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arxiv: 2606.28247 · v1 · pith:32LN6KXZnew · submitted 2026-06-26 · 🌌 astro-ph.CO · gr-qc· hep-ph· hep-th

Cancellation of one-loop time dependence in superhorizon curvature perturbations from all scales

Keisuke Inomata This is my paper

Pith reviewed 2026-06-29 02:30 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-phhep-th
keywords one-loop correctionscurvature perturbationssuperhorizon conservationinflationδN formalismspatially-flat gaugeboundary termsloop integrals
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The pith

The apparent time dependence of the one-loop curvature power spectrum cancels when all loop scales and boundary terms are combined consistently.

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

The paper establishes that superhorizon curvature perturbations remain conserved at one-loop order in inflation, without net time dependence, when loop integrals cover all wavenumbers rather than assuming a scale hierarchy. It achieves this by applying the δN formalism to connect inflaton fluctuations to curvature perturbations and retaining boundary terms that previous treatments omitted. A reader cares because this removes a potential inconsistency in one-loop predictions for the primordial power spectrum that seeds cosmic structure. The result applies in spatially-flat gauge and treats both k much larger than q and k comparable to or smaller than q on equal footing.

Core claim

In spatially-flat gauge, the one-loop curvature power spectrum on superhorizon scales shows no residual time dependence once the nonlinear δN relation between inflaton fluctuations and curvature perturbations is used and all contributions, including boundary terms from loop integrals over every scale, are retained consistently.

What carries the argument

The δN formalism that nonlinearly relates inflaton fluctuations to curvature perturbations, applied while integrating loop modes over all wavenumbers without a k ≫ q assumption and retaining boundary terms.

If this is right

  • Superhorizon curvature perturbations are conserved at one-loop order for any loop wavenumber.
  • The curvature power spectrum can be computed without restricting to k much larger than the external mode q.
  • Boundary terms must be kept to obtain a time-independent result.
  • The cancellation holds in spatially-flat gauge when the δN relation is used throughout.

Where Pith is reading between the lines

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

  • Similar all-scale consistency might remove time dependence in higher-loop or multi-field calculations.
  • Gauge-invariant observables could require explicit retention of boundary contributions whenever nonlinear field redefinitions are involved.
  • Numerical checks of the power spectrum at fixed superhorizon times could test whether the cancellation persists beyond analytic approximations.

Load-bearing premise

The nonlinear mapping from inflaton fluctuations to curvature perturbations given by the δN formalism stays valid at one-loop order with no additional gauge or interaction corrections.

What would settle it

An explicit recomputation of the one-loop power spectrum that keeps the k ≲ q regime and boundary terms yet still finds uncancelled time dependence on superhorizon scales.

Figures

Figures reproduced from arXiv: 2606.28247 by Keisuke Inomata.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagrams for the one-vertex contribution, [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagrams for the tadpole contribution, [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Diagram for the cut-in-the-middle contribution, [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diagrams for the cut-in-the-side contribution, [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

We show the conservation of the superhorizon curvature perturbations at one-loop level in spatially-flat gauge, including contributions from loop wavenumbers on all scales. In contrast to previous works, we do not assume a hierarchy $k \gg q$ between the wavenumber of the loop integral, $k$, and that of the power spectrum, $q$, and we explicitly include the regime $k \lesssim q$. Taking into account the nonlinear relation between the inflaton fluctuation and the curvature perturbation with the $\delta N$ formalism, we show that the apparent time dependence of the one-loop curvature power spectrum cancels once all contributions, including boundary terms, are combined consistently.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that the apparent time dependence in the one-loop superhorizon curvature power spectrum cancels when all loop contributions (including from modes with k ≲ q) are combined consistently with boundary terms, using the nonlinear δN relation between inflaton fluctuations and ζ in spatially-flat gauge, without assuming a scale hierarchy.

Significance. If the cancellation holds after all scales and terms are included, the result would confirm conservation of superhorizon ζ at one-loop order and strengthen the reliability of δN-based calculations of loop corrections to the power spectrum and non-Gaussianity. The absence of free parameters in the final cancellation and the explicit treatment of the full k-range are strengths.

major comments (1)
  1. [δN formalism application (abstract and main derivation sections)] The central cancellation result rests on the assumption that the δN mapping from inflaton fluctuation to ζ remains exact at one-loop order with no additional gauge or interaction corrections when the loop integral includes modes with k ≲ q. This assumption is load-bearing for the claim that all contributions combine to cancel time dependence, yet the manuscript does not provide an explicit verification or counterterm analysis for the non-hierarchical regime.
minor comments (1)
  1. Clarify the precise definition and handling of boundary terms in the one-loop integrals to ensure reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of the δN formalism's applicability across all scales. We address the single major comment below.

read point-by-point responses

  1. Referee: [δN formalism application (abstract and main derivation sections)] The central cancellation result rests on the assumption that the δN mapping from inflaton fluctuation to ζ remains exact at one-loop order with no additional gauge or interaction corrections when the loop integral includes modes with k ≲ q. This assumption is load-bearing for the claim that all contributions combine to cancel time dependence, yet the manuscript does not provide an explicit verification or counterterm analysis for the non-hierarchical regime.

    Authors: The δN formalism supplies the exact, nonlinear relation between the inflaton fluctuation δφ (in spatially flat gauge) and the curvature perturbation ζ on uniform-density slices; this relation follows directly from the definition of ζ and holds order by order without further gauge or interaction corrections at one-loop level. Our calculation implements this mapping uniformly for every mode in the loop integral, with no scale hierarchy imposed and with the regime k ≲ q treated on the same footing as k ≫ q. All time-dependent pieces, including those arising from the nonlinear expansion and the associated boundary terms, are retained and shown to cancel identically once the full set of contributions is assembled. Because the cancellation is obtained by direct, consistent application of the δN relation over the complete k range, a separate counterterm analysis is not required at this perturbative order; the result is parameter-free and follows from the gauge-invariant construction itself. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation uses established external δN formalism

full rationale

The paper's central result is a perturbative cancellation of apparent time dependence in the one-loop superhorizon curvature power spectrum when all scales (including k ≲ q) and boundary terms are included. This is obtained by applying the standard δN formalism to relate inflaton fluctuations to ζ, which is an independent, externally established relation from prior literature and not derived or fitted inside the paper. No self-citation chains, self-definitional reductions, fitted inputs renamed as predictions, or ansatzes smuggled via author citations are present in the described derivation. The calculation is self-contained against external benchmarks once the δN mapping is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, new entities, or detailed axioms beyond standard tools; full paper would be required for exhaustive ledger.

axioms (2)
  • domain assumption δN formalism applies to relate inflaton fluctuations to curvature perturbations at one-loop level
    Invoked in abstract as the method to account for nonlinear relations
  • domain assumption Spatially-flat gauge is suitable for the one-loop calculation of curvature perturbations
    Specified in abstract as the gauge used

pith-pipeline@v0.9.1-grok · 5636 in / 1341 out tokens · 60963 ms · 2026-06-29T02:30:37.654223+00:00 · methodology

discussion (0)

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Reference graph

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