Cosmological scalar perturbations in Horndeski-like gravity

Dani de Boe; Fabiano F. Santos; Jackson Levi Said

REVIEW 4 major objections 5 minor 58 references

Cosmological scalar perturbations in Horndeski-like gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Stability and entropy bounds narrow the viable parameter space of this Horndeski-like dark energy model.

desk verdict A checkable extension of the tensor-sector analysis to scalars, with entropy bounds as a useful new tool, but the gamma <= 0 derivation needs rewriting and a few typos must be fixed. read the letter →

arxiv 2501.04524 v2 pith:Z6HVZ4ZD submitted 2025-01-08 gr-qc astro-ph.CO

Dani de Boe , Fabiano F. Santos , Jackson Levi Said This is my paper · ORCID

classification gr-qcastro-ph.CO

keywords Horndeskigravityscalarperturbationsentropyboundsdarkenergyfirst-orderformalismgravitationalslipapparenthorizonstabilityconditions

topics Dark Energy

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

The paper studies a two-dilaton scalar-tensor model of dark energy, in which a Horndeski-like Lagrangian is supplemented by a second scalar field. It derives the quadratic action for scalar perturbations and imposes that both kinetic and gradient coefficients stay positive, yielding stability bounds on the model's couplings. It then applies the second law to entropy from particle production and from the apparent horizon, obtaining independent entropy bounds that further restrict the parameter space. The central claim is that these entropy bounds can constrain this modified gravity model without computing positivity bounds on flat spacetime, and that they complement the earlier tensor-sector stability bound.

What carries the argument

The argument is carried by three pieces. First, the quadratic action for scalar perturbations, $S^{(2)}_S = \int dt\,d^3x\,a^3\big(G_S \dot{\zeta}^2 - \frac{F_S}{a^2}(\vec{\nabla}\zeta)^2\big)$, whose positivity conditions $F_S > 0$ and $G_S > 0$ define stability against ghosts and gradient instabilities. Second, the first-order formalism: a superpotential $W(\phi)$ with $\dot{\phi} = -W_\phi$ and $H = W$, chosen as $W(\phi) = \cos^{2/3}(3\phi/2)$ so that background equations reduce to first-order ODEs and all bounds become functions of $W$ and its derivatives. Third, the two entropy functionals: the particle-production rate $\Gamma_\phi$ entering $\dot{S}_{\mathrm{in}} = \Gamma_\phi S_{\mathrm{in}}$, and the apparent-horizon entropy $S = H^{-2}(1 - \xi/4)$ with $\xi = (\alpha + \gamma\Lambda)/\alpha$. Positivity of each rate or of the entropy change translates into the coupling bounds.

What would settle it

Turn on $\delta\chi \neq 0$ in the same background and recompute the quadratic action numerically: if any point inside the claimed allowed region ($\gamma/\alpha \lesssim 1+1.25\alpha$, $\gamma \leq 0$, $3\alpha \geq \gamma\Lambda$) gives $F_S \leq 0$ or $G_S \leq 0$, the stability bound as stated fails. Conversely, finding a point with $\gamma > 0$ that satisfies the full two-field stability and entropy conditions would falsify the paper's exclusion of $\gamma > 0$.

Watch

Extended reading notes

Core claim

For the action (2) with background superpotential $W(\phi) = \cos^{2/3}(3\phi/2)$, the paper claims the scalar sector is stable only if $\gamma/\alpha \lesssim 1 + 1.25\alpha$ with $\alpha \geq 0$, while the tensor sector requires $\gamma/\alpha \lesssim 1.75$. Demanding that the entropy from particle production obey $\dot{S}_{\mathrm{in}} \geq 0$ gives the bound $\gamma \leq 0$, and the apparent-horizon entropy condition gives $3\alpha \geq \gamma\Lambda$. Together these limits constrain the three couplings $(\alpha,\gamma,\Lambda)$ more tightly than the stability conditions alone, and they hold on a Friedmann–Robertson–Walker background rather than in flat spacetime. The paper also proposes the gravitational slip minus one, divided by the apparent-horizon entropy, as a cosmological replacement for the black-hole shear-viscosity-to-entropy ratio.

