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Tidal Love Numbers of Neutron Stars in Horndeski Theories

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper shows that in scalar-tensor theories the $1/r^3$ coefficient in the asymptotic perturbation fields contains a contamination independent of the tidal Love numbers, and derives corrected formulas for the minimally coupled scalar…

desk verdict Careful, important resolution of the 1/r^3 extraction ambiguity in scalar-tensor Love numbers; the parity assumption is the main soft spot, but the internal consistency check largely covers it. read the letter →

arxiv 2501.07998 v2 pith:GUFTPM7N submitted 2025-01-14 gr-qc hep-phhep-th

Robin Fynn Diedrichs , Shinji Tsujikawa , Kent Yagi This is my paper · ORCID
classification gr-qchep-phhep-th
keywords tidalLovenumbersneutronstarsscalar-tensortheoriesHorndeskigravityeffectivefieldtheoryasymptoticexpansionspontaneousscalarizationgravitationalwaves
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper addresses how tidal Love numbers of neutron stars should be extracted from perturbation theory in Horndeski scalar-tensor theories, the most general scalar-tensor theories with second-order field equations. It establishes that the standard identification of the $1/r^3$ term in the asymptotic expansion of the perturbation fields with the quadrupolar tidal response is contaminated: an additional contribution, independent of the Love numbers, appears at the same order. The authors compute this contamination for a minimally coupled massless scalar field and for the DEF model (a scalar-tensor theory with nonminimal coupling $F(\phi)=e^{-\beta\phi^2/(2M_{\rm Pl}^2)}$), and derive corrected extraction formulas. They find that neglecting the contamination shifts the DEF Love numbers by up to about 15% for $\beta = -6$ and by a few percent for $\beta = -4.5$, which matters for gravitational-wave parameter estimation.

What carries the argument

The load-bearing tool is an effective-field-theory point-particle action in isotropic coordinates. The bulk plus background worldline action is invariant under inversion of both fields and couplings, $\hat\varphi \to -\hat\varphi$, $\hat\phi \to -\hat\phi$, $M \to -M$, $Q \to -Q$, which together with dimensional analysis forces the perturbation fields $\delta\hat\varphi$ and $\delta\hat\phi$ of an object with zero Love numbers to contain only even powers of the radial coordinate (Eq. (4.43)). Transforming the EFT expansion back to Schwarzschild coordinates and matching through the gauge-invariant combination $\Psi = H_0 - f'\delta\phi/(f\phi')$ converts that even-power structure into explicit $1/r^3$ contamination terms, which are then separated from the tidal contributions $\lambda_{hh}$, $\lambda_{h\phi}$, and $\lambda_{\phi\phi}$ read off from the point-particle tidal action.

What would settle it

Compute the five-worldline-coupling diagrams of Fig. 1c explicitly in isotropic coordinates: the paper's parity argument predicts they vanish, so any nonzero result would falsify the even-power structure (4.43) and with it the contamination formulas (4.65)-(4.66).

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Extended reading notes

Core claim

The central claim is that in scalar-tensor theories the coefficient of the $1/r^3$ term in the asymptotic expansion of the metric and scalar perturbations is not purely tidal. Writing $H_0(r) = H_{0,-2} r^2 + \dots + H_{0,3}/r^3 + \dots$ and $\delta\phi(r) = \delta\phi_{-2} r^2 + \dots + \delta\phi_3/r^3 + \dots$, the paper splits $H_{0,3} = H^\lambda_{0,3} + H^0_{0,3}$ and $\delta\phi_3 = \delta\phi^\lambda_3 + \delta\phi^0_3$, where the superscript $\lambda$ contains the tidal Love numbers and the superscript $0$ is a non-tidal part built only from the mass $M$ and scalar charge $\phi_1$. For the minimally coupled scalar this non-tidal part is $H^0_{0,3} = H_{0,-2}(3 M^3 \phi_1^2/(5 M_{\rm Pl}^2) - 3 M \phi_1^4/(80 M_{\rm Pl}^4))$, with the same expression for $\delta\phi^0_3$ in terms of $\delta\phi_{-2}$. The DEF-model generalization is given in Eqs. (5.25)-(5.26), and the corrected Love-number extraction formulas in Eqs. (5.32)-(5.35).

