de Sitter Infrared Physics and Stochastic Inflation

de Sitter infrared statements are meaningful only after the patch, state, field algebra, observable, gauge, regulator, order of limits, and duration are fixed. Stochastic inflation is a controlled leading-long-wavelength description in specified scalar regimes; it is not a universal replacement for in-in QFT, a solution of the gravitational observable problem, or an exact equilibrium description of finite slow roll.

Helpful background. Langevin fields and noise supplies stochastic normalization; influence functionals supplies reduced dynamics; cosmological loops and secular growth supplies in-in power counting; and Bunch–Davies and alpha-state diagnostics supplies the free-state distinction.

Infrared and stochastic control parameters

In the spatially flat patch,

$$ds^2=dt^2-e^{2Ht}d\mathbf x^2 =a^2(\eta)(d\eta^2-d\mathbf x^2), \qquad a=-\frac1{H\eta}.$$

With the site’s curvature convention $R_{\rm site}=-12H^2$, a scalar governed by $P=\Box+m^2+\xi R$ has

$$M^2=m^2-12\xi H^2, \qquad \nu^2=\frac94-\frac{M^2}{H^2}.$$

The standard massive Euclidean/Bunch–Davies state is distinct from the $M^2=0$ minimally coupled zero-mode problem. For $0<M^2\ll H^2$, the long-field relaxation time scales as $H^2/M^2$ e-folds; taking $M\to0$ before or after that time gives different answers. Allen’s state classification and zero-mode analysis make this scope distinction explicit Allen 1985, §§II–IV, pp. 3138–3147.

For a light spectator with fixed $H$, the leading sharp-window stochastic equation in cosmic time is

$$d\phi_L=-\frac{V'(\phi_L)}{3H}\,dt +\frac{H^{3/2}}{2\pi}\,dW_t, \qquad \mathbb E[dW_t^2]=dt.$$

Equivalently, with $\langle\zeta(t)\zeta(t')\rangle=H^3\delta(t-t')/(4\pi^2)$,

$$\partial_t P =\frac1{3H}\partial_\phi(V'P) +\frac{H^3}{8\pi^2}\partial_\phi^2P.$$

The coefficient is tied to a specified state and moving coarse-graining window. Smooth windows generate finite correlation time and calculable matching corrections. Under fixed $H$, zero probability current, no volume weighting, and normalizability, the stationary solution is

$$P_{\rm eq}(\phi)=\mathcal N \exp\!\left[-\frac{8\pi^2V(\phi)}{3H^4}\right].$$

These formulas were established for light scalar long modes and their leading infrared dynamics Starobinsky and Yokoyama 1994, §§II–III, Eqs. (2.1)–(3.7). Every extension in this chapter states which assumptions survive.

For a light quartic scalar, the diagrammatic correspondence with in-in QFT is established at leading infrared order, not for every gradient or unequal-time observable Garbrecht et al. 2015, §§III–V, Eqs. (35)–(75).

The structure map separates free-state and zero-mode questions from interacting resummation, scalar coarse-graining, open-system corrections, first passage, and gravitational observables.

Free-field state selection, zero modes, interacting logarithms, resummation, stochastic coarse-graining, open-system dynamics, and relational gravity are distinct calculations with different control parameters. Schematic; not to scale.

Routes through the chapter

1. de Sitter infrared regimes fixes patches, observables, regulators, and limit order.
2. Euclidean and Bunch–Davies free fields derives the standard massive scalar state.
3. Massless zero modes isolates the minimally coupled obstruction and restricted observables.
4. Interacting infrared logs identifies perturbative nonuniformity by object and symmetry.
5. Graviton infrared claims tests coordinate correlators against relational observables.
6. Infrared resummation compares large-N, stochastic, and diagrammatic mass scales.
7. Stochastic coarse-graining 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 restores colored noise, memory, and influence kernels.
10. Stationary and first-passage observables imposes current and boundary conditions.
11. QFT–stochastic matching states the renormalized observable subset that agrees.
12. Quasi-de Sitter validity separates slow-variation, finite-duration, and evidence errors.

Domain and failure conditions

This is the canonical comparison table for the chapter. Its evidence status is current through 10 August 2026 where a dispute is consequential.

Problem	Patch, state, and object	Regulator or split	Controlled output	Decisive check	Failure or handoff
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

The validity map emphasizes wrong-order limits, gauge-dependent graviton claims, unjustified white-noise or stationary approximations, and overextended QFT–stochastic equivalence.

Each infrared or stochastic result is licensed only for its declared observable, limit order, coarse-graining hierarchy, and evidence class. Schematic; not to scale.

References

• Allen, B., “Vacuum States in de Sitter Space,” Physical Review D 32, 3136–3149 (1985), doi:10.1103/PhysRevD.32.3136.
• Garbrecht, B., F. Gautier, G. Rigopoulos, and Y. Zhu, “Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de Sitter Space,” Physical Review D 91, 063520 (2015), doi:10.1103/PhysRevD.91.063520.
• 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.