On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes

The concepts of Lorentz invariance of local (flat space) physics, and unitarity of time evolution and the S-matrix, are famously rigid and robust, admitting no obvious consistent theoretical deformations, and confirmed to incredible accuracy by experiments. But neither of these notions seem to appear directly in describing the spatial correlation functions at future infinity characterizing the "boundary" observables in cosmology. How then can we see them emerge as {\it exact} concepts from a possible ab-initio theory for the late-time wavefunction of the universe? In this letter we examine this question in a simple but concrete setting, for the perturbative wavefunction in a class of scalar field models where an ab-initio description of the wavefunction has been given by "cosmological polytopes". Singularities of the wavefunction are associated with facets of the polytope. One of the singularities -- corresponding to the "total energy pole" -- is well known to be associated with the flat-space scattering amplitude. We show how the combinatorics and geometry of this {\it scattering facet} of the cosmological polytope straightforwardly leads to the emergence of Lorentz invariance and unitarity for the S-matrix. Unitarity follows from the way boundaries of the scattering facet factorize into products of lower-dimensional polytopes, while Lorentz invariance follows from a contour integral representation of the canonical form, which exists for any polytope, specialized to cosmological polytopes.