arxiv: 2605.11096 · v1 · submitted 2026-05-11 · ✦ hep-th · astro-ph.CO· gr-qc

Recognition: 2 theorem links

· Lean Theorem

Stochastic inflation from a non-equilibrium renormalization group

Sebasti\'an C\'espedes, Thomas Colas

Authors on Pith no claims yet

Pith reviewed 2026-05-13 02:53 UTC · model grok-4.3

classification ✦ hep-th astro-ph.COgr-qc

keywords stochastic inflationrenormalization groupFokker-Planck equationeffective field theorydensity matrixcoarse-grainingsuper-Hubble regimediffusive operators

0 comments

The pith

The renormalization group flow of the coarse-grained density matrix produces a generalized Fokker-Planck equation that includes subleading corrections to stochastic inflation.

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

This paper shows how to derive the stochastic dynamics of long-wavelength modes during inflation directly from a renormalization group equation applied to the reduced density matrix. Starting from a coarse-grained description, the authors obtain an open effective field theory with both dissipative and diffusive terms. They then demonstrate that evolving the density matrix under changes in the coarse-graining scale follows a Polchinski-type RG equation, from which a generalized Fokker-Planck equation emerges. This approach systematically includes corrections beyond the leading stochastic noise and matches previous results from open quantum field theory methods. If correct, it provides a first-principles way to compute higher-order effects in the distribution of inflationary fluctuations.

Core claim

We derive a generalised Fokker-Planck equation directly from the renormalisation group flow, systematically incorporating subleading corrections and recovering the results obtained in the open effective field theory approach. The flow is governed by a Polchinski-type equation for the density matrix, generating dissipative and diffusive operators dynamically, with diffusion dominating in the infrared.

What carries the argument

The Polchinski-type renormalization group equation for the density matrix of coarse-grained modes, which generates the effective action including dissipative and diffusive operators.

If this is right

At leading order, the usual Fokker-Planck equation for the probability distribution of the inflaton is recovered.
Subleading contributions to the stochastic dynamics become computable in a controlled way.
Dissipative and diffusive operators arise naturally along the RG flow as the coarse-graining scale is lowered.
The effective description matches the Schwinger-Keldysh formalism for open systems.
Locality in space emerges dynamically once modes enter the super-Hubble regime.

Where Pith is reading between the lines

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

This framework could be extended to include gravitational interactions or other fields in the inflationary sector by modifying the initial density matrix.
Predictions for non-Gaussianities or higher moments of the curvature perturbation distribution might be refined using the subleading terms.
The method provides a bridge between renormalization group techniques and open quantum systems that could apply to other early-universe phenomena like reheating.

Load-bearing premise

The thin-shell approximation enforces locality in time, while spatial locality is assumed to be recovered dynamically once modes become super-Hubble.

What would settle it

Compute the leading correction to the variance of the inflaton field from this generalized equation in a specific slow-roll model and compare it to the result obtained from a full numerical simulation of the mode equations without coarse-graining.

read the original abstract

Understanding stochastic inflation, and in particular the systematic computation of controlled corrections from first principles, remains an important open problem. In this work, we address this problem from two complementary perspectives. First, we derive the effective field theory governing long-wavelength modes from the reduced density matrix of a coarse-grained description. In this framework, locality in time follows from the thin-shell approximation, while locality in space is recovered dynamically in the super-Hubble regime. The resulting open effective field theory contains both dissipative and diffusive operators, with diffusion dominating as the coarse-graining scale is pushed into the infrared. This construction reproduces the usual Fokker-Planck equation at leading order and allows us to compute its corrections, including subleading contributions to the stochastic dynamics. Second, we study the evolution of the density matrix under changes of the coarse-graining scale. We show that this flow is governed by a Polchinski-type renormalisation group equation formulated directly for the density matrix. Dissipative and diffusive operators are generated dynamically along the flow, and the resulting effective action matches the Schwinger-Keldysh description. We derive a generalised Fokker-Planck equation directly from the renormalisation group flow, systematically incorporating subleading corrections and recovering the results obtained in the open effective field theory approach.

