Steinmann Relations and the Wavefunction of the Universe

TL;DR

The paper investigates how causality and unitarity—embodied by Steinmann relations in flat space—manifest in cosmology through the wavefunction of the universe. Using cosmological polytopes, it shows that the scattering facet encodes flat-space Steinmann relations via its boundary structure and that similar Steinmann-type constraints apply directly to the wavefunction itself. The authors derive precise combinatorial conditions under which sequential cuts vanish and demonstrate how the canonical form’s factorization enforces these constraints. These results provide a geometric, first-principles route to constrain the analytic structure of cosmological correlators and suggest avenues for bootstrapping the wavefunction across FRW backgrounds.

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.

Figures (7)

• Figure 1: On the left, we show a Feynman graph that contributes to the wavefunction of the universe. On the right, we depict the associated reduced graph, which is obtained from the Feynman graph by suppressing the external edges.
• Figure 2: Examples of cosmological polytopes obtained from the intersection of two triangles. The images in the left column illustrate the intersection of pairs of triangles at either one or two midpoints, while the corresponding convex hulls are depicted in the middle column. The column on the right shows the associated reduced graphs.
• Figure 3: An example of a codimension-one face of the scattering facet, which is associated with an individual cut of the scattering amplitude. In the diagram on the left, we mark the vertices that do not appear on the scattering facet $\mathcal{S}_{\mathcal{G}}$ by , and the additional vertices that get eliminated when we intersect it with the facet corresponding to $\mathfrak{g}$ by . In the diagram on the right, we depict the vertices which do contribute to $\mathcal{P}_{\mathcal{G}} \cap \mathcal{W}_{\mathfrak{g}}$ by open circles. The vertices marked by and are associated with the scattering facets $\mathcal{S}_{\mathfrak{g}}$ and $\mathcal{S}_{\bar{\mathfrak{g}}}$, respectively. The remaining vertices, marked by , are associated with the simplex $\Sigma_{\centernot{\mathcal{E}}}$.
• Figure 4: An example of a pair of subgraphs that correspond to partially overlapping momentum channels, and their realization as faces on the scattering facet.
• Figure 5: The intersection of the pair of facets depicted in Figure \ref{['fig:ovch']}. This intersection factorizes into four lower-dimensional scattering facets $\mathcal{S}_{\mathfrak{g}_1 \cap \mathfrak{g}_2}$, $\mathcal{S}_{\mathfrak{g}_1 \cap \bar{\mathfrak{g}}_2}$, $\mathcal{S}_{\bar{\mathfrak{g}}_1 \cap \mathfrak{g}_2}$ and $\mathcal{S}_{\bar{\mathfrak{g}}_1\cap\bar{\mathfrak{g}}_2}$, whose vertices are respectively depicted by the markings , , , and . The remaining vertices, denoted by , identify the simplex $\Sigma_{\centernot{\mathcal{E}}}$. This represents an example of an intersection that satisfies condition \ref{['eq:SteinRel2']}, yet that does not form a codimension-two facet of $\mathcal{S}_{\mathcal{G}}$.