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Tidal effects in gravitational waves from neutron stars in scalar-tensor theories of gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that in scalar-tensor theories the net tidal contribution to the gravitational-wave phase of a neutron-star inspiral is generally smaller than in general relativity, even though it involves three distinct Love numbers.

desk verdict The derivation is sound but the headline formula (Eq. 83) has a sign error in the GR limit: the Lambda-tilde term should be negative, which makes the paper's central ready-to-use phase unusable until corrected. read the letter →

arxiv 2412.06620 v3 pith:OP6U5BSY submitted 2024-12-09 gr-qc

Gastón Creci , Iris van Gemeren , Tanja Hinderer , Jan Steinhoff This is my paper · ORCID
classification gr-qc
keywords scalar-tensorgravityneutronstartidaldeformabilityLovenumbersgravitational-wavephasepost-Newtonianapproximationscalarizationbinaryinspiral
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 aims to establish that tidal effects in gravitational waves from neutron-star binaries are more varied, and in a specific sense weaker, in scalar-tensor gravity than in general relativity. Working to first post-Newtonian order and first order in finite-size effects, the authors derive a ready-to-use Fourier-domain phase formula for the inspiral in which scalar, tensor, and mixed scalar-tensor tidal deformabilities enter at different powers of the frequency parameter. Because those terms carry opposite signs, the net tidal phase correction in scalar-tensor theory is generally smaller than the corresponding general-relativistic correction for the same binary. The result matters because template models built on a single GR-like tidal deformability would miss the scalar and mixed tidal signatures and could misread the neutron-star equation of state.

What carries the argument

The load-bearing structure is a skeletonized two-body action in which each neutron star carries three families of adiabatic, even-parity tidal Love numbers $\lambda_S^\ell$, $\lambda_T^\ell$, and $\lambda_{ST}^\ell$, responding respectively to scalar tides, tensor tides, and the cross-response between the two. These enter through tidal fields built from derivatives of the near-zone scalar and gravitational potentials, produce effective tidal coefficients $\zeta_\ell$, $\bar{\zeta}_\ell$, and $\tilde{\zeta}_\ell$ in the Lagrangian and in the radiative multipoles, and are converted into phasing by adiabatic energy balance between the binding energy and the combined scalar-plus-tensor flux. The final object is the stationary-phase Fourier phase of Eq. (83), with the inspiral split into dipole-driven and quadrupole-driven frequency domains; the quadrupole-driven version is the one relevant for current ground-based gravitational-wave detectors.

What would settle it

A concrete check is to run a numerical-relativity simulation of an equal-mass, equal-charge scalarized neutron-star binary, where $S_-=0$ so Eq. (83) reduces to the $c_3 x^3$ and $c_5 x^5$ terms: if the extracted tidal phase has the opposite sign or the wrong frequency scaling, the sign structure collapses. A second check is to recompute the phase after adding the Sec. 2.2 dynamical-friction force; if the 0.1-rad tidal difference shifts by a comparable amount, the phase model is incomplete.

Watch

Extended reading notes

Core claim

The paper's central claim is that the tidal part of the Fourier-domain gravitational-wave phase in the quadrupolar-driven inspiral regime is $$\psi_{\rm tid} = \frac{3}{128\eta $x^{{5/2}}$}\left[c_2 S_- $x^{2}$ + c_3 $x^{3}$ + c_4 S_-^2\left(\log x - \frac{2}{3}\right)$x^{4}$ + \left(\frac{39}{2\$alpha^{5}$\$xi^{2}$}\tilde{\Lambda} + c_5\right)$x^{5}$\right],$$ where $x=(G\alpha M\omega)^{2/3}$, $S_-=(q_A-q_B)/(2\sqrt{\alpha})$, $\alpha=1+q_A q_B$, $\xi=1+S_+^2\alpha/6$, and $\tilde{\Lambda}$ is the mass-weighted combination of quadrupolar tensor deformabilities with the same functional form as in GR. The coefficients $c_2$ and $c_4$ vanish for equal-mass, equal-charge binaries; $c_3$ involves scalar dipolar Love numbers; $c_5$ and $\tilde{\Lambda}$ involve quadrupolar tensor, scalar, and mixed Love numbers; and in the general-relativistic limit only the $\tilde{\Lambda}$ term survives. Because the scalar and scalar-tensor contributions enter with opposite sign from the tensor ones, the authors conclude that the net tidal phase in scalar-tensor gravity is generally smaller in magnitude than in GR, while still depending on all three types of Love numbers.

