An Infinite Set of Ward Identities for Adiabatic Modes in Cosmology

TL;DR

This paper derives an infinite set of new consistency relations for cosmological correlation functions, linking soft modes to symmetry transformations, extending known relations to higher orders and verifying them in slow-roll inflation.

Contribution

It introduces an infinite tower of Ward identities for adiabatic modes, revealing new constraints on cosmological correlators beyond previously known relations.

Findings

Recover Maldacena's consistency relations at n=0

Derive conformal consistency relations at n=1

Verify n=2 identities in slow-roll inflation

Abstract

We show that the correlation functions of any single-field cosmological model with constant growing-modes are constrained by an infinite number of novel consistency relations, which relate (N+1)-point correlation functions with a soft-momentum scalar or tensor mode to a symmetry transformation on N-point correlation functions of hard-momentum modes. We derive these consistency relations from Ward identities for an infinite tower of non-linearly realized global symmetries governing scalar and tensor perturbations. These symmetries can be labeled by an integer n. At each order n, the consistency relations constrain - completely for n=0,1, and partially for n>= 2 - the q^n behavior of the soft limits. The identities at n=0 recover Maldacena's original consistency relations for a soft scalar and tensor mode, n=1 gives the recently-discovered conformal consistency relations, and the identities for n>= 2 are new. As a check, we verify directly that the n=2 identity is satisfied by known correlation functions in slow-roll inflation.