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Gravitational self-interactions in minimal inflation decohere super-Hubble scalar fluctuations at a rate growing like (aH/k)^5, faster than earlier results, with the dominant contribution coming from the nonlocal cubic interaction obtained

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2026-08-04 21:40 UTC pith:34V5J4NG

load-bearing objection Careful Open-EFT machinery and a systematic upgrade of [26], but the leading decoherence rate is not trustworthy: the Kossakowski matrix from the paper's own correlators has a negative eigenvalue, so the claimed Lindblad evolution and the 8π coefficient cannot stand. the 3 major comments →

arxiv 2509.07769 v2 pith:34V5J4NG submitted 2025-09-09 gr-qc astro-ph.COhep-phhep-th

Inflationary Decoherence from the Gravitational Floor

C.P. Burgess , R. Holman , Greg Kaplanek This is my paper
classification gr-qc astro-ph.COhep-phhep-th
keywords primordial decoherencegravitational floorinflationary perturbationspurityopen quantum systemsMarkovian limitnonlocal cubic interactionssuper-Hubble modes
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.

The paper tries to establish a floor for how fast primordial quantum fluctuations decohere purely from the gravitational self-interactions that any inflationary model must contain. By including all leading-order gravitational interactions among scalar metric fluctuations, not just those among super-Hubble modes, the authors find that the dominant decohering vertex is a nonlocal cubic interaction obtained by eliminating the lapse and shift constraints. They compute a purity loss for a super-Hubble mode of momentum k, roughly 1 − (8πϵ1 H^2/(5 M_p^2))(aH/k)^5, which grows like e^{5Ht} and is much faster than the (aH/k)^3 rate reported earlier. The paper also shows that super-Hubble evolution is automatically Markovian, with non-Markovian corrections suppressed by powers of k/(aH), and argues that the leading purity loss is UV finite. If correct, gravitational decoherence alone would make primordial fluctuations look classical within a few e-folds after horizon exit, dimming prospects for measuring their quantum origin.

Core claim

The paper's central claim is that the leading gravitational decoherence of long-wavelength scalar curvature perturbations in minimal single-field inflation is governed by the nonlocal cubic self-interaction obtained by solving the lapse and shift constraints, not by the local derivative interaction used previously. Starting from a pure Bunch-Davies vacuum, tracing out short-wavelength scalar modes, and working at second order in the cubic interaction (order H^2/M_p^2), the reduced density matrix evolves in Lindblad form for super-Hubble modes. The resulting purity evolution is ∂zγk = 8πϵ1 H^2/(z^6 M_p^2) with z = k/(aH), integrating to γk(z) = 1 − 8πϵ1 H^2/(5 M_p^2 z^5) + ⋯ (Section 6);

What carries the argument

The machinery is the purity γk of the reduced density matrix of each Mukhanov-Sasaki mode, evolved with a second-order time-convolutionless (TCL2) Nakajima-Zwanzig equation. The environment is traced out from an initial Bunch-Davies vacuum, and the small parameter z = k/(aH) organizes everything: correlators of environment operators are evaluated on a deformed time contour; expanding fields at η′ in terms of η via kernels W, X, Y, Z and Taylor expanding ϱ(η′) about η shows that all n≥1 terms are suppressed by z, making the evolution Markovian and reducible to Lindblad form. Power counting in z isolates the dominant coefficient: Re[J^{ζζ}_{k0}] ~ 20π/z^6 with a positive sign, giving the leadi

Load-bearing premise

The quoted rate is a gravitational floor only if the leading UV-divergent pieces from the scalar sector cancel against tensor-mode and counterterm contributions leaving the finite coefficient; the paper states in Section 7.3 that this tensor calculation has not yet been explicitly verified.

What would settle it

Perform the tensor-mode environment calculation to the same order and sum it with the scalar and counterterm contributions; the (aH/k)^5 floor stands only if the result is UV finite with the coefficient quoted in Section 6 (8π) or Section 7.1 (3π). A residual divergence or a different finite remainder in the leading z^-6 term would falsify the claim.

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

If this is right

  • Perturbation theory for the purity breaks down when z^5 ~ H^2/(4π^2 M_p^2), which for observationally allowed parameters happens roughly five e-foldings after horizon exit; late-time predictions require the Lindblad resummation the paper begins.
  • Given the Markovian Lindblad form, the purity can in principle be evolved to the end of inflation; the paper's estimates indicate the state is strongly decohered by then, so any quantum-origin signal in the CMB would be heavily suppressed.
  • The leading coefficient is universal with respect to the split between system and environment, depending neither on k_UV nor on other details of the observable window.
  • Non-Markovian corrections are suppressed by powers of k/(aH), so the rotating-wave approximation is unnecessary; the Markovian limit follows from the super-Hubble expansion alone.
  • Although UV divergences show up in intermediate correlation functions, they do not enter the leading purity, consistent with the effective-field-theory expectation that decoherence cannot be a counterterm effect.

