Stochastic Inflation at NNLO

TL;DR

This work develops a systematic, EFT-based approach (SdSET) to Stochastic Inflation in de Sitter space, enabling the first NNLO corrections to the stochastic dynamics of a light scalar. By performing one-loop matching of a UV $\lambda\phi^4$ theory onto SdSET, the authors derive a two-loop anomalous dimension that generates a universal non-Gaussian noise term and compute the resulting NNLO Fokker-Planck equation, equilibrium distribution, and relaxation eigenvalues. The results reveal how UV physics feeds into IR stochastic evolution through operator mixing, and demonstrate a consistent separation between Wilson coefficients and initial conditions beyond tree level. This framework sharpens the connection between stochastic inflation, IR behavior of cosmological correlators, and EFT methods, with potential implications for non-Gaussianity and eternal inflation scenarios.

Abstract

Stochastic Inflation is an important framework for understanding the physics of de Sitter space and the phenomenology of inflation. In the leading approximation, this approach results in a Fokker-Planck equation that calculates the probability distribution for a light scalar field as a function of time. Despite its successes, the quantum field theoretic origins and the range of validity for this equation have remained elusive, and establishing a formalism to systematically incorporate higher order effects has been an area of active study. In this paper, we calculate the next-to-next-to-leading order (NNLO) corrections to Stochastic Inflation using Soft de Sitter Effective Theory (SdSET). In this effective description, Stochastic Inflation manifests as the renormalization group evolution of composite operators. The leading impact of non-Gaussian quantum fluctuations appears at NNLO, which is presented here for the first time; we derive the coefficient of this term from a two-loop anomalous dimension calculation within SdSET. We solve the resulting equation to determine the NNLO equilibrium distribution and the low-lying relaxation eigenvalues. In the process, we must match the UV theory onto SdSET at one-loop order, which provides a non-trivial confirmation that the separation into Wilson-coefficient corrections and contributions to initial conditions persists beyond tree level. Furthermore, these results illustrate how the naive factorization of time and momentum integrals in SdSET no longer holds in the presence of logarithmic divergences. It is these effects that ultimately give rise to the renormalization group flow that yields Stochastic Inflation.

Figures (7)

• Figure 1: Visualization of the Kramers-Moyal expansion. The top panel shows the probability of "hopping" from $\phi$ to $\phi'$, $W(\phi'|\phi)$ or from $\phi'$ to $\phi$, $W(\phi|\phi')$, or equivalently any other point such as $\phi"$. On the bottom, we see the process in terms of the probability $\widetilde{W}(\Delta\phi|\phi)$ to hop from a specific starting point $\phi$ by a distance $\Delta\phi$.
• Figure 2: Diagrams for the one-loop matching, as computed in the UV theory. The horizontal line indicates a surface on constant conformal time $\tau_0$ on which our in-in correlators are evaluated. Left: One-loop power spectrum. Right: One-loop trispectrum.
• Figure 3: Tree level pentaspectrum
• Figure 4: One loop correction to $\varphi_+^2$ that looks like an anomalous dimension ($\varphi_+^2 \to \varphi_+^2$). We start from the tree level trispectrum and integrate over two of the fields to form the composite operator $\varphi_{+}^2$. Left: The Witten diagram with a boundary at future infinity. Right: The Feynman diagram with the same momentum flow.
• Figure 5: Diagram of contribution one loop contribution to $\Gamma_{2,4}$ ($\varphi_+^2 \to \varphi_+^4$). We start from the tree level pentaspectrum (6 points function) and integrate over two of the fields to form the composite operator $\varphi_{+}^2$. Left: The Witten diagram with a boundary at future infinity. Right: The Feynman diagram with the same momentum flow.