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Gravity in de Sitter space may make the out-of-time-ordered correlator grow, in apparent violation of the quantum bound on chaos.

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2026-08-02 06:07 UTC pith:MWQ2F6GO

load-bearing objection Technically serious and largely explicit dS OTOC computation, but the headline growth claim rests on a large-diffeomorphism contribution whose gauge-invariant status is not yet established; the authors concede the dressing question is open. the 3 major comments →

arxiv 2607.13137 v1 pith:MWQ2F6GO submitted 2026-07-14 hep-th gr-qc

Out-of-time-ordered Correlators in de Sitter Revisited

Alexey Milekhin , Vladimir Narovlansky , Jiuci Xu This is my paper
classification hep-th gr-qc PACS 04.60.-m04.62.+v
keywords de SitterOTOCeikonal approximationlarge diffeomorphismsquantum bound on chaosLyapunov exponentgraviton propagatorIR divergence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that the out-of-time-ordered correlator (OTOC) of scalar fields in de Sitter space, computed in the eikonal approximation to gravity, can grow at early times instead of decaying, in apparent violation of the quantum bound on chaos. The growth is traced to large diffeomorphisms in the graviton propagator—pure-gauge modes that do not decay at infinity and produce a 'time advance' effect characteristic of de Sitter shockwaves. At tree level the OTOC grows as 1 + C G_N ℓ^{-d} e^{t/ℓ} with C>0, giving the maximal Lyapunov exponent 2π/β_dS, while in a graviton-mass-regulated double-scaling limit the leading correction grows with twice the maximal exponent. The authors also identify a loop-level infrared divergence in the vector (L=1) sector of the graviton propagator and show that regularizing it forces all odd powers of Newton's constant to vanish in simple operator configurations. If correct, the result implies that perturbative quantum gravity in de Sitter behaves qualitatively differently from AdS black holes and challenges the universality of the chaos bound.

Core claim

Using the eikonal approximation for gravitational scattering of massive scalars around de Sitter space in any dimension, the authors compute the 4-point OTOC of operators placed along an observer's worldline. They find that the perturbative OTOC initially grows: in the unregulated massless calculation, D/(G13G24) = 1 + C G_N ℓ^{-d} e^{t/ℓ} with C>0 (eq. 5.67), i.e., growth with the maximal Lyapunov exponent 2π/β_dS, driven by the L=0 angular mode; in the m_g-regulated double-scaling limit, only even powers of G_N/m_g² survive and the first correction grows as e^{4πt/β_dS} (eqs. 5.28–5.30). The authors emphasize that this growth is mediated by large diffeomorphisms in the graviton propagator—

What carries the argument

The central object is the eikonal phase, the exponent of the resummed ladder diagrams, written as an integral of the graviton propagator contracted with the null momenta of two nearly lightlike scalar trajectories. The argument hinges on the decomposition of the graviton propagator into transverse-traceless, vector, and scalar (longitudinal) parts, and specifically on the pure-diffeomorphism piece ∇µ∇ν∇µ′∇ν′ G^{−−}_{m²=−d} (eq. 4.21), a tachyon-mass regularized scalar propagator whose Z log Z growth yields the L=0 'time advance' contribution; and the L=1 vector zero mode of the transverse sphere, which causes a logarithmic IR divergence that is regularized by a graviton mass m_g. In the regu

Load-bearing premise

The load-bearing premise is that the pure-diffeomorphism part of the graviton propagator contributes physically to a gauge-invariant OTOC; the paper itself notes in §6 that it did not properly discuss the gravitational dressing of the external operators, and if that dressing cancels or renormalizes these large gauge modes—as the L=1 vector sector is canceled by constraints—the growth disappears.

