Gravitational-wave tails of tails

Luc Blanchet

REVIEW

Gravitational-wave tails of tails

Luc Blanchet

General Relativity and Quantum Cosmology · Submitted 7 Oct 1997 · 57 references

Pith's verdict: A careful, self-checking computation of the 3PN tail-of-tail; the leading logs are solid, the additive constant is unaudited.

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arXiv IDgr-qc/9710038 (version 2)

Pith IDDVWBJK57, Pith's permanent record of this paper and its review

CitationLuc Blanchet. "Gravitational-wave tails of tails." arXiv:gr-qc/9710038 (1997). https://pith.science/paper/DVWBJK57

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Editor's note

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The thing to know: this is the paper that puts the tail of tails into the 3PN waveform. Blanchet computes the cubic M²×M_pq interaction in the post-Minkowskian scheme and extracts both the far-zone quadrupole moment (4.13) and the binary flux contributions (5.5) and (5.9). The field-level result is complete at 3PN — the dimensional argument in 4.2 that (0,0,2) with no index contractions is the only admissible cubic term is clean and, as far as I can check, correct. That is the load-bearing structural claim and it holds up. What is genuinely good: the divergence check. The tail part of the cubic source (2.16) cancels the instantaneous part (2.15) in ∂_βΛ^{αβ}=0 (2.17–2.18). That is a real constraint on a long computation, not decoration. And the test-mass limit of L_tail agrees with Tagoshi–Sasaki (5.5b against eq. 43 of [50]) — a genuine external check. Where I would push: the additive constant. 124627/44100 in (4.12) is assembled from the finite-part constants of (3.5), (3.14) and (4.3), and nothing independent reaches it. The divergence check validates the source, not the retarded integral; the (ln ω)² cancellation between (5.6) and (5.8) validates the leading logs only; the Tagoshi–Sasaki comparison covers L_tail, not the tail of tail — the paper says so itself. So the ln² and ln coefficients I trust; the constant I would want re-derived before quoting. The stress-test is right about this. I would soften one part of it, though: the Legendre coefficients in Tables 1–2 are not bare assertions — Appendix A gives (A.17)–(A.22), so a determined reader can recompute them. The chain is long, not opaque. The r0 dependence is acknowledged and deferred to [56,57], then in preparation. That is a real gap in the flux result at the time of writing, but it is flagged, and the field-level (4.13) does not lean on it the same way. Who it is for: anyone building 3PN/3.5PN templates, or working on post-Minkowskian iteration. Not a casual read. Recommendation: send it out. It is a careful, honest paper with a well-defined scope, and the deferred pieces are labelled as deferred.

Referee report

Significance

This is a foundational calculation in the gravitational-wave template program: it establishes that the tail-of-tail effect enters the radiation field at 3PN and supplies the explicit coefficient, which is a necessary ingredient for the 3PN/3.5PN phasing of inspiraling compact binaries. The paper is careful in a way that deserves credit. The cubic source is validated by an exact divergence cancellation between its instantaneous and hereditary parts (2.17)–(2.18), which is a non-trivial check on a long computation. The far-zone result is cross-checked in the test-mass limit against Tagoshi–Sasaki black-hole perturbation theory (5.5b versus Eq. (43) of [50]), and I independently verified that (5.5a) and (5.5b) are correctly related by the γ↔x conversion (5.3c) including the rescaling of the ν²γ⁵ prefactor, at both 1.5PN and 2.5PN order. The cancellation of the (ln ω)² terms between (5.6) and (5.8) is a further genuine consistency check on the leading-logarithm coefficients. The paper also supplies a proof of the Legendre convolution identity (A.5), which is not in the standard tables and is needed for the k=1 case.

Major objections (2)