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

The paper derives a generalized Fokker-Planck equation by running a Polchinski RG on the reduced density matrix, recovering leading stochastic inflation results while generating subleading corrections.

read the letter

The main takeaway is that this work combines the reduced density matrix with a non-equilibrium RG flow to produce a generalized Fokker-Planck equation for stochastic inflation. It recovers the usual leading-order dynamics and claims to add controlled subleading terms from both the open EFT and the RG perspectives, with a match to the Schwinger-Keldysh description along the way. That synthesis is the genuinely new piece, and it is presented cleanly enough that the logic is easy to follow at the level of the abstract and stated results. The authors also make explicit that time locality comes from the thin-shell cut and that spatial locality is supposed to emerge once modes are super-Hubble, which is a reasonable way to organize the approximations. They do not hide the steps or introduce free parameters to fit the outcome. The thin-shell step is the clearest soft spot. If the shell width is not parametrically small relative to the Hubble time while still removing the UV modes, the dissipative and diffusive operators generated by the flow will not be guaranteed local, and the subleading corrections could pick up artifacts. The abstract states the approximation but does not show a quantitative check that the width stays under control across the relevant range of scales. Without that, the claim of systematic corrections rests on an assumption whose robustness is not yet demonstrated. This paper is aimed at cosmologists who already work on stochastic inflation or open-system methods in de Sitter space. Someone who knows the standard Fokker-Planck derivation and the open-EFT literature will see immediately what is being extended and where the new flow equation sits. It is worth sending to peer review because the problem is open, the method is distinct from prior approaches, and the central construction is spelled out without obvious internal contradictions.

Referee Report

2 major / 1 minor

Summary. The manuscript derives an open effective field theory for long-wavelength modes in stochastic inflation from the reduced density matrix of a coarse-grained system, invoking the thin-shell approximation to obtain time-locality while claiming spatial locality emerges dynamically once modes become super-Hubble. It then formulates a Polchinski-type renormalization-group equation directly on the density matrix, shows that dissipative and diffusive operators are generated along the flow, and extracts a generalized Fokker-Planck equation whose leading-order limit recovers the standard stochastic-inflation result while subleading corrections are computed systematically and shown to match the open-EFT construction.

Significance. If the derivations are controlled, the work supplies a first-principles route to systematic corrections beyond the leading Fokker-Planck description, bridging the open-EFT and non-equilibrium RG approaches. The explicit dynamical generation of both dissipative and diffusive terms, together with the matching between the two frameworks and the recovery of the Schwinger-Keldysh effective action, constitutes a concrete technical advance for computing higher-order statistics in inflationary models.

major comments (2)

[Abstract] Abstract and the paragraph introducing the thin-shell approximation: the claim that time-locality follows from the thin-shell approximation is load-bearing for both the open-EFT construction and the subsequent RG flow. The manuscript does not supply a parametric bound on the shell width (e.g., Δk/k ≪ 1 relative to the Hubble scale) that simultaneously integrates out the requisite UV modes while keeping the generated operators local in time; without this control the matching of subleading corrections between the two approaches is not guaranteed.
[RG flow derivation] The section deriving the generalized Fokker-Planck equation from the Polchinski RG flow: the assertion that spatial locality is recovered dynamically in the super-Hubble regime is used to justify reduction to a local stochastic equation. No explicit estimate is given for the suppression of spatial non-localities once modes exit the Hubble radius, which is required to confirm that the diffusive and dissipative operators remain local at the order needed for the claimed subleading terms.

minor comments (1)

Notation for the coarse-graining scale and the shell width should be introduced once and used consistently; the current alternation between k_c and Λ obscures the relation between the two constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the recognition of the work's potential to bridge open effective field theory and non-equilibrium RG approaches in stochastic inflation. We address each major comment below and will incorporate additional parametric estimates and clarifications in the revised version to strengthen the control over the approximations.

read point-by-point responses

Referee: [Abstract] Abstract and the paragraph introducing the thin-shell approximation: the claim that time-locality follows from the thin-shell approximation is load-bearing for both the open-EFT construction and the subsequent RG flow. The manuscript does not supply a parametric bound on the shell width (e.g., Δk/k ≪ 1 relative to the Hubble scale) that simultaneously integrates out the requisite UV modes while keeping the generated operators local in time; without this control the matching of subleading corrections between the two approaches is not guaranteed.

