We study finite-coupling effects of QFT on a rigid de Sitter (dS) background taking the O(N) vector model at large N as a solvable example. Extending standard large N techniques to the dS background, we analyze the phase structure and late-time four-point functions. Explicit computations reveal that the spontaneous breaking of continuous symmetries is prohibited due to strong IR effects, akin to flat two-dimensional space. Resumming loop diagrams, we compute the late-time four-point functions of vector fields at large N, demonstrating that their spectral density is meromorphic in the spectral plane and positive along the principal series. These results offer highly nontrivial checks of unitarity and analyticity for cosmological correlators.