Load-bearing premise

The load-bearing premise is that the second scalar field $\chi$ is a background field with $\delta\chi = 0$, so the scalar sector carries only one propagating degree of freedom; the paper also neglects matter in computing the particle-production rate, an additional assumption that could shift the entropy bounds if it fails.

Editorial extensions

If this is right

If the paper is right, the allowed region shrinks to $\gamma \leq 0$ and $3\alpha \geq \gamma\Lambda$ together with $\gamma/\alpha \lesssim 1 + 1.25\alpha$; with $\alpha \geq 0$ this forces $\Lambda$ to be positive when $\gamma < 0$.
The apparent-horizon bound involves $\Lambda$ directly, so entropy arguments constrain a parameter that the scalar and tensor stability conditions leave untouched.
For $\gamma \lesssim 0.01$ the gravitational slip approaches the $\Lambda$CDM value, so observable deviations from general relativity in this model require larger $|\gamma|$.
The gravitational slip to entropy ratio $(\tilde{\gamma}-1)/S$ behaves differently from $\eta/S$ and may be a practical transport-like diagnostic in cosmology, where shear viscosity is difficult to define.

Reading between the lines

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

Editorial inference: if $\delta\chi \neq 0$ were turned on, the scalar sector would carry two propagating degrees of freedom; the quadratic action and the stability coefficients would change, potentially opening parts of the parameter space that the present paper excludes.
Editorial inference: the conclusions section writes the apparent-horizon bound as $3\alpha \leq \gamma\Lambda$, but the derivation in Section V and the condition $\dot{S}\geq 0$ give $3\alpha \geq \gamma\Lambda$; the former appears to be a sign typo rather than a competing claim.
Editorial inference: combining the particle-production bound $\gamma \leq 0$ with the positivity-bound result $\gamma = 0$ would eliminate the negative-$\gamma$ region entirely, making the model effectively trivial in $\gamma$; the paper stops short of drawing that conclusion.
Editorial inference: a numerical scan of $F_S, G_S$ over $\phi \in [0.1,0.5]$ for $\delta\chi \neq 0$ would be the natural first test of whether the claimed bounds are stable against the truncation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This is a legitimate step in the authors' program on Horndeski-like gravity. It extends the tensor-sector stability analysis from their earlier paper to scalar perturbations for a two-scalar action, then uses entropy bounds to constrain the coupling space. The first-order formalism with the superpotential W = cos^(2/3)(3φ/2) gives analytic expressions for stability coefficients, entropy, and gravitational slip, which is a clean way to get concrete constraints without numerics. The motivation—entropy bounds on FRW as a complement to flat-space positivity bounds—is sensible. Credit is due: the main derivations are checkable, and the paper is upfront about its assumptions (δχ = 0, matter neglected, late times, QSA unverified).

The soft spots are mostly fixable, but they are real. The biggest is in Section IV. The text says 'Using that ρdot + 3H(ρ+p) = 0' and then presents a nonzero Γφ. As written it's a contradiction. The intended reading is that ρφ, pφ are only the α+V part of the full fluid, so Γφ measures the transfer from the γ sector. That interpretation is plausible, but the paper needs to state it explicitly and justify why the second law applies to this subsystem alone. Right now the γ ≤ 0 bound is not fully supported without that step. The sign flip for the apparent-horizon bound (3α ≥ γΛ in Sec V, 3α ≤ γΛ in Conclusions) is a clear typo, as is the inconsistent scalar stability bound (γ vs γ/α). The slip-to-entropy ratio is an interesting proposal, but the figures use parameters that violate GW170817; the paper acknowledges this and frames it as a future tool, so it's minor.

On balance, I think the central claim—that entropy bounds can constrain (α, γ, Λ) in a way comparable to positivity bounds—is defensible after revision. The contradictions are in exposition, not in the algebra. The paper is incremental, not groundbreaking, but it is honest and useful for those working on modified gravity phenomenology. I would send it to peer review and ask for a careful rewrite of Section IV and a fix of the sign/notation errors.

Best.

