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From Locality and Unitarity to Cosmological Correlators

Sadra Jazayeri, Enrico Pajer, David Stefanyszyn

TL;DR

The work advances a boostless bootstrap program for cosmology by introducing the Manifestly Local Test (MLT) and partial-energy recursion, enabling the direct construction of wavefunction coefficients and cosmological correlators from locality, unitarity, and symmetries rather than explicit time evolution. The MLT enforces manifest locality for massless scalars and gravitons (with extensions to massive fields), fixing a wide class of 3-point shapes and revealing connections to EFT operators in inflation. The partial-energy recursion, combined with the Cosmological Optical Theorem (COT), yields exchange 4-point functions up to a boundary term, which is then fixed by locality, again reproducing bulk results up to contact interactions that map to local quartic operators. Across examples including graviton exchange and EFTI interactions, the approach recovers known results and clarifies the role of boundary terms and imaginary late-time contributions, providing a systematic, on-shell framework for cosmological trispectra and beyond. The methods hold promise for multifield scenarios, spinning particles, and loop-level generalizations, offering a route to constrain inflationary physics from fundamental principles alone.

Abstract

In the standard approach to deriving inflationary predictions, we evolve a vacuum state in time according to the rules of a given model. Since the only observables are the future values of correlators and not their time evolution, this brings about a large degeneracy: a vast number of different models are mapped to the same minute number of observables. Furthermore, due to the lack of time-translation invariance, even tree-level calculations require an increasing number of nested integrals that quickly become intractable. Here we ask how much of the final observables can be "bootstrapped" directly from locality, unitarity and symmetries. To this end, we introduce two new bootstrap tools to efficiently compute cosmological correlators/wavefunctions. The first is a Manifestly Local Test (MLT) that any $n$-point (wave)function of massless scalars or gravitons must satisfy if it is to arise from a manifestly local theory. When combined with a sub-set of the recently proposed Bootstrap Rules, this allows us to compute explicitly all bispectra to all orders in derivatives for a single scalar. Since we don't invoke soft theorems, this can also be extended to multi-field inflation. The second is a partial energy recursion relation that allows us to compute exchange correlators. Combining a bespoke complex shift of the partial energies with Cauchy's integral theorem and the Cosmological Optical Theorem, we fix exchange correlators up to a boundary term. The latter can be determined up to contact interactions using unitarity and manifest locality. As an illustration, we use these tools to bootstrap scalar inflationary trispectra due to graviton exchange and inflaton self-interactions.

From Locality and Unitarity to Cosmological Correlators

TL;DR

The work advances a boostless bootstrap program for cosmology by introducing the Manifestly Local Test (MLT) and partial-energy recursion, enabling the direct construction of wavefunction coefficients and cosmological correlators from locality, unitarity, and symmetries rather than explicit time evolution. The MLT enforces manifest locality for massless scalars and gravitons (with extensions to massive fields), fixing a wide class of 3-point shapes and revealing connections to EFT operators in inflation. The partial-energy recursion, combined with the Cosmological Optical Theorem (COT), yields exchange 4-point functions up to a boundary term, which is then fixed by locality, again reproducing bulk results up to contact interactions that map to local quartic operators. Across examples including graviton exchange and EFTI interactions, the approach recovers known results and clarifies the role of boundary terms and imaginary late-time contributions, providing a systematic, on-shell framework for cosmological trispectra and beyond. The methods hold promise for multifield scenarios, spinning particles, and loop-level generalizations, offering a route to constrain inflationary physics from fundamental principles alone.

Abstract

In the standard approach to deriving inflationary predictions, we evolve a vacuum state in time according to the rules of a given model. Since the only observables are the future values of correlators and not their time evolution, this brings about a large degeneracy: a vast number of different models are mapped to the same minute number of observables. Furthermore, due to the lack of time-translation invariance, even tree-level calculations require an increasing number of nested integrals that quickly become intractable. Here we ask how much of the final observables can be "bootstrapped" directly from locality, unitarity and symmetries. To this end, we introduce two new bootstrap tools to efficiently compute cosmological correlators/wavefunctions. The first is a Manifestly Local Test (MLT) that any -point (wave)function of massless scalars or gravitons must satisfy if it is to arise from a manifestly local theory. When combined with a sub-set of the recently proposed Bootstrap Rules, this allows us to compute explicitly all bispectra to all orders in derivatives for a single scalar. Since we don't invoke soft theorems, this can also be extended to multi-field inflation. The second is a partial energy recursion relation that allows us to compute exchange correlators. Combining a bespoke complex shift of the partial energies with Cauchy's integral theorem and the Cosmological Optical Theorem, we fix exchange correlators up to a boundary term. The latter can be determined up to contact interactions using unitarity and manifest locality. As an illustration, we use these tools to bootstrap scalar inflationary trispectra due to graviton exchange and inflaton self-interactions.

Paper Structure

This paper contains 37 sections, 173 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of the results
  3. Notation and conventions
  4. A review of the Cosmological Optical Theorem
  5. A quantum scalar field in de Sitter
  6. Wavefunction of the universe
  7. Kinematics
  8. The Cosmological Optical Theorem (COT)
  9. A Manifestly Local Test (MLT) for $n$-point functions
  10. Manifest locality from singularities
  11. Manifest locality from the bulk representation
  12. The Manifestly Local Test for massive fields
  13. Bootstrapping 3-point functions using the MLT
  14. Comparison with the previous literature
  15. Self-interactions of a massless scalar
  16. ...and 22 more sections

Figures (3)

  • Figure 1: A single exchange diagram with external scalars exchanging a generic field.
  • Figure 2: Triple exchange $6$-point diagram for scalars cut into the product of two $4$-point exchange diagrams.
  • Figure 3: The countour integrals adopted for relating the value of $\tilde{\psi}(z)$ at the origin to its residues and an associated boundary term at infinity.