The Large Scale Structure Bootstrap: perturbation theory and bias expansion from symmetries

TL;DR

This paper explores how symmetries, especially extended Galilean invariance, determine the structure of perturbation theory kernels in large-scale structure, impacting bias expansion and model-independent analyses in cosmology.

Contribution

It identifies which perturbation theory coefficients are fixed by symmetries up to third order and extends the understanding of bias expansion in non-standard cosmological models.

Findings

Only three coefficients are not fixed by symmetries for dark matter up to third order.

Seven coefficients are undetermined for generic biased tracers, aligning with existing bias expansions.

Extended Galilean invariance constrains the analytic structure of kernels as external momenta vanish.

Abstract

We investigate the role played by symmetries in the perturbative expansion of the large-scale structure. In particular, we establish which of the coefficients of the perturbation theory kernels are dictated by symmetries and which not. Up to third order in perturbations, for the dark matter density contrast (and for the dark matter velocity) only three coefficients are not fixed by symmetries and depend on the particular cosmology. For generic biased tracers, where number/mass and momentum conservation cannot be imposed in general, this number rises to seven in agreement with other bias expansions discussed in the literature. A crucial role in our analysis is provided by extended Galilean invariance, which follows from diffeomorphism invariance in the non-relativistic limit. We identify a full hierarchy of extended Galilean invariance constraints, which fix the analytic structure of the perturbation theory kernels as the sums of an increasing number of external momenta vanish. Our approach is especially relevant for non-standard models that respect the same symmetries as $\Lambda$CDM and where perturbation theory at higher orders has not been exhaustively explored, such as dark energy and modified gravity scenarios. In this context, our results can be used to systematically extend the bias expansion to higher orders and set up model independent analyses.