What would settle it

Compute the OTOC using an explicitly gauge-invariant definition with operators dressed to an observer worldline (as the abstract promises but Section 6 admits is not done), and check whether the coefficient C in D/(G13G24) = 1 + C G_N ℓ^{-d} e^{t/ℓ} (eq. 5.67) remains positive; if dressing removes the L=0 diffeomorphism contribution, C would vanish or change sign. Alternatively, compute the same regulated eikonal phase with a different IR regulator (e.g., deforming to a conical defect in dS3) and compare the Lyapunov exponents; if the 4π/β_dS scaling is regulator-dependent, the claim is not ro

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The gravitational OTOC in de Sitter can grow initially for any bulk scalar masses, contradicting the quantum bound on chaos in the regularized 'bound on chaos' configuration.
  • In the mass-regulated double-scaling limit, all odd powers of Newton's constant vanish, so the leading correction grows with twice the maximal Lyapunov exponent, e^{4πt/β_dS}.
  • The eikonal phase in de Sitter is sensitive to large diffeomorphisms and to the choice of integration contour, unlike AdS where only the physical transverse-traceless part contributes.
  • The loop-level IR divergence in the vector sector cannot be removed by dressing external operators; the authors suggest deforming away from empty de Sitter (e.g., a conical defect in dS3) may be needed to resolve it.
  • The initial growth is caused by time advance rather than time delay, consistent with earlier geodesic-approximation results but now derived from a first-principles diagrammatic computation.

Where Pith is reading between the lines

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

  • If a complete gauge-invariant construction of the OTOC with observer-dressed operators cancels the large-diffeomorphism modes (as the L=1 vector mode is canceled by constraints at tree level), the predicted growth—the paper's headline—would disappear; the paper concedes in §6 that it did not properly discuss this dressing.
  • The claimed violation of the bound on chaos may be an artifact of the eikonal approximation's treatment of zero modes rather than a genuine property of quantum gravity in de Sitter; a fully non-perturbative computation (e.g., in a microscopic model such as the doubled SYK model with equal-energy constraint) could test this.
  • The result suggests operators in de Sitter become 'simpler' under time evolution, opposite to the usual operator-growth picture of chaos; if physical, it could indicate that de Sitter holography requires dressing that creates complicated operators.
  • The gauge-dependence of the loop-phase divergence implies the mass-regulator procedure may not be unique; verifying that different regularizations (e.g., conical defect deformation) yield the same Lyapunov exponent would strengthen the claim.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper revisits the computation of four-point out-of-time-order correlators (OTOCs) in de Sitter space in the eikonal approximation. It derives the eikonal phase from resummed graviton exchange (Section 3), identifies an infrared divergence in the vector sector of the graviton propagator (Section 4.3), and computes the OTOC both with and without a graviton-mass regulator (Sections 5.3 and 5.5). The headline results are: (i) at tree level the unregulated OTOC grows as 1 + C G_N ℓ^{-d} e^{t/ℓ} with C>0 in the pole–antipole and chaos-bound configurations (eq. (5.67)), i.e. with the maximal Lyapunov exponent 2π/β_dS; and (ii) in the regulated double-scaling limit only even powers of G_N/m_g^2 survive, giving a leading correction e^{4πt/β_dS} (eqs. (5.28), (5.30)). The authors trace the tree-level growth to an L=0 large-diffeomorphism term in the graviton propagator (eq. (4.21)) and interpret the result as challenging the applicability of the MSS bound on chaos.

Significance. The computation is explicit and largely self-contained: the wavefunctions (5.19), the master formula (5.3), and the analytic continuations leading to (5.24) and (5.67) are spelled out, and the sign of the tree-level growth is traced to the negative eigenvalue of the L=0 transverse Green function (5.62). If the growth survives a fully gauge-invariant treatment, it would be a striking counterexample to the universality of the bound on chaos and would support the idea that large diffeomorphisms can be physically observable in de Sitter. However, the result hinges on an unproven assumption about observer dressing, and the loop-level/regulated claims are not backed by a direct loop calculation or a unitary regulator. Thus the significance is potentially high but currently conditional.

