Non-Trivial Checks of Novel Consistency Relations

TL;DR

Single-field inflation imposes an infinite set of consistency relations governing the squeezed limits of correlation functions, interpretable as Ward identities for spatial diffeomorphisms. The authors derive and test these identities up to cubic order in the soft momentum $q$ within slow-roll inflation with arbitrary sound speed $c_s$, computing new 3-point functions including $\langle 2\zeta,1\gamma\rangle$ and $\langle 2\gamma,1\zeta\rangle$ and verifying the Ward identities for all scalar/tensor combinations. A key technical insight is that potentially dangerous $1/c_s^3$ terms cancel in the required linear combinations, leaving results consistent with derivatives of the two-point functions. The work strengthens confidence in single-clock inflation predictions, clarifies the structure of higher-order soft limits, and suggests avenues to probe multi-field or non-adiabatic vacua with future observations.

Abstract

Single-field perturbations satisfy an infinite number of consistency relations constraining the squeezed limit of correlation functions at each order in the soft momentum. These can be understood as Ward identities for an infinite set of residual global symmetries, or equivalently as Slavnov-Taylor identities for spatial diffeomorphisms. In this paper, we perform a number of novel, non-trivial checks of the identities in the context of slow-roll single field inflationary models with arbitrary sound speed. We focus for concreteness on identities involving 3-point functions with a soft external mode, and consider all possible scalar and tensor combinations for the hard-momentum modes. In all these cases, we check the consistency relations up to and including cubic order in the soft momentum. For this purpose, we compute for the first time the 3-point functions involving 2 scalars and 1 tensor, as well as 2 tensors and 1 scalar, for arbitrary sound speed.