Cosmological Polytope & Wavefunctions

• Cosmological polytope is a convex polytope built from Feynman diagram data that encodes the perturbative cosmological wavefunction as its canonical form.
• Its combinatorial and polyhedral structure, featuring triangulations and Tutte polynomial relations, enables explicit computation of physical integrands.
• The geometry links singularity structures to flat-space limits and quantum correlators, illustrating a unified approach in cosmology and quantum field theory.

A cosmological polytope is a positive geometry canonically associated to any Feynman diagram contributing to the wavefunction of the universe in scalar FRW-type cosmologies. This construction, introduced by Arkani-Hamed, Benincasa, and Postnikov, encodes the perturbative cosmological wavefunction as the canonical form of a convex polytope defined via combinatorial and graphical data. The singularity structure, representation theory, and positive geometry of these polytopes mirror and generalize the role of the associahedron and amplituhedron in flat-space scattering amplitudes. They underpin a rich connection between graph theory, combinatorics, convex geometry, and quantum field-theoretic correlators, and have motivated a series of advances in toric geometry, Ehrhart theory, and polyhedral combinatorics with deep implications for the calculation and structure of cosmological observables.

1. Definition and Construction

Given a (possibly disconnected, with loops/multiedges/isolated vertices) undirected graph $G=(V,E)$, the cosmological polytope $\mathcal{C}_G \subset \mathbb{R}^{V \cup E}$ is constructed as the convex hull of three “edge-points” for each $f=ij \in E$: $$e_i + e_j - e_f,\quad e_i - e_j + e_f,\quad -e_i + e_j + e_f$$ where $e_i$ and $e_f$ are the standard basis vectors indexed by vertices and edges respectively. If the graph has no isolated vertices, the dimension of the lattice polytope is $|V| + |E| - 1$; otherwise, a vertex for each isolated site is included.

More generally, the relation to ternary Newton polytopes is exact: every cosmological polytope is a special case of the Graev construction: the convex hull in $\mathbb{R}^{|V| + |E|}$ of vectors associated to each edge-vertex-edge triple $[i,f,j]$ as $(\pm e_i) + (\pm e_j) + (\pm e_f)$ with the appropriate sign patterns (Lavrov, 8 Sep 2025). The entire facial and combinatorial structure is determined by the underlying ternary relation.

2. Canonical Forms and Wavefunction Integrands

The cosmological polytope $\mathcal{C}_G$ is a positive geometry: it possesses a unique top-form, the canonical form $\Omega_{\mathcal{C}_G}$, with logarithmic singularities on the facets and unit residues. In explicit coordinates, this form can be written

$$\Omega_{\mathcal{C}_G} = \prod_{v=1}^{|V|} dx_v \prod_{e=1}^{|E|} dy_e \cdot \Psi_G(x, y)$$

where $(x_v, y_e)$ are energy variables associated to the vertices and edges. The universal rational function $\Psi_G(x,y)$ matches, up to normalization, the Feynman time-integral or "old-fashioned perturbation theory" expression for the contribution of diagram $G$ to the wavefunction of the universe. The canonical form is thus a direct geometric encoding of the physical integrand (Arkani-Hamed et al., 2017, Benincasa, 2019).

Triangulations of $\mathcal{C}_G$ correspond bijectively to different physical representations: bulk (time-ordering) integrals, old-fashioned perturbation theory, and novel contour-integral (Aomoto-type) parametrizations. Each simplex in a regular unimodular triangulation contributes a rational term to the total wavefunction expression (Juhnke-Kubitzke et al., 2023).

3. Combinatorial and Polyhedral Structure

The face lattice and combinatorial invariants of cosmological polytopes are governed by structural rules:

• Faces: Faces correspond to subsets of the vertex set satisfying star and cycle constraints. The face associated to a connected subgraph $g \subset G$ is given by the intersection with the hyperplane $$\sum_{v \in V_g} x_v + \sum_{e \in E_g^{\text{ext}}} y_e = 0$$, encoding energy conservation for $g$ (Kühne et al., 2022, Benincasa et al., 2024).
• Minimal non-simplicial faces are associated to vertex-faces (cross-polytopes at vertices) and cycle-faces (cyclic subgraphs) (Kühne et al., 2022).
• Triangulation and Gröbner Structure: The toric ideal $I_{\mathcal{C}_G}$ of $\mathcal{C}_G$ admits a squarefree initial ideal, yielding a regular unimodular triangulation (Juhnke-Kubitzke et al., 2023, Benjes et al., 17 Mar 2025). This triangulation is in bijection with decorated acyclic edge-sets of $G$, and its count is $2^{|E|} T_G(2,1)$, where $T_G$ is the Tutte polynomial (Benjes et al., 17 Mar 2025).
• Ehrhart Theory: The Ehrhart $h^*$-polynomial of $\mathcal{C}_G$ is a specialization of the Tutte polynomial: $$h^*(\mathcal{C}_G;z) = (1+z)^{|E| - r}(2z)^r T_G\left(\frac{1+3z}{2z}, 1\right)$$ where $r = |V| - \#\text{components}$ (Bruckamp et al., 2024, Benjes et al., 17 Mar 2025). Explicit formulas exist for multitrees and multicycles; in particular, for a simple $n$-cycle, $h^* = (1+3z)^n - (2z)^n$.

