The properties of physical processes reflect themselves in the structure of the relevant observables. This idea has been largely exploited for the flat space S-matrix, whose analytic structure is determined by locality and unitary, the two pillars which our current understanding of nature is based on. In this context, it has been possible to find new mathematical structures whose properties turn out to be the ones we ascribe to scattering processes in flat-space, so that both unitarity and locality can be viewed as emergent from some more fundamental structure. However, the S-matrix does certainly not exhaust all the physical information contained in a theory and thus an interesting question to ask is whether and how we can get a similar and deeper understanding of other observables.
In this talk I will report on some recent progress towards such a formulation for quantities such as the universe wave-function and late-time correlators. In particular, I will focus on the perturbative representations of such quantities, their analytic properties and their geometrical interpretation as volumes of certain new polytopes. I will consider a specific toy model which allows to extend most of the discussion to other set-ups than the cosmological one, such as flat and AdS space-times.