Load-bearing premise

The model assumes that the inspiral's orbital decay is driven only by scalar and tensor radiation reaction, with no comparable environmental force; the paper's own estimate in Sec. 2.2 finds that dynamical friction from the scalar cloud around each star is about one tenth of the gravitational-wave flux near 200 Hz, which is the same order of magnitude as the tidal phase differences it predicts.

Editorial extensions

If this is right

  • Inspiral templates in scalar-tensor gravity must include all three Love numbers, because the scalar and mixed tidal terms scale as $x^2$, $x^3$, $x^4$, and $x^5$ and cannot be absorbed into a single GR-like deformability.
  • Equal-mass, equal-charge binaries have $S_-=0$, which switches off the dipolar tidal terms $c_2S_-x^2$ and $c_4S_-^2x^4$, so systems with maximum scalar-charge asymmetry are the ones that maximize those terms.
  • For the surveyed equations of state and couplings, the dipole-driven regime ends below about 3 Hz, so ground-based detectors should be modelled with the quadrupole-driven phase formula (83).
  • The scalar-tensor Love number $\lambda_{ST}$ contributes at quadrupolar order at a level comparable to or larger than the pure scalar Love number, so it cannot be neglected once scalar tides are included.
  • The smaller net tidal phase in scalar-tensor gravity creates a potential degeneracy with a softer equation of state in GR, but the much larger point-particle phase difference between the theories should resolve it in most cases.

Reading between the lines

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

  • If the sign structure of Eq. (83) holds in more complete calculations, then parameter estimation with GR templates containing a single tidal deformability would map a scalar-tensor signal onto a biased equation-of-state measurement, and the size of that bias is a direct injection-study prediction.
  • Beyond the paper, the same sign logic suggests a null test for modified gravity: a measured tidal phase whose frequency dependence cannot be fit by any single $\tilde{\Lambda}$ value would indicate the presence of the $x^2$, $x^3$, or $x^4$ terms.
  • At deci-Hertz frequencies, where the dipole-driven regime becomes accessible, the tidal terms grow with different frequency powers, so future space-based detectors could separate the scalar and mixed Love numbers cleanly.
  • The paper's own dynamical-friction estimate implies that scalar-cloud back-reaction may shift the phase by an amount comparable to the tidal signal, so the next step is to model that environment and test whether the predicted ST-minus-GR tidal difference survives.
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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

3 major / 4 minor

Summary. The paper computes tidal contributions to the gravitational-wave phase of inspiraling neutron-star binaries in massless scalar-tensor theories, working to first order in a combined post-Newtonian and small-finite-size expansion. The authors derive a skeletonized action with three types of tidal Love numbers, compute scalar and tensor energy fluxes and the associated Fourier-domain phase in both dipole-driven and quadrupole-driven regimes, and provide ready-to-use expressions for the tidal phase coefficients. They then apply these results to Gaussian scalar-tensor couplings with the WFF1, SLy, and H4 equations of state, surveying parameter space and presenting case studies that lead to the claim that the net tidal GW phase signature in scalar-tensor theories is smaller than in GR because contributions from different Love numbers enter with opposite signs and different frequency scalings.

