arxiv: 2507.07676 · v2 · submitted 2025-07-10 · 🌀 gr-qc

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

Eve Dones , Laura Bernard This is my paper

Pith reviewed 2026-05-19 05:47 UTC · model grok-4.3

classification 🌀 gr-qc

keywords scalar-tensor theoriestidal deformabilitypost-Newtonian expansiongravitational wave fluxwaveform phasingneutron star binariesmemory effects

0 comments p. Extension

The pith

Tidal deformations in scalar-tensor gravity correct the energy flux at 2PN and waveform phasing at 1.5PN beyond the leading dipolar term.

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

The paper calculates the imprint of tidal deformations on gravitational and scalar radiation from neutron star binaries in scalar-tensor theories. These deformations arise from both the companion star and the scalar field and depend on the stars' internal structure. Working in the adiabatic approximation with an adapted post-Newtonian multipolar-post-Minkowskian formalism, the authors derive the corrections to the total energy flux at next-to-next-to-leading order, equivalent to 2PN beyond the leading dipolar contribution. At this order three independent deformabilities—scalar, tensorial, and mixed—enter the signal. The work also supplies the full set of waveform amplitude modes, including memory terms, to relative 1.5PN order.

Core claim

Within scalar-tensor theories the tidal corrections to the total energy flux, accounting for both gravitational and scalar radiation, reach NNLO, or 2PN order past the leading dipolar tidal term. Three distinct tidal deformabilities contribute at this accuracy. The full waveform amplitude modes, gravitational and scalar, together with the memory (m=0) modes, are obtained to N^{1.5}LO.

What carries the argument

The post-Newtonian multipolar-post-Minkowskian formalism adapted to scalar-tensor theories, applied inside the adiabatic approximation to capture tidal deformations of neutron stars.

If this is right

Both gravitational and scalar radiation channels receive tidal corrections at the stated orders.
Scalar, tensorial, and mixed scalar-tensorial deformabilities each produce independent contributions to the flux and phasing.
The derived amplitude modes include all gravitational, scalar, and memory terms through relative 1.5PN.
These corrections become measurable with the precision expected from next-generation gravitational-wave detectors.

Where Pith is reading between the lines

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

Incorporating these terms into template banks could tighten bounds on scalar-tensor parameters while simultaneously constraining neutron-star equations of state.
Relaxing the adiabatic assumption would require a dynamical-tide extension that couples orbital motion to stellar oscillation modes.
The same formalism might be reused to compute tidal effects in other modified-gravity theories that introduce additional radiative degrees of freedom.

Load-bearing premise

The orbital period remains much longer than the internal dynamical time of each neutron star so that the adiabatic approximation remains valid.

What would settle it

High-precision numerical simulations of binary neutron star mergers in scalar-tensor gravity that extract the phasing correction at 2PN and compare it directly with the analytic expression.

read the original abstract

Within scalar-tensor (ST) theories, neutron stars in binary systems experience tidal deformations caused by both their companion and the scalar field. These deformations are strongly correlated to the star's internal structure and composition. Accurately modeling their imprint on the emitted gravitational waves will be essential for interpreting the high-precision data expected from future detectors and for disentangling potential signatures of modified gravity from those arising due to the properties of neutron star matter. Using the post-Newtonian multipolar-post-Minkowskian formalism adapted to ST theories, and working within the adiabatic approximation, we compute the tidal corrections to the total energy flux, accounting for both gravitational and scalar radiation, and to the waveform phasing, at the next-to-next-to-leading order (NNLO). This corresponds to second post-Newtonian (2PN) order beyond the leading-order dipolar tidal contribution. At this accuracy, three independent types of tidal deformability (scalar, tensorial, and mixed scalar-tensorial) contribute to the signal. We also derive the full waveform amplitude modes, including gravitational and scalar modes, as well as the memory (m=0) ones, to the $\text{N}^{1.5}\text{LO}$ (i.e. to relative 1.5PN order).

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.

