Consistency Relations for the Conformal Mechanism

TL;DR

The paper derives consistency relations for theories with spontaneously broken conformal symmetry and a linearly realized de Sitter subalgebra, showing how a soft Goldstone mode π constrains N-point functions through a background-wave approach. It provides a concrete equal-time master relation that decomposes into contributions from broken generators and verifies these relations with explicit correlator calculations. The authors uncover observational consequences, including a novel one-loop π exchange that enhances the collapsed limit of the four-point function and induces anisotropy and stochastic signatures in spectator fields, potentially observable as scale-dependent bias and μ-distortion signals. They also address subtleties in conformal weight assignment and off-shell ambiguities, establishing Ward-identity–based, on-shell formulations to ensure robust, model-independent predictions within the conformal mechanism.

Abstract

We systematically derive the consistency relations associated to the non-linearly realized symmetries of theories with spontaneously broken conformal symmetry but with a linearly-realized de Sitter subalgebra. These identities relate (N+1)-point correlation functions with a soft external Goldstone to N-point functions. These relations have direct implications for the recently proposed conformal mechanism for generating density perturbations in the early universe. We study the observational consequences, in particular a novel one-loop contribution to the four-point function, relevant for the stochastic scale-dependent bias and CMB mu-distortion.

Figures (3)

• Figure 1: Schematic representation of the consistency relations: $(N+1)$-point functions with a soft external $\pi$ leg are related to $N$-point functions.
• Figure 2: Factorization of an $(N+M)$-point function via an exchange of a soft $\pi$.
• Figure 3: The four-point function of $\chi$'s with an exchange of one and two soft $\pi$'s.