An Infinite Set of Ward Identities for Adiabatic Modes in Cosmology

TL;DR

This work establishes an infinite tower of Ward identities for adiabatic modes in cosmology, derived from non-linearly realized residual diffeomorphisms acting on scalar and tensor perturbations. Each order n in the tower constrains the soft-limit behavior at order q^n of N+1-point functions via a symmetry transformation on N-point functions, with n=0 reproducing Maldacena's original relations and n=1 yielding conformal (and linear-gradient) relations, while n≥2 provides new constraints. The identities are derived non-perturbatively using Noether charges and the in-in formalism, and are shown to hold regardless of horizon crossing; explicit checks include the n=2 tensor relation tested against slow-roll results and a separate non-trivial check on ζζζ contributions. Collectively, these Ward identities offer a comprehensive, symmetry-based framework to test single-field cosmologies against soft-limit correlator data, strengthening the theoretical consistency checks for inflationary scenarios.

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.