Lagrangian Formulation of Stochastic Inflation

• Stochastic inflation is formulated by modeling long-wavelength modes with a Langevin equation that incorporates quantum-induced stochastic noise.
• The Hamiltonian phase-space extension introduces both field and momentum variables, allowing quantification of decoherence using phase space area (Δk) and von Neumann entropy.
• This approach connects the quantum-to-classical transition of cosmological perturbations with observable CMB acoustic oscillations and nonlinear interaction effects.

Stochastic inflation, in its Lagrangian formulation, represents the long-wavelength (super-Hubble) evolution of inflationary fluctuations as a dynamical system subject to stochastic noise induced by quantum fluctuations of the inflaton and metric perturbations. This approach systematically bridges the quantum field-theoretic description of inflationary perturbations with their effective, classicalized stochastic evolution—enabling a nonperturbative treatment of infrared effects, decoherence, and quantum-to-classical transition of cosmological perturbations.

1. Standard Lagrangian (Starobinsky) Approach

The foundational method is the Starobinsky stochastic inflation framework, in which the infrared (coarse-grained) inflaton field $\phi(t, x)$ obeys a Langevin equation: $$\dot{\phi}(t, x) = -\frac{1}{3H} V'(\phi) + F(t, x)$$ where $V'(\phi)$ is the gradient of the potential, $H$ the Hubble parameter, and $F(t, x)$ a stochastic source (white noise) that models the continual influx of quantum fluctuations from sub-Hubble to super-Hubble scales. The noise is characterized by

$$\langle F(t, x) F(t', x) \rangle = \frac{H^3}{4\pi^2} \delta(t - t')$$

This formulation captures leading-order infrared divergences and secular growth in inflationary correlators, but it does not account for conjugate momentum or a full phase-space description. There is no mechanism to compute a phase space area or a quantitative measure of decoherence.

2. Hamiltonian Phase-Space Generalization and Decoherence

To address these limitations, the classical phase space (Hamiltonian) formulation extends the coarse-graining to include both the field $\phi$ and its canonical momentum $\pi_\phi$. The Hamiltonian equations for a free field (ignoring spatial gradients) are: $$\begin{aligned} \dot{\phi} &= \frac{1}{a^3} \pi_\phi + F_1(t, x) \ \dot{\pi}_\phi &= a^3 \frac{\nabla^2}{a^2} \phi + F_2(t, x) \end{aligned}$$ Here, the noise terms $F_1$ and $F_2$ are sourced by integrating out sub-Hubble modes, with their correlators determined by the choice of coarse-graining window function.

A crucial outcome of the phase space description is the ability to define the phase space area for a mode $k$: $$\Delta_k^2 = 4 \left[ \langle |\phi(k)|^2 \rangle \langle |\pi_\phi(k)|^2 \rangle - \langle \frac{1}{2} \{ \phi(k), \pi_\phi(k) \} \rangle^2 \right]$$ For a free Gaussian state, $\Delta_k = 1$, corresponding to zero von Neumann entropy. Growth of $\Delta_k > 1$ signals decoherence, with the Gaussian von Neumann entropy given by: $$S_g = \int \frac{d^3k}{(2\pi)^3} \left\{ \frac{\Delta_k+1}{2} \ln\left(\frac{\Delta_k+1}{2}\right) - \frac{\Delta_k-1}{2} \ln\left(\frac{\Delta_k-1}{2}\right) \right\}$$ This formalism quantifies the quantum-to-classical transition and allows direct computation of the entropy increase due to interactions.

3. Noise Structure: Instant Quantum Noise versus Classical White Noise

The stochastic noise injected into the phase space system has a quantum origin tied to the horizon crossing of modes. The formalism distinguishes between two cases:

• Classical white noise: Noise sources are commuting classical variables. For a quantum harmonic oscillator, this provides continuous kicks that steadily increase phase space area: $\Delta \sim 1 + \mathrm{const} \cdot t$.
• Instant quantum noise: Noise sources act as delta-function kicks in time and may not commute, reflecting their operator nature. In stochastic inflation, horizon crossing leads to such instant noise with correlators engineered by the time-derivative of the coarse-graining window. For a free field, this maintains $\Delta_k = 1$ for super-Hubble modes—no decoherence in the free, non-interacting case, even with stochastic injection.

4. Role of Interactions and Growth of Entropy

When self-interactions (e.g., $V(\phi) = \lambda \phi^4/4!$) are included, the phase space Hamiltonian equations acquire nonlinear terms,

$$\begin{aligned} \dot{\phi} &= \frac{1}{a^3} \pi_\phi + F_1(t, x) \ \dot{\pi}_\phi &= a \nabla^2 \phi + F_2(t, x) - a^3 \frac{\lambda}{3!} \phi^3 \end{aligned}$$

Stochastic noise is typically treated as classical (random Gaussian variables), but the nonlinearity in the equations induces mode coupling among long-wavelength degrees of freedom so that higher-order (non-Gaussian) correlators, otherwise inaccessible, become dynamically important.

The consequence is that the phase space area $\Delta_k$ grows above unity and the Gaussian von Neumann entropy $S_g$ increases, quantitatively capturing the onset and degree of decoherence. Since decoherence is driven entirely by these nonlinear interactions, its growth can be linked to physically meaningful observables.

5. Connection to CMB Acoustic Oscillations

The degree of decoherence during inflation is not merely a formal property—it has direct phenomenological implications. Specifically, the entropy transfer from quantum to classical perturbations affects the statistical properties of primordial fluctuations, including those that seed cosmic microwave background (CMB) acoustic oscillations. A higher decoherence level (larger $S_g$, greater $\Delta_k$) can lead to observable modifications in CMB acoustic peak structure, such as shifts in the first peak position or changes in the relative amplitudes of secondary peaks, through modifications of the phase space distribution of scalar perturbations.

6. Pedagogical Illustration: Quantum Stochastic Harmonic Oscillator

A quantum harmonic oscillator subject to both deterministic and stochastic forces provides an analytic testbed for the method. Classical white noise induces a linear time-dependent phase space area, mimicking ordinary Brownian motion. For instant quantum noise (delta-function in time, with nontrivial commutator), there is an immediate jump to a nonzero phase space area reflecting a quantum "kick." In stochastic inflation, the noise injected at horizon crossing for each mode aligns with this "instant quantum noise," but for free fields in de Sitter space, the minimal state area is preserved and pure states are maintained until interactions are included.

7. Summary Table: Lagrangian versus Hamiltonian Formulations

Feature	Lagrangian (Starobinsky)	Hamiltonian Phase Space
Coarse-grained variable(s)	$\phi$	$(\phi, \pi_\phi)$
Noise injection	Only $\phi$ equation	Both $\phi$ and $\pi_\phi$ equations
Decoherence quantification	Not accessible	Via phase space area $\Delta_k$ and $S_g$
Interactions and entropy growth	Not resolved (mean field only)	Decoherence/entropy grows through nonlinearity
CMB connection	Infrared correlators only (no entropy)	Decoherence affects CMB acoustic structure

8. Significance and Outlook

The classical phase space (Hamiltonian) formulation of stochastic inflation enables first-principles and quantitative study of decoherence of cosmological perturbations. It extends the powerful stochastic method beyond the Lagrangian (single-field) regime, providing a concrete, computable measure of entropy growth and quantum-to-classical transition. This approach is essential for precision cosmology, as it links foundational aspects of quantum field theory in curved spacetime with the statistical properties of primordial fluctuations observable in the CMB. The method also provides a systematic framework for further generalizations to multi-field inflation, non-Gaussianity, and scenarios with significant nonlinear evolution (Weenink et al., 2011).