REVIEW
Improved filters for gravitational waves from inspiralling compact binaries
General Relativity and Quantum Cosmology · Submitted 18 Aug 1997 · 46 references
Pith's verdict: The Padé-resummation construction is the real contribution and it holds up in the test-mass limit; the finite-mass robustness tables are partly self-referential and the abstract states the 96.5% number more broadly than the body supports.
T0 · AI review
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Referee report
The paper
The paper claims
Rewriting the post-Newtonian series as rational approximants, rather than extending it, lets fifth-order gravitational-wave templates match the true inspiral signal above 96.5 percent.
The reading
What carries the argument
Two substitutions carry the argument. The first is the energy function $$e(x) = -1 + left(frac{E_{tot}^2 - $m_1^{2}$ - $m_2^{2}$}{2 m_1 m_2}right)^2,$$ which in the test-mass limit equals $-x(1-4x)/(1-3x)$ and therefore has a simple pole at the light ring and a minimum at the last stable orbit. Its diagonal Padé approximant built from only the second post-Newtonian coefficients reproduces this exact expression, and hence reproduces $v_{lso} = 1/sqrt{6}$ from data that a Taylor series cannot use. The second is the factored flux $$f(v) = left(1 - frac{v}{v_{pole}}right) F(v),$$ with the pole removed and the logarithmic terms normalized to $v_{lso}$ and separated, so that the remaining series is amenable to Padé resummation. The waveform is then $h(v) = C v^2 cos 2Phi(v)$ with $t(v)$ and $Phi(v)$ obtained by numerically integrating $E'[e^P_n]$ and $F[f^P_n]$ and inverting $v(t)$; overlaps are computed by fast Fourier transform against the initial-LIGO noise spectrum and maximized over time, phase and masses.
Extended reading notes
Core claim
The central claim is that the standard map from post-Newtonian coefficients to waveform is a poor one, and that a better map exists using the same input data. Instead of inserting the Taylor approximants $E^T_n$ and $F^T_n$ directly into the energy-balance phasing integral, the authors pass through two intermediate functions: a symmetric energy function $e(x)$ built from the Mandelstam variable $s$, and a flux function $f(v)$ with the light-ring pole factored out and the logarithms separated. Successive approximants of $e$ and $f$ are then taken as diagonal or subdiagonal Padé approximants rather than truncated power series, and the waveform is rebuilt from those. In the test-mass limit, where the exact energy and a numerical exact flux are known, the resulting sequence is reported to converge faster and essentially monotonically, while the Taylor sequence oscillates: for a $1.4$–$10$ solar-mass system the Taylor overlap falls to about $0.71$ at order $v^8$ and recovers only to about $0.86$ at $v^{11}$. The P-sequence reaches overlaps above $0.965$ already at order $v^5$ for all but one of the systems tested, which the authors translate into more than 90 percent of otherwise detectable events surviving the template mismatch, against 10–15 percent for Taylor templates. They also report smaller parameter biases, and a prediction that the last stable circular orbit sits at a higher frequency when the mass ratio is turned on than earlier estimates had suggested.
Load-bearing premise
The claim rests on assuming that switching on the mass ratio only smoothly deforms the test-mass behaviour, so that convergence lessons learned for a test particle carry over to comparable-mass binaries, and the finite-mass check partly defines its own exact target.
Editorial extensions
If this is right
- At fifth post-Newtonian order the P-approximants give overlaps above 96.5 percent for the systems tested, so more than 90 percent of otherwise detectable events survive the template mismatch, against roughly 10–15 percent for Taylor templates.
- The P-sequence converges monotonically rather than oscillating, so adding post-Newtonian orders improves the template instead of sometimes degrading it.
- Parameter estimation improves alongside detection: total-mass biases stay under 3.5 percent and often under 1 percent, where Taylor templates that do reach high overlap can carry biases of tens of percent.
- The construction places the last stable orbit at a higher frequency as the mass ratio grows, giving about 2043 Hz for a $1.4$–$1.4$ solar-mass neutron-star binary, in contrast to earlier estimates of about 1420 Hz.
