Cosmohedra

TL;DR

Cosmohedra introduce a new geometric object that unifies the combinatorics of the cosmological wavefunction for $\mathrm{Tr}(\phi^3)$ theory by blowing up associahedra into cosmohedra and organizing contributions via graph associahedra. The framework provides a canonical-form-based route to compute the wavefunction at tree level and extends to loop orders through loop cosmohedra and loop graph associahedra, with permuto-cosmohedra offering a simple, though larger, simple-polytope realization for extraction. It also connects to cosmological correlahedra, proposing a higher-dimensional geometry that captures full correlators by sandwiching cosmohedra and associahedra. The work outlines explicit 5–7 point and loop examples, characterizes facet factorization, and suggests directions toward a stringy formulation of cosmology and deeper relations to kinematic flow and surface-based pictures. Overall, cosmohedra provide a promising, self-contained geometric language for cosmological observables and their diagrammatic structure, with rich avenues for future exploration.

Abstract

It has been a long-standing challenge to find a geometric object underlying the cosmological wavefunction for Tr($φ^3$) theory, generalizing associahedra and surfacehedra for scattering amplitudes. In this note we describe a new class of polytopes -- "cosmohedra" -- that provide a natural solution to this problem. Cosmohedra are intimately related to associahedra, obtained by "blowing up" faces of the associahedron in a simple way, and we provide an explicit realization in terms of facet inequalities that further "shave" the facet inequalities of the associahedron. We also discuss a novel way for computing the wavefunction from cosmohedron geometry that extends the usual connection with polytope canonical forms. We illustrate cosmohedra with examples at tree-level and one loop; the close connection to surfacehedra suggests the generalization to all loop orders. We also briefly describe "cosmological correlahedra" for full correlators. We speculate on how the existence of cosmohedra might suggest a "stringy" formulation for the cosmological wavefunction/correlators, generalizing the way in which the Minkowski sum decomposition of associahedra naturally extend particle to string amplitudes.

Figures (19)

• Figure 1: Associahedron (left) and cosmohedron (right) at 6-points
• Figure 2: (Left) Triangulation of the $\vec{k}$$6$-gon and the respective dual cubic diagram. (Right) Russian doll on the momentum $6$-gon and respective tubing. Associated with each Russian doll, $\mathcal{R}$, a factor of 1 over the product of the perimeters of the subpolygons entering in $\mathcal{R}$. Any russian doll always contains the full polygon whose perimeter is the sum of the $|\vec{k}_i|$ which we call the total energy $E_t$.
• Figure 3: 5(left) and 6(right) point graph associahedron. When drawing the graph we omit the external legs to make manifest that for the purpose of the combinatorics of tubings what matters is the topology of the graph with just the internal edges.
• Figure 4: (Left) 7-point graph associahedra for triangulations from the cyclic classes $\{(1,3),(1,4),(4,6),(1,6)\},\, \{(1,3),(3,5),(1,5),(1,6)\}$. (Right) 7-point graph associahedra for triangulations from the cyclic classes $\{(1,3),(1,4),(1,5),(1,6)\},$$\{(1,3),(1,4),(1,5),(5,7)\},$$\{(1,3),(1,4),(4,7),(5,7)\},\, \{(1,3),(3,7),(3,6),(4,6)\}$.
• Figure 5: (Left) Associahedron (pentagon) and cosmohedron (decagon) at 5-points. (Right) $5$-point cosmohedron with respective labelling of facets in terms of relevant sub-polygons.