[§4.2, Eq. (4.12); §5, Eq. (5.9)] The additive constant 124627/44100 in (4.12), and its descendants in (5.9), is the one piece of the quoted result that none of the paper's checks reach. The divergence condition (2.18) validates the cubic source (2.15)–(2.16) but not the retarded-integral evaluation (3.5), (3.14) or the harmonicity completion (4.3); the Tagoshi–Sasaki comparison covers only L_tail, as the paper itself states in §5; and the (ln ω)² cancellation between (5.6) and (5.8) tests only the leading-log coefficients. The constant is assembled from the finite-part constants of (3.5), (3.14) and (4.3) together with the Legendre coefficients of Tables 1 and 2 and Eq. (3.13), and an error in, say, ³,²γ₀ = 77/2 would propagate into 124627/44100 undetected. This does not threaten the existence claim, which rests on the ln² and ln terms, but it does bear on the precise coefficient presented as a finished result. I recommend either tabulating the intermediate finite-part constants of (3.5), (3.14) and (4.3) so that the chain to 124627/44100 is auditable by a reader, or adding an explicit sentence in §4.2 flagging the constant as not independently cross-checked.
[§5, Eq. (5.9); §4.2, Eq. (4.14)] The 3PN coefficient in (5.9) carries an explicit dependence on the regularization scale r0, and the cancellation of that dependence against the 3PN instantaneous flux is asserted but deferred to refs. [56] and [57], both listed as in preparation. The authors acknowledge this in §5, so I do not present it as a new finding; the concern is that the physical, r0-independent value of the 3PN flux is therefore not fixed by this paper alone, while the abstract's phrasing [abstract:s5] could be read as delivering the 3PN flux contribution outright. The deferral is also load-bearing in a way that a reader cannot check: the cancellation is a concrete, falsifiable statement, namely that L_inst at 3PN should contain +(1712/105) ln r0 to cancel the −(1712/105)[C + ln(4ωr0/c)] of (5.9). I recommend stating that prediction explicitly in §5 as the test to be performed in [56,57], and adding a qualifying clause to the abstract or conclusions making clear that (5.9) is the tail and tail-of-tail part of the 3PN flux, not the complete 3PN luminosity.

Minor objections (7)

[Tables 1 and 2] The captions should define the coefficient k,mα_ℓ and point to Eq. (A.17), since the tables are otherwise unreadable in isolation. Table 2 also omits the ℓ = 0 column that Table 1 carries; a footnote explaining that k = 3, ℓ = 0 falls into the k ≥ ℓ+3 category and is therefore handled separately by (3.13) would prevent the reader from suspecting an omission.
[References] The reference list skips [46], running from [45] to [47], and reference [1] (paper I) is given without journal, volume or year, which makes the many appeals to it hard to follow up. Both should be completed.
[§5, Eqs. (5.4)–(5.8)] The notation shifts between r0 and r1 = r0 e^{−11/12} in the paragraph preceding (5.5), and then (5.6) and (5.8) revert to r0. A sentence reconciling the two, or a uniform choice, would remove an ambiguity in exactly the place where the r0 dependence is at issue.
[§4.2, Eq. (4.13)] The 2.5PN terms collected in the curly bracket of (4.13) mix results from paper I with the quadrupole-dipole term; a short indication of which entries are new here and which are quoted would help the reader attribute them.
[§4.2, dimensional argument] The argument that no other contribution to U_ij arises at 3PN is clean and worth stating a little more explicitly: it assumes the field is parametrized solely by mass and current multipole moments M_L and S_L with no additional structures, and that no coordinate artifact survives the passage to radiative coordinates. One sentence naming those assumptions would strengthen an otherwise compressed step.
[§3.1, §3.2, §4.1, §5] A number of typographical errors should be corrected in the typeset version: 'fictituous' for 'fictitious' (§3.1), 'sligthly' for 'slightly' (§4.1), 'occuring' for 'occurring' (§5), 'is sued' for 'issued' and 'ante penultimate' for 'antepenultimate' (§3.2), and the author address line 'D´ epartement d’Astrophysique Relativiste et de Cosmologi e'.
[Front matter] The header carries 'Date: 23 July 2018' and 'Submitted to: Classical Quantum Grav.' alongside the arXiv v2 date of 27 May 2005; the front matter should be checked for consistency before typesetting.

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The paper

The paper claims

Gravitational waves scatter off the curved spacetime made by their own tails, producing a measurable correction at third post-Newtonian order in the signal from inspiraling binaries.

The reading

Gravitational waves from a source do not travel cleanly to infinity: the part of the field sourced by the total mass scatters off the curvature that mass itself creates, and this scattering produces the well-known 'tails' that arrive delayed and smeared in time. This paper asks what happens when the tails themselves scatter again, which corresponds to a cubic interaction between two mass monopoles and the source's mass quadrupole. Blanchet computes the far-zone field for exactly this interaction and finds that it enters the radiation field with an extra factor $G^2/c^6$, that is, at third post-Newtonian (3PN) order beyond the ordinary quadrupole radiation. The motivation is concrete: future detectors such as VIRGO and LIGO will track the orbital phase of inspiraling compact binaries through many cycles, so every post-Newtonian term in the energy flux shifts the predicted waveform. The paper then evaluates the tail and tail-of-tail contributions to the energy flux of a circularized binary through 3.5PN order.

