Cosmohedra

TL;DR

Cosmohedra are a new class of polytopes that geometrically encode the cosmological wavefunction for Tr(φ^3) theory, extending associahedra and surfacehedra, and potentially leading to a stringy formulation of cosmological correlators.

Contribution

Introduction of cosmohedra as a novel geometric object underlying the cosmological wavefunction, generalizing associahedra and surfacehedra, with explicit facet inequalities and computational methods.

Findings

Cosmohedra relate to associahedra via face blow-ups.

Explicit facet inequalities for cosmohedra are provided.

Examples at tree-level and one loop demonstrate the concept.

Abstract

It has been a long-standing challenge to find a geometric object underlying the cosmological wavefunction for Tr($\phi^3$) 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.