Single-Field Consistency Relations of Large Scale Structure

TL;DR

The paper derives relativistic consistency relations for large-scale structure in CDM and $\Lambda$CDM by showing that long-wavelength perturbations, which were outside the sound horizon since inflation, act as diffeomorphisms. They construct adiabatic modes à la Weinberg and propagate them to include short-scale modes, deriving second-order squeezed-limit metrics and density perturbations. The central result is a set of squeezed-limit relations: $\langle \Phi_{\vec q} \delta_{\vec k_1}(\eta_1) \cdots \delta_{\vec k_n}(\eta_n) \rangle'_q = P_{\Phi}(q) \sum_a \mathcal{O}_a \langle \delta_{\vec k_1}(\eta_1) \cdots \delta_{\vec k_n}(\eta_n) \rangle'$, with operators encoding dilations, special conformal transformations, and time translations, including both inflationary initial conditions and late-time GR effects. The relations hold non-perturbatively in short-scale dynamics and generalize from matter domination to $\Lambda$CDM, providing a powerful, model-independent test of single-field inflation and the equivalence principle in the late universe. They remain valid in the nonlinear regime, including bias, and offer avenues for tests with galaxy and halo clustering, while deviations would signal multi-field dynamics or modified gravity.

Abstract

We derive consistency relations for the late universe (CDM and ΛCDM): relations between an n-point function of the density contrast δand an (n+1)-point function in the limit in which one of the (n+1) momenta becomes much smaller than the others. These are based on the observation that a long mode, in single-field models of inflation, reduces to a diffeomorphism since its freezing during inflation all the way until the late universe, even when the long mode is inside the horizon (but out of the sound horizon). These results are derived in Newtonian gauge, at first and second order in the small momentum q of the long mode and they are valid non-perturbatively in the short-scale δ. In the non-relativistic limit our results match with (Kehagias and Riotto '13) and (Peloso and Pietroni '13). These relations are a consequence of diffeomorphism invariance; they are not satisfied in the presence of extra degrees of freedom during inflation or violation of the Equivalence Principle (extra forces) in the late universe.