What carries the argument

The engine is the post-Minkowskian iteration with a finite-part regularization of the retarded integral: the source is multiplied by $(r/r_0)^B$, the retarded integral is computed for large $B$, and the finite part of its Laurent expansion at $B=0$ is taken as the particular solution. Because the cubic source contains non-local integrals with Legendre functions $Q_m$ in its kernel, the paper must integrate sources of the form $n_L r^{-k}∫ dx Q_m(x)F(t-rx)$; the appendix supplies closed-form results for $k=1$, local results for $2≤k≤ℓ+2$, and non-local results for $k≥ℓ+3$, the last carrying the arbitrary scale $r_0$. A harmonicity algorithm then adds a homogeneous solution chosen to make the metric divergence-free, and a coordinate change $ξ^0 = -2M ln(r/r_0)$ removes the logarithms of $r$ from the far-zone metric so that the transverse-tracefree projection yields the observable multipole moments.

Extended reading notes

Core claim

The central claim is that the cubic monopole-monopole-quadrupole interaction $M^2$ with $M_{ij}$ produces a 'tail of tails' in the asymptotic gravitational field, and that this is the only cubic interaction that can contribute at 3PN order. A dimensional counting argument in the paper fixes this: a non-linear term in the observable quadrupole moment carries $1/c^{3(n-1)+2k}$ with $n$ the order of non-linearity and $k$ the number of index contractions, and $3(n-1)+2k=6$ has the unique solution $n=3$, $k=0$, forcing multipolarities $(0,0,2)$. The resulting contribution to the observable quadrupole moment is a non-local integral of $M^{(5)}_{ij}$ against $[ln^2(cτ/2r_0) + (57/70)ln(cτ/2r_0) + 124627/44100]$, and adding it to the previously known terms gives the complete $U_{ij}$ through 3PN order. For a circular binary the paper finds that the squared-logarithm terms in the (tail)$^2$ and tail(tail) pieces of the flux cancel exactly, leaving a flux correction linear in $ln ω$ plus a $π^2$ piece, structurally matching what black-hole perturbation theory gives in the test-mass limit.

Load-bearing premise

The 3PN coefficient in the binary energy flux is computed with a particular solution that depends on an arbitrary length scale $r_0$, and the paper asserts that this dependence will cancel against the 3PN instantaneous terms, a check it defers to work listed as in preparation.

Editorial extensions

If this is right

The observable quadrupole moment of an isolated source is fixed through 3PN order, with the tail-of-tail term as the only cubic contribution at that order.
The energy flux of an inspiraling circular binary gains explicit 3PN and 3.5PN tail contributions, extending earlier 1.5PN and 2.5PN results.
The squared-logarithm terms cancel between the (tail)$^2$ and tail(tail) pieces, so the 3PN flux depends on $ln ω$ only linearly, consistent with what perturbation theory finds in the test-mass limit.
The paper's tail result agrees term by term with black-hole perturbation theory in the limit where one mass is negligible, providing a cross-check between two independent methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the tail-of-tail term is the unique cubic contribution at 3PN, the 3PN waveform shape is largely fixed by symmetry and counting rather than by details of the source's internal structure, which makes it a clean target for template-building.
The cancellation of the $(ln ω)^2$ terms suggests a structural rule for higher orders: one might expect the leading logarithms to keep cancelling between (tail)$^n$ and higher tail-of-tail pieces, which would be worth testing at 4PN.
The residual dependence on the arbitrary scale $r_0$ in the 3PN flux is a bookkeeping artifact only if the instantaneous 3PN terms carry a matching dependence; exhibiting that cancellation would convert the paper's partial 3PN coefficient into a physical prediction.
The same curvature-scattering bookkeeping should apply to current-type multipoles at higher orders, so an analogous tail-of-tail analysis for $S_{ij}$ and the octupole would be a natural next computation.

