Out-of-time-ordered Correlators in de Sitter Revisited

We study the 4-point out-of-time-ordered correlator (OTOC) of scalar fields using the eikonal approximation to gravity around a de Sitter background. It can be defined in a gauge-invariant way by dressing the operators to an observer. We consider de Sitter space in any dimension and any bulk masses of the scalar fields. At tree level we find the maximal Lyapunov exponent of $2 \pi/\beta_{\rm dS}$. However, we describe an IR problem in this calculation at the loop level associated with the vector part of the graviton propagator. Regularizing this divergence leads to the vanishing of all odd powers in Newton's constant in the asymptotic expansion, so that the leading answer comes with twice the maximal Lyapunov exponent in simple operator configurations. We find that the perturbative OTOC in de Sitter can initially grow, which is not allowed by the quantum bound on chaos. Interestingly, we find that this growth is mediated by large diffeomorphisms in the graviton propagator.