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arxiv: 2310.19679 · v2 · submitted 2023-10-30 · 🌀 gr-qc

Tidal effects up to next-to-next-to leading post-Newtonian order in massless scalar-tensor theories

Laura Bernard , Eve Dones , Stavros Mougiakakos This is my paper

Pith reviewed 2026-05-24 06:14 UTC · model grok-4.3

classification 🌀 gr-qc
keywords tidal effectspost-Newtonian orderscalar-tensor theoriesconservative dynamicsconserved quantitiesEinstein-scalar-Gauss-Bonnettwo-body problemgravitational waves
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The pith

The conservative dynamics and ten conserved quantities for tidal effects in spinless binary systems are computed at next-to-next-to-leading post-Newtonian order in massless scalar-tensor theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes the tidal contributions to the conservative two-body dynamics at NNLO in the post-Newtonian expansion within massless scalar-tensor theories for spinless sources. Both a Fokker Lagrangian method and the PN-EFT formalism are used to obtain these results, and the ten associated conserved quantities are derived at the same order. The calculations are then extended to the case of Einstein-scalar-Gauss-Bonnet gravity. These expressions matter because they supply the building blocks needed to model gravitational-wave signals from compact binaries at the precision required by next-generation detectors.

Core claim

In massless scalar-tensor theories the conservative dynamics of a gravitationally bound two-body system that includes tidal effects is obtained at next-to-next-to-leading post-Newtonian order for spinless sources; the ten conserved quantities are likewise determined at NNLO; and the same results hold when the theory is extended to Einstein-scalar-Gauss-Bonnet gravity.

What carries the argument

The Fokker Lagrangian together with the PN-EFT formalism applied to the two-body problem with tidal interactions in scalar-tensor theories.

If this is right

  • The derived dynamics supplies the conservative sector for gravitational-wave phasing at NNLO in these theories.
  • The ten conserved quantities fix the integrals of motion that govern the orbital evolution at this order.
  • The extension supplies the corresponding tidal sector in Einstein-scalar-Gauss-Bonnet gravity.
  • These expressions can be inserted into waveform models used for the science case of future gravitational-wave detectors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same computational route could be followed for the dissipative sector or for massive scalar fields, yielding a complete set of NNLO predictions.
  • The conserved quantities at NNLO may permit closed-form expressions for the radial and angular motion that can be tested against numerical evolutions in scalar-tensor theories.
  • Because the paper already reaches Einstein-scalar-Gauss-Bonnet gravity, analogous calculations become feasible for other higher-curvature extensions that reduce to general relativity in the appropriate limit.

Load-bearing premise

The sources are spinless and the scalar field is massless.

What would settle it

A direct numerical check that the binding energy or orbital frequency expressions obtained from the Fokker Lagrangian disagree with those from the PN-EFT calculation at NNLO would falsify the claimed results.

Figures

Figures reproduced from arXiv: 2310.19679 by Eve Dones, Laura Bernard, Stavros Mougiakakos.

Figure 1
Figure 1. Figure 1: Diagrams contributing at order G2m2 1λ (0) 2 (1 + v 2 + v 4 ) F ig.(1a) = − 1+ + nv1(nv1 + 2nv2) − nv2 2 + v 2 2 2 + r 2  2na1 na2 − a1a2 2 − na2 1  + + 2r  (a1v2 − a1v1)(nv1 − nv2) − na1(v 2 2 + v 2 1 − 2v1v2 − 7 2 (nv2 1 + nv2 2 ) + 6nv1 nv2)  + v 4 2 8 + nv2 2 (3nv2 1 − v 2 2 2 ) + nv2 1 (2v1v2 − 3 v 2 2 2 ) + nv1 nv2(v 2 2 + 2v 2 1 − 2v1v2 − 6nv2 1 )− − 3 4r 2 (4µ (0) 2 + 4ν (0) 2 − c (0) 2 ) F ig.… view at source ↗
Figure 2
Figure 2. Figure 2: Diagrams contributing at order G3m2 1m2λ (0) 2 (1 + v 2 ) F ig.(2.1a) =2 ˜d (2) 1 ˜d (1) 2 ˜d (1) 1 h 2+ +  v1v2 − 2v 2 2 + 5nv2(2nv2 − 3nv1) i F ig.(2.1b) =2+ + v1v2 + 4v 2 1 + 2v 2 2 + 5nv2(2nv2 − 3nv1) F ig.(2.1c) = − 8v1v2 F ig.(2.2a) = − ˜d (2) 1 x1 h 2+ +  v1v2 − 2v 2 2 + 5nv2(2nv2 − 3nv1) i F ig.(2.2b) = − 2  (v1v2 − 9nv1 nv2) + 2(v 2 1 + nv2 1 )  F ig.(2.2c) =2 ˜d (2) 1 ˜d (1) 1 (v1v2 − nv1 n… view at source ↗
Figure 3
Figure 3. Figure 3: Diagrams contributing at order G3m3 1λ (0) 2 (1 + v 2 ) F ig.(3.1a) = ˜f (2) 1 ˜f (2) 0 ˜d (1) 1 h 2+ +  2nv2 1 + 2nv2 2 − 9nv1 nv2 − v 2 1 − v 2 2 + v1v2 i F ig.(3.1b) =3+ + 1 2  v 2 2 − 2nv2 2 + 9v 2 1 + 6nv2 1 + 3v1v2 − 11nv1 nv2  F ig.(3.1c) = − 4  v1v2 + 2nv2 1  F ig.(3.1d) = − 4  v 2 1 − nv2 1  F ig.(3.2a) = − ˜d (1) 1 x1 h 2+ +  2nv2 1 + 2nv2 2 − 9nv1 nv2 − v 2 1 − v 2 2 + v1v2 i F ig.(3.2… view at source ↗
Figure 4
Figure 4. Figure 4: Diagrams contributing at order G4m4 1λ (0) 2 F ig.(4a) = − 2 ˜f (2) 2 ˜f (2) 0 ( ˜d (1) 1 ) 2 F ig.(4b) = − 6 ˜f (2) 1 ˜f (2) 0 ˜d (1) 1 15 [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diagrams contributing at order G4m2 1m2(m1 + m2)λ (0) 2 16 [PITH_FULL_IMAGE:figures/full_fig_p040_5.png] view at source ↗
read the original abstract