Load-bearing premise

The calculation rests on the claim that for an object with vanishing Love numbers the perturbation fields in isotropic coordinates contain only even powers of the radial coordinate, which follows from inversion symmetry plus dimensional analysis.

Editorial extensions

If this is right

  • In any scalar-tensor theory whose exterior reduces to Einstein gravity with a massless minimally coupled scalar, the corrected formulas (4.72)-(4.73) must be used to extract $\lambda_{hh}$, $\lambda_{h\phi}$, and $\lambda_{\phi\phi}$ from the $1/r^3$ coefficients.
  • For DEF neutron stars with the DD2 equation of state, using the raw $1/r^3$ coefficients instead of the corrected formulas shifts $\Lambda_{hh}$ by up to about 3%, $\Lambda_{\phi\phi}$ by up to about 5%, and $\Lambda_{h\phi}$ by up to 15% for $\beta=-6$; for $\beta=-4.5$ the shifts are a few percent or less.
  • The corrected extraction makes $\lambda_{h\phi}$ obtained from the gravitational channel agree with that from the scalar channel to within numerical accuracy, whereas the uncorrected extraction disagrees by up to 13%.
  • On the GR branch of DEF the mixed Love number $\Lambda_{h\phi}$ vanishes as parity requires, while $\Lambda_{\phi\phi}$ stays nonzero, and spontaneous scalarization strongly enhances $\Lambda_{\phi\phi}$ for $\beta=-6$.
  • For scalar-Gauss-Bonnet gravity the inversion symmetry is broken, so the contamination cannot be fixed by the same parity argument and requires explicit diagrammatic computation.

Reading between the lines

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

  • The same even-power argument would apply to dipolar and higher-multipole Love numbers in massless scalar-tensor theories, suggesting the ambiguity is not specific to the quadrupole and could be resolved by the same EFT matching.
  • Because the contamination terms are built from the mass $M$ and the scalar charge $\phi_1$, their fractional importance grows with compactness and scalar charge, so published DEF Love-number tables that ignored them may need revision for strongly scalarized stars.
  • A practical diagnostic for future work would be to check whether the two independent extractions of $\lambda_{h\phi}$ agree: if they disagree at the percent level or more, the non-tidal $1/r^3$ contamination is present and must be subtracted.
  • For massive scalar fields the contamination would be exponentially suppressed at infinity, so the correction is relevant mainly for massless or extremely light scalars, which could guide which scalar-tensor models need the corrected extraction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper analyzes the extraction of quadrupolar tidal Love numbers for neutron stars in Horndeski scalar-tensor theories. Building on the perturbation equations of Ref. [57], the authors derive linear static even-parity perturbation equations and show that the asymptotic 1/r^3 coefficients of the metric and scalar perturbations contain, in addition to the tidal Love numbers, a non-tidal contribution that is independent of the Love numbers. Using an effective field theory (EFT) point-particle approach, they compute this contamination for a minimally coupled scalar field (Sec. IV) and for the Damour–Esposito–Farèse (DEF) model (Sec. V), obtaining closed-form expressions for the corrected Love numbers. They numerically compute the DEF Love numbers with the DD2 equation of state and demonstrate that neglecting the contamination changes the extracted Love numbers by up to 15% for β = −6. They also discuss why the same method cannot be straightforwardly applied to scalar-Gauss-Bonnet gravity (Sec. VI).