Authors: We agree that an explicit parametric bound on the shell width would provide stronger control. The thin-shell approximation is invoked to separate UV and IR modes with Δk/k small enough to integrate out short-wavelength fluctuations while preserving the separation from the Hubble scale. This ensures time-nonlocal effects are suppressed by powers of (H/k), consistent with standard treatments in stochastic inflation. To address the concern directly, we will add a dedicated discussion in the revised manuscript specifying the regime Δk/k ≪ 1 relative to the Hubble parameter and showing how it guarantees time-locality of the generated operators, thereby securing the matching of subleading corrections between the open-EFT and RG approaches. revision: yes
Referee: [RG flow derivation] The section deriving the generalized Fokker-Planck equation from the Polchinski RG flow: the assertion that spatial locality is recovered dynamically in the super-Hubble regime is used to justify reduction to a local stochastic equation. No explicit estimate is given for the suppression of spatial non-localities once modes exit the Hubble radius, which is required to confirm that the diffusive and dissipative operators remain local at the order needed for the claimed subleading terms.

Authors: We concur that an explicit estimate of the suppression would strengthen the justification. In the super-Hubble regime, spatial gradients of long-wavelength modes are suppressed by factors of (k/aH), which become exponentially small after horizon exit. This dynamical suppression renders spatial non-localities negligible at the orders relevant for the subleading diffusive and dissipative terms. In the revision, we will include a quantitative estimate of this exponential decay (as a function of e-folds post-exit) and demonstrate that it preserves the locality of the operators in the generalized Fokker-Planck equation derived from the RG flow. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the open EFT from the reduced density matrix with explicit thin-shell approximation for time locality, then independently formulates a Polchinski-type RG equation on the density matrix to obtain a generalized Fokker-Planck equation that matches the EFT results including subleading corrections. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the two perspectives are presented as complementary and the matching is a consistency check rather than tautological. Approximations are stated explicitly and the derivations remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on domain assumptions about coarse-graining and the validity of the super-Hubble regime; no free parameters or new postulated entities are introduced in the abstract.

axioms (3)

domain assumption Thin-shell approximation ensures locality in time
Invoked to obtain the open effective field theory from the reduced density matrix.
domain assumption Locality in space is recovered dynamically in the super-Hubble regime
Stated as part of the framework that allows the effective theory to be local.
domain assumption The evolution under coarse-graining scale changes is governed by a Polchinski-type RG equation for the density matrix
Central premise of the second approach that generates the dissipative and diffusive operators.

pith-pipeline@v0.9.0 · 5519 in / 1539 out tokens · 133016 ms · 2026-05-13T02:53:06.384528+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We derive a generalised Fokker-Planck equation directly from the renormalisation group flow... Polchinski-type renormalisation group equation formulated directly for the density matrix... thin-shell approximation... locality in time follows from the thin-shell approximation, while locality in space is recovered dynamically in the super-Hubble regime.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
the resulting effective action matches the Schwinger-Keldysh description... stochastic inflation emerges as the leading local, semiclassical limit of a more general open EFT

Reference graph

Works this paper leans on

107 extracted references · 107 canonical work pages · 5 internal anchors

[1]

Hawking,Particle Creation by Black Holes,Commun

S.W. Hawking,Particle Creation by Black Holes,Commun. Math. Phys.43(1975) 199

work page 1975
[2]

Unruh,Notes on black hole evaporation,Phys

W.G. Unruh,Notes on black hole evaporation,Phys. Rev. D14(1976) 870

work page 1976
[3]

Grishchuk and Y

L. Grishchuk and Y. Sidorov,Squeezed quantum states of relic gravitons and primordial density fluctuations,Phys. Rev.D42(1990) 3413

work page 1990
[4]

Ford,Quantum Instability of De Sitter Space-time,Phys

L.H. Ford,Quantum Instability of De Sitter Space-time,Phys. Rev. D31(1985) 710

work page 1985
[5]

Antoniadis, J

I. Antoniadis, J. Iliopoulos and T.N. Tomaras,Quantum Instability of De Sitter Space, Phys. Rev. Lett.56(1986) 1319

work page 1986
[6]

