Infrared dynamics of a light scalar field in de Sitter

TL;DR

Mirbabayi analyzes infrared dynamics of a light scalar in de Sitter by projecting to the static patch and a low-energy sector. He demonstrates that, for a subhorizon-averaged observable, long-time evolution is Markovian and governed by a Fokker-Planck equation with diffusion 1/(8π^2) and drift from the potential, reproducing stochastic-inflation results in a static-patch setting. The equilibrium distribution is p_eq(φ) ∝ e^{-(8π^2/3) V(φ)}, and the framework provides a controlled perturbative path to include corrections while preserving FP-type relaxation. This connects static-patch thermalization to the Starobinsky-Yokoyama stochastic approach and offers a clear, observer-agnostic view of infrared scalar dynamics in de Sitter.

Abstract

Inertial observers in de Sitter are surrounded by a horizon and see thermal fluctuations. To them, a massless scalar field appears to follow a random motion but any attractive potential, no matter how weak, will eventually stabilize the field. We study this thermalization process in the static patch (the spacetime region accessible to an individual observer) via a truncation to the low frequency spectrum. We focus on the distribution of the field averaged over a subhorizon region. At timescales much longer than the inverse temperature and to leading order in the coupling, we find the evolution to be Markovian, governed by the same Fokker-Planck equation that arises when the theory is studied in the inflationary setup.

Figures (5)

• Figure 1: Left: We study the evolution of $\phi$ smeared over a subhorizon region in the static patch (shaded). Right: In the stochastic approach of Starobinsky and Yokoyama the field is smeared over a superhorizon region of fixed physical size.
• Figure 2: The Penrose diagram for de Sitter spacetime connected to a Euclidean hemisphere along the $\tau=0$ slice (the dotted line). A 2-sphere is suppressed. Its radius, given by $\sin\chi$, expands and then shrinks from the right (north pole) to the left (south pole). The future horizon of an inertial observer at the north pole is shown with a red dashed line.
• Figure 3: The Hartle-Hawking state for the static patch is given by the path integral over a cut 4-sphere (the shaded region). A 2-sphere is suppressed. Its radius is $1$ at the center, but it shrinks to $0$ and caps off the geometry at the outer edge. The boundary conditions imposed on the two sides of the cut determine the arguments of the density matrix $\rho_{HH}[{\boldsymbol \varphi}_L,{\boldsymbol \varphi}_R]$.
• Figure 4: Left: $p_{HH}(\varphi)$ is obtained by gluing back all degrees of freedom on the two sides of the cut except for the average field $\bar{\phi}$ which is set to $\varphi$ on both sides. Right: The equilibrium distribution of $\bar{\phi}$.
• Figure 5: We consider a perturbed state obtained by a projection into $\bar{\phi}=\varphi_B$ (the red dot) at $t=i\pi$. Hence the path integral for the diagonal element of the reduced density matrix $p(t,\varphi|\varphi_B)$ is symmetric. It consists of a sum over the product of matrix elements of $U(-i\pi+t)$ and $U^{-1}(i\pi+t)$.