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The paper claims that relaxing either of two standard assumptions of stochastic inflation—the sharp step-function window or the Bunch-Davies vacuum—turns Gaussian white noise into colored noise, and a non-Bunch-Davies initial state addition

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2026-08-03 15:22 UTC pith:TVZVZP7L

load-bearing objection The colored-noise results are real, but the non-Gaussianity claim is not: the connected four-point cumulant vanishes at O(ε), so the headline overreaches. the 2 major comments →

arxiv 2512.17070 v2 pith:TVZVZP7L submitted 2025-12-18 gr-qc astro-ph.COhep-th

Deviations from Gaussian White Noise in Stochastic Inflation

Zahra Ahmadi , Mahdiyar Noorbala This is my paper
classification gr-qc astro-ph.COhep-th MSC 83F05
keywords stochastic inflationnoise power spectrumcolored noisenon-Gaussian noiseinitial statewindow functionde Sitter spacetimeinstantaneous power spectrum
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.

Stochastic inflation models the classical long-wavelength field as driven by random noise, and the standard setup yields Gaussian white noise. This paper isolates which assumptions produce that simplicity by violating them one at a time: leaving exact de Sitter spacetime only makes the white noise's amplitude time-dependent; smoothing the sharp cutoff (a Heaviside window) produces stationary colored noise with finite memory; and replacing the Bunch-Davies vacuum by a small two-particle admixture produces non-stationary colored noise that the paper argues is also non-Gaussian. The signatures are explicit power spectra: eq. (4.12) for the window-function case and eq. (5.20) for the initial-state case. A sympathetic reader would care because noise is the central input to stochastic inflation, and any memory or non-Gaussianity changes the Fokker-Planck dynamics and the statistics of superhorizon perturbations.

Core claim

The central discovery is that the white, Gaussian character of the stochastic-inflation noise is not robust: it disappears as soon as either the window function or the initial state is perturbed. For a massless field on exact de Sitter with a piecewise-linear window of width δ, the noise correlator becomes a peaked function of the e-fold separation with memory ΔN± and power spectrum (H/2π)² [cosh ΔN± − cos(ωΔN±)]/[(cosh ΔN± − 1)(ω²+1)], which is flat only in the sharp-cutoff limit. For the initial state, only a superposition of two-particle states (eq. 5.8), not a Bogoliubov transformation, yields a nontrivial O(ε) correction; the result is non-stationary and its instantaneous power spectrum

What carries the argument

The noise operator ξ_φ = (1−ε)∫ d³k/(2π)³ κW'(κ) φ̂_k is the central object; its two-point anticommutator in the chosen state, evaluated at a single patch, is the noise correlator. The argument turns on how the derivative of the window function localizes the integrand: a sharp cutoff gives W'(κ)=δ(κ−1), hence a delta-correlated white noise, while a broad window spreads each mode's contribution over a band of times, producing memory. For the initial-state deviation, the correlator's nontrivial part is proportional to Re[C0* C2(k1,k2) φ_{k1}(N1) φ_{k2}(N2)], and the spectral analysis uses the instantaneous (Wigner-Ville) power spectrum for non-stationary processes.

Load-bearing premise

The non-Gaussianity claim for non-Bunch-Davies states is not computed: the paper asserts that Gaussianity is lost for the two-particle state (5.8), so if the connected four-point noise cumulant vanished at O(ε) for this free field, that part of the thesis would collapse; the explicit colored-noise spectra also rest on C2 being real and depending only on |q|, and on keeping only first order in ε.

What would settle it

Compute the connected four-point correlation function of ξ_φ in the state (5.8) with a sharp cutoff and exact de Sitter background, to first order in ε. If the connected cumulant vanishes identically, the paper's central non-Gaussianity claim fails. A simpler check at the two-point level: measure or compute the instantaneous power spectrum (5.20); if it shows no frequency dependence beyond what the leading-order white term already has, the colored-noise claim for initial states fails.

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

If this is right

  • If the window function is not a sharp step, the coarse-grained field's Langevin equation acquires memory over ΔN± e-folds; the dynamics are no longer Markovian and require a generalized Fokker-Planck treatment.
  • If the initial state contains two-particle admixtures of size ε, the noise is non-stationary and colored, with frequency-dependent instantaneous power (5.20), so predictions derived from white noise are modified at order ε.
  • Bogoliubov-type vacua, despite containing many particles of the original basis, do not produce colored noise; only superpositions of two-particle states do.
  • Deviations from exact de Sitter, by themselves, only multiply the white noise by a time-dependent amplitude and leave Gaussianity untouched.
  • The window-function example shows high-frequency noise components decay as 1/(ω²+1) with oscillatory features; a broad class of smooth windows falls off as exp(-2|ΔN|), so any non-sharp cutoff yields a colored spectrum.