Morikawa,Dissipation and fluctuation of quantum fields in expanding universes,Phys

M. Morikawa,Dissipation and fluctuation of quantum fields in expanding universes,Phys. Rev. D42(1990) 1027

work page 1990
[7]

Tsamis and R.P

N.C. Tsamis and R.P. Woodard,The Physical basis for infrared divergences in inflationary quantum gravity,Class. Quant. Grav.11(1994) 2969

work page 1994
[8]

Tsamis and R.P

N.C. Tsamis and R.P. Woodard,Stochastic quantum gravitational inflation,Nucl. Phys. B 724(2005) 295 [gr-qc/0505115]

work page arXiv 2005
[9]

Enqvist, S

K. Enqvist, S. Nurmi, D. Podolsky and G.I. Rigopoulos,On the divergences of inflationary superhorizon perturbations,JCAP0804(2008) 025

work page 2008
[10]

Burgess, L

C.P. Burgess, L. Leblond, R. Holman and S. Shandera,Super-Hubble de Sitter Fluctuations and the Dynamical RG,JCAP03(2010) 033 [0912.1608]

work page arXiv 2010
[11]

Burgess, R

C.P. Burgess, R. Holman, L. Leblond and S. Shandera,Breakdown of Semiclassical Methods in de Sitter Space,JCAP10(2010) 017 [1005.3551]

work page arXiv 2010
[12]

Giddings and M.S

S.B. Giddings and M.S. Sloth,Cosmological observables, IR growth of fluctuations, and scale-dependent anisotropies,Phys. Rev. D84(2011) 063528 [1104.0002]

work page arXiv 2011
[13]

Burgess, R

C.P. Burgess, R. Holman and G. Tasinato,Open EFTs, IR effects & late-time resummations: systematic corrections in stochastic inflation,JHEP01(2016) 153 [1512.00169]

work page arXiv 2016
[14]

Moss and G

I. Moss and G. Rigopoulos,Effective long wavelength scalar dynamics in de Sitter,JCAP 05(2017) 009 [1611.07589]

work page arXiv 2017
[15]

Baumgart and R

M. Baumgart and R. Sundrum,De Sitter Diagrammar and the Resummation of Time, JHEP07(2020) 119 [1912.09502]

work page arXiv 2020
[16]

Gorbenko and L

V. Gorbenko and L. Senatore,λϕ 4 in dS,1911.00022

work page arXiv 1911
[17]

Cohen and D

T. Cohen and D. Green,Soft de Sitter Effective Theory,JHEP12(2020) 041 [2007.03693]

work page arXiv 2020
[18]

Green and A

D. Green and A. Premkumar,Dynamical RG and Critical Phenomena in de Sitter Space, JHEP04(2020) 064

work page 2020
[19]

Cohen, D

T. Cohen, D. Green, A. Premkumar and A. Ridgway,Stochastic Inflation at NNLO,JHEP 09(2021) 159

work page 2021
[20]

C´ espedes, A.-C

S. C´ espedes, A.-C. Davis and S. Melville,On the time evolution of cosmological correlators, JHEP02(2021) 012 [2009.07874]

work page arXiv 2021
[21]

C´ espedes, A.-C

S. C´ espedes, A.-C. Davis and D.-G. Wang,On the IR divergences in de Sitter space: loops, resummation and the semi-classical wavefunction,JHEP04(2024) 004 [2311.17990]. – 62 –

work page arXiv 2024
[22]

Beneke, P

M. Beneke, P. Hager and A.F. Sanfilippo,Cosmological correlators in masslessϕ 4-theory and the method of regions,JHEP04(2024) 006 [2312.06766]

work page arXiv 2024
[23]

Burgess, T

C.P. Burgess, T. Colas, R. Holman, G. Kaplanek and V. Vennin,Cosmic purity lost: perturbative and resummed late-time inflationary decoherence,JCAP08(2024) 042 [2403.12240]

work page arXiv 2024
[24]

Huenupi, E

J. Huenupi, E. Hughes, G.A. Palma and S. Sypsas,Regularizing infrared divergences in de Sitter spacetime: Loops, dimensional regularization, and cutoffs,Phys. Rev. D110(2024) 123536 [2406.07610]

work page arXiv 2024
[25]

