Real-time observables in de Sitter thermodynamics

We study real-time finite-temperature correlators for free scalars of any mass in a $d{S}_{d+1}$ static patch in any dimension. We show that whenever the inverse temperature is a rational multiple of the inverse de Sitter temperature, certain Matsubara poles of the symmetric Wightman function disappear. At the de Sitter temperature, we explicitly show how the Lorentzian thermal correlators can all be obtained by analytic continuations from the round $S^{d+1}$. We establish the precise relation between the Harish-Chandra character for $SO(1,d+1)$ and the integrated spectral function, providing a novel dynamical perspective on the former and enabling generalizations. Furthermore, we study scalars with exceptional non-positive masses. We provide a physical picture for the distinctive structures of their characters. For the massless case, we perform a consistent static patch quantization, and find the unique $S^{d+1}$ correlator that analytically continues to the correlators in the quantum theory.