- Extending the energy and flux to order $v^6$ would push maximized overlaps above 98 percent and diagonal overlaps above 99.5 percent, cutting the loss of detectable events below 1 percent.
- The stationary phase approximation is shown to be too crude for ranking waveform families, so overlap studies built on it should be redone with numerical Fourier transforms.
Reading between the lines
- The Padé step is a general recipe for any truncated series whose exact form is known to have a nearby pole: rational approximants can reconstruct that pole from few terms, which is exactly what happens here when second-order data alone reproduce the exact test-mass energy function and its last stable orbit.
- Because the finite-mass robustness test defines its own fiducial 'exact' flux as the order-$v^{11}$ P-approximant, the test measures internal consistency of the construction rather than accuracy against an independent target; an independent finite-mass flux would settle the question the paper cannot.
- If the predicted upward shift of the last-stable-orbit frequency with mass ratio is correct, template banks for comparable-mass binaries should be truncated at higher frequencies than earlier estimates implied, which changes both bank spacing and the frequency coverage of searches.
- The paper's diagnosis suggests that other post-Newtonian observables with known singularity structure, such as the angular momentum or higher harmonics of the waveform, could be resummed by the same two-step transformation rather than by adding orders.
Original abstract
The order of the post-Newtonian expansion needed, to extract in a reliable and accurate manner the fully general relativistic gravitational wave signal from inspiralling compact binaries, is explored. A class of approximate wave forms, called P-approximants, is constructed based on the following two inputs: (a) The introduction of two new energy-type and flux-type functions e(v) and f(v), respectively, (b) the systematic use of Pade approximation for constructing successive approximants of e(v) and f(v). The new P-approximants are not only more effectual (larger overlaps) and more faithful (smaller biases) than the standard Taylor approximants, but also converge faster and monotonically. The presently available O(v/c)^5-accurate post-Newtonian results can be used to construct P-approximate wave forms that provide overlaps with the exact wave form larger than 96.5% implying that more than 90% of potential events can be detected with the aid of P-approximants as opposed to a mere 10-15 % that would be detectable using standard post-Newtonian approximants.
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
The order of the post-Newtonian expansion needed, to extract in a reliable and accurate manner the fully general relativistic gravitational wave signal from inspiralling compact binaries, is explored. A class of approximate wave forms, called P-approximants, is constructed based on the following two inputs: (a) The introduction of two new energy-type and flux-type functions e(v) and f(v), respectively, (b) the systematic use of Pade approximation for constructing successive approximants of e(v) and f(v). The new P-approximants are not only more effectual (larger overlaps) and more faithful (smaller biases) than the standard Taylor approximants, but also converge faster and monotonically. The presently available O(v/c)^5-accurate post-Newtonian results can be used to construct P-approximate wave forms that provide overlaps with the exact wave form larger than 96.5% implying that more than 90% of potential events can be detected with the aid of P-approximants as opposed to a mere 10-15 % that would be detectable using standard post-Newtonian approximants.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The order of the post-Newtonian expansion needed, to extract in a reliable and accurate manner the fully general relativistic gravitational wave signal from inspiralling compact binaries, is explored. | background 0.56 | 0.62 | 0.04 | unclear 0.85 |
| s1 | A class of approximate wave forms, called P-approximants, is constructed based on the following two inputs: (a) The introduction of two new energy-type and flux-type functions e(v) and f(v), respectively, (b) the systematic use of Pade approximation for constructing successive approximants of e(v) and f(v). | method 0.86 | 0.92 | 0.06 | asserted 1.00 |
| s2 | The new P-approximants are not only more effectual (larger overlaps) and more faithful (smaller biases) than the standard Taylor approximants, but also converge faster and monotonically. | result 0.98 | 0.90 | 0.39 | asserted 1.00 |
| s3 | The presently available O(v/c)^5-accurate post-Newtonian results can be used to construct P-approximate wave forms that provide overlaps with the exact wave form larger than 96.5% implying that more than 90% of potential events can be detected with the aid of P-approximants as opposed to a mere 10-15 % that would be detectable using standard post-Newtonian approximants. | result 0.77 | 0.86 | 0.72 | asserted 0.94 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
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- R cross-paper relations
References
How later papers used it
73 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.
