Holographic timelike complexity for de Sitter

Abstract

We investigate the recent proposal of holographic volume complexity for timelike subregions \cite{Alishahiha:2025xml} in the framework of static patch holography for de Sitter spacetime. Using the stretched-horizon prescription, we compute the timelike subregion complexity as a function of the subregion duration for pure de Sitter and Schwarzschild de Sitter geometries. In pure de Sitter spacetime, the timelike subregion complexity displays exponential growth for short durations, and hyperfast growth near a maximal duration, paralleling the features of spacelike volume complexity \cite{Jorstad:2022mls}. For Schwarzschild de Sitter, when the stretched horizon is near the cosmological horizon, the behavior broadly remains similar to pure de Sitter. However, when the stretched horizon is near the black hole horizon, the hyperfast growth for long durations is replaced by nonlinear growth regime. Along the way, we also compute the corresponding timelike holographic entanglement entropy for de Sitter and Schwarzschild de Sitter.