Referee Report

4 major / 5 minor

Summary. The manuscript studies cosmological scalar perturbations in a Horndeski-like gravity with two scalar fields, using the ADM formalism and a first-order superpotential description. After assuming that the additional scalar χ has no perturbation (δχ = 0) and choosing the superpotential W(φ) = cos^(2/3)(3φ/2), it computes the scalar stability conditions F_S > 0 and G_S > 0, compares them with tensor stability, and derives entropy bounds from particle production and from the apparent horizon. It also proposes the gravitational slip (minus one) to apparent-horizon entropy ratio as a diagnostic analogous to η/S for black holes. The advertised parameter-space constraints are γ/α ≲ 1 + 1.25α from the scalar sector, γ/α ≲ 1.75 from the tensor sector, γ ≤ 0 from particle-production entropy, and 3α ≥ γΛ from apparent-horizon entropy. The paper is exploratory in style and repeatedly defers important checks, such as the validity of the quasi-static approximation and the inclusion of a non-trivial δχ, to future work.

Significance. If the central claims held, the paper would provide a tractable example in which stability conditions and thermodynamic entropy considerations narrow the allowed couplings of a Horndeski-like theory without invoking positivity bounds, and it would introduce a new diagnostic ratio (γ_slip − 1)/S. The analytical superpotential solution and the explicit formulas for F_S, G_S, and the entropy quantities are a useful starting point, and the comparison between scalar and tensor stability bounds is a worthwhile exercise. However, the particle-production section contains an internal contradiction that directly undermines the advertised γ ≤ 0 bound, and the apparent-horizon bound is stated with opposite inequalities in the derivation and the conclusions. Because these two entropy bounds are central to the paper's parameter-space claims, the results are not currently established as stated.

major comments (4)

[Section IV, Eqs. (26)-(27)] The text states 'Using that ρ̇ + 3H(ρ + p) = 0 it easily follows that the particle production rate is: Γ_φ = ...' and then gives a nonzero Γ_φ in Eq. (27). If the full dark-energy fluid obeys the continuity equation, then Γ_φ as defined in Eq. (26) is identically zero, and Eq. (28) gives dS_in/dt = 0, so the bound Γ_φ ≥ 0 and hence γ ≤ 0 do not follow. The apparent resolution is that ρ_φ and p_φ in Eq. (26) are a truncated subset of the full effective fluid of Eq. (17), omitting the γ-dependent terms; but then Γ_φ measures the non-conservation of that artificial split, and the second-law bound dS_in/dt ≥ 0 requires an open-system justification that the paper does not provide. Please derive Γ_φ from the actual scalar-field equation with an explicit interaction source, or clearly state that the result is a conditional bound under a specific effective-fluid split and justify why that split is physical.
[Section V vs. Section VI] The apparent-horizon entropy bound is derived in Section V as 3α ≥ γΛ (the text near Eq. (30) and the following paragraph), but the Conclusions state 'The result from the computation of the entropy of the apparent horizon is the bound 3α ≤ γΛ.' These are opposite inequalities. Since this is one of the two advertised entropy constraints, the sign error must be corrected and the consistent inequality must be used throughout the paper, including in the abstract and conclusions.
[Sections I and II] The scalar-sector results, including the quadratic action (5), the coefficients F_S and G_S in Eqs. (10)-(11), and the derived bound γ/α ≲ 1 + 1.25α, all rely on the assumption δχ = 0 stated at the end of Section I and repeated in Section II. The paper gives no physical justification for this truncation beyond simplicity. If δχ is switched on, the quadratic action acquires additional terms and the stability conditions change; the conclusions should be explicitly framed as conditional on this assumption, or the regime in which it applies should be specified and justified.
[Section II] The central scalar stability claim γ/α ≲ 1 + 1.25α is announced without showing the algebraic steps that reduce the conditions F_S > 0 and G_S > 0 to this inequality. Please include the explicit inequalities in terms of the superpotential and the interval φ ∈ [0.1, 0.5], or provide a derivation in a supplementary file, so that the reader can verify the bound and its dependence on the assumed field range.

minor comments (5)

