Single-Field Consistency Relations of Large Scale Structure. Part II: Resummation and Redshift Space

TL;DR

The paper develops a nonperturbative, resummed framework for single-field consistency relations in Large Scale Structure, valid even when long-wavelength modes with momentum $q$ are strongly coupled to short scales. It extends the theory to multiple soft legs, soft loops, and soft internal lines, and then translates the relations into redshift space without relying on the single-stream approximation. A key result is that equal-time correlators remain insensitive to soft modes due to the equivalence principle, while unequal-time correlators can exhibit soft-mode effects; perturbative checks, including tree-level trispectra and 1-loop power spectra, confirm the consistency relations. The redshift-space version reveals how long modes induce both translations and line-of-sight displacements, offering robust tests for multifield inflation or violations of the equivalence principle using squeezed-limit signals.

Abstract

We generalize the recently derived single-field consistency relations of Large Scale Structure in two directions. First, we treat the effect of the long modes (with momentum q) on the short ones (with momentum k) non-perturbatively, by writing resummed consistency relations which do not require k/q δ_q << 1. These relations do not make any assumptions on the short-scales physics and are extended to include (an arbitrary number of) multiple long modes, internal lines with soft momenta and soft loops. We do several checks of these relations in perturbation theory and we verify that the effect of soft modes always cancels out in equal-time correlators. Second, we write the relations directly in redshift space, without assuming the single-stream approximation: not only the long mode affects the short scales as a homogeneous gravitational field, but it also displaces them by its velocity along the line-of-sight. Redshift space consistency relations still vanish when short modes are taken at equal time: an observation of a signal in the squeezed limit would point towards multifield inflation or a violation of the equivalence principle.

Figures (2)

• Figure 1: Two diagrams that contribute to the tree-level trispectrum. Left: $T_{1122}$. Right: $T_{1113}$.
• Figure 2: Two diagrams that contribute to the 1-loop power spectrum. Left: $P_{22}$. Right: $P_{31}$.