Bootstrapping Inflationary Correlators in Mellin Space

Abstract

We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdS${}_{d+1}$ and dS${}_{d+1}$ is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dS${}_{d+1}$ from known results for Witten diagrams in AdS${}_{d+1}$ by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdS${}_{d+1}$ and dS${}_{d+1}$ in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin-$\ell$. The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general $d$ and generic scaling dimensions, in particular massive, light and (partially-)massless fields. When $d=3$ we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin-$\ell$ field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.

83+19 pages, 8 figures

References in corpus (93)

• Gauge Theory Correlators from Non-Critical String Theory
• The Large N Limit of Superconformal Field Theories and Supergravity
• Non-Gaussian features of primordial fluctuations in single field inflationary models
• Correlation functions in the CFT(d)/AdS(d+1) correpondence
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Implications of Conformal Invariance in Field Theories for General Dimensions
• Holography from Conformal Field Theory
• Quasi-Single Field Inflation and Non-Gaussianities
• Writing CFT correlation functions as AdS scattering amplitudes
• Spinning Conformal Correlators
• On graviton non-Gaussianities during inflation
• Analyticity in Spin in Conformal Theories
• Conformal Regge theory
• Partial Masslessness of Higher Spins in (A)dS
• Conserved Currents and the Energy Momentum Tensor in Conformally Invariant Theories for General Dimensions
• Spinning Conformal Blocks
• On Soft Limits of Inflationary Correlation Functions
• Implications of conformal invariance in momentum space
• A Natural Language for AdS/CFT Correlators
• Emergent Spacetime and Holographic CFTs
• Spinning AdS Propagators
• Effective field theory approach to quasi-single field inflation and effects of heavy fields
• Non-Gaussianity as a Particle Detector
• Two-point Functions and Quantum Fields in de Sitter Universe
• Unitarity and the Holographic S-Matrix
• Loops in AdS from Conformal Field Theory
• Towards Feynman rules for Mellin amplitudes in AdS/CFT
• A Mellin space approach to the conformal bootstrap
• Analyticity and the Holographic S-Matrix
• Quasi-Single Field Inflation with Large Mass
• Conformal Bootstrap in Mellin Space
• De Sitter Musings
• A Mellin Space Approach to Cosmological Correlators
• CMB from CFT
• Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
• Schwinger-Keldysh Diagrammatics for Primordial Perturbations
• Lecture notes on interacting quantum fields in de Sitter space
• Late-time Structure of the Bunch-Davies De Sitter Wavefunction
• Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman's Master Integrals
• Symmetries of Celestial Amplitudes
• Cubic interactions of massless higher spins in (A)dS: metric-like approach
• Higher-spin Interactions from CFT: The Complete Cubic Couplings
• Conformal Invariance and the Four Point Scalar Correlator in Slow-Roll Inflation
• Higher-Spin Interactions: four-point functions and beyond
• Double-Trace Deformations, Mixed Boundary Conditions and Functional Determinants in AdS/CFT
• Heavy-Lifting of Gauge Theories By Cosmic Inflation
• A Study of Quantum Field Theories in AdS at Finite Coupling
• Partially Massless Fields During Inflation
• Arbitrary Spin Representations in de Sitter from dS/CFT with Applications to dS Supergravity
• Constraints from Conformal Symmetry on the Three Point Scalar Correlator in Inflation
• Ward Identities for Scale and Special Conformal Transformations in Inflation
• Quasi Single Field Inflation in the non-perturbative regime
• Celestial Amplitudes: Conformal Partial Waves and Soft Limits
• Spinning Witten Diagrams
• Renormalised CFT 3-point functions of scalars, currents and stress tensors
• Massive Gravity on de Sitter and Unique Candidate for Partially Massless Gravity
• On the cubic interactions of massive and partially-massless higher spins in (A)dS
• Mixed-symmetry fields in de Sitter space: a group theoretical glance
• Generating functions of (partially-)massless higher-spin cubic interactions
• BCFW for Witten Diagrams
• No-Go Theorems for Unitary and Interacting Partially Massless Spin-Two Fields
• On the Polyakov-Mellin bootstrap
• Spinning AdS Loop Diagrams: Two Point Functions
• Momentum space approach to crossing symmetric CFT correlators
• Shapes of gravity: Tensor non-Gaussianity and massive spin-2 fields
• Conformal Operators for Partially Massless States
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• Planck-Suppressed Operators
• Evaluation of conformal integrals
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• Momentum-space conformal blocks on the light cone
• The group approach to AdS space propagators
• Simplifying large spin bootstrap in Mellin space
• New relation for AdS amplitudes
• Momentum space conformal three-point functions of conserved currents and a general spinning operator
• Exact Correlators from Conformal Ward Identities in Momentum Space and the Perturbative $TJJ$ Vertex
• Scale-dependence of Non-Gaussianity in the Curvaton Model
• Anatomy of Geodesic Witten Diagrams
• Group theoretical approach to quantum fields in de Sitter space I. The principal series
• Anomalous Dimensions from Crossing Kernels
• The large $N_c$ limit of four-point functions in N=4 super-Yang-Mills theory from anti-de Sitter Supergravity
• Geodesic Diagrams, Gravitational Interactions & OPE Structures
• S-matrix singularities and CFT correlation functions
• AdS Weight Shifting Operators
• On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
• The General 3-Graviton Vertex ($TTT$) of Conformal Field Theories in Momentum Space in $d=4$
• Heavy Spinning Particles from Signs of Primordial Non-Gaussianities: Beyond the Positivity Bounds
• Double-Trace Deformations of Conformal Correlations
• Looking for partially-massless gravity
• Fermions in Geodesic Witten Diagrams
• Feynman rules for higher-spin gauge fields on AdS${}_{d+1}$
• Quantum Theory on Lobatchevski Spaces
• Effective Theory of Squeezed Correlation Functions

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