Chaos and the Emergence of the Cosmological Horizon

TL;DR

This paper explores the algebraic structure of observables in a two-dimensional de Sitter universe with two quantized observers, revealing how chaos and horizon emergence relate to observer uncertainty and de Sitter expansion.

Contribution

It demonstrates the transition of observer algebras from commuting type II$_1$ factors to non-commuting type I non-factors when observers have finite mass, and computes chaos exponents exceeding the usual bound.

Findings

Observers' algebras become non-commutative with finite mass.

Lyapunov exponent exceeds the de Sitter chaos bound.

Cosmological horizon emerges in the large mass limit.

Abstract

We construct algebras of diff-invariant observables in a global de Sitter universe with two observers and a free scalar QFT in two dimensions. We work in the strict $G_N \rightarrow 0$ limit, but allow the observers to have an order one mass in cosmic units. The observers are fully quantized. In the limit when the observers have infinite mass and are localized along geodesics at the North and South poles, it was shown in previous work \cite{CLPW} that their algebras are mutually commuting type II$_1$ factors. Away from this limit, we show that the algebras fail to commute and that they are type I non-factors. Physically, this is because the observers' trajectories are uncertain and state-dependent, and they may come into causal contact. We compute out-of-time-ordered correlators along an observer's worldline, and observe a Lyapunov exponent given by $\frac{4 \pi}{\beta_{\text{dS}}}$, as a result of observer recoil and de Sitter expansion. This should be contrasted with results from AdS gravity, and exceeds the chaos bound associated with the de Sitter temperature by a factor of two. We also discuss how the cosmological horizon emerges in the large mass limit and comment on implications for de Sitter holography.