[Eq. (18)] The hypergeometric function is written as '2F1'; the standard notation is {}_2F_1.
[Section II] There is a typo in the sentence 'contrain the allowed value' (should be 'constrain'), and later 'the the no-ghost' contains a duplicated article.
[Eqs. (21)-(23)] The symbol γ is used both for the coupling constant and, via ˜γ, for the gravitational slip; although the paper defines ˜γ, the notation remains confusing in equations such as (21)-(23). Consider renaming the slip to a symbol such as η_slip to avoid ambiguity.
[Eq. (14)] The two equations in (14) are presented as a system, but no derivation is shown for either; please clarify how they follow from the Friedmann equations and the scalar-field equations.
[Section IV] The statement that the bound γ ≤ 0 is 'consistent with the tensor and scalar stability constraint' is misleading: the stability bounds allow positive γ, so the entropy bound is stricter rather than merely consistent. Please rephrase.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central constraints are derived from the stated action, a solved superpotential, and external thermodynamic postulates; the particle-production bound has an internal consistency flaw but is not a fitted input renamed as a prediction.

full rationale

The paper's derivation chain is largely self-contained rather than circular. The scalar and tensor stability bounds follow algebraically from the quadratic action (5) and the analytic superpotential W(phi)=cos^{2/3}(3 phi/2), which the paper solves from the first-order equation 2 W W_phi_phi + W_phi^2 + 3 W^2 = 0 rather than importing as an unexamined ansatz. The apparent-horizon entropy bound 3 alpha >= gamma Lambda follows by differentiating the imported entropy formula S = H^{-2}(1 - xi/4); that formula is an input from prior work, several items of which are self-citations, but the bound is a direct algebraic consequence and is not identical to the formula itself. The positivity bounds cited from the authors' earlier paper [16] are used only for comparison, not to derive the new constraints. The most serious issue is in Section IV: Eq. (26) defines Gamma_phi by the non-conservation of (rho_phi, p_phi), while the text invokes the full-fluid identity rho_dot + 3H(rho+p)=0 and obtains the nonzero Eq. (27). Taken literally, the cited continuity equation would give Gamma_phi=0, and the nonzero result comes from applying the continuity logic to a truncated subsystem that omits the gamma-dependent terms. This undermines the gamma <= 0 particle-production bound as a rigorous derivation, and the paper itself flags caveats about neglecting matter and placing bounds on total entropy. However, this is an internal consistency/correctness problem rather than a circularity: no parameter is fitted to data, no prediction is equivalent to its input by construction, and the central stability constraints do not reduce to a self-citation. The self-citations provide background formalism and a superpotential solution, but the paper's main inequalities are obtained by substituting that solution into independently stated stability and entropy conditions.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central constraints are derived analytically from the model action plus the chosen superpotential. The main free choices are the integration constants, the field range, and the entropy normalization. No new fields, particles, or forces are introduced. The analysis leans on prior results by the same authors (superpotential solution [55], holographic entropy [26,41-45]), but those serve as inputs rather than as the constrained outputs.

free parameters (3)

c1, c2 (superpotential integration constants) = c1 = 1, c2 = 0
Chosen by hand to select the late-time accelerating branch; all derived bounds depend on this choice.
field interval phi in [0.1, 0.5] = 0.1 <= phi <= 0.5
Restricted to the range where the solution gives w < -1/3 (cosmic acceleration); bounds reported only on this interval.
Sin(phi0) initial particle-production entropy = positive but unspecified
Normalization of Sin; the combined entropy bound Eq. (32) depends on its value.

assumptions (8)