Significance. If correct, this work provides the first systematic analytic treatment of quadrupolar tidal effects in scalar-tensor inspirals, showing that all three Love numbers enter the GW phase with distinct frequency scalings and cannot be absorbed into a single GR-like deformability. The derivation is careful and largely self-contained: the double expansion in PN and tidal parameters is performed explicitly, the SPA phase is built from the energy balance, and the authors verify the GR limit and cross-check their numerical evaluation of beta(phi) against a finite-difference derivative at the 0.37% level. The physical insight that sign cancellations can reduce the net tidal phase, together with the clear parameter-space study, makes the paper a useful methodological contribution for template-building and theory-specific tests beyond GR.

major comments (3)
  1. [Sec. 4.5.3, Eq. (83)] The coefficient of x^5 in the ready-to-use phase formula is printed with a positive sign: (39/(2 α^5 ξ^2)) Λ̃ + c5. This is inconsistent with the derivation in Sec. 4.5.2. In the GR limit (q=0, S_−=0, α=ξ=1, c3=c5=0), using ζ2=2 λ_T, ζ̃2=2 m λ_T, M=2m, η=1/4 in Eq. (82d) gives ρ^{nd,2}_tid = - (39/2) Λ̃, so the x^5 term in Eq. (83) should be negative to reproduce the standard GR tidal phase ψ_tid = -(3/(128 η))(39/2) Λ̃ x^5. The printed plus sign yields a positive GR tidal phase, contradicting the standard result cited in Refs. [42,86] and the negative λ_T contribution shown in the authors' own Fig. 9. Because the sign of this dominant term drives the cancellation argument that leads to the abstract's central claim of smaller net ST tidal effects, Eq. (83) and its restatement in Sec. 6 must be corrected, and the consequences for the discussion of sign structure should be re-examined.
  2. [Sec. 2.2, Eq. (10)] The authors estimate F_DF/F_GW ~ O(10^-1) at 200 Hz for a representative equal-mass SLy system, yet the phase evolution in Sec. 4.5 accounts only for scalar and tensor radiation reaction. The ST-vs-GR tidal phase differences presented in Fig. 8 are at the 0.1 rad level, so an environmental force of the estimated relative size is expected to produce a phase contribution of comparable order. The manuscript appropriately defers environmental effects to future work, but the abstract's claim that net tidal GW imprints in ST gravity are smaller than in GR applies to the modeled two-body system, not to the complete physical system. The authors should add a quantitative estimate of the environmental phase or otherwise explicitly qualify the scope of the central claim.
  3. [Sec. 2.2, Eq. (8)] The scalar tidal expansion is performed under the assumption (L_ϕ/r) ≪ 1 with L_ϕ ~ 6 R_body, but this condition is not well satisfied in the frequency band where the numerical results are presented. For a 1.4+1.4 M_sun SLy system at 200 Hz, the orbital separation is only a few times L_ϕ, and the QD-phase results in Figs. 8 and 9 are shown up to 500 Hz or 1 kHz. The paper should either quantify the error incurred by truncating the scalar-cloud multipole expansion at these separations or restrict the claimed quantitative accuracy to frequencies where the expansion parameter remains small.
minor comments (4)
  1. [Sec. 5.5.3] There are several typographical errors: "dubble" should be "double", "Tabel 1" should be "Table 1", and "induvidual" should be "individual". In Sec. 3 the term "Newtionian" should be "Newtonian".
  2. [Sec. 2.2, Eq. (9)] The phrase "lnΛ denotes the Coulomb algorithm" appears to be a typo; it should read "Coulomb logarithm".
  3. [Table 1] The header of Table 1 is garbled in the current text ("contribution to largest zero sign magnitude for phase coefficients for for NS-NS") and should be reformatted for clarity.
  4. [Appendix C.2, Eq. (D.1)] The dipolar-driven phase formula in Appendix C.2 is labeled Eq. (D.1), which conflicts with the appendix labeling (the formula appears in Appendix C.2). Please renumber or relabel for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (83) is derived from an action whose Love numbers are computed independently (with published code) in the authors' prior work, not fitted to the GW phase here.