Graph Family	$h^*$-polynomial	Volume (simplices)
$n$-path	$(1+3z)^n$	$4^n$
$n$-cycle	$(1+3z)^n - (2z)^n$	$4^n-2^n$
Multitree; $a_i$	$\prod_i [(1+z)^{a_i} + 2a_i z (1+z)^{a_i-1}]$	see text

Volumes of these polytopes count the number of rational terms in the canonical form decomposition, thus directly measuring the complexity of computing $\Psi_G$ (Bruckamp et al., 2024).

4. Physical Significance: Wavefunctions and Correlators

The entire singularity structure of the cosmological polytope is in correspondence with physical poles of $\Psi_G$. Residues on codimension-one facets encode flat-space factorizations; higher-codimension faces encode higher-order poles and nested factorizations, and the total-energy facet encodes the flat-space S-matrix (Arkani-Hamed et al., 2017, Benincasa, 2019). Symmetries (projective deformations) of the polytope descend to Ward identities acting on $\Psi_G$, including conformal and dilation operators in AdS/dS settings.

Extensions to weighted cosmological polytopes recover in-in and correlator integrands as canonical forms on orientation-reversed subdivisions, providing a unified geometric description of correlator factorization, Steinmann relations, and vanishing conditions (Benincasa et al., 2024). Push-forward and contour representations yield integral prescriptions generalizing Feynman parameterizations.

Generalizations to massive and spinning states are formalized via generalized polytopes constructed by "collisions" and degenerations, aligned with the structure of higher-order differential recursions in the bulk (Benincasa, 2019).

5. Triangulations, Tutte–Theory, and Algorithmic Aspects

Recent work establishes that every cosmological polytope admits "good" triangulations: regular unimodular subdivisions that enable algorithmic extraction of the canonical form directly from graphical data (Benjes et al., 17 Mar 2025). The set of maximal simplices is enumerated via acyclic subgraphs, with combinatorial control furnished by the Tutte polynomial.

These triangulations, along with the half-open decomposition, provide explicit and computable connections between the facial structure, the $h^*$-polynomial, and the Laurent expansion coefficients of the wavefunction. Such results settle conjectures on explicit volume formulas and Tutte specializations, giving a complete, uniform classification of all cosmological polytopes and a diagnosis of their combinatorial complexity (Benjes et al., 17 Mar 2025).

6. Relation to Broader Positive-Geometry and Polytope Theories

Cosmological polytopes generalize the positive geometry paradigm of the associahedron (for amplitudes) to the full wavefunction and in-in context of cosmology (Arkani-Hamed et al., 2017, Arkani-Hamed et al., 2024). The cosmohedron and correlahedron/correlatron further interpolate between associahedra, surfacehedra, and cosmological correlators (Arkani-Hamed et al., 2024, Figueiredo et al., 24 Jun 2025). Each step introduces refined combinatorial “shaving” prescriptions, giving rise to intricate hybrid polytopes combining chord and subpolygon data, with canonical forms that reproduce explicit physical quantities.

Under the lens of Graev's construction, the entire cosmological polytope program is seen as a subtheory of ternary-relation Newton polytopes, opening the route to importing classical results from convex geometry and finite-metric polytope theory (Lavrov, 8 Sep 2025). This unification both streamlines combinatorial proofs and enables the use of computational convex geometry packages to enumerate faces, $h^*$-vectors, and canonical forms for arbitrary input graphs.

7. Current Directions and Open Problems

Key areas of active research involve extending the tutted-combinatorial framework to wider classes of graphs (e.g. 2-connected, planar, or higher genus), exploring explicit $h^*$- and $f$-vector characterizations, and classifying higher-codimension faces and their relation to nested factorization and the analytic structure of cosmological correlators (Bruckamp et al., 2024, Benjes et al., 17 Mar 2025, Kühne et al., 2022). Other directions include the geometric interpretation of more general correlators (cosmohedra, correlatrons) (Arkani-Hamed et al., 2024, Figueiredo et al., 24 Jun 2025), connections to stringy polytope expansions, loop-level polytope constructions, and algorithmic approaches leveraging the positive geometry underlying quantum field-theoretic observables. The precise physical meaning and combinatorics of zeroes implied by the geometry, logarithmic concavity of coefficients, and connections to Lorentzian polynomials and matroid theory, remain open and promising for further investigation.