de Sitter Infrared Physics and Stochastic Inflation

Light fields in de Sitter spacetime can accumulate long-wavelength fluctuations over time. Infrared resummation and stochastic inflation organize that accumulation in specified regimes, with state choice and the order of limits kept explicit.

Enter this chapter

Begin with de Sitter Infrared Regimes: States, Observables, Gauges, and Limits to identify the field, state, observable, and infrared limits of the calculation.

Helpful background. Langevin Field Equations and Noise and System–Environment Splits and Influence Functionals develop the definitions used by the chapter’s opening pages.

Topics

1. de Sitter Infrared Regimes: States, Observables, Gauges, and Limits — Fixes patches, observables, regulators, and limit order.
2. Euclidean and Bunch–Davies Free Fields in de Sitter — Derives the standard massive scalar state.
3. Massless Zero Modes and the Invariant-State Obstruction — Isolates the minimally coupled obstruction and restricted observables.
4. Interacting Infrared Logs and Secular Regimes — Identifies perturbative nonuniformity by object and symmetry.
5. Graviton Infrared Claims and Relational Observables — Tests coordinate correlators against relational observables.
6. Infrared Resummation and Dynamical Mass — Compares large-N, stochastic, and diagrammatic mass scales.
7. Stochastic Coarse-Graining and Its Derivation from QFT — Derives drift and noise from a moving split.
8. Langevin and Fokker–Planck Dynamics — Fixes calculus, measure, moments, and normalization.
9. Open-System Noise and Dissipation in de Sitter — Restores colored noise, memory, and influence kernels.
10. Stationary Distributions and First-Passage Observables — Imposes current and boundary conditions.
11. QFT–Stochastic Matching and Renormalization — States the renormalized observable subset that agrees.
12. Quasi-de Sitter Validity and Evidence Status — Separates slow-variation, finite-duration, and evidence errors.

How the ideas fit together

A moving separation between short and long wavelengths transfers fluctuations into a coarse-grained field. In suitable scalar regimes this yields a stochastic equation and a probability evolution that can resum leading infrared effects Starobinsky and Yokoyama 1994, §§ II–III.

A stationary probability distribution requires normalizability and compatible boundary conditions. Its existence does not settle a finite-duration slow-roll problem, a massless zero-mode obstruction, or a gravitational gauge question. Matching stochastic observables to renormalized QFT and controlling the late-time limit remain independent tasks.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Problem	Patch, state, and object	Regulator or split	Controlled output	Decisive check	Failure or further method needed
Infrared classification	Declared patch, state, field or local composite	Mass, volume, or invariant prescription	Limit-specific correlator	Exchange limit order	Noncommuting limits forbid a universal statement
Massive free field	Euclidean/BD state, $M^2>0$	Euclidean continuation and $i0$	Hadamard Wightman function	Canonical jump and coincidence form	Does not cover the minimal massless zero mode
Minimal zero mode	Global mode algebra or shift-invariant subalgebra	Small mass and finite volume kept distinct	Derivative or difference observables	Regulator agreement on the same algebra	No standard invariant Fock vacuum for $\phi$
Interacting logs	Specified in-in observable and interaction	UV subtraction plus declared IR regulator	Fixed-order secular interval	Compare $\lambda N^p$ with unity	A large log signals nonuniform perturbation, not automatically instability
Graviton infrared	Gauge, dressing, finite operational region	Residual-gauge and smearing prescription	Relational or curvature observable	Gauge/dressing comparison	Coordinate two-point growth alone is insufficient
Resummed scalar	Coupling, $N$, mass, equal- or unequal-time object	Large-N, 2PI, stochastic, or RG scheme	Method-specific infrared scale	Translate conventions and observables	Equal-time agreement does not prove universal dynamics
Coarse-grained scalar	BD-like short modes and window $W(k/k_c)$	$k_c=\epsilon aH$ with hierarchy	Leading drift and noise	Window variation after matching	White noise is not exact for a smooth split
Fokker–Planck evolution	Field coordinate, measure, Itô/Stratonovich choice	Time step and field grid	Normalized moments or density	Drift conversion and ensemble agreement	Convention-dependent density is rejected
Open-system correction	Declared system/environment and initial state	Influence kernels and correlation time	Noise, dissipation, memory	Markov hierarchy and covariance positivity	KMS fluctuation–dissipation is not generic
Stationary or first passage	Fixed $H$, normalizable current condition, boundaries	Absorbing/reflecting domain	Equilibrium density or hitting distribution	Relaxation and boundary variation	No volume-weighted eternal-inflation measure claim
QFT matching	Same renormalized in-in observable and order	UV scheme plus stochastic matching scale	Leading-IR equality for a stated subset	Unequal-time and gradient tests	One variance does not establish full-theory equivalence
Quasi-de Sitter transport	$H(t)$, slow parameters, state, duration	Local window and time-dependent matching	Finite-time prediction with two errors	Vary duration and formalism	Exact-de Sitter equilibrium may never be reached

Where to go next

Continue to Cosmological Analyticity and Bootstrap to distinguish wavefunction coefficients, in/in correlators, and scattering amplitudes.

References

• Starobinsky, A. A., and J. Yokoyama, “Equilibrium State of a Self-Interacting Scalar Field in the de Sitter Background,” Physical Review D 50, 6357–6368 (1994), doi:10.1103/PhysRevD.50.6357.