full rationale

The central phase formula (83) is obtained from the adiabatic energy-balance integral (71) using tidal contributions to the binding energy (38) and to the scalar and tensor fluxes (63), (67), all derived from the tidal action (17) and the multipole decompositions (51), (57). The Love numbers λ_S, λ_T, λ_ST enter as inputs taken from the authors' earlier paper [45], where they are obtained by solving the linearized scalar-tensor field equations for static, tidally perturbed neutron stars; the present paper states that it extends the publicly available Mathematica code of [45] to compute β, so the prior result is code-reproduced and not a fit to the GW phase being predicted. The only other same-author citation [49] is used for the validity of the dipole-driven regime boundary and is likewise an independent derivation. No quantity in the claimed prediction is defined in terms of the predicted phase, and the GR limit of Eq. (83) is checked against the standard result. The paper itself flags the dynamical-friction environmental effect (Eq. (10)) as a caveat; that is a completeness limitation rather than a circular reduction. A separate apparent sign inconsistency between the printed plus sign in Eq. (83) and the negative ρ^{nd,2}_tid from Eq. (82d) is a correctness concern, not a circularity, because it does not make the derivation tautological.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central claim is a relationship between phase and imported Love numbers, so it inherits the accuracy of those inputs. The theory parameters beta_0 and phi_infinity are chosen, not derived. No new particle or force is invented; the scalar field is part of the ST framework. The main uncharged assumption is that the phase evolution is set by radiation reaction alone, while the authors' own dynamical-friction estimate suggests otherwise.

free parameters (2)
  • beta_0 (Gaussian coupling parameter) = -4.5 and -6 (case-study choices)
    Chosen by hand to produce scalarized neutron stars; beta_0 = -6 is explicitly stated to be ruled out by binary-pulsar observations, so it is an illustrative rather than realistic parameter.
  • phi_infinity (cosmological scalar field value) = 10^-3
    Set at the Cassini upper bound on alpha_0, not fitted to the GW data; it controls the strength of the scalar coupling.
assumptions (8)
  • domain assumption The Damour-Esposito-Farese coupling A(phi)=exp(beta_0 phi^2/2) with massless scalar field describes the relevant ST theories.
    Sec. 2.1 and Sec. 5.1.1; the paper specializes to this coupling for case studies while claiming broader applicability.
  • domain assumption The two-body motion is quasi-circular and adiabatic, with energy balance E-dot = -F and the stationary-phase approximation for the Fourier phase.
    Sec. 4.5, Eqs. (68)-(71); all phase results depend on this.
  • domain assumption Finite-size effects are captured by the skeletonized action (17) with static, even-parity tidal multipoles to linear order; only l = 1 and l = 2 are retained in the fluxes.
    Sec. 2.4 and Sec. 4.4; the central phase formula (83) includes only these multipoles.
  • domain assumption Love numbers lambda_S, lambda_T, lambda_ST and scalar charges q, beta from the companion paper [45] are correct and applicable.
    Sec. 5.1.2 imports them; the phase coefficients are written in terms of these inputs.
  • domain assumption The scalar cloud energy is concentrated within L_phi ~ 6 R_body, so R/r and L_phi/r are small expansion parameters.
    Sec. 2.2 and Fig. 2; the numerical sweep finds 90 percent of ADM energy within about 10 R_body.
  • ad hoc to paper Dynamical friction from the scalar environment is omitted from the phase evolution, despite an estimate F_DF/F_GW ~ O(10^-1) at 200 Hz.
    Sec. 2.2, Eq. (10) and the following paragraph; the authors flag this as future work, so the phase model is incomplete at the percent-to-ten-percent level in that regime.
  • ad hoc to paper The PN/tidal approximation is valid only before merger; waveforms are cut at benchmark frequencies of 500 Hz or 1 kHz.
    Sec. 5.3 and Sec. 5.5.2; this is a stated limitation.
  • standard math Standard identities for symmetric trace-free tensors and Faà di Bruno's formula are used without proof.
    Appendix A and Eq. (50); these are background mathematical tools.

