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The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds

Daniel Baumann, Carlos Duaso Pueyo, Austin Joyce, Hayden Lee, Guilherme L. Pimentel

TL;DR

This work systematizes the cosmological bootstrap by deriving spin-raising and weight-raising operators from conformal field theory in embedding space, showing that all inflationary correlators can be generated from a single scalar-exchange seed. By translating weight- and spin-raising operators into Fourier space, the authors obtain explicit spinning exchange solutions for arbitrary spin and exterior weights, and connect de Sitter boundary data to inflationary observables through soft-limit and slow-roll expansions. The approach yields compact, universal formulas for inflationary bispectra from massive and (partially) massless exchanges, including detailed results for spin-1, spin-2, and higher spins, and illuminates the role of PM fields in shaping distinctive non-Gaussian signatures. Overall, the paper strengthens the theoretical foundation of the cosmological bootstrap, provides a streamlined methodology, and opens avenues to test heavy-field content in the early universe via generalized bispectrum shapes and their angular structures.

Abstract

A key insight of the bootstrap approach to cosmological correlations is the fact that all correlators of slow-roll inflation can be reduced to a unique building block---the four-point function of conformally coupled scalars, arising from the exchange of a massive scalar. Correlators corresponding to the exchange of particles with spin are then obtained by applying a spin-raising operator to the scalar-exchange solution. Similarly, the correlators of massless external fields can be derived by acting with a suitable weight-raising operator. In this paper, we present a systematic and highly streamlined derivation of these operators (and their generalizations) using tools of conformal field theory. Our results greatly simplify the theoretical foundations of the cosmological bootstrap program.

The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds

TL;DR

This work systematizes the cosmological bootstrap by deriving spin-raising and weight-raising operators from conformal field theory in embedding space, showing that all inflationary correlators can be generated from a single scalar-exchange seed. By translating weight- and spin-raising operators into Fourier space, the authors obtain explicit spinning exchange solutions for arbitrary spin and exterior weights, and connect de Sitter boundary data to inflationary observables through soft-limit and slow-roll expansions. The approach yields compact, universal formulas for inflationary bispectra from massive and (partially) massless exchanges, including detailed results for spin-1, spin-2, and higher spins, and illuminates the role of PM fields in shaping distinctive non-Gaussian signatures. Overall, the paper strengthens the theoretical foundation of the cosmological bootstrap, provides a streamlined methodology, and opens avenues to test heavy-field content in the early universe via generalized bispectrum shapes and their angular structures.

Abstract

A key insight of the bootstrap approach to cosmological correlations is the fact that all correlators of slow-roll inflation can be reduced to a unique building block---the four-point function of conformally coupled scalars, arising from the exchange of a massive scalar. Correlators corresponding to the exchange of particles with spin are then obtained by applying a spin-raising operator to the scalar-exchange solution. Similarly, the correlators of massless external fields can be derived by acting with a suitable weight-raising operator. In this paper, we present a systematic and highly streamlined derivation of these operators (and their generalizations) using tools of conformal field theory. Our results greatly simplify the theoretical foundations of the cosmological bootstrap program.

Paper Structure

This paper contains 36 sections, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation and conventions
  4. De Sitter Four-Point Functions
  5. Boundary Correlators
  6. Scalar Seed Functions
  7. Spin-Exchange Solutions
  8. Inflationary Correlators
  9. CFTs in Embedding Space
  10. Projective Null Cone
  11. Tensors in Embedding Space
  12. Conformal Correlators
  13. Exchange of Spinning Particles
  14. Spin-Raising Operator
  15. Raising Internal Spin
  16. ...and 21 more sections

Figures (3)

  • Figure 1: Schematic illustration of the spin-raising and weight-shifting procedure studied in this paper.
  • Figure 2: Illustration of particle creation and decay producing correlations on the future boundary of the de Sitter spacetime. The momentum dependence of the boundary correlators encodes the time dependence of the processes in the bulk.
  • Figure 3: Illustration of the action of Lorentz transformations (and re-scalings) on points living on the Euclidean section of the lightcone in embedding space.