From the flat-space S-matrix to the Wavefunction of the Universe

TL;DR

The paper investigates how the cosmological wavefunction at late times can be encoded in cosmological polytopes, revealing that the boundaries of these polytopes, especially the scattering facet, carry the flat-space S-matrix information. It shows that residues at all poles factorize into products of lower-point flat-space amplitudes and wavefunctions, and that many facets are isomorphic to the scattering facet or admit triangulations into Lorentz-invariant propagator products. The authors demonstrate that the wavefunction can be reconstructed from flat-space data plus a symmetry set, at tree level, and that loop wavefunctions arise via a projection (tree-loop map) from tree-level structures. This geometric framework provides a new, highly structured link between cosmological observables and flat-space scattering, with potential implications for reconstructing cosmological correlators from fundamental S-matrix data in a controlled toy-model setting.

Abstract

The physical information encoded in the cosmological late-time wavefunction of the universe is tied to its singularity structure and its behaviour as such singularities are approached. One important singularity is identified by the vanishing of the total energy, where the wavefunction reduces to the physics of scattering in flat space. In this paper, we discuss the behaviour of the perturbative wavefunction as its other singularities are approached and the role played by the flat-space scattering, in the simplified context of the class of toy models admitting a first principle definition in terms of cosmological polytopes. The problems then translates into the analysis of the structure of its facets, one of which -- the scattering facet -- beautifully encodes the flat-space S-matrix. We show that all the boundaries of the cosmological polytope encode information about the flat-space physics. In particular, a subset of its facets turns out to have a similar structure as the scattering facet, with the vertices which can be grouped together to form lower dimensional scattering facets. The other facets admit one (and only one) triangulation in terms of products of lower dimensional scattering facets. As a consequence, the whole perturbative wavefunction can be represented as a sum of product of flat-space scattering amplitudes. Finally, we turn the table around and ask whether the knowledge of the flat-space scattering amplitudes suffices to reconstruct the wavefunction of the universe. We show that, at least for our class of toy models, this is indeed the case at tree level if we are also provided with a subset of symmetries that the wavefunction ought to satisfy. Once the tree cosmological polytopes are reconstructed, the loop ones can be obtained as a particular projection of them.

Figures (14)

• Figure 1: Loops from trees. The projection of the double square pyramid related to the tree-level three-site graph, which is generated by intersecting two triangles on a midpoint in one of their intersectable edges, through a cone with origin in ${\bf O}\,\equiv\,{\bf x}_2-{\bf x}_1$ maps it into the truncated tetrahedron, which encodes the one-loop two site graph.
• Figure 2: Facets of images of $\mathcal{P}_{\mathcal{A}_{\mathcal{G}}}$. The facets identified by an hyperplane $\mathcal{W}_{\bar{v}I}\,=\,\tilde{x}_{\bar{v}}{\bf\tilde{X}}_{\bar{v}I}+\ldots$ are shown for some of the polytopes generated as morphisms on $\mathcal{P}_{\mathcal{A}_{\mathcal{G}}}$. A given hyperplane of this type identified the facet of different polytopes: their sum provides a triangulation of the facet of the sum $\hat{\mathcal{P}}_{\mathcal{G}}$. In this Figure, a complete example is given by third pictures in both lines: they provide a triangulation of the facets at $\sum_{j=4}^{7}x_j+x_1+y_{12}\,=\,0$. The sum of the second and the fourth facets in each line are part of a triangulation of the facet $x_5+x_6+x_7+y_{15}\,=\,0$.
• Figure 3: Poles location in the $\varepsilon$-plane for a generic one-parameter family of wavefunctions $\Psi_{\hbox{\tiny $\mathcal{G}$}}(\varepsilon)$ (left) and for the three-site graph $\Psi_3(\varepsilon)$ at tree level. The contour can be closed either in the left half-plane, enclosing with counter-clockwise orientation the poles which are function of $x_i$; or in the right half-plane, enclosing the poles which are function of $x_j$ with clockwise orientation. The quantities $z$'s are different sums of $x$'s and $y$'s, which do not involve neither $x_i$ nor $x_j$.
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