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Cite this review

Pith. "Pith review of Tidal effects in gravitational waves from neutron stars in scalar-tensor theories of gravity." pith.science (2026). https://pith.science/paper/OP6U5BSY

@misc{pith2026241206620,
  author       = {Pith},
  title        = {Pith review of: Tidal effects in gravitational waves from neutron stars in scalar-tensor theories of gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OP6U5BSY}},
  note         = {Machine review of arXiv:2412.06620}
}
read the original abstract

We compute tidal signatures in the gravitational waves (GWs) from neutron star binary inspirals in scalar-tensor gravity, where the dominant adiabatic even-parity tidal interactions involve three types of Love numbers that depend on the matter equation of state and parameters of the gravitational theory. We calculate the modes of the GW amplitudes and the phase evolution in the time and frequency domain, working up to first order in the post-Newtonian and small finite-size approximations. We also perform several case studies to quantify the dipolar and quadrupolar tidal effects and their parameter dependencies specialized to Gaussian couplings. We show that various tidal contributions enter with different signs and scalings with frequency, which generally leads to smaller net tidal GW imprints than for the same binary system in General Relativity.

Figures

Figures reproduced from arXiv: 2412.06620 by the authors.

Figure 1
Figure 1. Schematic illustration of the binary systems of NS A and B and their re [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Ratio of the ADM energy as function of the radial distance over the to [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Charge-mass curves (top row), and tidal deformabilties (bottom row) in [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Results for the parameter β characterizing the variation of the scalar charge with the field at its cosmological value as a function of mass. The results are com￾puted in the Einstein frame for three EoSs (WFF1, SLy, and H4) and for β0 = −4.5 (left column) and β0 = −6 …
Figure 5
Figure 5. Figure 5: The plus polarization of the time-domain waveform for a BH-NS system [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Estimated transition frequency (72) for a NS-NS system between dipole￾and quadrupole-driven regimes of the inspiral in the MA − MB parameter space for β0 = −4.5 and the SLy EoS. The color scale indicates the value of the estimated transition frequency in Hz. 5.4.1 Esti…
Figure 7
Figure 7. Figure 7: Contour plots of two of the coefficients [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Left panel: Difference between GR and ST Fourier phase as a function of frequency for NS-NS binaries. The blue lines correspond to the point particle (pp) contributions and the red ones to the tidal contributions. The solid, dashed and dotted lines indicate the differe…
Figure 9
Figure 9. Figure 9: Different contributions of the tidal deformabilities to the Fourier phase [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]
Figure 10
Figure 10. Figure 10: Contour plots of the ci coefficients defined in (83), in the MA − MB pa￾rameter space for β0 = −4.5 and the SLy EoS for a BH-NS system (top) and NS-NS system (middle and bottom). The plots for the other equations of state are qualita￾tively similar. 48 [PITH_FULL_IMA…
Figure 11
Figure 11. Figure 11: Different contributions of the tidal deformabilities to the Fourier phase [PITH_FULL_IMAGE:figures/full_fig_p049_11.png]

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Forward citations

Cited by 3 Pith papers

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

  1. Dynamical Love Numbers for Black Holes and Beyond from Shell Effective Field Theory

    hep-th 2025-12 accept novelty 8.0 of 10

    A shell-based EFT computes scalar Love numbers for Schwarzschild black holes through O(G^9) and conjectures an all-orders Riemann-zeta structure.

  2. Tidal Love Numbers of Neutron Stars in Horndeski Theories

    gr-qc 2025-01 accept novelty 7.0 of 10

    In scalar-tensor theories, the 1/r^3 term used to extract neutron star tidal Love numbers contains a Love-number-independent contamination, computed here for minimally coupled and DEF scalar fields.

  3. Tidal deformation and strain accumulation of solid compact stars

    astro-ph.HE 2026-07 conditional novelty 5.5 of 10

    Solid strangeon stars of 1.4 Msun differ by ~40% in tidal deformability from fluid counterparts and release up to 10^46 erg via central-peaking strain fracture at hundreds of Hz.

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