Chaos and complementarity in de Sitter space

Abstract

We consider small perturbations to a static three-dimensional de Sitter geometry. For early enough perturbations that satisfy the null energy condition, the result is a shockwave geometry that leads to a time advance in the trajectory of geodesics crossing it. This brings the opposite poles of de Sitter space into causal contact with each other, much like a traversable wormhole in Anti-de Sitter space. In this background, we compute out-of-time-order correlators (OTOCs) to asses the chaotic nature of the de Sitter horizon and find that it is maximally chaotic: one of the OTOCs we study decays exponentially with a Lyapunov exponent that saturates the chaos bound. We discuss the consequences of our results for de Sitter complementarity and inflation.

Figures (7)

• Figure 1: Penrose diagram of de Sitter space. By complexifying the static time coordinate, we can cover each of the four static patches. The flow of the timelike Killing vector $\partial_t$ is indicated with arrows in each patch.
• Figure 2: Penrose diagram of the shockwave geometry \ref{['eq:discontinuousshock']} created by a highly boosted particle that travels along the past horizon $v=0$ (the blue line). There is a discontinuity in the coordinate $\tilde{u}$ by an amount $\alpha$ which brings the left and right static patch into causal contact with each other.
• Figure 3: The geodesic (green) connecting the operators $V_L(0)$ and $V_R(0)$ in the shockwave geometry \ref{['eq:continuousshock']} created by the operator $W_R(t)$. Due to the shockwave (blue) the geodesic is shifted by an amount $\alpha$ along the past horizon.
• Figure 4: The OTOC $F(t_w)$ calculated using the geodesic approximation which is valid for $m^2\ell^2\gg 1$. It develops oscillations after the scrambling time $t>t_*=\ell \log(S_{dS})$. To plot the figure we took $G_N = 1/2, \ell = 1$ and $m^2\ell^2=10$.
• Figure 5: The out-of-time-order correlator $\braket{V_L(0)W_R(t)V_R(0)W_R(t)}$ for $|g(t=0)|=10,\ell=1$.