Positivity in the effective field theory of cosmological perturbations

Gen Ye; Yun-Song Piao

REVIEW 3 major objections 5 minor 2 cited by

Positivity in the effective field theory of cosmological perturbations

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that a $c_T=1$ beyond-Horndeski EFT in an expanding universe obeys positivity bounds with leading $O(H^2/\Lambda^2)$ corrections, which can be stronger or weaker than their flat-space counterparts.

desk verdict This paper has a useful EFT derivation but its advertised H^2/Λ^2 positivity bounds rest on an unproven and likely subleading amplitude decomposition. read the letter →

arxiv 1908.08644 v2 pith:NCPEC225 submitted 2019-08-23 hep-th astro-ph.COgr-qchep-ph

Gen Ye , Yun-Song Piao This is my paper · ORCID

classification hep-thastro-ph.COgr-qchep-ph

keywords positivityboundsEFTofcosmologicalperturbationsbeyond-Horndeskislow-rollinflationGoldstonescatteringunitarityandcausalityc_T=1corrections

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 sets out to extend positivity bounds — the inequalities that a low-energy effective theory must satisfy if it is to come from a unitary, causal, local ultraviolet completion — from flat spacetime to the effective field theory of cosmological perturbations. Using a shift-symmetric beyond-Horndeski theory with gravitational-wave speed $c_T=1$ as a concrete example, it derives two coefficient inequalities, equations (22) and (23), that incorporate the leading cosmological corrections of order $H^2/\Lambda^2$. Because these corrections can enter with either sign, the cosmological bounds can be stronger or weaker than the corresponding Minkowski bounds. The application to slow-roll inflation shows that potential-driven inflation suppresses the beyond-Horndeski coupling $B_X$, effectively pushing the theory back to general relativity, while kinetically driven inflation can keep it alive. If valid, the bounds give model-independent consistency conditions on inflation models with beyond-Horndeski operators.

What carries the argument

The load-bearing mechanism is the assumed amplitude decomposition (20), $A = A_{\min}\,\delta^{(4)}(\Sigma p) + A_{\cos}\,\rho(\Delta E)\,\delta^{(3)}(\Sigma p)$, which isolates a Minkowski-like part that obeys ordinary energy-momentum conservation and a cosmological remainder of order $H^2/M^2$ whose effect is argued to be subdominant at zero energy mismatch. Positivity is then imported from the standard Cauchy-integral argument of flat-space dispersion relations: analyticity, crossing symmetry, the Froissart-Martin bound, and the optical theorem yield $A''(s\to 0)\ge 0$ and positivity of $t$-derivatives (equations (14) and (15)). Applied to the tree amplitude (21) of the Goldstone Lagrangian (11), these inequalities become the coefficient bounds (22) and (23). A second ingredient is the Goldstone action itself: the decoupling-limit Lagrangian (11) with cubic coefficients $\alpha_i$ and quartic coefficients $\beta_i$ encodes the beyond-Horndeski couplings $B(X)$, $G_2(X)$, $G_3(X)$ in specific combinations, and the $H^2/\Lambda^2$ corrections in those coefficients are what turn the flat-space bound into a cosmological one.

What would settle it

Compute the exact tree-level $2\to 2$ amplitude with the Hankel mode functions (16) without imposing the decomposition (20): if the cosmological term $\rho(\Delta E=0)$ contributes at the same order as the Minkowski delta function, the inequalities (22) and (23) do not follow.

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Extended reading notes

Core claim

The central claim is that, for a $c_T=1$ shift-symmetric beyond-Horndeski EFT around a slowly varying FRW background, the tree-level $2\to 2$ Goldstone amplitude (21) satisfies the positivity inequalities (22) and (23), which are the standard flat-space bounds corrected at leading order by $O(H^2/\Lambda^2)$. These inequalities constrain $B_X$, $G_{2XX}$, $G_{3X}$, and higher coefficient derivatives; in the slow-roll limit they force $B_X$ to vanish up to slow-roll-suppressed corrections, implying the potential-driven de Sitter limit of the theory reduces to general relativity. The authors deliberately separate the amplitude into an explicitly Minkowski part, which conserves energy and momentum, and a cosmological correction of order $H^2/M^2$, which they argue can be neglected in the positivity argument because the delta-function peak dominates. They state clearly that this decomposition (20) is assumed, not derived, and that the bounds are trustworthy only if the assumptions of section III B hold. The paper also shows how the $H^2/\Lambda^2$ corrections to the bounds can be either positive or negative, so the cosmological positivity windows can be wider or narrower than flat space.

