A non-perturbative construction of the de Sitter late-time boundary

We propose a new approach for constructing the late-time conformal boundary of quantum field theory in de Sitter spacetime. A boundary theory which consists of a continuous family of primary operators residing on unitary irreducible representations, the principal series. These boundary operators exhibit two-point functions that include contact terms alongside standard CFT two-point functions. We introduce a bulk-to-boundary expansion in which a bulk operator, when pushed to the boundary, is represented as an integral over boundary operators. The kernel of this integral is related to the K\"all\'en-Lehmann spectral density, and we examine the convergence of the expansion by deriving the spectral density's large dimension limit. Additionally, we derive an inversion formula for the bulk-to-boundary expansion, where, given a bulk theory, the boundary operator content is constructed as an integral of the bulk operator times the bulk-to-boundary propagator. We verify the inversion formula by recovering the boundary two-point function and reproducing perturbation theory. Along the way, we define an operator that generates both the bulk-to-boundary and free bulk-to-bulk propagators from the boundary two-point function, proving to be a powerful tool for simplifying de Sitter diagrams.