Desk Editor's Note private letter to a colleague

This paper supplies the 2PN tidal flux and 1.5PN phasing corrections in massless scalar-tensor theories, including scalar and mixed channels, but rests on the standard adiabatic treatment without extra checks.

read the letter

The key takeaway is that Dones and Bernard have worked out the NNLO tidal corrections to the energy flux at 2PN and the waveform phasing at 1.5PN in massless scalar-tensor gravity. These include the scalar, tensorial, and mixed deformability contributions to both gravitational and scalar radiation channels, which were not in the earlier literature they cite. They also give the amplitude modes and memory terms to relative 1.5PN. The calculation adapts the multipolar-post-Minkowskian formalism to ST theories under the adiabatic approximation, which is the usual route for these expansions. That extension is the main new piece and it looks methodical on the surface. The work is relevant for anyone building waveform models that need percent-level accuracy for neutron-star binaries in next-generation detectors, where modified-gravity signals might appear. They build directly on established PN methods rather than starting from scratch, which keeps the derivation traceable. The soft spot is the adiabatic assumption itself. The stress-test note points out that the massless scalar could shift the star's internal dynamical frequencies through back-reaction, so the orbital timescale might not stay cleanly separated from the internal one in all cases. The paper does not appear to add any non-adiabatic estimates or regime checks, which leaves that limitation unaddressed even though it is standard in the field. No obvious fitting parameters or circular reductions show up, and the citation pattern follows the usual PN and tidal literature. This is specialized material for people already working on PN expansions or tidal effects in alternatives to GR. A reader who needs the explicit higher-order terms for template development will find it useful. It is worth sending to a serious referee because the order is high enough to matter for future data analysis, even if the adiabatic part will probably draw comments.

Referee Report

1 major / 2 minor

Summary. The manuscript adapts the post-Newtonian multipolar-post-Minkowskian formalism to massless scalar-tensor theories and, working in the adiabatic approximation, computes tidal corrections to the total energy flux (gravitational plus scalar radiation) at next-to-next-to-leading order, corresponding to 2PN beyond the leading dipolar tidal term. It also derives the waveform phasing at 1.5PN order and the full set of amplitude modes, including gravitational, scalar, and memory (m=0) contributions to relative 1.5PN order. Three independent tidal deformabilities (scalar, tensorial, and mixed) enter at this accuracy.

Significance. If the derivations hold, the results supply the higher-order tidal terms required for accurate waveform modeling in scalar-tensor gravity. This is directly relevant for future high-precision gravitational-wave observations that aim to separate modified-gravity signatures from neutron-star equation-of-state effects. The systematic inclusion of scalar radiation and mixed deformabilities is a clear technical advance over existing GR tidal calculations.

major comments (1)

The adiabatic approximation is invoked throughout to treat the stellar response (including the scalar profile) as instantaneous. However, the massless scalar introduces an additional long-range degree of freedom whose back-reaction on the stellar interior can shift the effective dynamical frequencies. When these frequencies become comparable to the orbital frequency the derived 2PN tidal corrections (scalar, tensorial, and mixed) cease to be valid. The manuscript should supply explicit bounds on the regime of applicability or a concrete test of the approximation at the claimed order.

minor comments (2)

The abstract states that three independent types of tidal deformability contribute; their definitions, notation, and relations to the stellar structure should be introduced with explicit equations in the main text rather than deferred.
Ensure that all lower-order terms recovered from the new calculation are explicitly cross-checked against existing results in the literature for both GR and scalar-tensor theories.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for identifying an important caveat concerning the adiabatic approximation. We address the major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: The adiabatic approximation is invoked throughout to treat the stellar response (including the scalar profile) as instantaneous. However, the massless scalar introduces an additional long-range degree of freedom whose back-reaction on the stellar interior can shift the effective dynamical frequencies. When these frequencies become comparable to the orbital frequency the derived 2PN tidal corrections (scalar, tensorial, and mixed) cease to be valid. The manuscript should supply explicit bounds on the regime of applicability or a concrete test of the approximation at the claimed order.