Load-bearing premise

The load-bearing premise, stated by the authors rather than derived, is that the amplitude separates into a Minkowski-like part conserving energy and momentum and a cosmological correction that can be ignored in the positivity argument, with Minkowski-like asymptotic states and flat-space external legs assumed throughout.

Editorial extensions

If this is right

For slow-roll inflation driven by a potential, the corrected bounds imply $B_X$ must vanish at leading order, so the $c_T=1$ beyond-Horndeski EFT effectively reduces to general relativity.
In kinetically driven inflation, where the slow-roll parameter $\epsilon$ need not be small, the same bounds leave room for nonzero $B_X$ and hence for genuine beyond-Horndeski dynamics.
The bounds can be stronger than their flat-space counterparts in some regions of the parameter space and weaker in others, so future data or theory that fixes the coefficients could indicate which case nature realizes.
Because the leading correction is quadratic in $H/\Lambda$, the bounds do not distinguish between an expanding and a contracting universe at this order.
The $H^2/\Lambda^2$ corrections carry coefficients of order $10^2$, so even $H/\Lambda\sim 0.1$ can substantially modify the allowed parameter space compared with the Minkowski limit.

Reading between the lines

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

One extension left implicit is that the same decomposition and positivity argument should carry over to other single-field EFTs of cosmological perturbations, yielding inequality constraints on whatever derivative couplings appear once the Goldstone action is constructed.
The bounds may eventually arbitrate between nonsingular bounce models built from beyond-Horndeski operators: the paper notes such models evade the bounds only because $\varphi$ dependence and $\ddot{\varphi}$ violate its assumptions, so a version of the argument that keeps those terms could decide whether these models admit unitary, causal, local UV completions.
A concrete testable prediction of the assumed decomposition is that the subleading $H^2/M^2$ correction must be peaked at zero energy mismatch; a more complete computation of the exact cosmological propagator could confirm or falsify this, which would sharpen or overturn the inequalities.
Combined with future cosmological parameter measurements, bounds like (22) and (23) could be turned around to map out which regions of scalar-tensor theory space are UV-completable, effectively using data as a probe of the ultraviolet completion.

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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

You should know two things about arXiv:1908.08644. First, it does something new: it works out the Goldstone Lagrangian for a c_T=1 beyond-Horndeski EFT on an FRW background and writes down positivity inequalities that include H^2/Λ^2 corrections to the Wilson coefficients. Those terms aren't in the earlier Horndeski positivity papers [40,41]. Second, the key move that gets these bounds is an assumed amplitude decomposition, Eq. (20), which the authors openly state they did not derive. That assumption is load-bearing, and the abstract's "we derive such bounds" goes beyond what the body concedes.

What's good: the algebraic derivation of the Goldstone action from the covariant Lagrangian is careful, and the final coefficients in Appendix C are explicit and reproducible. The application to slow-roll inflation, concluding that B_X is suppressed, is interesting if the bounds hold. The paper is honest about the obstructions in curved spacetime; section III.B is a fair account of why a robust positivity bound is hard.

The soft spot: the central inequalities (22) and (23) are only as solid as Eq. (20). The authors note that A_cos is O(H^2/M^2) and then drop it because δ(0) dominates ρ(0). But in a regulated finite-volume treatment δ(0) is a finite volume factor, and the analytic structure of ρ is unknown. More importantly, the retained H^2/Λ^2 corrections are parametrically smaller than the neglected H^2/M^2 piece, since M^2 ≪ Λ^2. So the paper's "cosmological correction of order H^2/Λ^2" is not the leading cosmological correction to the amplitude; the uncomputed one is larger. Without a derivation of (20) or a controlled treatment of A_cos, the bounds don't follow from unitarity, causality, and locality. This is not a minor caveat; it directly affects the headline result.

That said, the paper deserves a serious referee. The underlying question—whether positivity bounds can be adapted to cosmological backgrounds—is important, and the Goldstone Lagrangian calculation is useful in its own right. A referee should push the authors to either justify (20) or present the results as conditional on an explicit assumption, and to correct the abstract. If that revision happens, the bounds could be a useful consistency check for beyond-Horndeski model builders. As it stands, I'd be cautious about citing the bounds as derived.