Where Pith is reading between the lines

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

  • The tensor-mode contribution is not computed here; if it multiplies the scalar result by a factor of three as in the earlier subset calculation, the (aH/k)^5 growth survives but the total coefficient changes. Reconciling that coefficient with the 8π/3π discrepancy between Sections 6 and 7.1 of the paper is an immediate next step.
  • The identification of constraint-obtained nonlocal cubic vertices as the dominant decoherence mechanism suggests analogous constraint-sourced interactions may dominate decoherence in other gravitational open-system settings, such as horizon-induced decoherence.
  • Completing the resummation and computing γk at the end of inflation would give a concrete observable target: whether residual purity survives or γk → 0, which would sharpen predictions for any future attempt to detect quantum signatures.
  • A testable extension is to repeat the calculation for long-wavelength tensor modes and for post-inflationary evolution; the paper notes the slow-roll approximation is not conceptually required, so the same methods could track whether decoherence persists through reheating and horizon re-entry.

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 / 3 minor

Summary. This paper re-examines the decoherence of super-Hubble scalar curvature perturbations in single-field inflation, using only gravitational cubic self-interactions at leading order in H/M_p. Treating short-wavelength modes as the environment, it re-derives the time-convolutionless master equation, argues that the evolution becomes Markovian for z = k/(aH) << 1, computes the environmental correlators with an iε-regulated contour, and extracts small-z Laurent expansions. The headline result is a perturbative purity decay γ ≈ 1 − 8π ε_1 H^2/(5 M_p^2) z^{−5} (Eq. 6.8/6.9), growing like (aH/k)^5, faster than the (aH/k)^3 rate of earlier work. The authors attribute the enhancement to nonlocal cubic interactions obtained by solving the constraints, claim that the leading coefficient is UV finite, and argue that remaining divergences cancel against tensor-mode and counterterm contributions that are not computed here.

Significance. If correct, the result gives a concrete, much stronger gravitational decoherence floor for primordial fluctuations and identifies the nonlocal constraint interactions as the dominant channel. The paper is well organized, transparent about its setup, and useful in its detailed treatment of the contour regularization and super-Hubble power counting. It also honestly flags several open issues. However, the current manuscript contains an internal positivity contradiction in the advertised Lindblad equation, an inconsistency in the quoted central coefficient, and an unverified tensor-sector cancellation. These issues are load-bearing for the claimed rate and for the late-time resummation program.

major comments (3)
  1. [§5.2, Eqs. (5.18)–(5.21); §4.2.2, Eq. (4.27); Appendix D] The claim in §4.2.2 that positivity of the Kossakowski matrix is automatic is contradicted by the paper's own small-z expressions. Using Re[J_ζζ] ≈ 20π/z^6, Re[J_ζp + J_pζ] ≈ 8π/z^5, Im[J_ζp − J_pζ] ≈ 20π/z^4, and Re[J_pp] ≈ −24π/z^4, the matrix (4.27) has h_11 ∝ 40π/z^6 > 0 but h_22 ∝ −24π/z^4 < 0, and the determinant (D.11) is negative. The smaller eigenvalue is therefore negative, so the master equation is not a valid Lindblad generator and the late-time resummation in §6.2 lacks its stated foundation. This is not cosmetic: if the sign of Re[J_pp] is corrected to +12π/z^4, the z^{−6} bracket in (6.7) changes from 16π to 40π, and the central coefficient in (6.8)/(6.9) changes from 8π to 20π. The authors must either prove positivity with corrected correlators or identify which approximation has failed.
  2. [Abstract; §7.3; Eq. (6.8)] The abstract and introduction state that the leading purity evolution is UV finite and that divergent scalar-sector pieces cancel against tensor-mode and counterterm contributions. However, §7.3 explicitly says: “We expect the same to be true here but have not yet explicitly verified this.” The tensor-mode calculation is not performed, and the finite remainder left by the required cancellation could modify the leading coefficient. Since the “gravitational floor” interpretation depends on omitted competitors not changing the leading result, this is an unverified load-bearing assumption. The manuscript should either provide the tensor computation or restrict its claims to the scalar-sector contribution.
  3. [§7.1, Eq. (7.1) vs §6.1, Eq. (6.9)] The main result is quoted inconsistently. Equation (6.9) gives γ ≈ 1 − 8π ε_1 H^2/(5 M_p^2) z^{−5}, while Eq. (7.1) states γ ≈ 1 − 3π ε_1 H^2/(5 M_p^2) (aH/k)^5. The coefficients differ by a factor 8/3. Because this is the headline quantitative claim, the correct coefficient must be identified and used consistently in the summary.
minor comments (3)
  1. [Abstract and §7.1 heading] There are typographical errors: “approrpriate” in the abstract and “F or inflationary primordial fluctuations” in the §7.1 heading.
  2. [§4.2.2, final bullet] The bullet stating that the Lindblad eigenvalues “are positive within the domain of validity” should be revised to reflect the positivity issue identified above; otherwise the summary is internally inconsistent.
  3. [§6.2, Eq. (6.10)] The definition λ := H^2/(4π^2 M_p^2) is later referred to as “the core underlying perturbative expansion parameter,” but the text does not connect it explicitly to the earlier semiclassical parameter GH^2 = H^2/(8π M_p^2). A brief clarifying sentence would help.