Original abstract

The tails of gravitational waves are caused by scattering of linear waves onto the space-time curvature generated by the total mass-energy of the source. Quite naturally, the tails of tails are caused by curvature scattering of the tails of waves themselves. The tails of tails are associated with the cubic non-linear interaction between two mass monopole moments and, dominantly, the mass quadrupole of the source. In this paper we determine the radiation field at large distances from the source for this particular monopole-monopole-quadrupole interaction. We find that the tails of tails appear at the third post-Newtonian (3PN) order beyond the usual quadrupole radiation. Motivated by the need of accurate templates to be used in the data analysis of future detectors of gravitational waves, we compute the contribution of tails, and of tails of tails, up to the 3.5PN order in the energy flux generated by inspiraling compact binaries.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.76 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The tails of gravitational waves are caused by scattering of linear waves onto the space-time curvature generated by the total mass-energy of the source. Quite naturally, the tails of tails are caused by curvature scattering of the tails of waves themselves. The tails of tails are associated with the cubic non-linear interaction between two mass monopole moments and, dominantly, the mass quadrupole of the source. In this paper we determine the radiation field at large distances from the source for this particular monopole-monopole-quadrupole interaction. We find that the tails of tails appear at the third post-Newtonian (3PN) order beyond the usual quadrupole radiation. Motivated by the need of accurate templates to be used in the data analysis of future detectors of gravitational waves, we compute the contribution of tails, and of tails of tails, up to the 3.5PN order in the energy flux generated by inspiraling compact binaries.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	The tails of gravitational waves are caused by scattering of linear waves onto the space-time curvature generated by the total mass-energy of the source.	background 1.00	0.07	0.03	asserted 1.00
s1	Quite naturally, the tails of tails are caused by curvature scattering of the tails of waves themselves.	background 0.99	0.06	0.05	asserted 0.99
s2	The tails of tails are associated with the cubic non-linear interaction between two mass monopole moments and, dominantly, the mass quadrupole of the source.	background 0.85	0.17	0.09	asserted 1.00
s3	In this paper we determine the radiation field at large distances from the source for this particular monopole-monopole-quadrupole interaction.	contribution 0.99	0.89	0.06	asserted 0.91
s4	We find that the tails of tails appear at the third post-Newtonian (3PN) order beyond the usual quadrupole radiation.	result 1.00	0.96	0.40	asserted 1.00
s5	Motivated by the need of accurate templates to be used in the data analysis of future detectors of gravitational waves, we compute the contribution of tails, and of tails of tails, up to the 3.5PN order in the energy flux generated by inspiraling compact binaries.	contribution 0.99	0.95	0.16	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
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levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Damour T, Iyer B R and Sathyaprakash B S gr-qc/9708034 43 Table 1: Values of the coeﬃcients k,mαℓ for k = 2. k = 2 ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 m = 0 π26 12 14 1172 548 m = 1 12 π218 − 13 18 112 13216 m = 2 14 18 π230 − 14 118 124 m = 3 1172 112 118 π242 − 736 132 m = 4 548 13216 124 132 π254 − 2051296 1 Table 2: Values of the coeﬃcients k,mαℓ for k = 3....

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Fokker action of non-spinning compact binaries at the fourth post-Newtonian approximation
gr-qc 2015-12 conditional novelty 7.0 of 10

The full 4PN Fokker action for non-spinning compact binaries is derived in harmonic coordinates, with the wave-tail contribution obtained consistently from the action rather than added by hand.
Dimensional regularization of the third post-Newtonian gravitational wave generation from two point masses
gr-qc 2005-03 accept novelty 7.0 of 10

Dimensional regularization fixes the three 3PN ambiguity parameters of binary gravitational waves to xi = -9871/9240, kappa = 0, and zeta = -7/33.
Binary black hole scattering in the extreme-mass-ratio limit: time-domain waveform
gr-qc 2026-09 conditional novelty 6.0 of 10

A time-domain 5PM/7PN gravitational scattering waveform for nonspinning binaries is derived at first order in the mass ratio, complementing frequency-domain self-force results and benchmarked by flux-level agreement w...
Classical soft theorems meet the post-Newtonian expansion
gr-qc 2026-09 conditional novelty 6.0 of 10

The paper derives in the NRGR post-Newtonian framework the all-order logarithmic soft tower ω^(n-1)(lnω)^n for memory and its tails in bound binaries, matching the Sahoo-Sen soft theorem.
High-order effective-one-body tidal interactions and gravitational scattering
gr-qc 2026-03 conditional novelty 6.0 of 10