Significance. If the central derivation holds, the paper resolves a previously overlooked ambiguity in computing tidal Love numbers in scalar-tensor theories. The result that the 1/r^3 asymptotic coefficient is not purely tidal is conceptually important and has direct implications for gravitational-wave parameter estimation. The paper's strengths include the derivation of explicit, ready-to-use formulas (Eqs. (4.72)–(4.73), (5.32)–(5.35)), the validation of the numerical extraction against Hinderer's exact GR formula (Fig. 3), and the nontrivial internal consistency check in Fig. 5, where the mixed Love number λ_hϕ extracted via two independent channels agrees to better than 0.07% after correction. The presentation is detailed and the appendices provide extensive asymptotic expansions that would be useful for future work.

major comments (2)
  1. [Sec. IVB, Eq. (4.43)] The even-power parity assumption in Eq. (4.43) is load-bearing for the central claim, but it is asserted rather than proven. The inversion symmetry (4.42) is verified only for the truncated linearized point-particle action (4.23), not for the full action (4.22) or for generic operators such as the p φ^2/M_Pl^2 term introduced in Eq. (3.2). Such operators break the symmetry and could in principle contribute to the 1/r^3 coefficient, altering the non-tidal parts (4.65)–(4.66) and the resulting Love-number formulas (5.32)–(5.35). The footnote justifying the truncation rules out some classes of diagrams but does not explicitly exclude p-type operators at the relevant order. I request a diagram-level proof, or at least a clear statement of the conditions under which Eq. (4.43) holds. The numerical agreement in Fig. 5 is reassuring, but it tests only the DEF model with the specific PP action used and does not constitute a general proof.
  2. [Sec. IVC, Eqs. (4.72)–(4.73)] The universality claim stated after Eq. (4.73)—that the result applies to any scalar-tensor theory reducing to Einstein gravity with a minimally coupled scalar in vacuum—is conditional on the specific truncation of the point-particle action in Eq. (4.23). If higher-derivative or nonlinear worldline operators are present, the inversion symmetry (4.42) is generically broken and the decomposition into tidal and non-tidal parts may need to be revisited. The authors should either prove that such operators cannot contribute to the 1/r^3 coefficient (for example by explicit diagrammatic counting) or explicitly restrict the statement of universality to the class of models where those operators are absent. This clarification is necessary because the paper's main phenomenological conclusions for the DEF model rely on this decomposition.
minor comments (6)
  1. [Sec. II, first paragraph] There is a typo: "neither neither G4,X nor G5 vanish" should read "neither G4,X nor G5 vanish".
  2. [Sec. IVB, footnote 3] The footnote says the calculation is "not directly comparable to the expansion in Eq. (2.7)", but Eq. (2.7) is a background equation; the intended reference is likely Eq. (4.7).
  3. [Abstract] The phrase "O(1 ∼ 10) %" is nonstandard notation; it should be written as "O(1–10)%".
  4. [Sec. IVC, paragraph after Eq. (4.63)] The text refers to "the coefficients H0,3 and δϕ3 of Eqs. (4.13) and (4.14)", but Eqs. (4.13)–(4.14) are expansions; the wording should be "the coefficients H0,3 and δϕ3 in Eqs. (4.13) and (4.14)".
  5. [Fig. 5] The axis label uses the symbol φ (Greek phi without subscript) in "|λhφ/ ˆλhφ − 1|"; for consistency with the text, the subscript should be the scalar-field symbol ϕ.
  6. [Sec. I, last paragraph of introduction] A comma is missing: "as long as one uses Schwarzschild coordinates combined with the Regge-Wheeler gauge — in scalar-tensor theory" should have a comma after "gauge".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-tidal 1/r^3 terms are computed from background data and checked against an external GR benchmark, not fitted from the Love numbers.

