Soft-Pion Theorems for Large Scale Structure

TL;DR

This work reframes large-scale structure consistency relations as nonperturbative Ward identities arising from residual diffeomorphisms, with the velocity acting as the soft pion. By systematically mapping symmetries from $oldsymbol{ζ}$-gauge to Newtonian gauge and exploiting the adiabatic mode condition, the authors derive an infinite tower of scalar and tensor consistency relations, including the Newtonian translation relation of Kehagias–Riotto and Peloso–Pietroni as the sub-Hubble limit of a special conformal transformation. They examine the robustness of these relations under realistic galaxy dynamics and biasing, identifying conditions (e.g., infrared-divergent nonlocal bias or velocity bias) that can violate them, while showing that a local bias model and adiabatic large-scale flows preserve the relations. The paper also provides a simple fluid Lagrangian that captures LSS dynamics in the Newtonian limit and clarifies how relativistic diffeomorphisms generate the same constraints in GR, highlighting the practical importance of these symmetry-based relations for testing initial conditions and the equivalence principle on cosmological scales.

Abstract

Consistency relations -- which relate an N-point function to a squeezed (N+1)-point function -- are useful in large scale structure (LSS) because of their non-perturbative nature: they hold even if the N-point function is deep in the nonlinear regime, and even if they involve astrophysically messy galaxy observables. The non-perturbative nature of the consistency relations is guaranteed by the fact that they are symmetry statements, in which the velocity plays the role of the soft pion. In this paper, we address two issues: (1) how to derive the relations systematically using the residual coordinate freedom in the Newtonian gauge, and relate them to known results in $ζ$-gauge (often used in studies of inflation); (2) under what conditions the consistency relations are violated. In the non-relativistic limit, our derivation reproduces the Newtonian consistency relation discovered by Kehagias \& Riotto and Peloso & Pietroni. More generally, there is an infinite set of consistency relations, as is known in $ζ$-gauge. There is a one-to-one correspondence between symmetries in the two gauges; in particular, the Newtonian consistency relation follows from the dilation and special conformal symmetries in $ζ$-gauge. We probe the robustness of the consistency relations by studying models of galaxy dynamics and biasing. We give a systematic list of conditions under which the consistency relations are violated; violations occur if the galaxy bias is non-local in an infrared divergent way. We emphasize the relevance of the adiabatic mode condition, as distinct from symmetry considerations. As a by-product of our investigation, we discuss a simple fluid Lagrangian for LSS.