Chaos and the Emergence of the Cosmological Horizon

TL;DR

This work analyzes diff-invariant observables in a global de Sitter background with two fully quantized observers and a free scalar field in two dimensions, focusing on how observer dynamics alter the algebraic structure of the static patches. In the strict $G_N\to0$ (semiclassical) limit with infinite observer masses, the right and left algebras commute and form type $II_1$ factors, compatible with a horizon interpretation; away from this limit, the algebras become noncommuting type $I$ objects, reflecting state-dependent trajectories and possible causal contact. The authors compute OTOCs along an observer’s worldline and find a Lyapunov exponent of $\lambda_L=\frac{4\pi}{\beta_{\text{dS}}}$, signaling chaos that exceeds the de Sitter chaos bound by a factor of two, and they show how the cosmological horizon emerges in the large-mass (semiclassical) regime as a limiting horizon structure. The results provide a bottom-up perspective on de Sitter holography, showing how horizon- and chaos-related features arise from non-factorizing observer algebras and illustrating tensions with AdS intuition, with implications for the nature of possible holographic duals in de Sitter space.

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 $\frac{4 π}{β_{\text{dS}}}$, 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.

Figures (7)

• Figure 1: There are observers located at the North and South poles of de Sitter space (labeled $N$ and $S$), and each observer carries a clock. We sometimes refer to the North (resp. South) observer as the right (resp. left) observer. The dots represent the points along the observers' worldlines where the clocks read zero. The boost isometry that preserves the wedges moves the dots along the worldlines, and we work in a gauge where the right dot is fixed at the location shown. The location of the left dot is given by $p$, measured in units of proper time. In the text, we use a time coordinate $t$ that measures proper time along the North and South pole geodesics and always increases to the future, or toward the top of the diagram.
• Figure 2: A geodesic through the Euclidean hemisphere that connects the North and South poles of de Sitter. This geodesic represents a semiclassical Euclidean path integral calculation of the Hartle-Hawking state.
• Figure 3: A Penrose diagram of dS$_2$. An $SO^+(2,1)$-invariant function of $(t_L,\theta_L,t_R,\theta_R)$ can be gauge-fixed such that $(t_L,\theta_L)$ is fixed to the blue dot (which we take to be the origin, $(0,0)$), and $(t_R,\theta_R)$ is integrated along the red lines only.
• Figure 4: The center vertical line connecting points 1 and 4 is the North Pole. Each red dot represents one of the four operators in the OTOC. As we evolve along the observer's worldline to compute the OTOC, the observer travels from point 1 back to point 1 following the blue arrows. The red dots are at future and past infinity because we are working in the limit $\tau \rightarrow \infty$. The distances labeled by $x_+$ and $x_-$ are greatly exaggerated because \ref{['eq:6.31']} will be dominated by order $\frac{1}{\sqrt{\Lambda}}$ values of $x_+$ and $x_-$.
• Figure 5: The Hermitian operator $\hat{P}^+$ is defined to be the symmetry generator whose flow lines are shown above. This isometry is used to define the $x_+$ coordinate along future infinity. A constant shift of $x_+$ moves an operator along a flow line. Some examples of flow lines are given in red. We label two arbitrary points $X,Y$ on the center line (the North Pole), which is the observer's worldline before backreaction is taken into account. Note that the flow lines are transverse to the observer's worldline. When a transformation labeled by $x_+$ is applied to point $X$ only, the change in the geodesic distance between $X$ and $Y$ is of order $x_+^2$ for small $x_+$, due to symmetry. Next, note that when $x_+$ is large enough, point $X$ will be null-separated from point $Y$. Because $x_+$ is proportional to $e^\tau$, this indicates that for a certain choice of kinematics, the OTOC will initially grow and then decay.