Circularity Check

0 steps flagged

No significant circularity: the central decoherence rate is computed from explicit correlator integrals and is not inserted as an input.

full rationale

The paper's load-bearing result, eqs. (6.8)–(6.9), is obtained by (i) writing down the standard single-field inflationary action and the cubic interactions from the externally established Maldacena classification [44] (reproduced in Appendix A), (ii) deriving the TCL2/Lindblad evolution in the text and Appendix D, and (iii) evaluating the environmental correlators explicitly in Section 5, with the small-z Laurent expansions given in (5.18)–(5.21). The final purity rate follows by substitution into (6.4)/(6.7); no term in the input side already contains the 8π coefficient or the (aH/k)^5 growth. Self-citations to [26] concern the standard open-EFT/Nakajima-Zwanzig framework and the comparison in Appendix C actually corrects [26], which shows the new result is not inherited from the cited paper. The gravitational-floor claim does rely on an unverified assumption about tensor-mode and counterterm cancellations; the paper explicitly says in §7.3 'We expect the same to be true here but have not yet explicitly verified this.' That is an admitted limitation/completeness gap, not circular reasoning. Similarly, the asserted positivity of the Kossakowski matrix (4.27) is a consistency claim; if the explicit coefficients (5.18)–(5.21) yield a negative determinant, that would be a calculational error, not an input–output circularity. Overall, the derivation is self-contained for the quantity it actually computes, and the self-citations are not load-bearing for the central coefficient.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

No new particles, forces, or dimensions are introduced; the environment consists of standard scalar metric perturbations and, in future work, tensor modes. The two listed free parameters are regulators or coarse-graining choices that drop out of the leading purity coefficient. The main load-bearing axiom is the unverified cancellation with tensor modes.

free parameters (2)
  • system/environment split scale k_UV
    Coarse-graining scale separating observable long-wavelength system from unobserved short-wavelength environment. It enters subleading corrections and regulators but cancels in the leading z^(-6) purity coefficient, so it is not a fitted parameter of the central claim.
  • contour regulator epsilon = taken to 0
    Maldacena i-epsilon deformation used to project onto Bunch-Davies and regulate coincident divergences. Not a physical parameter; the final leading coefficient is epsilon-independent.
axioms (4)
  • domain assumption Single-field slow-roll inflation with action (2.1) and approximately de Sitter background.
    The calculation is restricted to minimal single-clock models; the result is not derived for multifield or non-slow-roll models.
  • domain assumption Initial state factorizes as Bunch-Davies vacuum at eta_in -> -infinity via the Maldacena i-epsilon contour.
    Assumed in section 3.1 and section 4.2; it is the standard vacuum choice but is not derived.
  • ad hoc to paper Purity computed in the zeta basis is gauge independent; the co-moving gauge interaction Hamiltonian suffices.
    Stated after eq (2.31) as an inference from [44]'s bispectrum check, not proven for the purity observable.
  • ad hoc to paper Divergences from the scalar sector cancel against tensor-mode and counterterm contributions, leaving the leading coefficient unchanged.
    The abstract says divergences must cancel but the tensor calculation is in progress (section 7.3); this is load-bearing for the UV-finite leading result.

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We re-examine the decoherence rate of primordial fluctuations within minimal inflationary models, using only the gravitational interactions required for the underlying fluctuation-generation mechanism itself. Since gravity provides the weakest interactions the result provides a plausible floor on the rate of primordial decoherence. Previous calculations ({\tt arXiv:2211.11046}) did so using only a subset of these interactions, motivated by assuming both system and environment were super-Hubble. We extend this by including the effects on super-Hubble modes of {\it all} gravitational interactions amongst scalar fluctuations at leading order in $H/\Mp$ (and so need not restrict the decohering environment to being super-Hubble). We show how the decohering evolution becomes Markovian for super-Hubble modes, without the need to appeal to truncations (like the `rotating wave' approximation) that are often used in optics but can be inapprorpriate for cosmology. We find that the dominant contribution comes from the nonlocal cubic interactions obtained by solving the constraints. We find a decoherence rate that grows in the super-Hubble regime {\it faster} than found earlier and identify its leading divergent and finite parts. We argue why the divergent parts must cancel with other competing contributions -- such as decoherence due to environmental tensor modes -- that are partially computed elsewhere (and for which a complete calculation is in progress). We discuss the steps required to resum this result to late times and briefly discuss why they are more complicated than for earlier calculations.

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.