Referee Report

3 major / 5 minor

Summary. This paper aims to extend positivity bounds from flat-space EFTs to the EFT of cosmological perturbations. The authors consider a shift-symmetric, c_T=1 beyond-Horndeski theory on an FRW background, derive the Goldstone Lagrangian (11) and the tree-level 2→2 amplitude (21), and then propose a decomposition of the cosmological amplitude into a Minkowski-like piece A_min and a cosmological correction A_cos (Eq. (20)). Applying standard Minkowski dispersion relations to A_min yields the inequalities (22) and (23), which contain O(H^2/Λ^2) corrections. The paper discusses applications to slow-roll inflation and the GR/Galileon limit, and concludes that the cosmological bounds can be either stronger or weaker than their flat-space counterparts. Crucially, the authors explicitly state that Eq. (20) is assumed, not derived, and that the positivity argument is applied only to A_min.

Significance. The goal of obtaining positivity constraints on the EFT of cosmological perturbations is important and timely, and the explicit computation of the Goldstone Lagrangian and tree-level amplitude for a c_T=1 beyond-Horndeski theory is a useful technical contribution. The paper is also unusually candid about the assumptions that underpin its central result. However, as it stands, the claimed derivation of the bounds is conditional on an unproven amplitude decomposition and on neglecting a cosmological term that is parametrically larger than the retained corrections. If the assumptions (18)–(20) were justified, the bounds could provide nontrivial constraints on beyond-Horndeski models; but the paper does not supply that justification, so the central claim is not established in its current form.

major comments (3)

[Section III-B, Eq. (20)] The decomposition iA = iA_min δ^(4)(Σp) + iA_cos ρ(ΔE) δ^(3)(Σp) is explicitly assumed rather than derived, as the authors state. The positivity argument is then applied only to A_min, and the inequalities (22) and (23) are extracted from it. However, the Cauchy integrals in (12)–(13) require analyticity and boundedness of the full amplitude, not just of A_min. If A_cos has branch cuts, poles, or other non-analyticities in the complex s-plane, its contribution to the dispersion integral will not cancel and can alter A''(s) at the values used for the bounds. Therefore, until (20) is justified and the A_cos contribution to (12)–(13) is shown to be negligible, the bounds (22) and (23) do not follow from unitarity, causality, and locality alone.
[Section III-B, around Eq. (20) and Section III-C] The neglect of A_cos is justified by the claim that δ(0) dominates ρ(ΔE = 0). This is not a sufficient argument. A_cos is estimated to be of order H^2/M^2, where M is the scattering energy with M^2 ≪ Λ^2, whereas the cosmological corrections explicitly retained in (22) and (23) are of order H^2/Λ^2. Since H^2/M^2 ≫ H^2/Λ^2, the neglected contribution is parametrically larger than the terms the bounds claim to compute. In a finite-volume regularization, δ(0) is a large volume factor and ρ(ΔE) is a nontrivial distribution whose contribution to the dispersion integrals can affect A''(s) at the same or larger order. The paper does not demonstrate that A_cos is harmless, so the O(H^2/Λ^2) coefficients in (22) and (23) are not protected from O(H^2/M^2) corrections.
[Abstract and Section IV Conclusion] The abstract states that the paper derives cosmological positivity bounds, and the conclusion claims that 'the leading cosmological correction to positivity bounds indeed comes at H^2/Λ^2'. The body of the paper, however, concedes that the key decomposition (20) is assumed, and the H^2/Λ^2 corrections are computed in the Lagrangian coefficients and in A_min, not in the full amplitude or in the positivity proof itself. The claims in the abstract and conclusion overstate what has been demonstrated. If the assumptions are to be retained, the results should be presented as conditional bounds—valid provided (18)–(20) hold and provided the A_cos contributions to the dispersion integrals are negligible—and the abstract and conclusion should be revised accordingly.

minor comments (5)

[Section III-D] Typo: 'cosmolgical' should read 'cosmological'.
[Section III-A] Typo: 'compatifying' should read 'compactifying'.
[Appendix A] Typo: 'guage' should read 'gauge'.
[Eq. (19)] The notation in the integrand, in particular the factor '1/2' and the treatment of the branch of the gamma function, is not fully defined; please specify the assumptions on the integration contour and the principal branch used.
[Section II, after Eq. (11)] The statement that the coefficients of the final Lagrangian do not contain B(X) or Q(X), followed by a parenthetical reference to Eq. (B12) which contains B(X), is confusing and should be clarified or rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the positivity bounds are algebraic constraints on Lagrangian coefficients, and the one assumed decomposition is an explicitly flagged validity premise, not an input-output identity.