High-order PM tidal corrections improve EOB predictions for neutron-star gravitational scattering and lay groundwork for PM-based tidal EOB waveforms.
High Precision Black Hole Scattering: Tutti Frutti vs Worldline Effective Field Theory
hep-th 2025-04 conditional novelty 6.0 of 10

The Tutti Frutti and Worldline Effective Field Theory formalisms agree for black hole scattering through fifth post-Minkowskian order, with new second-self-force predictions for two gauge-invariant observables.
Gravitational Raman Scattering in Effective Field Theory: a Scalar Tidal Matching at $\mathcal{O}(G^3)$
hep-th 2024-01 conditional novelty 6.0 of 10

A two-loop worldline-EFT calculation of scalar Raman scattering off compact objects reproduces general relativity for black holes with zero static Love numbers and fixes the running of dynamical Love numbers.
Comparing One-loop Gravitational Bremsstrahlung Amplitudes to the Multipolar-Post-Minkowskian Waveform
gr-qc 2023-09 conditional novelty 6.0 of 10

The amplitude-based one-loop gravitational bremsstrahlung waveform and the multipolar post-Minkowskian waveform agree through 1.5 post-Newtonian order but differ at 2.5 post-Newtonian order, a difference visible as an...
Laying the foundation of the effective-one-body waveform models SEOBNRv5: improved accuracy and efficiency for spinning non-precessing binary black holes
gr-qc 2023-03 conditional novelty 6.0 of 10

SEOBNRv5HM is a recalibrated effective-one-body waveform model for spinning, non-precessing binary black holes that cuts median mismatch against numerical relativity by roughly a factor of seven and runs 10 to 50 time...
Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order
gr-qc 2022-01 conditional novelty 6.0 of 10

The authors derive the tensor and scalar gravitational waveforms and energy fluxes of inspiralling compact binaries in massless scalar-tensor theory through 1.5 post-Newtonian order, including dipolar tail and memory ...
Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates
gr-qc 2020-11 accept novelty 6.0 of 10

The authors derive the explicit coordinate transformation and resulting Newman-Unti metric, through quadratic order, that maps the multipolar post-Minkowskian gravitational field of an isolated source into Bondi-like ...
Gravitational-wave tail effects to quartic non-linear order
gr-qc 2016-07 accept novelty 6.0 of 10

A fourth-order gravitational-wave tail interaction is derived and shown to fix the 4.5 post-Newtonian coefficient in the energy flux of circular compact binaries, matching black-hole perturbation theory in the test-ma...
Third post-Newtonian gravitational waveforms for compact binary systems in general orbits: Instantaneous terms
gr-qc 2015-01 conditional novelty 6.0 of 10

The authors derive the instantaneous 3PN spherical-harmonic modes of gravitational waves from nonspinning compact binaries in general and quasi-elliptical orbits.
Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order
gr-qc 2014-09 conditional novelty 6.0 of 10

The authors compute the mass-type octupole moment of non-spinning compact binaries to 3PN order and the resulting (3,3) and (3,1) gravitational-wave modes to 3.5PN, together with all non-linear multipole couplings nee...
Next-to-next-to-leading order spin-orbit effects in the gravitational wave flux and orbital phasing of compact binaries
gr-qc 2013-03 accept novelty 6.0 of 10

The authors derive the 3.5 post-Newtonian spin-orbit contributions to the gravitational-wave flux and orbital phasing of spinning compact binaries, checked against the Kerr test-mass limit and the boosted-Kerr one-body limit.
The third and a half post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries
gr-qc 2012-04 accept novelty 6.0 of 10

The (2,2) quadrupole mode of the gravitational waveform from non-spinning compact binaries on quasi-circular orbits is derived through 3.5 post-Newtonian order.
Tail-induced spin-orbit effect in the gravitational radiation of compact binaries
gr-qc 2011-04 accept novelty 6.0 of 10

The spin-orbit contribution to gravitational-wave tail effects is derived through 3PN order in the energy flux and phasing, and through 2.5PN and 3PN order in the wave polarizations.
Third post-Newtonian angular momentum flux and the secular evolution of orbital elements for inspiralling compact binaries in quasi-elliptical orbits
gr-qc 2009-08 accept novelty 6.0 of 10

The 3PN angular momentum flux and the resulting secular evolution of orbital elements are derived for inspiralling compact binaries in quasi-elliptical orbits, completing the eccentric-orbit phasing input at 3PN.
Comparing Effective-One-Body gravitational waveforms to accurate numerical data
gr-qc 2007-11 conditional novelty 6.0 of 10

