Cosmohedra (2412.19881v1)

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.

• The paper introduces cosmohedra as a novel class of geometric objects that extend associahedra to represent nested subpolygon structures in cosmological wavefunctions.
• It derives cosmohedra by 'blowing up' associahedra faces with extra inequalities, thereby capturing the intricate combinatorial structures underlying cosmological diagrams.
• The framework lays a foundation for refined quantum cosmology models, paralleling geometric approaches used in particle scattering amplitudes.

An Examination of "Cosmohedra"

In the paper "Cosmohedra," the authors introduce a novel class of geometric objects termed "cosmohedra," offering a unified framework for understanding the cosmological wavefunction within Tr$(\phi^3)$ theory. This foundational work extends the geometric conceptualization of scattering amplitudes—as encoded by associahedra—to the field of cosmology, presenting a multidimensional plot that captures the intricate combinatorial structures of cosmological wavefunctions.

Summary and Key Findings

At the heart of this paper is the proposition of cosmohedra as higher-complexity counterparts to associahedra. Whereas traditional associahedra encode the combinatorics of scattering amplitudes through non-overlapping chords, cosmohedra capture the richer structure of nested subpolygons essential to cosmological wavefunctions. The cosmohedra are introduced as being derived from associahedra through a combinatorial procedure akin to "blowing up" its faces, which effectively transitions the geometric representation from scattering amplitudes to cosmological configurations.

In constructing cosmohedra, the paper delineates a precise geometric embedding, illustrating that the faces of associahedra are augmented by introducing additional inequalities that reflect the nested structure of subpolygons. This novel structure supports both the theoretical description and practical calculation of the full wavefunction, which extends beyond isolated diagrams to encompass a comprehensive summation over potential contributions.

An integral mathematical development is the introduction of graph associahedra, which illuminate the contribution of individual Feynman diagrams to the wavefunction's structure. For example, at six points, distinct topologies such as chains and stars showcase variations leading to five and six-term graph associahedra, respectively. This partitioning is instrumental in handling the factorial growth in complexity reminiscent of the transition from tree-level to one-loop amplitudes in particle physics.

Theoretical Implications and Future Directions

The cosmohedron's construction represents a significant milestone in bridging the cosmological wavefunction with the established combinatorial geometry of particle interactions. While associahedra grounded particle amplitudes in familiar geometric structures, cosmohedra promise a description that is similarly rich and detailed for cosmological phenomena. This work opens pathways for further exploration, notably the potential for a "stringy" formulation of cosmological functions parallel to string amplitudes' extension of particle interactions.

Another critical implication focuses on the geometric extraction of the cosmological wavefunction. By operating within permuto-cosmohedra—a variant that admits simpler geometric properties while retaining essential combinatorial features—the canonical extraction process is refined. The article's future directions will likely explore leveraging these polytopes for comprehensive models of quantum cosmology.

Conclusion

The paper has meticulously laid out a framework wherein cosmohedra serve as geometric organizers of the cosmological wavefunction. Through rigorous combinatorial definitions and geometric interpretations, the authors not only advance the understanding of cosmological observables but also provide a robust platform for future developments. As explorations continue, the cosmohedron stands to offer foundational insight into the geometric nature of the cosmos, potentially mirroring the success seen in particle scattering and amplitudes. The marriage of combinatorial geometry with cosmology that this paper proposes is poised to redefine how fundamental interactions are conceived in the vast cosmological context.