing Modes of Post-Newtonian Waveforms From Inspiralling Compact Binaries in Circular Orbit · Amplitude comparison · 0.80
Cites as support 5 papers
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In addition, the choice of equal spins also yields an important sub-family of configurations: we found in [ref] that it is possible to rather accurately model any non-precessing binaries with unequal spins using essentially only a mass-weighted sum of the spins of the binary, motivated by numerical evidence from [ref] , and PN theory [ref] .
Inspiral-merger-ringdown waveforms for black-hole binaries with non-precessing spins · Summary of numerical simulations · 0.97
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In addition, the choice of equal spins also yields an important sub-family of configurations: we found in [ref] that it is possible to rather accurately model any non-precessing binaries with unequal spins using essentially only a mass-weighted sum of the spins of the binary, motivated by numerical evidence from [ref] , and PN theory [ref] .
Matched Filtering of Numerical Relativity Templates of Spinning Binary Black Holes · Summary of numerical simulations · 0.97
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In addition, the choice of equal spins also yields an important sub-family of configurations: we found in [ref] that it is possible to rather accurately model any non-precessing binaries with unequal spins using essentially only a mass-weighted sum of the spins of the binary, motivated by numerical evidence from [ref] , and PN theory [ref] .
Exploring black hole superkicks · Summary of numerical simulations · 0.97
Cites as background 19 papers
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This question was first addressed for equal-mass nonspinning waveforms in [ref] , and later with increasing levels of precision in [ref] .
Where post-Newtonian and numerical-relativity waveforms meet · Late inspiral comparison of NR and PN waveforms · 0.84
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This question was first addressed for equal-mass nonspinning waveforms in [ref] , and later with increasing levels of precision in [ref] .
High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions · Late inspiral comparison of NR and PN waveforms · 0.84
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This question was first addressed for equal-mass nonspinning waveforms in [ref] , and later with increasing levels of precision in [ref] .
Comparison between numerical relativity and a new class of post-Newtonian gravitational-wave phase evolutions: the non-spinning equal-mass case · Late inspiral comparison of NR and PN waveforms · 0.84
Discussion
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