In this article, we study the tidal effects in the gravitationally bound two-body system at next-to-next-to leading post-Newtonian order for spin-less sources in massless scalar-tensor theories. We compute the conservative dynamics, using both a Fokker Lagrangian approach and effective field theory with the PN-EFT formalism. We also compute the ten conserved quantities at the same NNLO order. Finally, we extend our results from simple ST theories to Einstein-scalar-Gauss-Bonnet gravity. Such results are important in preparation of the science case of the next generation of gravitational wave detectors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper computes tidal effects and conservative dynamics in spinless two-body systems at next-to-next-to-leading post-Newtonian (NNLO) order in massless scalar-tensor theories. It employs both a Fokker Lagrangian approach and PN effective field theory (EFT) as independent methods, derives the ten conserved quantities at NNLO, and extends the results to Einstein-scalar-Gauss-Bonnet gravity. The work is positioned as preparation for next-generation gravitational-wave detectors.

Significance. If the derivations hold, the dual-method cross-check supplies a valuable internal consistency test for NNLO tidal contributions in scalar-tensor theories, while the explicit extension to Einstein-scalar-Gauss-Bonnet broadens applicability. These higher-order results in modified gravity are directly relevant for waveform modeling and strong-field tests with future detectors such as LISA or the Einstein Telescope.

minor comments (2)
  1. [Abstract] The abstract states that ten conserved quantities are computed at NNLO but does not name them; an explicit list or forward reference to the relevant section would improve readability.
  2. [Introduction] In the discussion of the two methods, a short paragraph contrasting the new scalar-tensor terms with the corresponding GR expressions at the same order would help readers assess the size of the modifications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of the dual-method cross-check, and the recommendation for minor revision. No specific major comments were provided in the report, so we have no points requiring detailed rebuttal or revision at this stage. We are happy to incorporate any minor editorial suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent methods

full rationale

The paper restricts scope explicitly to spinless sources and massless scalar in ST theories, computes conservative dynamics and ten conserved quantities at NNLO via two distinct approaches (Fokker Lagrangian and PN-EFT) presented as cross-checks, then extends results to ESGB. No equations reduce by construction to fitted parameters, no load-bearing self-citation chains, and no ansatz or uniqueness imported from prior author work. The dual-method construction supplies independent internal consistency without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The massless scalar and spinless-source assumptions are implicit domain restrictions rather than new postulates.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Tidal effects in the total flux and waveform in massless scalar-tensor theories to, respectively, relative 2PN and 1.5PN orders

    gr-qc 2025-07 unverdicted novelty 6.0

    Derives next-to-next-to-leading tidal corrections to flux and phasing in scalar-tensor gravity using adapted post-Newtonian multipolar-post-Minkowskian methods under the adiabatic approximation.

  2. Dynamical Tidal Response of Non-rotating Black Holes: Connecting the MST Formalism and Worldline EFT

    gr-qc 2025-11 unverdicted novelty 5.0

    Renormalized dynamical tidal response functions for non-rotating black holes in GR carry inevitable ambiguities from renormalization scheme and flow initial condition, yielding scheme-dependent dynamical tidal Love nu...

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · cited by 2 Pith papers · 21 internal anchors

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    on-shell

    is the starting point to develop the machinery of the PN-EFT forma lism. To do so, we also perform a Kaluza-Klein decomposition of the metric as ˜gµν =e 4 φg ˜Mpl ( − 1 2 Aj/ ˜Mpl 2Ai/ ˜Mpl e − 8 φg ˜Mplγij − 4AiAj/ ˜M 2 pl ) , (28) whereγij =δij +σij/ Λ. In the following, we will work with the Kaluza-Klein gravitational field s, (φ g, Ai, σij), instead of...