major comments (3)
  1. [Section 6 / Abstract] Section 6 states: 'we did not properly discuss the gravitational dressing of the external operators.' This is not a side remark: the abstract promises a gauge-invariant definition by dressing to an observer, and the entire positive-growth result depends on large-diffeomorphism pieces (eq. (4.21) for the L=0 mode; §4.4 for the L=1 sector) whose contribution to a dressed observable is never computed. In parallel, §4.3 shows that the tree-level L=1 divergence is removed by imposing matter constraints. Without a dressed-observable computation showing that the L=0 and L=1 large-diffeomorphism phases survive with the same coefficient, the growth in eqs. (5.67) and (5.30) is not established for a gauge-invariant OTOC.
  2. [Section 4.4 / eq. (5.67)] The paper admits in §4.4 that the answer is 'sensitive to the integration contour choice' for general kinematics, and the Discussion repeats that the growing contribution makes the result contour-sensitive. Equation (5.67) is obtained through a specific analytic continuation (ϵ_1=-π, etc., and e^{t41/ℓ}→-i e^{t/ℓ}). If the leading large-diffeomorphism term is contour-dependent, then the sign C_massless>0 and the coefficient in (5.67) are not uniquely determined by the calculation. The authors need to specify the physical contour selected by the observer worldline or by the dressing procedure and show that (5.67) is either contour-independent or the correct physical prescription.
  3. [Sections 4.3, 4.5, 5.3] The loop-level claim (Abstract; §1.1; eqs. (5.28)–(5.30)) is not supported by an explicit loop computation. The logarithmic divergence of the vector part of the propagator (4.14) is identified in the tree-level eikonal integral; the statement that it becomes a 'genuine divergence' at loop level because it lives in internal graviton propagators (Discussion) is an assertion. Moreover, the m_g regulator of §4.5 is a non-unitary massive-graviton deformation, and the double-scaling limit is taken with G_N e^{t/ℓ}/m_g^2 fixed. Since the claimed λ_L=4π/β_dS result is derived entirely within this regulator, a reader cannot distinguish a physical Lyapunov exponent from a regulator artifact. A direct loop calculation retaining the dressing, or a unitary/invariant regularization, is required.
minor comments (5)
  1. [Section 5.2, eq. (5.11)] The notation t*_i appears in ψ_1 and ψ_2 without an explicit definition. It should be defined near eq. (5.11), e.g. as the antipodal time shift t* = t + iπℓ, to avoid confusion with the Euclidean regulators ϵ_i used later.
  2. [Section 5.5] Typo: 'pode–antipode configuration' should be 'pole–antipode configuration'.
  3. [Figures 6 and 7] The horizontal axes are labeled by -iμ or μ, but the ranges and the meaning of Re(f) in these plots are not fully explained in the captions. The values are also extremely large (e.g. ~10^12); a brief statement about normalization and the role of the ϵ regulator would help.
  4. [References] Reference [31] is listed as 'work in progress' (2026). This is not a citable reference for the present paper; it should be removed or replaced by a preprint number if available.
  5. [Section 6 / [4]] The discussion of the MSS bound would benefit from a more explicit statement of why the bound is expected to apply to the dS static-patch observer setup, given that there is no asymptotic boundary and the dressing is not specified. The reference to the doubled SYK model is suggestive but no quantitative comparison is made.

Circularity Check

0 steps flagged

No circularity: OTOC growth is a direct evaluation of stated propagator/wavefunction inputs; dressing and contour caveats are robustness gaps, not definitional circularity.

full rationale

The paper contains no fitted parameters and no ex-post adjustments. The eikonal phase is computed from the de Sitter graviton propagator (eqs. 4.15, 4.21, 4.32, using the external propagator results [22-24]); the wavefunctions are Fourier transforms of the free de Sitter two-point function (5.11, 5.19); and the OTOC is assembled with the standard eikonal/OTOC machinery of [17,18]. The two headline results follow by direct evaluation: the tree-level growth (5.66)-(5.67) comes from the sign of the L=0 eigenvalue in the transverse Green function, and the regulated even-power selection rule (5.27)-(5.30) follows from the angular integral (E.3), with the e^{2t/ell} factor arising arithmetically from the surviving k=1 term. No parameter is adjusted to match the quoted growth; the mass regulator and double-scaling limit are stated kinematic choices, and the paper itself labels the massless result as reliable only at tree level ('Because of the underlying IR issue... reliable only at tree level', Sec. 5). The principal caveats are physical robustness issues rather than circularity: Sec. 4.4 admits 'sensitivity to the integration contour choice' and Sec. 6 admits 'we did not properly discuss the gravitational dressing of the external operators.' These undermine the gauge-invariance of the observable but do not make the derivation identical to its inputs. Self-citations ([2], [4], [30]) appear only in the interpretive discussion of the doubled-SYK model and the bound-on-chaos tension; they are not load-bearing for the computation. The derivation is therefore self-contained in the sense relevant to circularity: the asserted growth is a computed consequence of stated propagator and wavefunction inputs, not a restatement of those inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 2 invented entities

