On the Initial State and Consistency Relations

TL;DR

This work analyzes how non-Bunch-Davies initial states affect inflationary consistency relations derived from diffeomorphism invariance. By formulating the problem in both 4d in-in and 3d fixed-time path integrals, it demonstrates that the initial wavefunctional must satisfy a Slavnov-Taylor identity, paralleling the bulk action. The authors identify two distinct violation channels: (i) initial local non-Gaussianity directly spoiling the analytic soft limit, and (ii) Gaussian initial states (e.g., Bogoliubov) that modify non-equal-time propagators, mapping analytic 4d vertices to non-analytic 3d vertices and breaking the expected $q$-scaling. These results sharpen the conditions under which inflationary consistency relations hold and suggest observable tests and extensions to large-scale structure and conformal-like cosmologies.

Abstract

We study the effect of the initial state on the consistency conditions for adiabatic perturbations. In order to be consistent with the constraints of General Relativity, the initial state must be diffeomorphism invariant. As a result, we show that initial wavefunctional/density matrix has to satisfy a Slavnov-Taylor identity similar to that of the action. We then investigate the precise ways in which modified initial states can lead to violations of the consistency relations. We find two independent sources of violations: i) the state can include initial non-Gaussianities; ii) even if the initial state is Gaussian, such as a Bogoliubov state, the modified 2-point function can modify the q->0 analyticity properties of the vertex functional and result in violations of the consistency relations.