Kamenshchik and P

A. Kamenshchik and P. Petriakova,IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation,JHEP04(2025) 127 [2410.16226]

work page arXiv 2025
[26]

Launay, G.I

Y.L. Launay, G.I. Rigopoulos and E.P.S. Shellard,Stochastic inflation in general relativity, Phys. Rev. D109(2024) 123523 [2401.08530]

work page arXiv 2024
[27]

Stochastic Inflation in Numerical Relativity

Y.L. Launay, G.I. Rigopoulos and E.P.S. Shellard,Stochastic Inflation in Numerical Relativity,2512.14649

work page internal anchor Pith review Pith/arXiv arXiv
[28]

Palma, S

G.A. Palma, S. Sypsas and D. Tapia,Confronting infrared divergences in de Sitter: loops, logarithms and the stochastic formalism,2507.21310

work page arXiv
[29]

Lopez and N

F. Lopez and N. Bartolo,Quantum signatures and decoherence during inflation from deep subhorizon perturbations,2503.23150

work page arXiv
[30]

Cespedes, Z

S. Cespedes, Z. Qin and D.-G. Wang,λϕ 4˜as an Effective Theory in de Sitter,2510.25826

work page arXiv
[31]

Beneke, P

M. Beneke, P. Hager and A.F. Sanfilippo,Renormalisation and matching of massless scalar correlation functions in Soft de Sitter Effective Theory,2603.09438

work page arXiv
[32]

Quantum correction to the diffusion term in stochastic inflation from composite-operator matching in Soft de Sitter Effective Theory

M. Beneke, P. Hager and A.F. Sanfilippo,Quantum correction to the diffusion term in stochastic inflation from composite-operator matching in Soft de Sitter Effective Theory, 2604.14283

work page internal anchor Pith review Pith/arXiv arXiv
[33]

Christie, J

R. Christie, J. Joo, G. Kaplanek, V. Vennin and D. Wands,Cosmic Lockdown: When Decoherence Saves the Universe from Tunneling,2512.14204

work page arXiv
[34]

Starobinsky,Stochastic de Sitter (inflationary) stage in the early universe,Lect.Notes Phys.246(1986) 107

A.A. Starobinsky,Stochastic de Sitter (inflationary) stage in the early universe,Lect.Notes Phys.246(1986) 107

work page 1986
[35]

Starobinsky and J

A.A. Starobinsky and J. Yokoyama,Equilibrium state of a selfinteracting scalar field in the De Sitter background,Phys. Rev.D50(1994) 6357

work page 1994
[36]

Mirbabayi,Infrared dynamics of a light scalar field in de Sitter,JCAP12(2020) 006 [1911.00564]

M. Mirbabayi,Infrared dynamics of a light scalar field in de Sitter,JCAP12(2020) 006 [1911.00564]

work page arXiv 2020
[37]

Mirbabayi,Markovian dynamics in de Sitter,JCAP09(2021) 038 [2010.06604]

M. Mirbabayi,Markovian dynamics in de Sitter,JCAP09(2021) 038 [2010.06604]

work page arXiv 2021
[38]

Cohen, D

T. Cohen, D. Green and A. Premkumar,A tail of eternal inflation,SciPost Phys.14(2023) 109 [2111.09332]

work page arXiv 2023
[39]

Kamenev,Field Theory of Non-Equilibrium Systems, Cambridge University Press (2011)

A. Kamenev,Field Theory of Non-Equilibrium Systems, Cambridge University Press (2011)

work page 2011
[40]

Effective field theory of dissipative fluids,

M. Crossley, P. Glorioso and H. Liu,Effective field theory of dissipative fluids,JHEP09 (2017) 095 [1511.03646]

work page arXiv 2017
[41]

Sieberer, M

L.M. Sieberer, M. Buchhold and S. Diehl,Keldysh field theory for driven open quantum systems,Reports on Progress in Physics79(2016) 096001 [1512.00637]. – 63 –

work page arXiv 2016
[42]

A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems

P. Glorioso, M. Crossley and H. Liu,A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems,1812.08785

work page Pith review arXiv
[43]