The computation is parameter-free in the sense that no quantity is fitted to data; the scalar masses are physical inputs scanned over in Figs. 6–7. The free parameters are the regulator m_g and the gauge parameters (α, β), both expected to drop out of physical statements (the paper argues the former defines a limit; the latter cancel at tree level but leave an α-proportional L=1 divergence at loop level). The axioms are dominated by domain assumptions: the known graviton propagator results [22–25], the eikonal/ladder-diagram dominance [17,18], the physicality of large-diffeomorphism contributions, and the non-cancellation of the loop IR divergence — the last two carrying the weight of the paper's interpretation. No invented physical entities are proposed; the massive-graviton regulator is a deformation, and 'large diffeomorphisms' are boundary modes already present in the theory, though their role as mediators of the OTOC growth is the paper's interpretation and lacks an independent falsifiable handle.

free parameters (3)
  • graviton mass regulator m_g = m_g → 0, G_N e^{t/ℓ}/m_g² fixed
    Introduced in §4.5 (eq. 4.31) to regulate the vector-sector IR divergence. The double-scaling limit makes L=1 dominate and produces the even-power expansion and λ_L = 4π/β (eqs. 5.28, 5.30). The m_g→0 limit is below the Higuchi bound, so the regulated theory is non-unitary; the paper neither flags this nor analyzes its effect.
  • gauge-fixing parameters (α, β) = none (probes α, β dependence)
    Defined in eq. 4.13. The L=1 IR divergence is ∝ α and diverges for any α (§4.3); the paper argues cancellation for the physical L=0, but the loop-level cancellation is not demonstrated. These are free parameters of the computation, not of a final physical answer.
  • Euclidean time-shift regulators ϵ_i = ϵ = 0.001 in Figs. 6–7; ϵ_i = {-π, -π/2, 0, π/2}ℓ in the chaos-bound configuration
    Imaginary shifts (5.29) defining the OTOC time orderings. They are analytic-continuation prescriptions, not fitted values, but the sign and reality of f depend on them, so they are part of what the central claim rests on.
axioms (6)
  • domain assumption Graviton propagator on dS as given by [22–25]
    The paper states it uses existing propagator results and the bi-tensor decomposition adapted to dS (App. B, F); the eikonal phase integrals over these propagators in d=3,4,5 are the paper's own but are not shown (§4.3).
  • domain assumption Eikonal approximation: generalized ladder diagrams dominate and exponentiate for large center-of-mass energy in graviton exchange
    Adopted from [17, 18] in §3.1. The paper itself notes the eikonal approximation fails for scalar exchange (§4.1, citing [19–21]); its validity for graviton exchange in dS is assumed throughout.
  • ad hoc to paper Observer dressing, invoked to make the OTOC gauge-invariant, does not alter the eikonal phase or external-line couplings used here
    Abstract and §1 set up the gauge-invariant definition by dressing to an observer, but §6 admits the dressing is not implemented; the central growth claim depends on this premise.
  • domain assumption Large diffeomorphisms that do not decay at infinity contribute a non-vanishing, physical shift to the eikonal phase in dS (unlike AdS)
    Argued in §3.2 and §4.4; the L=0 growth term is a large-diffeomorphism contribution (eq. 4.21), and the paper concedes this makes the answer contour-sensitive (§4.4).
  • ad hoc to paper The vector-sector IR divergence in the graviton propagator survives to genuine loop-level divergences in the OTOC
    The central motivation for the regulator (§1.1, §4.3, §6). No loop diagram is computed and no gauge-invariant cancellation analysis is given; the claim is the transfer of a propagator-level divergence to loops.
  • standard math Standard QFT analytic continuation (iϵ prescriptions; Euclidean rotation (3.24); Bunch–Davies wavefunction conventions)
    Used in §3.1 (eq. 3.24) and throughout §5 to evaluate Fourier transforms and momentum integrals; standard but not proved in the text.
invented entities (2)
  • Massive graviton regulator (m_g) no independent evidence
    purpose: Regularize the vector-sector IR divergence; defines the double-scaling limit that yields the even-power expansion and λ_L = 4π/β_dS
    Introduced by hand in §4.5 (eq. 4.31). It is a formal deformation, not a physical dS mode; no external falsifiable prediction is attached, and below the Higuchi bound it is non-unitary.
  • Large diffeomorphisms acting as physical carriers of the eikonal phase and OTOC growth no independent evidence
    purpose: Source of the L=0 growth (time advance) and of the L=1 IR divergence in the propagator
    The paper identifies these boundary graviton modes as the mechanism (§4.4) but supplies no independent handle: the contribution is contour-sensitive (§4.4) and the dressing that would fix their physical charge is not implemented (§6).