Authors: We agree that the adiabatic approximation requires careful justification when a massless scalar field is present, as its long-range character can in principle couple to stellar interior modes. In the manuscript the approximation is adopted consistently with the post-Newtonian ordering, under the assumption that the orbital frequency remains well below the characteristic frequencies of the stellar response (including scalar-induced modes). To meet the referee’s request we will add a new paragraph in Section II (or a dedicated subsection) that supplies explicit bounds on the regime of validity. These bounds will be expressed in terms of the ratio of orbital frequency to the lowest scalar-tensor stellar oscillation frequency, using typical neutron-star compactness and the leading-order scalar charge. We will also note that corrections arising from non-adiabatic effects enter only at orders higher than those computed here. A full time-dependent numerical test lies outside the scope of the present analytic work but is flagged as a natural extension. revision: yes

Circularity Check

0 steps flagged

Direct perturbative expansion with no reduction to inputs by construction

full rationale

The paper adapts the multipolar-post-Minkowskian formalism to massless scalar-tensor theories and performs an explicit perturbative calculation of tidal corrections to the energy flux and waveform phasing at NNLO (2PN beyond leading dipolar) under the adiabatic approximation. No equations or steps reduce the claimed results to fitted parameters, self-defined quantities, or load-bearing self-citations by construction; the derivation is presented as a standard order-by-order expansion whose outputs are independent of the inputs once the formalism and approximation are accepted. The adiabatic assumption is an external modeling choice whose validity is separate from circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the adiabatic approximation for tidal response and on the adaptation of the multipolar-post-Minkowskian formalism to massless scalar-tensor theories; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Adiabatic approximation for neutron-star tidal deformations
Invoked to close the system of equations for the tidal response in binary systems.
domain assumption Validity of the post-Newtonian multipolar-post-Minkowskian formalism when extended to massless scalar-tensor theories
Used as the calculational framework for both gravitational and scalar radiation.

pith-pipeline@v0.9.0 · 5767 in / 1401 out tokens · 47146 ms · 2026-05-19T05:47:13.320738+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

Using the post-Newtonian multipolar-post-Minkowskian formalism adapted to ST theories, and working within the adiabatic approximation, we compute the tidal corrections to the total energy flux... at the next-to-next-to-leading order (NNLO). This corresponds to second post-Newtonian (2PN) order beyond the leading-order dipolar tidal contribution.
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We also derive the full waveform amplitude modes, including gravitational and scalar modes, as well as the memory (m=0) ones, to the N^{1.5}LO (i.e. to relative 1.5PN order).

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

34 extracted references · 34 canonical work pages · 14 internal anchors

[1]

, α (2 + γ) = 2(1 − ζ) β2 = ζ α2 (1 − 2s1)2 (λ1 (1 − 2s2) + 2ζs ′

work page
[2]

, β+ = β1+β2 2 , β− = β1−β2 2 . 2PN δ1 = ζ(1−ζ) α2 (1 − 2s1)2 , δ2 = ζ(1−ζ) α2 (1 − 2s2)2 , Degeneracy δ+ = δ1+δ2 2 , δ− = δ1−δ2 2 , 16δ1δ2 = γ2(2 + γ)2 χ1 = ζ α3 (1 − 2s2)3 λ2 − 4λ2 1 + ζλ 1 (1 − 2s1) − 6ζλ 1s′ 1 + 2ζ 2s′′ 1 , χ2 = ζ α3 (1 − 2s1)3 λ2 − 4λ2 1 + ζλ 1 (1 − 2s2) − 6ζλ 1s′ 2 + 2ζ 2s′′ 2 , χ+ = χ1+χ2 2 , χ− = χ1−χ2 2 . 4 Table I: Parameters fo...

work page
[3]

vi + 1 c2 ψ(0)(1 − 2s) + 1 2 v2 + 3V vi − 4V i # + O 1 c4 , (5.2a) P i tidal = λ(0) c4

Here, the unit vector normal to the orbital plane ℓ = ℓ0 is constant due to the planar nature of the motion. Finally, anticipating on our results, we introduce the following redefinition of the phase variable ψ ≡ ϕ − 2(1 − ζ) α x3/2 " 1 + 8ζx3 1 − ζ ˜λ(0) + + √α(S− ˜Λ(0) − + S+ ˜Λ(0) + ) ¯γ # log(4ωτ0) + γE − 11 12 , (4.11) that is similar to the one prop...

work page 2020
[4]

Neutron-star mergers in scalar-tensor theories of gravity

E. Barausse, C. Palenzuela, M. Ponce, and L. Lehner, Neutron-star mergers in scalar-tensor theories of gravity, Phys. Rev. D 87, 081506 (2013), arXiv:1212.5053 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[5]

