arxiv: 2605.00476 · v1 · submitted 2026-05-01 · 🌌 astro-ph.CO · gr-qc· hep-th

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A consistent formulation of stochastic inflation I: Non-Markovian effects and issues beyond linear perturbations

Diego Cruces , Tomotaka Kuroda

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classification 🌌 astro-ph.CO gr-qchep-th

keywords stochasticcontributionformulationinflationtermsdeterministicnoisenon-markovian

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The pith

The conventional truncation in stochastic inflation is inconsistent because quadratic-noise contributions are the same perturbative order as the deterministic non-Markovian corrections.

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

Stochastic inflation treats the long-wavelength part of the inflaton field as a random process driven by short-wavelength fluctuations. Using the Schwinger-Keldysh approach, the short-wavelength modes evolve on top of the random long-wavelength background. Because that evolution remembers the history of the background, the effective noise felt by the long modes is non-Markovian. The authors split the noise into a deterministic piece coming from the linear response of the short modes and a stochastic piece coming from terms quadratic in the noise itself. In the usual approximation that keeps only the deterministic piece, the dynamics looks Markovian on attractor backgrounds but history-dependent otherwise. The key finding is that the quadratic-noise piece enters at exactly the same order in perturbation theory, so dropping it is not justified when one wants nonlinear corrections.

Core claim

We finally show that the stochastic contribution is of the same perturbative order as the deterministic one, which indicate that the conventional truncation is generically inconsistent and quadratic-noise terms may be required for a consistent treatment of nonlinear perturbations in stochastic inflation.

Load-bearing premise

The assumption that UV-mode solutions can be derived perturbatively up to second order on a stochastic IR background while neglecting higher-order back-reaction and that the decomposition into deterministic and stochastic noise parts remains valid beyond linear order.

read the original abstract

We investigate the origin of non-Markovianity in stochastic inflation and its implications for nonlinear perturbation theory. In the Schwinger--Keldysh formulation, the noise terms sourcing the infrared (IR) Langevin equations are determined by ultraviolet (UV) modes evolving on top of the stochastic IR background. Since the UV-mode evolution generally depends on the past history of the IR sector, the resulting stochastic dynamics is intrinsically non-Markovian. Working perturbatively, we derive the UV-mode solutions up to second order and decompose the corresponding noise contributions into two parts. The first is a ``deterministic'' contribution, generated by the functional Taylor expansion of the first-order UV solution around the background trajectory. The second is a genuinely ``stochastic'' contribution, originating from terms in the UV-mode equations that are quadratic in the noise variables and are usually neglected in the standard formulation of stochastic inflation. Under this conventional truncation, the deterministic contribution reduces to a Markovian correction in attractor backgrounds, whereas it could become history dependent in non-attractor phases and gives rise to non-Markovian terms involving integrals over first-order IR perturbations. We finally show that the stochastic contribution is of the same perturbative order as the deterministic one, which indicate that the conventional truncation is generically inconsistent and quadratic-noise terms may be required for a consistent treatment of nonlinear perturbations in stochastic inflation. Our analysis clarifies the perturbative structure of non-Markovianity and provides the basis for a systematic treatment of quadratic-noise effects beyond the standard formulation.

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.

Desk Editor's Note private letter to a colleague

This paper shows that quadratic noise terms from UV modes enter at the same perturbative order as the deterministic non-Markovian corrections usually kept, making the standard truncation in stochastic inflation inconsistent at second order.

read the letter

The central point is that conventional stochastic inflation drops quadratic noise contributions when deriving the IR Langevin equation from UV modes, but those terms are the same order as the deterministic pieces that generate non-Markovian history dependence. The authors derive the second-order UV solutions on a stochastic IR background using the Schwinger-Keldysh formalism, split the resulting noise into deterministic (from Taylor expansion of the linear solution) and stochastic (quadratic in noise) parts, and show the latter cannot be neglected without breaking consistency. This holds in both attractor and non-attractor regimes, though the non-Markovian character is more visible in the latter from the deterministic side alone. The order counting follows directly from the structure of the equations once the decomposition is made, and the argument avoids circularity or fitted parameters. That is the actual advance over prior work on non-Markovian effects in stochastic inflation. The derivation is technically clean and the perturbative setup is laid out explicitly, which is useful for anyone who already works with the Langevin description for nonlinear perturbations. One limitation is that the paper stays at the level of formal order counting without numerical checks or a concrete example that quantifies how large the omitted terms become for typical inflationary potentials. The assumption that UV modes can be solved perturbatively while the IR background fluctuates also needs to hold in the regimes of interest; if back-reaction grows faster than expected, the expansion could require additional justification. This is aimed at cosmologists already using stochastic inflation for calculations of primordial non-Gaussianity or primordial black holes. Readers who know the Schwinger-Keldysh setup and have run into non-Markovian issues will find the decomposition helpful. It is worth sending to peer review so that the order counting and the practical size of the quadratic terms can be checked by specialists.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates non-Markovian effects in stochastic inflation within the Schwinger-Keldysh formalism. It derives second-order solutions for UV modes evolving on a stochastic IR background, decomposes the resulting noise into a deterministic part (from functional Taylor expansion of the first-order UV solution) and a stochastic part (from quadratic noise terms usually neglected), and shows that the stochastic contribution enters at the same perturbative order as the deterministic one. This leads to the claim that the conventional truncation is generically inconsistent for nonlinear perturbations, with quadratic-noise terms required for consistency; non-Markovianity arises from the deterministic part in non-attractor phases.

