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Cosmological Polytopes and the Wavefunction of the Universe

Nima Arkani-Hamed, Paolo Benincasa, Alexander Postnikov

TL;DR

This work introduces cosmological polytopes as a geometric framework for the wavefunction of the universe in toy scalar models with time-dependent couplings. It shows that diagrammatic wavefunction integrands are canonical forms on these polytopes, unifying bulk time-integral and boundary perturbation theory representations through polytope triangulations. The approach reveals how polytope symmetries map to wavefunction symmetries such as conformal invariance and demonstrates that, upon integration, certain cases produce polylogarithmic structures with symbols determined by polytope projections. Overall, the paper proposes a time-independent, positive-geometry picture of cosmological correlators with potential computational and conceptual advantages for understanding early-universe physics.

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.

Cosmological Polytopes and the Wavefunction of the Universe

TL;DR

This work introduces cosmological polytopes as a geometric framework for the wavefunction of the universe in toy scalar models with time-dependent couplings. It shows that diagrammatic wavefunction integrands are canonical forms on these polytopes, unifying bulk time-integral and boundary perturbation theory representations through polytope triangulations. The approach reveals how polytope symmetries map to wavefunction symmetries such as conformal invariance and demonstrates that, upon integration, certain cases produce polylogarithmic structures with symbols determined by polytope projections. Overall, the paper proposes a time-independent, positive-geometry picture of cosmological correlators with potential computational and conceptual advantages for understanding early-universe physics.

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 theory in , 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.

Paper Structure

This paper contains 4 sections, 21 equations, 2 figures.

Table of Contents

  1. Cosmological Time and The Wavefunction of the Universe
  2. Computations of the Wavefunction
  3. The Model
  4. "Bulk" Representation as Time Integral

Figures (2)

  • Figure 1: Two-site chain, three-site chain, four-site star and two-site one-loop graphs.
  • Figure 2: Total energy pole. Upon performing analytic continuation in the energy space, it is possible to reach the point where the sum of the all energies vanishes. From the time perspective, this is equivalent to carry the interaction at early time, so that it is infinitely far away from the boundary, which appears to be effectively absent from the perspective of the early time process. The energy conservation is restored and the process is related to scattering in flat space-time.