On scrambling, tomperature and superdiffusion in de Sitter space

TL;DR

This work analyzes the de Sitter static patch using simple two-point functions in the probe limit to uncover transport and information-theoretic properties. It shows that perturbations spread superdiffusively on the transverse sphere and that scrambling cannot complete faster than a time of order log(1/G_N) for free bulk fields, due to reflections from the cosmological horizon pole. The paper discusses the interplay between thermodynamic temperature and tomperature, highlighting observer-dependent decay rates and caveats related to boundary conditions and deformations. Collectively, the results suggest stringent constraints on any holographic dual of the static patch and motivate a future OTOC calculation to more directly quantify scrambling in de Sitter space.

Abstract

This paper investigates basic properties of the de Sitter static patch using simple two-point functions in the probe approximation. We find that de Sitter equilibrates in a superdiffusive manner, unlike most physical systems which equilibrate diffusively. We also examine the scrambling time. In de Sitter, the two-point functions of free fields do not decay for sometime because quanta can reflect off the pole of the static patch. This suggests a minimum scrambling time of the order $\log(1/G_N)$, even for perturbations introduced on the stretched horizon, indicating fast scrambling inside de Sitter static patch. We also discuss the interplay between thermodynamic temperature and inverse correlation time, sometimes called "tomperature".

Figures (8)

• Figure 1: Illustrations of scrambling using Penrose diagrams of anti-de Sitter -- black hole (Left) and de Sitter (Center and Right). Left: in anti-de Sitter the signal (purple) falls from the boundary inside the black hole horizon (red). Once it is inside the stretched horizon it is scrambled. Center: naive picture in de Sitter. The signal starts at the stretched horizon (black) and quickly falls inside the cosmological horizon. Right: our proposal for how free bulk fields scramble in de Sitter. A part of the initial perturbation falls towards the pole first. This takes a long time (of order $2 R_{dS} \log(1/G_N)$), this is why the scrambling takes at least this time.
• Figure 2: A null geodesic (purple) connecting two points on the stretched horizon (black), resulting in a very large correlation.
• Figure 3: Behavior of the two-point function of a massive scalar field in 3-dimensional de Sitter. Left: operators localized on the transverse sphere. Right: operators uniformly smeared over the sphere.
• Figure 4: The size distribution for which we prove eq. (\ref{['eq:bound_comp']}). There is a delta-function at $s=1$, then some distribution for $1<s<\Lambda$ for which we remain agnostic and after $s>\Lambda$ it is monotonically decreasing.
• Figure 5: The illustration of how the maximally entangled state is prepared and how we split the system after acting with $U$.