Slavnov-Taylor Identities for Primordial Perturbations

TL;DR

This work derives a unifying master identity for cosmological consistency relations from the Slavnov-Taylor identity for spatial diffeomorphisms, showing that all adiabatic, soft-limit relations follow from a single equation valid for any soft momentum $\vec{q}$. By differentiating this master identity with respect to $\vec{q}$ and taking $|\vec{q}|\to 0$, the authors recover the dilation and special conformal (SCT) relations at leading orders and organize higher-order relations through a model-dependent, analytic transverse piece $A_{ij}$ that vanishes at leading orders. The locality/analyticity assumption on the vertex functional is identified as the precise condition ensuring the universality of these relations in standard single-field inflation, while signaling their breakdown in nonlocal or non-adiabatic models. The derivations are presented in both fixed-time 3D path-integral and full 4D in-in formalisms, with implications for extensions to large-scale structure and multi-field contexts.

Abstract

Correlation functions of adiabatic modes in cosmology are constrained by an infinite number of 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. They constrain, at each order n, the q^n behavior of the soft limits. In this paper we show that all consistency relations derive from a single, master identity, which follows from the Slavnov-Taylor identity for spatial diffeomorphisms. This master identity is valid at any value of q and therefore goes beyond the soft limit. By differentiating it n times with respect to the soft momentum, we recover the consistency relations at each q order. Our approach underscores the role of spatial diffeomorphism invariance at the root of cosmological consistency relations. It also offers new insights on the necessary conditions for their validity: a physical contribution to the vertex functional must satisfy certain analyticity properties in the soft limit in order for the consistency relations to hold. For standard inflationary models, this is equivalent to requiring that mode functions have constant growing-mode solutions. For more exotic models in which modes do not "freeze" in the usual sense, the analyticity requirement offers an unambiguous criterion.