Uses its method or tool 3 of 26 papers with this label · 1 by the same authors
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Invoked the mini-max overlap detailed in this paper, along with another work's amoeba routine, to compute the associated fitting factors.
Gravitational waves from compact binaries inspiralling along post-Newtonian accurate eccentric orbits: Data analysis implications (2007) · 1.00 · paraphrased: the licence does not allow quoting
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Took the minimum to the minimax match of this paper as the minimax match at a given polar angle.
Observing mergers of non-spinning black-hole binaries (2010) · 0.90 · paraphrased: the licence does not allow quoting
By the same authors
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Terminated other approximants, except EOB, at the extremum of this paper's P-approximant energy function.
Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors (2009) (same authors) · 0.85 · paraphrased: the licence does not allow quoting
How the literature uses it
64 arXiv papers cite this in 132 sentences
Papers citing it · 87 sentences without a confident label
Builds on it 1 paper
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We are interested in generalizing the procedure presented in [ref] for maximizing the match [ref] with respect to the initial orbital phase constants between a target or exact (label X) waveform and a template or approximate (label A) waveform.
Observing mergers of non-spinning black-hole binaries · Appendix: Generalized Phase Maximization · 0.93
Uses its method or tool 23 papers
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We quantify our results by computing fitting factors (FF) following prescriptions detailed in Refs. [ref] , and inherent systematic errors in the estimated value of $m$ and the symmetric mass-ratio, $η$, for the
Gravitational wave data analysis implications of TaylorEt inspiral approximants for ground-based detectors: the non-spinning case · Introduction · 1.00
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We extensively employ Ref. [ref] to compute the fully maximized overlaps with the help of their minimax overlap.
On the ability of various circular inspiral templates to capture inspiral gravitational waves from compact binaries having tiny orbital eccentricities · Performances of various circular templates against GW signals from compact binaries in inspiralling eccentric orbits · 1.00
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We calculate the mismatch between strains $h_1$ and $h_2$ as [ref] [equation] with the maximum evaluated with respect to time-shifts, orbital-phase shifts and polarization angles.
General relativistic simulations of the quasi-circular inspiral and merger of charged black holes: GW150914 and fundamental physics implications · Supplemental material · 0.99
Compares with it 2 papers
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This situation is somehow reminiscent of what found in Ref. [ref] when analyzing the PN-expanded energy flux of a test-mass on circular orbits around Schwarzschild.
Towards a gravitational self force-informed effective-one-body waveform model for nonprecessing, eccentric, large-mass-ratio inspirals · GSF-informed potentials · 0.93
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This table compares in a quantitative way the result of the “untuned” Padé resummation ($=1/√(3)$) of Ref. [ref] to the result of the “$$-tuned” Padé resummation described here.
The Effective One Body description of the Two-Body problem · Resummation of $F̂^Taylor$ using a one-parameter family of Padé approximants: tuning $v_pole$ · 0.99
Cites it as background 11 papers
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Several authors have assessed whether the accuracy with which the formalism provides the waveforms is good enough for the purpose of detection and parameter estimation [ref] .
Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors · Introduction · 0.94
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In addition to the models and their underlying point-mass baselines, we also explore the performance of PN based models, in particular the and approximants (see e.g. [ref] ).
Matter imprints in waveform models for neutron star binaries: tidal and self-spin effects · Variable $f_max$ · 0.89
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Section [ref] discusses two additional resummation approaches based on minimizing Padé approximants of (10) an “improved” PN energy function (the $e$-method [ref] ) and (11) a function based on the orbital angular momentum (the $j$-method [ref] ).
Conservative self-force correction to the innermost stable circular orbit: comparison with multiple post-Newtonian-based methods · Summary · 0.84
Discussion
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