Systematic Renormalization of the Effective Theory of Large Scale Structure

TL;DR

This paper develops a systematic renormalization method for the effective theory of large scale structure, ensuring physical predictions by associating counterterms with UV-sensitive diagrams, and proves key scaling behaviors and simplifications.

Contribution

It introduces a comprehensive renormalization procedure for large scale structure, including local operator basis and all-order proofs of key scaling and counterterm practices.

Findings

Renormalization associates counterterms with UV-sensitive diagrams order by order.

Short-distance perturbations contribute as $k^2$ to density contrast and $k$ to momentum density.

Counterterms in the Euler equation suffice for correlators of density contrast at all orders.

Abstract

A perturbative description of Large Scale Structure is a cornerstone of our understanding of the observed distribution of matter in the universe. Renormalization is an essential and defining step to make this description physical and predictive. Here we introduce a systematic renormalization procedure, which neatly associates counterterms to the UV-sensitive diagrams order by order, as it is commonly done in quantum field theory. As a concrete example, we renormalize the one-loop power spectrum and bispectrum of both density and velocity. In addition, we present a series of results that are valid to all orders in perturbation theory. First, we show that while systematic renormalization requires temporally non-local counterterms, in practice one can use an equivalent basis made of local operators. We give an explicit prescription to generate all counterterms allowed by the symmetries. Second, we present a formal proof of the well-known general argument that the contribution of short distance perturbations to large scale density contrast $\delta$ and momentum density $\mathbf\pi(\mathbf k)$ scale as $k^2$ and $k$, respectively. Third, we demonstrate that the common practice of introducing counterterms only in the Euler equation when one is interested in correlators of $ \delta$ is indeed valid to all orders.