standard math Stability coefficients F_S, G_S, Sigma, Theta from Kobayashi et al. [27] apply to action (2)
The scalar and tensor no-ghost and no-gradient conditions are taken from the generalized G-inflation formalism.
domain assumption Only one propagating scalar: delta chi = 0, chi treated as a background field
Stated in Section II. The theory has two scalars; the stability and entropy results hold only for this frozen-field limit.
domain assumption Late-time background with matter and radiation neglected
Stated in Section II; the Friedmann equations (12) and superpotential equations (14) omit matter and radiation.
domain assumption First-order formalism: phi_dot = -W_phi, H = W, with W = W(phi) and chi = 0
The paper restricts to background solutions of this form, following [55]; this selects a narrow class of histories.
domain assumption Entropy of apparent horizon S = H^{-2}(1 - xi/4) with xi = (alpha + gamma Lambda)/alpha
Imported from the authors' prior holographic works [26,41-45]; the paper uses it as the bulk entropy.
domain assumption Particle-production thermodynamics dSin/dt = Gamma Sin from [7,46]
Assumes constant specific entropy and no interaction with other species; the paper notes the matter action is neglected in Gamma_phi.
domain assumption Quasi-static approximation (QSA) for gravitational slip formulas gamma_0 and gamma_infinity
Section III notes the QSA is model-dependent and is not verified in this work.
domain assumption Positivity bounds from [16] transported from Minkowski to FRW background
Used only for comparison; the paper itself cautions the transportation may not be valid.

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Cite this review

Pith. "Pith review of Cosmological scalar perturbations in Horndeski-like gravity." pith.science (2026). https://pith.science/paper/Z6HVZ4ZD

@misc{pith2026250104524,
  author       = {Pith},
  title        = {Pith review of: Cosmological scalar perturbations in Horndeski-like gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z6HVZ4ZD}},
  note         = {Machine review of arXiv:2501.04524}
}

abstract

Scalar-tensor theories are promising dark energy models. A promising scalar-tensor theory, called Horndeski-like gravity, is coming from the application of the Horndeski gravity in string theory and cosmology that takes into account two dilaton fields. In this work we study the stability of the scalar sector of this theory and compare it with that coming from the previously studied tensor sector. With the first-order formalism we investigate the allowed background solutions. Focusing on the background solution with a single scalar field, the entropy coming from particle production $S_{in}$ and that of the apparent horizon $S$ will be studied, which translates into \textit{entropy bounds}. These entropy bounds are compared with the stability of the scalar and tensor sector as well. The gravitational slip (minus one) to entropy ratio is also considered as a possible replacement for the usual shear viscosity to entropy ratio for black holes.

Figures

Figures reproduced from arXiv: 2501.04524 by the authors.

**Figure 1.** On the y-axis is related to the scalar field ϕ and on the x-axis is the time t in some units (here we have chosen c˜1 = −4 such that ϕ(0) = 0). The plot assumes the superpotential W(ϕ) = cos2/3 (3ϕ/2). Here the two branches of the solution are illustrated. of the Universe but rather some phase of the cosmic history in which cosmic acceleration occurs. Note also that a = exp − R W Wϕ dϕ = exp(cot(3ϕ/2)) for the su… view at source ↗

**Figure 2.** (γ0 − 1)/S versus ϕ for different values of γ and fixed α = Λ = 1. The values of γ plotted are: γ = 0.1 (dotted), γ = 0.01 (dashed), γ = 0.001 (dashdot) and γ = 0 (solid). The plot assumes the superpotential W(ϕ) = cos2/3 (3ϕ/2), canonical normalization of the scalar field and the equation for γ0 in the QSA [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** (γ∞ − 1)/S versus ϕ for different values of γ and fixed α = Λ = 1. The values of γ plotted are: γ = 0.1 (dotted), γ = 0.01 (dashed), γ = 0.001 (dashdot) and γ = 0 (solid). The plot assumes the superpotential W(ϕ) = cos2/3 (3ϕ/2), canonical normalization of the scalar field and the equation for γ∞ in the QSA. on the Minkowski background, but the usual assumption is to transport them to the FRW background. The trans… view at source ↗

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Reviewed August 10, 2026 · model on record in the stance chip above.

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(31) As commented on in [7] the sum of the entropy Sin and that of the apparent horizon Sh ≡ S, denoted Ssum = Sin + Sh, should be considered as well

we see that this translates into: Gef f= π 2 1 − ξ 4 . (31) As commented on in [7] the sum of the entropy Sin and that of the apparent horizon Sh ≡ S, denoted Ssum = Sin + Sh, should be considered as well. In the previous section, the condition for ˙Sin(t) ≥ 0 was de- rived. Let us consider the impact of the entropy bound ˙S(t) ≥ 0 on the allowed paramete...

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