Where Pith is reading between the lines

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

  • If the asserted non-Gaussianity survives a direct computation, stochastic-inflation simulations that assume Gaussian white noise will underweight rare large excursions, with potential consequences for primordial black hole abundance estimates.
  • The Mellin-transform link between C2 and the noise spectrum suggests an inverse problem: a measured noise color profile could be inverted to constrain the two-particle wavefunction of the inflaton's initial state.
  • A direct test of the paper's initial-state claim is to compute the connected four-point noise cumulant for the state (5.8) at O(ε); a vanishing result would overturn the non-Gaussian statement.
  • Relaxing the paper's simplifying assumptions—C2 real and depending only on |q|, and truncation at O(ε)—could yield partially stationary or direction-dependent noise, which would change the memory length and spectral shape.

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

2 major / 3 minor

Summary. The paper studies how the statistical properties of the noise in stochastic inflation deviate from Gaussian white noise when three standard assumptions are relaxed: exact de Sitter background, a sharp Heaviside window function, and the Bunch-Davies initial state. For a free test field the authors argue that (i) a quasi-de Sitter background keeps the noise white but makes its amplitude time-dependent; (ii) a non-sharp window function produces stationary Gaussian colored noise, with explicit correlator (4.11) and spectrum (4.12); and (iii) a small two-particle admixture to the initial state produces non-stationary colored noise, with the NLO instantaneous power spectrum (5.20). They further claim that such initial states also give non-Gaussian noise. The paper contains detailed appendices on noise correlators, instantaneous power spectra, and a Green's-function approximation for quasi-de Sitter mode functions.

Significance. If the results held as stated, the paper would provide a useful systematic classification of how the noise sector of stochastic inflation responds to relaxing the standard assumptions, with explicit analytic spectra that others could use. The window-function and initial-state two-point calculations are largely internally consistent, and the recovery of the white-noise limits (δ→0 and δ→1 in §4) is a good check. However, the paper's headline non-Gaussianity claim is not derived, and the concrete model in §5 contains a normalization inconsistency. Because these issues affect the abstract and the main conclusions, the manuscript needs substantial revision before the central claims are supported.

major comments (2)
  1. [§5, Eq. (5.17)] The claim that the state (5.8) yields non-Gaussian noise with a size "evidently proportional to ε" is not demonstrated, and the assertion is not correct at O(ε). The text itself admits "we have not calculated the higher order correlators." For the state (5.8), with |C0|²=1−O(ε²) and C2=O(ε), the O(ε) correction to any correlator is a cross term C0*C2⟨2|...|0⟩. Since the noise is linear in â and â†, Wick's theorem allows at most two contractions with the two-particle state, so the O(ε) term in log⟨e^{iJξ}⟩ is quadratic in the source J. A quadratic term only shifts the two-point function and cannot generate a connected four-point cumulant. Hence ⟨ξξξξ⟩_c is O(ε²), not O(ε). The authors should either compute the leading non-Gaussian cumulant or revise the abstract and conclusions to state that non-Gaussianity is subleading, appearing at O(ε²).
  2. [§5, Eq. (5.17)] The normalization stated for the model wavefunction is inconsistent. With C2(q1,q2)=√(2ε)/(πQ³) exp[-(q1+q2)/Q], and using ∫d³q e^{-q/Q}=8πQ³, one obtains (1/2)∫d³q1 d³q2 |C2|² = 64ε, not ε². Thus C2 is O(√ε), not O(ε), contradicting the general assumption (5.5). Consequently the NLO correlator (5.18) and the instantaneous power spectrum (5.20) do not have the claimed O(ε) scaling and their coefficients are not correct. To satisfy (1/2)∫|C2|²=ε², the prefactor should be ε/(4√2 π Q³) (up to an equivalent convention for the measure). This is a concrete, load-bearing error in the illustrative model and must be corrected and propagated.
minor comments (3)
  1. [§4, after Eq. (4.10)] The identification of the anticommutator (A.11) with a classical noise correlator requires the commutator (A.1) to be negligible in the σ→0 limit. For the window (4.1) this is not as immediate as in the sharp-cutoff case (A.6); a short estimate showing that [ξφ(N1),ξφ(N2)] is O(σ) in the massless dS example would make the classicality assumption explicit.
  2. [General] Typos and minor wording: "F rom" in the Contents heading; "Lagnevin" in §6 should be "Langevin"; "FLR W" in §2 should be "FLRW"; and "THe" appears in a few places. Please proofread.
  3. [§4, Eq. (4.12)] The power spectrum (4.12) correctly reduces to (H/2π)² at ω=0 and to the white-noise limit as δ→0. It would be helpful to state explicitly in the text or figure caption that the plotted spectrum is normalized to (H/2π)², since the vertical scaling is otherwise easy to misread.