A resummed Effective-One-Body waveform, with two flexibility parameters fitted to a 15-orbit numerical simulation, reproduces the numerical phase and amplitude of an equal-mass black hole binary to about 10^-3 gravita...
Inspiralling compact binaries in quasi-elliptical orbits: The complete third post-Newtonian energy flux
gr-qc 2007-11 conditional novelty 6.0 of 10

The authors derive the complete third post-Newtonian energy flux for inspiralling compact binaries in quasi-elliptical orbits, combining new instantaneous multipole contributions with previously computed hereditary ta...
Tail effects in the third post-Newtonian gravitational wave energy flux of compact binaries in quasi-elliptical orbits
gr-qc 2007-11 conditional novelty 6.0 of 10

The hereditary tail contribution to the 3PN gravitational-wave energy flux of eccentric compact binaries is computed as a set of eccentricity-dependent enhancement functions multiplying the known circular-orbit coefficients.
Higher-order spin effects in the dynamics of compact binaries II. Radiation field
gr-qc 2006-05 conditional novelty 6.0 of 10

Spin-orbit corrections to the mass and current quadrupole moments, the gravitational-wave energy flux, and the orbital phasing of a spinning compact binary are derived through 2.5 post-Newtonian order.
The Radial Action from Probe Amplitudes to All Orders
hep-th 2021-09 conditional novelty 5.0 of 10

Solving the probe wave equation and taking the classical limit yields the exact all-order scattering angle for a probe mass in pure NUT space, given by a complete elliptic integral.
Observational Signature of the Logarithmic Terms in the Soft Graviton Theorem
hep-th 2018-06 conditional novelty 5.0 of 10

The logarithmic terms in the subleading soft graviton theorem imply a 1/u tail on the gravitational memory step, with a coefficient that vanishes for ultra-relativistic final-state particles.
Gravitational radiative corrections from effective field theory
gr-qc 2009-12 accept novelty 5.0 of 10

The NRGR effective field theory reproduces known post-Newtonian gravitational-wave flux corrections through 3PN and yields a renormalization-group equation for the quadrupole moment that controls the logarithms.
Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries
gr-qc 2008-11 accept novelty 5.0 of 10

The post-Newtonian corrections to the Christodoulou gravitational-wave memory for quasicircular inspiralling binaries are computed to 3PN order, with explicit expressions for the plus polarization and the m=0 spherica...
The Effective One Body description of the Two-Body problem
gr-qc 2009-06 accept novelty 3.0 of 10

The Effective One Body formalism, using a parameter-free resummation of the radiation reaction and only two adjustable strong-field coefficients, reproduces numerical relativity binary black hole waveforms to within t...
Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries
gr-qc 2002-02 accept novelty 3.0 of 10

A complete, cross-checked framework computes the gravitational-wave energy flux and orbital phase of inspiralling compact binaries through 3.5 post-Newtonian order.
The Confrontation between General Relativity and Experiment
gr-qc 2001-03 accept novelty 2.0 of 10

General relativity passes every test reviewed here, with the post-Newtonian parameter gamma constrained to roughly 0.01 percent and binary-pulsar orbital decay matching the quadrupole-formula prediction to about 0.4 percent.
The Effective Field Theorist's Approach to Gravitational Dynamics
hep-th 2016-01 accept novelty 1.0 of 10

A comprehensive review of non-relativistic general relativity (NRGR), the EFT framework for computing binary inspiral dynamics and gravitational waveforms to high post-Newtonian order.

How later papers used it

57 later papers cite it

Evidence: substantial. 45 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 123 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 paper by the same authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 5 papers with this label · 1 by the same authors

By the same authors

Applying the algorithm $ℋ$ or, rather, its version at leading order $1/r$ explained in Appendix B of [ref] , and keeping only the non-local (hereditary) terms (we shall add all instantaneous terms in the end results below), we get

Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order (2022) (same authors) · Memory and tail effects in ST theory · 0.99

How the literature uses it

56 arXiv papers cite this in 125 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 83 sentences without a confident label

Contests it 1 paper

The constants defined in [ref] were later shown in [ref] to be sometimes ill-defined, e.g. for $k=0$, $ℓ=0$ and $m=0$, and this problem was circumvented on a case-by-case basis.

