On Soft Limits of Large-Scale Structure Correlation Functions

TL;DR

This paper investigates the soft limits of correlation functions in large-scale structure formation, deriving universal symmetry-based conditions and analyzing their validity beyond linear order, with implications for cosmological models.

Contribution

It re-derives universal consistency conditions using the eikonal approximation and critically examines the validity of equal-time relations beyond linear order in cosmology.

Findings

Consistency conditions hold at linear order but fail at next-to-leading order.

Equal-time relations approximate density fluctuations well beyond linear order.

Background method with curvature generalization improves understanding of soft mode effects.

Abstract

We study soft limits of correlation functions for the density and velocity fields in the theory of structure formation. First, we re-derive the (resummed) consistency conditions at unequal times using the eikonal approximation. These are solely based on symmetry arguments and are therefore universal. Then, we explore the existence of equal-time relations in the soft limit which, on the other hand, depend on the interplay between soft and hard modes. We scrutinize two approaches in the literature: The time-flow formalism, and a background method where the soft mode is absorbed into a locally curved cosmology. The latter has been recently used to set up (angular averaged) `equal-time consistency relations'. We explicitly demonstrate that the time-flow relations and `equal-time consistency conditions' are only fulfilled at the linear level, and fail at next-to-leading order for an Einstein de-Sitter universe. While applied to the velocities both proposals break down beyond leading order, we find that the `equal-time consistency conditions' quantitatively approximates the perturbative results for the density contrast. Thus, we generalize the background method to properly incorporate the effect of curvature in the density and velocity fluctuations on short scales, and discuss the reasons behind this discrepancy. We conclude with a few comments on practical implementations and future directions.