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    (49b) The expression for the NNLO tidal correction to the relative accele ration is displayed in App

    In the CM frame we get up to the NLO, ai CM, LO = α 2 ˜G2m c2r5 8ζ 1 − ζλ (0) + ni, (49a) ai CM, NLO = α 2 ˜G2 c4r5 ζ 1 − ζm [ ni ( 12(nv)2(δλ(0) − − 2νλ (0) + ) − 2v2(δλ(0) − + 3λ (0) + − 12νλ (0) + ) ) − 4(nv)vi(δλ(0) − +λ (0) + − 4νλ (0) + ) ] + α 3 ˜G3 c4r6 ζ 2(1 − ζ)m2ni [ − 80 ¯β −λ (0) − ¯γ + (80 ¯β + − 96¯γ − 47¯γ 2 − 52 ¯γν )λ (0) + ¯γ +δ ((80 ¯β...

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    (50b) Again, to lighten the main text, we have relegated the NNLO tidal con tribution to the CM energy in the App

    Energy In the center of mass frame, the conserved energy at NLO is given by ELO = α 2 ˜G2m2ν c2r4 2ζ 1 − ζλ (0) + , (50a) EN LO = α 2 ˜G2 c4r4 −ζ 2(1 − ζ)m2ν [ (nv)2(− 4δλ(0) − + 8νλ (0) + ) +v2( δλ(0) − + (− 1 + 2ν)λ (0) + ) ] + α 3 ˜G3 c4r5 ζ 1 − ζm3ν [ { − 8 ¯β − ¯γ + ( 16 ¯β + + ¯γ(2 + 3¯γ) ) δ 2¯γ } λ (0) − + { 16 ¯β + − ¯γ(10 + 3¯γ) 2¯γ − 8 ¯β −δ ¯γ...

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    (51c) VII

    Angular momentum Finally, the conserved angular momentum in the center of mass fram e is given by JLO = 0, (51a) JN LO = α 2 ˜G2 c4r3 ζ 1 − ζm2ν ( − δλ(0) − + (1 − 2ν)λ (0) + ) , (51b) J i N N LO = α 2 ˜G2 c6r3 −ζ 2(1 − ζ)m2ν ( 2(nv)2( δ(21 + 8ν)λ (0) − + (23 + 6ν − 20ν2)λ (0) + ) +v2( −δ(11 + 6ν)λ (0) − − (13 − 8ν + 18ν2)λ (0) + ) ) + α 3 ˜G3 c6r4 −ζ 2(1...

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    The starting point for the PNEFT machinery is the action (

    Feynman rules In this appendix, we give the Feynman rules that are relevant for th e computation of tidal effects at NNLO. The starting point for the PNEFT machinery is the action (

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    Since we know that at this sector we don’t encounter divergence s, we can readily set d = 3 for simplicity

    accompanied with the Kaluza-Klein decomposition of the metric ( 28). Since we know that at this sector we don’t encounter divergence s, we can readily set d = 3 for simplicity. Below are presented the Feynman rules with respect to the Kaluza- Klein modes and the canonically normalized scalar field ψ relevant for the computation at this order. The propagato...

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    − 2(∂t1∂t2 +∂t1∂t3 +∂t2∂t3) ) , k1 k2 k3 = − 8 i ˜Mpl ∂t1∂t2, k1 k2 k3i = − 2 i ˜Mpl ( k1(i∂t2) +k2(i∂t1 ) ) i , k1 k2 k3ij = 1 2cd × k1 k2 k3ij = − i ˜Mpl ( kk 1kl 2Qijkl +∂t1∂t2δij ) , k1 k2 k3 k4 = − i ˜M 2 pl x2 2 (k2 1 +k2 2 +k2 3 +k2 4) where Pijkl = − (δikδjl +δilδjk + (2 − cd)δijδkl), Qijkl =Iijkl − δijδkl and Iijkl =δilδkj +δikδjl. 19

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    Acceleration in the CM frame The NNLO tidal correction to the relative acceleration in the center of mass frame is given, after separating it in power of ˜G, by ai, (2) CM, NNLO =α 2 ˜G2 ζ 1 − ζm [ − 9(2 + ¯γ)2 4(1 − ζ)2r7 (c(0) + − 4ν(0) + )ni + 36 c2r7µ (0) + ni + 1 c6r5 ( vi ( (nv)v2( 4δ(18 +ν) λ (0) − − 4(− 2 +ν)(9 + 4ν)λ (0) + ) + (nv)3( − 6δ(23 + 2ν...

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    +v1v2 − 5nv1 nv2 ) Fig. (3. 2d) = − 8 ( v2 1 − 2nv2 1 ) (10) D. G4m4 1λ0 (2) ˜f (2) n The values of the diagrams are normalized with 2 G4m4 1λ0 (2) r6 ˜f (2) 0 ( ˜d(1) 1 )2. We have to take into account also the mirror images (1 ↔ 2). (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 4: Diagrams contributing at order G4m4 1λ (0) 2 Fig. (4a) = − 2 ˜f (2) 2 ˜f...

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