One-particle-irreducible consistency relations for cosmological perturbations

TL;DR

Problem: establish universal, quantum-mechanically exact relations between equal-time 1PI correlators Γ^(n) of the curvature perturbation in the squeezed limit. Approach: derive Ward identities from residual 3D conformal diffeomorphisms of the unitary-gauge action and formulate the constraints as relations for the 3D effective action Γ[ζ̄]. Main results: dilation and special-conformal consistency relations linking Γ^(n) to Γ^(n+1), with explicit forms Gamma^(n+1)(0, k1, ..., kn) = (3 − D_n) Gamma^(n)(k1, ..., kn) and 2 grad_{q→0} Gamma^(n+1)(q, k1, ..., −q − sum k) = sum_a S_a Gamma^(n)(k1, ..., k_n), demonstrated in slow-roll inflation and flat-space ghost condensate, and a flat-space counterexample highlighting limits. Significance: the framework does not rely on de Sitter isometries or horizon physics, applies to single-field dynamics beyond slow roll, clarifies off-shell behavior, and sets the stage for extensions to tensors and multi-field models.

Abstract

We derive consistency relations for correlators of scalar cosmological perturbations which hold in the "squeezed limit" in which one or more of the external momenta become soft. Our results are formulated as relations between suitably defined one-particle irreducible N-point and (N-1)-point functions that follow from residual spatial conformal diffeomorphisms of the unitary gauge Lagrangian. As such, some of these relations are exact to all orders in perturbation theory, and do not rely on approximate deSitter invariance or other dynamical assumptions (e.g., properties of the operator product expansion or the behavior of modes at horizon crossing). The consistency relations apply model-independently to cosmological scenarios where the time evolution is driven by a single scalar field. Besides reproducing the known results for single-field inflation in the slow roll limit, we verify that our consistency relations hold more generally, for instance in ghost condensate models in flat space. We comment on possible extensions of our results to multi-field models.