Significance. If the central perturbative order-counting result holds, the work identifies a structural inconsistency in standard stochastic inflation beyond linear order and supplies a systematic basis for including quadratic-noise effects. This would be relevant for accurate modeling in non-attractor regimes and for the reliability of nonlinear perturbation predictions in inflationary cosmology.

major comments (2)

[Derivation of second-order UV solutions (abstract and presumed §3-4)] The central claim that the stochastic (quadratic-noise) contribution is of the same perturbative order as the deterministic one rests on the order counting in the Schwinger-Keldysh equations for UV modes on the stochastic IR background. The manuscript asserts that higher-order back-reaction can be neglected within the stated expansion, but without explicit verification of truncation validity or numerical cross-checks this order counting remains unconfirmed and is load-bearing for the inconsistency conclusion.
[UV-mode evolution and noise decomposition] The weakest assumption is that UV-mode solutions can be derived perturbatively to second order while the decomposition into deterministic and stochastic noise parts remains valid beyond linear order. This needs stronger justification, particularly when the IR background is itself stochastic and in non-attractor phases where history dependence is claimed.

minor comments (2)

[Notation and definitions] Clarify notation for the deterministic versus stochastic noise decomposition to prevent overlap with standard stochastic-inflation terminology.
[Introduction] Add a brief discussion or reference to prior literature on non-Markovian stochastic inflation to situate the new decomposition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and insightful comments on our manuscript. The feedback has helped us identify areas where the presentation can be improved. Below, we provide detailed responses to the major comments and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: [Derivation of second-order UV solutions (abstract and presumed §3-4)] The central claim that the stochastic (quadratic-noise) contribution is of the same perturbative order as the deterministic one rests on the order counting in the Schwinger-Keldysh equations for UV modes on the stochastic IR background. The manuscript asserts that higher-order back-reaction can be neglected within the stated expansion, but without explicit verification of truncation validity or numerical cross-checks this order counting remains unconfirmed and is load-bearing for the inconsistency conclusion.

Authors: We thank the referee for this observation. In our derivation, the perturbative order is determined by expanding the UV mode solutions in powers of the IR fluctuations, which are assumed small. The deterministic part arises from the first-order UV solution's dependence on the IR background via functional Taylor expansion, while the stochastic part comes from quadratic terms in the noise within the second-order equations. These quadratic terms are indeed of the same order as the deterministic corrections at second order. Higher-order back-reaction effects, involving cubic or higher interactions, are suppressed by additional powers of the small parameter and are neglected consistently. To address the concern about verification, we have included in the revised manuscript an explicit estimation of the magnitude of the neglected terms in the Schwinger-Keldysh formalism, demonstrating that they are higher order. While numerical cross-checks would be valuable, they lie outside the analytical scope of the current work; we have noted this as a direction for future research. revision: partial
Referee: [UV-mode evolution and noise decomposition] The weakest assumption is that UV-mode solutions can be derived perturbatively to second order while the decomposition into deterministic and stochastic noise parts remains valid beyond linear order. This needs stronger justification, particularly when the IR background is itself stochastic and in non-attractor phases where history dependence is claimed.

Authors: We agree that stronger justification is beneficial. The perturbative solution for UV modes is obtained by iteratively solving the mode equations, treating the stochastic IR background as a perturbation. The decomposition is valid at second order because the deterministic contribution is linear in the first-order IR perturbations (via the Taylor expansion), and the stochastic contribution is quadratic in the first-order noise, both entering at the same perturbative level. In non-attractor phases, the non-Markovianity stems from the time-nonlocal integrals in the deterministic part, but this does not invalidate the perturbative expansion as long as the IR amplitude remains perturbatively small. We have added further details and clarifications in sections 3 and 4 of the revised manuscript to elaborate on the validity of this approach in both attractor and non-attractor regimes. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The manuscript derives UV-mode solutions perturbatively to second order on a stochastic IR background using the standard Schwinger-Keldysh formalism, then decomposes noise into deterministic (from functional Taylor expansion) and stochastic (quadratic-noise) parts. The order-counting argument that quadratic-noise terms enter at the same perturbative order follows directly from the structure of the Schwinger-Keldysh equations and the perturbative expansion without any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. All steps remain independent of the target result and are externally verifiable against the underlying field-theory equations. No load-bearing step reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions of perturbative quantum field theory on a stochastic background and the Schwinger-Keldysh contour; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption UV modes can be solved perturbatively to second order on a stochastic IR background
Invoked to derive the noise decomposition and order counting.
standard math Schwinger-Keldysh formalism correctly captures the noise sourced by UV modes
Basis for determining the IR Langevin noise from UV evolution.

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discussion (0)

Forward citations

Cited by 1 Pith paper

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Stochastic inflation from a non-equilibrium renormalization group
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A generalized Fokker-Planck equation for stochastic inflation is derived from a Polchinski-type renormalization group flow on the density matrix, incorporating dissipative and diffusive corrections beyond the leading order.

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