Abstract
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.
Highlights
The physics of the early Universe involves processes at ultrahigh energies, as the Hubble parameter during inflation can be as large as 1014 GeV
Our ability to understand physics at such scales depends in part on our understanding of the analytic structure of these quantities, which we expect to be constrained by basic physical principles such as unitarity and causality
We investigate how the Steinmann relations are encoded in the scattering facet, 1We emphasize that positive geometries can be defined and studied independently of any physical interpretation [24]
Summary
The physics of the early Universe involves processes at ultrahigh energies, as the Hubble parameter during inflation can be as large as 1014 GeV. Steinmann relations apply to any quantum field theory and even to individual Feynman integrals [20]; their implications can be subtle in cases involving massless external particles [5,20] It is not clear how these physical principles make their appearance in cosmological processes, where the relevant observables are cosmological correlators or, equivalently, the wavefunctions of the Universe which generate them. We demonstrate that the Steinmann relations emerge naturally from the face structure of the scattering facet, insofar as the codimension-one boundaries of this facet that correspond to partially overlapping momentum channels never intersect to form codimension-two boundaries This elucidates the mechanism by which flat-space causality emerges from the wavefunction of the Universe, complementing the existing understanding of how flat-space unitarity and Lorentz invariance emerge [21].
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