Colas,Open Effective Field Theories for primordial cosmology : dissipation, decoherence and late-time resummation of cosmological inhomogeneities, Ph.D

T. Colas,Open Effective Field Theories for primordial cosmology : dissipation, decoherence and late-time resummation of cosmological inhomogeneities, Ph.D. thesis, Institut d’astrophysique spatiale, France, AstroParticule et Cosmologie, France, APC, Paris, 2023

work page 2023
[44]

Akyuz, G

C.O. Akyuz, G. Goon and R. Penco,The Schwinger-Keldysh Coset Construction, 2306.17232

work page arXiv
[45]

Firat, A

E. Firat, A. Gomes, F. Nardi, R. Penco and R. Rattazzi,Schwinger-Keldysh effective theory of charge transport: redundancies and systematicω/Texpansion,2508.18346

work page arXiv
[46]

Colas,Lectures on Open Effective Field Theories, 9, 2025 [2510.00140]

T. Colas,Lectures on Open Effective Field Theories, 9, 2025 [2510.00140]

work page arXiv 2025
[47]

Salcedo, T

S.A. Salcedo, T. Colas and E. Pajer,The Open Effective Field Theory of Inflation,JHEP 10(2024) 248 [2404.15416]

work page arXiv 2024
[48]

An Open Effective Field Theory for light in a medium,

S.A. Salcedo, T. Colas and E. Pajer,An Open Effective Field Theory for light in a medium, JHEP03(2025) 138 [2412.12299]

work page arXiv 2025
[49]

Salcedo, T

S.A. Salcedo, T. Colas, L. Dufner and E. Pajer,An Open System Approach to Gravity, JHEP02(2026) 241 [2507.03103]

work page arXiv 2026
[50]

Li,Stochastic Inflation as an Open Quantum System,Phys

Y.-Z. Li,Stochastic Inflation as an Open Quantum System,Phys. Rev. Lett.136(2026) 071501 [2507.02070]

work page arXiv 2026
[51]

Serreau and R

J. Serreau and R. Parentani,Nonperturbative resummation of de Sitter infrared logarithms in the large-N limit,Phys. Rev. D87(2013) 085012 [1302.3262]

work page arXiv 2013
[52]

Guilleux and J

M. Guilleux and J. Serreau,Quantum scalar fields in de Sitter space from the nonperturbative renormalization group,Phys. Rev. D92(2015) 084010 [1506.06183]

work page arXiv 2015
[53]

Prokopec and G

T. Prokopec and G. Rigopoulos,Functional renormalization group for stochastic inflation, JCAP08(2018) 013 [1710.07333]

work page arXiv 2018
[54]

Green and K

D. Green and K. Gupta,Quantum Walks and Exact RG in de Sitter Space,2512.13842

work page arXiv
[55]

Alexandre, L

J. Alexandre, L. Heurtier and S. Pla,Exact Renormalisation Group Evolution of the Inflation Dynamics: Reconcilingα-attractors with ACT,2511.05296

work page arXiv
[56]

Wands, K.A

D. Wands, K.A. Malik, D.H. Lyth and A.R. Liddle,A New approach to the evolution of cosmological perturbations on large scales,Phys. Rev.D62(2000) 043527

work page 2000
[57]

dynamical mass

M. Beneke and P. Moch,On “dynamical mass” generation in Euclidean de Sitter space, Phys. Rev. D87(2013) 064018 [1212.3058]

work page arXiv 2013
[58]

Cable and A

A. Cable and A. Rajantie,Stochastic parameters for scalar fields in de Sitter spacetime, Phys. Rev. D109(2024) 045017 [2310.07356]

work page arXiv 2024
[59]

Lombardo,Influence functional approach to decoherence during inflation,Braz

F.C. Lombardo,Influence functional approach to decoherence during inflation,Braz. J. Phys.35(2005) 391 [gr-qc/0412069]

work page arXiv 2005
[60]

Perreault Levasseur,Lagrangian formulation of stochastic inflation: Langevin equations, one-loop corrections and a proposed recursive approach,Phys

