Cosmological Polytopes and the Wavefunction of the Universe

TL;DR

This paper introduces cosmological polytopes as a geometric framework to understand the wavefunction of the universe in cosmological models, linking positive geometry with quantum cosmology and revealing new representations and symmetries.

Contribution

It establishes a novel geometric approach using cosmological polytopes to analyze the wavefunction of the universe, connecting physics with intrinsic mathematical structures.

Findings

Cosmological polytopes encode the singularity structure of the wavefunction.

Triangulations of polytopes reproduce known and new wavefunction representations.

Polylogarithms associated with graphs are derived from the geometry of the polytopes.

Abstract

We present a connection between the physics of cosmological time evolution and the mathematics of positive geometries, roughly analogous to similar connections seen in the context of scattering amplitudes. We consider the wavefunction of the universe in a class of toy models of conformally coupled scalars (with non-conformal interactions) in FRW cosmologies. The contribution of each Feynman diagram to the wavefunction of the universe is associated with a certain universal rational integrand, which we identify as the canonical form of a "cosmological polytope", which have an independent, intrinsic definition, making no reference to physics. The singularity structure of the wavefunction for this model of scalars is common to all theories, and is geometrized by the cosmological polytope. Natural triangulations of the polytope reproduce the path-integral and "old-fashioned perturbation theory" representations of the wavefunction, and we also find new representations of the wavefunction with no extant physical interpretation. We show in suitable examples how symmetries of the cosmological polytope descend to symmetries of the wavefunction, (such as conformal invariance). In cases such as $\phi^3$ theory in $dS_4$, the final wavefunction obtained from integration of the rational functions gives rise to polylogarithms associated with every graph. We give an explicit expression for the symbol of these polylogs, which record the geometry of sequential projections of the cosmological polytope.