Dynamical scalarization of neutron stars in scalar-tensor gravity theories

C. Palenzuela, E. Barausse, M. Ponce, and L. Lehner, Dynamical scalarization of neutron stars in scalar-tensor gravity theories, Phys. Rev. D 89, 044024 (2014), arXiv:1310.4481 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[6]

´A. D. Kov´ acs and H. S. Reall, Well-posed formulation of Lovelock and Horndeski theories, Phys. Rev. D 101, 124003 (2020), arXiv:2003.08398 [gr-qc]. 22

work page arXiv 2020
[7]

W. E. East and J. L. Ripley, Evolution of Einstein-scalar-Gauss-Bonnet gravity using a modified harmonic formulation, Phys. Rev. D 103, 044040 (2021), arXiv:2011.03547 [gr-qc]

work page arXiv 2021
[8]

Cayuso, P

R. Cayuso, P. Figueras, T. Fran¸ ca, and L. Lehner, Self-Consistent Modeling of Gravitational Theories beyond General Relativity, Phys. Rev. Lett. 131, 111403 (2023), arXiv:2303.07246 [gr-qc]

work page arXiv 2023
[9]

Corman, L

M. Corman, L. Lehner, W. E. East, and G. Dideron, Nonlinear studies of modifications to general relativity: Comparing different approaches, Phys. Rev. D 110, 084048 (2024), arXiv:2405.15581 [gr-qc]

work page arXiv 2024
[10]

Post-Newtonian Theory for Gravitational Waves

L. Blanchet, Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Rel. 17, 2 (2014), arXiv:1310.1528 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[11]

Compact binary systems in scalar-tensor gravity: Equations of motion to 2.5 post-Newtonian order

S. Mirshekari and C. M. Will, Compact binary systems in scalar-tensor gravity: Equations of motion to 2.5 post-Newtonian order, Phys. Rev. D 87, 084070 (2013), arXiv:1301.4680 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[12]

Bernard, Dynamics of compact binary systems in scalar-tensor theories: Equations of motion to the third post-Newtonian order, Phys

L. Bernard, Dynamics of compact binary systems in scalar-tensor theories: Equations of motion to the third post-Newtonian order, Phys. Rev. D 98, 044004 (2018), arXiv:1802.10201 [gr-qc]

work page arXiv 2018
[13]

Dynamics of compact binary systems in scalar-tensor theories: II. Center-of-mass and conserved quantities to 3PN order

L. Bernard, Dynamics of compact binary systems in scalar-tensor theories: II. Center-of-mass and conserved quantities to 3PN order, Phys. Rev. D 99, 044047 (2019), arXiv:1812.04169 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[14]

R. N. Lang, Compact binary systems in scalar-tensor gravity. II. Tensor gravitational waves to second post-Newtonian order, Phys. Rev. D 89, 084014 (2014), arXiv:1310.3320 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[15]

R. N. Lang, Compact binary systems in scalar-tensor gravity. III. Scalar waves and energy flux, Phys. Rev. D 91, 084027 (2015), arXiv:1411.3073 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[16]

Gravitational waveforms in scalar-tensor gravity at 2PN relative order

N. Sennett, S. Marsat, and A. Buonanno, Gravitational waveforms in scalar-tensor gravity at 2PN relative order, Phys. Rev. D 94, 084003 (2016), arXiv:1607.01420 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[17]

Bernard, L

L. Bernard, L. Blanchet, and D. Trestini, Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order, JCAP 08 (08), 008, arXiv:2201.10924 [gr-qc]

work page arXiv
[18]

Trestini, Quasi-Keplerian parametrization for eccentric compact binaries in scalar-tensor theories at second post- Newtonian order and applications, Phys

D. Trestini, Quasi-Keplerian parametrization for eccentric compact binaries in scalar-tensor theories at second post- Newtonian order and applications, Phys. Rev. D 109, 104003 (2024), arXiv:2401.06844 [gr-qc]

work page arXiv 2024
[19]

Trestini, Gravitational waves from quasielliptic compact binaries in scalar-tensor theory to one-and-a-half post- Newtonian order, (2024), arXiv:2410.12898 [gr-qc]

