  The physical principles of causality and unitarity put strong constraints on
the analytic structure of the flat-space S-matrix. In particular, these
principles give rise to the Steinmann relations, which require that the double
discontinuities of scattering amplitudes in partially-overlapping momentum
channels vanish. Conversely, at cosmological scales, the imprint of causality
and unitarity is in general less well understood---the wavefunction of the
universe lives on the future space-like boundary, and has all time evolution
integrated out. In the present work, we show how the flat-space Steinmann
relations emerge from the structure of the wavefunction of the universe, and
derive similar relations that apply to the wavefunction itself. This is done
within the context of scalar toy models whose perturbative wavefunction has a
first-principles definition in terms of cosmological polytopes. In particular,
we use the fact that the scattering amplitude is encoded in the scattering
facet of cosmological polytopes, and that cuts of the amplitude are encoded in
the codimension-one boundaries of this facet. As we show, the flat-space
Steinmann relations are thus implied by the non-existence of codimension-two
boundaries at the intersection of the boundaries associated with pairs of
partially-overlapping channels. Applying the same argument to the full
cosmological polytope, we also derive Steinmann-type constraints that apply to
the full wavefunction of the universe. These arguments show how the
combinatorial properties of cosmological polytopes lead to the emergence of
flat-space causality in the S-matrix, and provide new insights into the
analytic structure of the wavefunction of the universe.