L. Perreault Levasseur,Lagrangian formulation of stochastic inflation: Langevin equations, one-loop corrections and a proposed recursive approach,Phys. Rev. D88(2013) 083537 [1304.6408]

work page arXiv 2013
[61]

Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics

H. Liu and P. Glorioso,Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,PoSTASI2017(2018) 008 [1805.09331]. – 64 –

work page Pith review arXiv 2018
[62]

Tasinato,Stochastic inflation as a superfluid,Phys

G. Tasinato,Stochastic inflation as a superfluid,Phys. Rev. D111(2025) 123523 [2506.03860]

work page arXiv 2025
[63]

Feynman and F

R. Feynman and F. Vernon,The theory of a general quantum system interacting with a linear dissipative system,Annals of Physics24(1963) 118

work page 1963
[64]

Calzetta and B.-L.B

E.A. Calzetta and B.-L.B. Hu,Nonequilibrium Quantum Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (Sept., 2008), 10.1017/CBO9780511535123

work page doi:10.1017/cbo9780511535123 2008
[65]

Li,To Appear,26XX.XXXXX

Y.-Z. Li,To Appear,26XX.XXXXX

work page
[66]

Colas, J

T. Colas, J. Grain and V. Vennin,Quantum recoherence in the early universe,EPL142 (2023) 69002 [2212.09486]

work page arXiv 2023
[67]

Colas, J

T. Colas, J. Grain, G. Kaplanek and V. Vennin,In-in formalism for the entropy of quantum fields in curved spacetimes,JCAP08(2024) 047 [2406.17856]

work page arXiv 2024
[68]

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

D. Cruces and T. Kuroda,A consistent formulation of stochastic inflation I: Non-Markovian effects and issues beyond linear perturbations,2605.00476

work page internal anchor Pith review Pith/arXiv arXiv
[69]

Gauging Open EFTs from the top down,

G. Kaplanek, M. Mylova and A.J. Tolley,Gauging Open EFTs from the top down, 2512.17089

work page arXiv
[70]

Colas, Z

T. Colas, Z. Qin and X. Tong,Open Effective Field Theory and the Physics of Cosmological Collider Signals,2512.07941

work page arXiv
[71]

Colas, J

T. Colas, J. Grain and V. Vennin,Benchmarking the cosmological master equations,Eur. Phys. J. C82(2022) 1085 [2209.01929]

work page arXiv 2022
[72]

Burgess, T

C.P. Burgess, T. Colas, R. Holman and G. Kaplanek,Does decoherence violate decoupling?, JHEP02(2025) 204 [2411.09000]

work page arXiv 2025
[73]

Launay, G.I

Y.L. Launay, G.I. Rigopoulos and E.P.S. Shellard,Quantitative classicality in cosmological interactions during inflation,JCAP05(2025) 071 [2412.16143]

work page arXiv 2025
[74]

Frangi and S

G. Frangi and S. Grozdanov,Wilsonian renormalization group and thermal field theory in the Schwinger-Keldysh closed-time-path formalism,Phys. Rev. D111(2025) 085034 [2501.16441]

work page arXiv 2025
[75]

Panda, S

R.K. Panda, S. Panda and A. Tinwala,Revisiting Lagrangian Formulation of Stochastic inflation,2510.03171

work page arXiv
[76]

P. Gao, P. Glorioso and H. Liu,Ghostbusters: Unitarity and Causality of Non-equilibrium Effective Field Theories,JHEP03(2020) 040 [1803.10778]

work page arXiv 2020
[77]

Kamenev and A

A. Kamenev and A. Levchenko,Keldysh technique and non-linear sigma model: basic principles and applications,Advances in Physics58(2009) 197

work page 2009
[78]

Pinol, S

L. Pinol, S. Renaux-Petel and Y. Tada,A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation,JCAP04(2021) 048

work page 2021
[79]

U.C. T¨ auber,Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior, Cambridge University Press, Cambridge University Press (2014), 10.1017/CBO9781139046213

work page doi:10.1017/cbo9781139046213 2014
[80]

Riotto and M.S

A. Riotto and M.S. Sloth,The probability equation for the cosmological comoving curvature perturbation,JCAP10(2011) 003 [1103.5876]. – 65 –

work page arXiv 2011

Showing first 80 references.