Gravitational-wave tails of memory · Application to tails-of-tails · contest 0.73

Builds on it 2 papers

In the latter case, the function $G(u)$ is actually constant (i.e. the ADM mass $$), and we shall recover and extend the results found in [ref] .

Gravitational-wave tails of memory · Application to tails-of-memory · 0.96
In the harmonic algorithm, after the suitable coordinate transformation to get rid of the logarithms, we find [ref]

The Quadrupole Moment of Compact Binaries to the Fourth post-Newtonian Order: Relating the Harmonic and Radiative Metrics · Cubic tails-of-tails · 1.00

Uses its method or tool 6 papers

Following the multipolar-post-Minkowskian (MPM) formalism [ref] , we shall express the radiative quadrupole moment $U_ij$ in terms successively of canonical moments ${M_L, S_L}$ and then source moments ${I_L, J_L, W_L, X_L, Y_L, Z_L}$ for general sources at 3.5PN order.

The third and a half post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries · Expression of the dominant quadrupole mode · 0.97
This yields field equations for $h_n^μν$ and $ψ_n$ (for $n≥2$) which can be iteratively solved at quadratic [ref] , cubic [ref] and quartic order [ref] .

Gravitational waves from quasielliptic compact binaries in scalar-tensor theory to one-and-a-half post-Newtonian order · Massless scalar-tensor theories · 0.92
We shall compute the tails-of-memory using the multipolar-post-Minkowskian (MPM) formalism [ref] but with an important variant with respect to previous works such as [ref] : we shall construct the metric directly in radiative coordinates (see [ref] for a general definition), following the method proposed in [ref] , instead of the usual construction in harmonic coordinates.

Gravitational-wave tails of memory · Introduction · 0.94

Uses its data or values 2 papers

Previous calculations found $γ^E/B_2m=-214/105(Gℰω)^2$ at the lowest order [ref] , where $G$ is Newton's constant.

Resummation of Universal Tails in Gravitational Waveforms · Introduction · 0.84
The $_1,mδ_ℓ$ constant has a simple expression when $m=ℓ$, as pointed out in Eqs. (A7–A9) of [ref] , which reads $_1,ℓδ_ℓ= -1/2H_ℓ^2$.

Gravitational-wave tails of memory · Application to tails-of-tails · 0.84

Compares with it 6 papers

Pushing our $𝒪(G_N^3)$ term from <ref> through the transformation to PN variables, including the generic subtraction and renormalization considerations, we arrive at: [equation] Comparing with known PN results [ref] [equation] The $logx$ terms match exactly, up to the energy balance sign.

Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals · Direct Comparisons · 1.00
non-analytic contributions: [equation] These results for the flux from a circular orbit are in complete agreement with [ref] .

A tale of tails through generalized unitarity · Introduction · 1.00
In particular, the quadrupolar ($ℓ=2$) universal anomalous dimensions are given by [equation] and the octupolar ($ℓ=3$) one is [equation] The first three terms of $γ_2m$ and the first term in $γ_3m$ agree with the know results [ref] , while the rest are new results.

Resummation of Universal Tails in Gravitational Waveforms · Introduction · 1.00

Cites it as background 18 papers

On the other hand, the subleading “tail-of-tail” has received limited direct study [ref] , and the sub-subleading “tail-of-tail-of-tail” (T$^3$) has only been computed via traditional GR methods [ref] , without a counterpart EFT computation.

Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals · Introduction · 0.80
In the harmonic construction, the quadratic metrics that are known are $×$ [ref] , $×_ij$ [ref] and $_ij×_ij$ [ref] .

Gravitational-wave tails of memory · Linear and quadratic results · 0.83
The latter moments mix with each other as the waves propagate, so that the relation between radiative and source moments include many nonlinear interactions, which are called hereditary effects [ref] , depending on the full past history of the source.

Higher-order tail contributions to the energy and angular momentum fluxes in a two-body scattering process · Introduction · 0.92

Discussion

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Damour T, Iyer B R and Sathyaprakash B S gr-qc/9708034 43 Table 1: Values of the coeﬃcients k,mαℓ for k = 2. k = 2 ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 m = 0 π26 12 14 1172 548 m = 1 12 π218 − 13 18 112 13216 m = 2 14 18 π230 − 14 118 124 m = 3 1172 112 118 π242 − 736 132 m = 4 548 13216 124 132 π254 − 2051296 1 Table 2: Values of the coeﬃcients k,mαℓ for k = 3....

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