[Paper Review] Cosmological Polytopes and the Wavefunction of the Universe

Nima Arkani–Hamed, Paolo Benincasa+1 more|arXiv (Cornell University)|Sep 8, 2017

Black Holes and Theoretical Physics10 references80 citations

TL;DR

The paper links the late-time wavefunction of the universe in scalar toy models to the canonical forms of cosmological polytopes, establishing a geometry-driven framework for perturbative wavefunctions and their singularities.

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 $ϕ^3$ 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.

Motivation & Objective

• Motivate and formalize a connection between cosmological time evolution and the wavefunction of the universe.
• Introduce cosmological polytopes as time-agnostic objects whose canonical forms yield wavefunction integrands.
• Show how triangulations and dual polytopes correspond to bulk time integrals and old-fashioned perturbation theory.
• Demonstrate symmetry and polylogarithmic structures in specific models, and relate polytope geometry to wavefunction properties.

Proposed method

• Define a class of toy scalar field models with time-dependent couplings in FRW cosmologies.
• Compute the wavefunction perturbatively, deriving universal rational integrands psi(x_v,y_e) associated with each Feynman diagram.
• Introduce cosmological polytopes from graphs, with vertices given by x_v and y_e basis vectors and 3E polytope vertices per graph edge.
• Use the canonical form of the polytope to obtain the wavefunction integrand with logarithmic singularities on all facets.
• Relate time-integral representations and old-fashioned perturbation theory via natural triangulations of the polytope and its dual.
• Discuss new representations of canonical forms via contour integrals and push-forwards applicable to the cosmological setting.
• Present symmetry implications, including conformal invariance in de Sitter contexts, and begin extracting polylogarithmic structures in specific theories.

Experimental results

Research questions

• RQ1 How can the wavefunction of the universe be recast as a canonical form of a positive geometry?
• RQ2 What is the precise relationship between cosmological time evolution and the geometric structure of cosmological polytopes?
• RQ3 How do different triangulations of the cosmological polytope relate to bulk time integrals and boundary perturbation theory?
• RQ4 What symmetries of the cosmological polytope descend to the wavefunction, and how do they appear in de Sitter correlators?
• RQ5 What is the nature of the final wavefunction in specific models, such as c6^3 theory in dS4, in terms of polylogarithms and their symbols?

Key findings

• Each Feynman diagram contributes a universal rational integrand that can be identified with the canonical form of a cosmological polytope.
• Triangulations of the polytope correspond to bulk time-integral representations, while dual-polytope triangulations reproduce old-fashioned perturbation theory.
• New polytope representations via contour-integrals and push-forwards yield alternative wavefunction expressions without direct physical interpretation.
• Symmetries of the cosmological polytope, such as conformal invariance in de Sitter space, manifest as symmetries of the wavefunction.
• In φ^3 theory in dS4, the integrated wavefunction is polylogarithmic with a calculable symbol encoding the geometry of sequential projections of the polytope.