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read the original abstract

We study the 4-point out-of-time-ordered correlator (OTOC) of scalar fields using the eikonal approximation to gravity around a de Sitter background. It can be defined in a gauge-invariant way by dressing the operators to an observer. We consider de Sitter space in any dimension and any bulk masses of the scalar fields. At tree level we find the maximal Lyapunov exponent of $2 \pi/\beta_{\rm dS}$. However, we describe an IR problem in this calculation at the loop level associated with the vector part of the graviton propagator. Regularizing this divergence leads to the vanishing of all odd powers in Newton's constant in the asymptotic expansion, so that the leading answer comes with twice the maximal Lyapunov exponent in simple operator configurations. We find that the perturbative OTOC in de Sitter can initially grow, which is not allowed by the quantum bound on chaos. Interestingly, we find that this growth is mediated by large diffeomorphisms in the graviton propagator.

Figures

Figures reproduced from arXiv: 2607.13137 by Alexey Milekhin, Jiuci Xu, Vladimir Narovlansky.

Figure 1
Figure 1. Figure 1: Observers in de Sitter. While this setup might ideally make it possible to discuss the OTOC at finite Newton’s constant GN , it is beneficial to start with GN → 0. It is particularly interesting to consider an observer whose mass is much larger than the de Sitter scale, so that the observer is localized, but much smaller than the Planck scale, so that it does not affect spacetime significantly. In this set… view at source ↗
Figure 2
Figure 2. Figure 2: Shockwaves in AdS and dS. We depict the shockwaves by heavy lines. To probe them [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Penrose diagram of global de Sitter space. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Generalized ladder diagrams used in the eikonal approximation. The horizontal lines [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two particles (blue and red) follow classical null geodesic trajectories and exchange [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plots of the coefficient f as a function of −iµ ∈ (−1, 1), parametrizing the mass of light scalars in three dimensional de Sitter in the complementary series representation. We regularize using ϵ = 0.001. In each case we show the real part which is the non-zero component (the other component may be non-zero because of the regulator and is small). (a) single-sided configuration, (b) two-sided configuration,… view at source ↗
Figure 7
Figure 7. Figure 7: Plots of the coefficient f as a function of µ ∈ (−3, 3), parametrizing the mass of heavy scalars in three dimensional de Sitter in the principal series representation. We regularize using ϵ = 0.001. In each case we show the real part which is the non-zero component (the other component may be non-zero because of the regulator and is small). (a) single-sided configuration, (b) two-sided configuration, (c) b… view at source ↗

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Forward citations

Cited by 2 Pith papers

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  2. Anti-scrambling and euclidean folds from observer correlators in de Sitter space

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