D. Trestini, Gravitational waves from quasielliptic compact binaries in scalar-tensor theory to one-and-a-half post- Newtonian order, (2024), arXiv:2410.12898 [gr-qc]

work page arXiv 2024
[20]

Bernard, Dipolar tidal effects in scalar-tensor theories, Phys

L. Bernard, Dipolar tidal effects in scalar-tensor theories, Phys. Rev. D 101, 021501 (2020), [Erratum: Phys.Rev.D 107, 069901 (2023)], arXiv:1906.10735 [gr-qc]

work page arXiv 2020
[21]

Creci, T

G. Creci, T. Hinderer, and J. Steinhoff, Tidal properties of neutron stars in scalar-tensor theories of gravity, Phys. Rev. D 108, 124073 (2023), [Erratum: Phys.Rev.D 111, 089901 (2025)], arXiv:2308.11323 [gr-qc]

work page arXiv 2023
[22]

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

L. Bernard, E. Dones, and S. Mougiakakos, Tidal effects up to next-to-next-to-leading post-Newtonian order in massless scalar-tensor theories, Phys. Rev. D 109, 044006 (2024), arXiv:2310.19679 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2024
[23]

W. D. Goldberger and I. Z. Rothstein, An Effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006), arXiv:hep-th/0409156

work page internal anchor Pith review Pith/arXiv arXiv 2006
[24]

Creci, I

G. Creci, I. van Gemeren, T. Hinderer, and J. Steinhoff, Tidal effects in gravitational waves from neutron stars in scalar- tensor theories of gravity, (2024), arXiv:2412.06620 [gr-qc]

work page arXiv 2024
[25]

The supplemental material ST Tides 2PN Radiation.m contains the GW energy fluxes, phase and waveform modes, includ- ing their 2PN tidal corrections beyond the leading-order dipolar contribution. A second supplemental material, available upon request only, contains intermediate results for the conservative dynamics on circular orbits, dissipative equations...

work page
[26]

C. M. Will and K. Nordtvedt, Jr., Conservation Laws and Preferred Frames in Relativistic Gravity. I. Preferred-Frame Theories and an Extended PPN Formalism, Astrophys. J. 177, 757 (1972)

work page 1972
[27]

C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, 2018)

work page 2018
[28]

D. M. Eardley, Observable effects of a scalar gravitational field in a binary pulsar., ”Astrophys. J. Lett.” 196, L59 (1975)

work page 1975
[29]

L. D. Landau and E. M. Lifschits, The Classical Theory of Fields, Course of Theoretical Physics, Vol. Volume 2 (Pergamon Press, Oxford, 1975)

work page 1975
[30]

Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries

M. Favata, Post-Newtonian corrections to the gravitational-wave memory for quasi-circular, inspiralling compact binaries, Phys. Rev. D 80, 024002 (2009), arXiv:0812.0069 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[31]

Hadamard Regularization

L. Blanchet and G. Faye, Hadamard regularization, J. Math. Phys. 41, 7675 (2000), arXiv:gr-qc/0004008

work page internal anchor Pith review Pith/arXiv arXiv 2000
[32]

Blanchet, On the multipole expansion of the gravitational field, Class

L. Blanchet, On the multipole expansion of the gravitational field, Class. Quant. Grav. 15, 1971 (1998), arXiv:gr- qc/9801101

work page arXiv 1971
[33]

K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, The 2.5PN gravitational wave polarisations from inspiralling compact binaries in circular orbits, Class. Quant. Grav. 21, 3771 (2004), [Erratum: Class.Quant.Grav. 22, 3115 (2005)], arXiv:gr-qc/0404085

work page internal anchor Pith review Pith/arXiv arXiv 2004
[34]

Can the post-Newtonian gravitational waveform of an inspiraling binary be improved by solving the energy balance equation numerically?

W. Tichy, E. E. Flanagan, and E. Poisson, Can the postNewtonian gravitational wave form of an inspiraling binary be improved by solving the energy balance equation numerically?, Phys. Rev. D 61, 104015 (2000), arXiv:gr-qc/9912075

work page internal anchor Pith review Pith/arXiv arXiv 2000