Circularity Check

0 steps flagged

No significant circularity; one minor non-load-bearing self-citation; main derivations are self-contained.

full rationale

The paper's central results are explicit computations from stated inputs, not reductions of outputs into inputs. Eq. (4.11) is the direct evaluation of the general vacuum correlator (A.11) with the piecewise-linear window (4.1) and dS mode function (4.8), and its Fourier transform (4.12) follows from the definition of the power spectrum for a stationary process (B.5). Eq. (5.11) is the NLO contraction of (5.6) for the state (5.8), and eq. (5.20) is the resulting instantaneous power spectrum (B.14) for the explicitly chosen C2 in (5.17). No parameter is fitted to data, and no result is back-substituted as an input. The only self-citation, ref. [90] (M. Noorbala), supports a side remark on classicality away from σ→0 (Section 3.2 and Appendix A.1) and is not load-bearing for the colored-noise or non-Gaussianity conclusions. The paper explicitly flags the non-Gaussianity claim as uncomputed — 'although we have not calculated the higher order correlators in this section, but it should be clear that Gaussianity is in general lost' (end of Section 5) — so the abstract's statement that 'changing the initial state yields a non-Gaussian noise' is an assertion with a missing proof, an evidentiary gap rather than a circular step. The assumptions that C2 is real and depends only on |q| are clearly stated toy-model inputs, not hidden restatements of the output. Therefore the derivation chain is self-contained and no significant circularity is present.

Axiom & Free-Parameter Ledger

5 free parameters · 8 axioms · 0 invented entities

The paper introduces no new particles, forces, or degrees of freedom. The conclusions rest on standard QFT in curved spacetime, the stochastic-inflation dictionary, and a set of explicitly chosen toy parameters (δ, ε, Q, σ, c). No constants are fitted to observational data.

free parameters (5)
  • δ = 0 < δ < 1 (illustrative)
    Half-width of the piecewise-linear window function; controls the memory length. Chosen by hand for the toy model, not fitted to data.
  • ε = ε ≪ 1 (illustrative)
    Small deviation from the Bunch-Davies vacuum. Sets the size of the colored/non-Gaussian initial-state correction; not fitted.
  • Q = Q > 0 (illustrative)
    Momentum decay scale in the exponential two-particle wavefunction C2; introduces a privileged time NQ. Chosen for analytic convenience, not fitted.
  • σ = σ ≪ 1 (chosen small)
    Coarse-graining ratio k/(aH). Standard stochastic-inflation parameter; many results are quoted in the σ→0 limit.
  • c = c = 1 in figures
    Constant in the toy scale factor of eqs. (3.8) and (C.17); used only to illustrate quasi-dS behavior.
axioms (8)
  • domain assumption Free scalar test field on a fixed FLRW background
    The action (2.1) and all calculations treat ϕ as a spectator with no backreaction on the metric.
  • standard math Standard canonical commutation relations and Wronskian normalization
    Eqs. (2.3)–(2.4) fix the mode-function normalization and operator algebra.
  • domain assumption Bunch-Davies vacuum as the reference state |0⟩
    The positive-frequency mode functions are chosen in the asymptotic past; deviations are measured relative to this state.
  • domain assumption Noise commutators are negligible in the σ→0 limit and the anticommutator equals twice the classical correlator
    This is the standard stochastic-inflation dictionary, used throughout; eqs. (A.6)–(A.8) and the discussion after eqs. (2.9)–(2.14).
  • ad hoc to paper Small deviation from Bunch-Davies: |C0|² = 1 − O(ε²) and |C_N| = O(ε)
    Eq. (5.5) truncates the state expansion at next-to-leading order; this is what makes the two-particle analysis tractable.
  • ad hoc to paper C2 depends only on |q1| and |q2|, and C0, C2 are taken real for the explicit power-spectrum example
    Imposed in section 5 after eq. (5.11) to make angular integrals tractable and to give a simple sign; not required by the general formalism.
  • domain assumption Slow-roll/quasi-dS approximations for mode functions and power spectra
    Section 3.2 and appendix C use Hankel-mode approximations and H(N_k)/2π for the field power spectrum; this is standard but approximate.
  • standard math Wick theorem applies in the vacuum state
    Used in appendix A.3 to argue Gaussianity of the noise when |Ψ⟩ = |0⟩.

pith-pipeline@v1.3.0-alltime-deepseek · 27095 in / 23338 out tokens · 235256 ms · 2026-08-03T15:22:43.755667+00:00 · methodology

0 comments
read the original abstract

Stochastic inflation is widely used as a framework to study scalar field perturbations on an inflationary spacetime in a classical manner. In Starobinsky's seminal work and most of the subsequent literature, stochastic inflation is driven by a white noise. This is a consequence of a number of assumptions about the background metric, the window function, and the initial state. Given that noise is the central object in this approach, it is worthwhile to investigate how the noise is modified upon relaxing some of these assumptions. We show that while deviation from an exact de Sitter background maintains the white character of the noise (only with a time-dependent amplitude), deviation from the Heaviside window function or the Bunch-Davies initial state can produce colored noise. We calculate the power spectrum and the memory of the noise for a toy model with a piecewise linear window function. We also show that, in order to produce a colored noise, the deviation from the Bunch-Davies vacuum should essentially be a sum of two-particle states. The resulting noise is non-stationary and we find its instantaneous power spectrum in a concrete example. Furthermore, while deviations from de Sitter background and sharp cutoff do not affect Gaussianity, changing the initial state yields a non-Gaussian noise.

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