Markovian Dynamics in de Sitter

TL;DR

This work analyzes the thermalization of a light scalar field from the viewpoint of a de Sitter static-patch observer. By identifying the smeared field $\bar{\phi}$ as the slow degree of freedom and tracing out the short-memory environment, the authors derive a Markovian master equation for the reduced distribution $p(t,\varphi)$ and develop a diagrammatic, $1/t_r$ expansion to compute corrections to the stochastic (Starobinsky–Yokoyama) picture. They show that relaxation exponents are robust to the precise smearing function and express them in terms of an effective potential $V_{\rm eff}$ that includes UV corrections; explicit next-to-leading order results are obtained for $\lambda\phi^4$ theory, with a clear mass-renormalization interpretation $m_R^2=m^2+6\lambda c_{UV}$. The findings clarify how horizon physics and local interactions yield Markovian, thermally equilibrating behavior for an open system in de Sitter space, with potential implications for cosmological perturbations and horizon thermodynamics.

Abstract

The equilibrium state of fields in the causal wedge of a dS observer is thermal, though realistic observers have only partial access to the state. To them, out-of-equilibrium states of a light scalar field appear to thermalize in a Markovian fashion. We show this by formulating a systematic expansion for tracing out the environment. As an example, we calculate the $O(λ)$ correction to the result of Starobinsky and Yokoyama for the relaxation exponents of $λφ^4$ theory.

Figures (10)

• Figure 1: Left: In the stochastic approach of Starobinsky the field is smeared over a superhorizon region of fixed physical size. The Poincaré patch is shaded. Right: We study the evolution of $\phi$ smeared over a subhorizon region in the static patch. With an abuse of notation, we have denoted the Poincaré patch time and the static patch time both by $t$ even though they are distinct except along the worldline of one observer.
• Figure 2: The initial state $\rho_0$ is obtained by the path integral of the free theory over Euclidean dS after projection onto $\bar{\phi} = \varphi_B$ in the midpoint of the thermal circle.
• Figure 3: Left: Interaction-picture fields combine in interaction vertices to build a third order field. Right: A contribution to $\left\langle \phi^2 \right\rangle$, correlating a fifth order and a first order field. The final insertion can be thought of as the $0^{th}$ vertex. It is the symmetrization vertex of the $0-4$ contraction. The symmetrization vertex for both $2-3$ and $2-4$ lines is vertex $1$.
• Figure 4: A $4^{th}$ order cluster contributing to $\partial_t \left\langle \bar{\phi}^2(t) \right\rangle$. Label $\delta$ indicates $\delta\phi$ at that end of the line. The two short-range lines on the left and the causal structure of the in-in diagram lead to an exponential suppression if any vertex is taken much earlier than $t$. The crossed legs indicate possible uncontracted $\bar{\phi}$ fields at the vertices.
• Figure 5: A time-folded correlator $\left\langle \bar{\phi}(t)V^{(2)}(\bar{\phi}(t_3)) V^{(1)}(\bar{\phi}(t_2)) \bar{\phi}(t) V^{(3)}(\bar{\phi}(t_1)) \right\rangle$ that results from a typical cluster.