full rationale

The central derivation takes the covariant shift-symmetric c_T=1 beyond-Horndeski Lagrangian (1), constructs the Goldstone EFT (11), computes the tree-level 2-to-2 amplitude (21), and applies the standard Minkowski positivity integrals (13)-(15) to obtain inequalities (22) and (23). The coefficients in these inequalities are functions of G2(X), G3(X), B(X) and their derivatives, together with H^2/Lambda^2 terms inherited from the background quantities epsilon and epsilon_H. No parameter is fitted to the bound, and no quantity in the inequalities is defined in terms of the bound itself. The paper explicitly identifies the only non-derivation step: 'We have to emphasize that we did not derive (20), but rather assumed it based on observations in this subsection.' That assumed decomposition A = A_min delta^(4)(sum p) + A_cos rho(Delta E) delta^(3)(sum p) is a premise about how cosmological corrections enter the amplitude, not the content of (22)-(23); the inequalities would fail to follow if the premise fails, but that is a validity and robustness concern, not circularity. The objection that the neglected A_cos ~ H^2/M^2 term may be larger than the retained H^2/Lambda^2 corrections is substantive, but it does not reduce the derivation to its inputs. The self-citations [15,17-20] appear only in the concluding discussion of bouncing models, where the authors state the bounds cannot be applied because 'such models display obvious phi-dependence and nonnegligible phi-double-dot around the bounce point, which invalidates the assumptions we used to derive the bounds here'; this is explicitly non-load-bearing. No uniqueness theorem is imported from the authors' own work, and no known empirical result is renamed as a derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data in this paper. The EFT coefficients B_X, G2XX, G3X and higher derivatives are the subjects of the bounds; H and Λ are physical scales, and ϵ, ϵH describe the background. The choice G2X = -1/2 is a normalization, not a fitted parameter. No new particles, forces, dimensions or conserved quantities are introduced; the Goldstone mode π is the standard EFT degree of freedom.

assumptions (6)

domain assumption The UV completion is unitary, causal, local and Lorentz-invariant, allowing analytic dispersion relations and the Froissart-Martin bound.
This is the standard positivity-bound input inherited from [23, 27, 36], invoked in Section IIIA.
domain assumption The background varies slowly enough that H^2/Λ^2 and ˙H/Λ^2 corrected Lagrangian coefficients can be treated as constants during scattering.
Stated in Section I and used throughout Section II to integrate out time dependence; requires |\ddot φ/(H \dot φ)| << 1.
ad hoc to paper The asymptotic state is Minkowski-like: |Ω> ∝ exp(-iT∫ HI dτ)|0>, neglecting particle production.
Equation (18) in Section IIIB; explicitly stated as an assumption to make crossing symmetry and positivity applicable.
ad hoc to paper The 2 to 2 amplitude decomposes as iA = iA_min δ^(4)(Σp) + iA_cos ρ(ΔE) δ^(3)(Σp), with A_cos of order H^2/M^2, and positivity is applied only to A_min.
Equation (20), Section IIIB; the paper states this is assumed, not derived. This is the load-bearing step for the cosmological bounds.
ad hoc to paper External legs are sub-Hubble and can be contracted using flat-space solutions e^{-ip·x}.
Section IIIB: 'we further assume contraction of all external legs yields the flat space solution.'
domain assumption Truncation to leading order in H/Λ is valid and the cubic and quartic interactions remain perturbative.
Section II and Appendix B; requires H/Λ < O(1) and S(2) dominance.

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Pith. "Pith review of Positivity in the effective field theory of cosmological perturbations." pith.science (2026). https://pith.science/paper/NCPEC225

@misc{pith2026190808644,
  author       = {Pith},
  title        = {Pith review of: Positivity in the effective field theory of cosmological perturbations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NCPEC225}},
  note         = {Machine review of arXiv:1908.08644}
}

abstract

Requiring the existence of a unitary, causal and local UV-completion places a set of positivity bounds on the corresponding effective field theories (EFTs). We discuss the obstructions and possibility in applying the positivity bound to cosmology, in particular the EFT of cosmological perturbations. Taking a $c_T=1$ beyond-Horndeski EFT as an illustrative example, we derive such bounds, which incorporate the cosmological correction of order $H^2/\Lambda^2$, $\Lambda$ being the cutoff scale. The derived bounds are applied to slow-roll inflation with beyond Horndeski operators. It is found that the cosmological positivity bounds may be either stronger or weaker than their flat space counterpart.

Figures

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**Figure 2.** FIG. 2: The constrained phase space of [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

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