full rationale

The paper's central decomposition (4.72)-(4.73) separates the 1/r^3 coefficients into a tidal part, proportional to the Wilson coefficients λij, and a non-tidal part, proportional to M and φ1 (Eqs. 4.65-4.66). The non-tidal part is obtained by matching the EFT even-power ansatz (4.43) to the asymptotic expansions (4.13)-(4.14), using only the background mass and scalar charge; no Love number is used as an input. The Love numbers are then extracted by solving the linear system (5.27)-(5.30) from two independent numerical perturbation profiles, and the corrected values are verified in two independent ways: Fig. 3 reproduces the exact Hinderer GR formula for β=0, and Fig. 5 shows that λhφ obtained from chφ agrees with λhφ obtained from cφh to better than 0.07% after correction. A wrong subtraction would generically spoil that agreement. The even-power parity assumption (4.43) is asserted rather than proven at diagram level, but this is an unproven assumption or correctness risk, not circularity: the parity structure is not defined in terms of the target Love numbers. The only overlap with prior work is Ref. [57] (Kase & Tsujikawa, one coauthor), which supplies the underlying Horndeski perturbation equations; those equations are prior independent input and are not equivalent to the paper's claim. The paper also explicitly leaves the analogous sGB diagram calculation for future work, an acknowledged limitation rather than a circular step. I therefore find no circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation has no fitted constants: the contamination terms are fixed by the background solution and the assumed EFT symmetries. The numerical examples use the DD2 equation of state from the literature and chosen values of the DEF coupling beta, which are inputs rather than free parameters tuned to produce the result. No new particles, forces, dimensions, or conserved quantities are introduced; the Wilson coefficients lambda_hh, lambda_hphi, and lambda_phiphi are standard EFT couplings, not invented entities.

assumptions (5)
  • domain assumption The even-parity static perturbation equations (2.23)-(2.28) from Kase and Tsujikawa (2022) are correct.
    The paper builds directly on Ref. [57] without re-deriving the second-order action; all perturbation coefficients are quoted from that prior work.
  • domain assumption A massless, static scalar field and static perturbations are sufficient for the adiabatic tidal limit.
    The paper assumes the orbital frequency is much smaller than the stellar mode frequency so time derivatives drop, and assumes the scalar is massless so the asymptotic fields fall as powers of 1/r without exponential suppression.
  • domain assumption The bulk plus background point-particle action preserves the inversion symmetry (4.42), and this symmetry forbids odd powers in isotropic-coordinate perturbations for vanishing Love numbers.
    This is the key structural input to the contamination calculation; it is argued from inspection of the action and dimensional analysis, not proven as a theorem.
  • domain assumption For the DEF model, the conformal transformation to the Einstein frame yields a minimally coupled scalar with the matter action transforming in the standard way.
    This standard DEF property is used in Sec. V.B to recycle the minimal-scalar formulas (4.72)-(4.73).
  • domain assumption The least-squares fit to order r^-8 introduces errors at the 0.01% level, as inferred from the GR Hinderer test.
    Used to assign numerical uncertainty; the paper argues from Fig. 3 that truncation error is below numerical noise.

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Pith. "Pith review of Tidal Love Numbers of Neutron Stars in Horndeski Theories." pith.science (2026). https://pith.science/paper/GUFTPM7N

@misc{pith2026250107998,
  author       = {Pith},
  title        = {Pith review of: Tidal Love Numbers of Neutron Stars in Horndeski Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GUFTPM7N}},
  note         = {Machine review of arXiv:2501.07998}
}
abstract

Precision measurements of the gravitational wave signal from compact binary inspirals allow us to constrain the internal structure of those objects via physical parameters such as the tidal Love numbers. In scalar-tensor theories, one typically finds new types of Love numbers that are usually not considered or simply absent in General Relativity, which further allows us to constrain deviations from General Relativity. Building upon previous results, we present the linear perturbation equations necessary to calculate static and even-parity tidal Love numbers in Horndeski theories, the most general scalar-tensor theories with second-order field equations of motion. We further focus on the quadrupolar Love numbers and demonstrate how these can be extracted from the asymptotic expansion of the perturbation fields. We find that there is a potential ambiguity in extracting the Love numbers in this way, which we resolve by performing supplementary calculations in the effective field theory framework. We show that, in the case of scalar-tensor theories, the tidal Love numbers are not directly given by the $1/r^3$ term in the asymptotic expansion of the perturbation fields, as there is an additional contribution to this term independent of the Love numbers. We calculate such a contribution for a minimally coupled scalar field and also for the Damour-Esposito-Far\`ese model. For the latter, we find that the Love numbers can differ by $\mathcal{O}(1 \sim 10)\,\%$, if this additional contribution is not taken into account.

Figures

Figures reproduced from arXiv: 2501.07998 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Left [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Left [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

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Forward citations

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