On the Velocity in the Effective Field Theory of Large Scale Structures

TL;DR

This paper calculates the renormalized velocity-related two-point functions in the Effective Field Theory of Large Scale Structures, revealing the necessity of new counterterms and clarifying the role of vorticity and momentum in the theory.

Contribution

It provides the first detailed computation of renormalized velocity correlators, including vorticity, and introduces counterterms to cancel divergences, clarifying velocity definitions in the EFT framework.

Findings

Vorticity power spectrum at leading order is fixed by symmetries.

New counterterms are required to cancel divergences in velocity correlators.

All three velocity-related correlators share the same structure with different coefficients.

Abstract

We compute the renormalized two-point functions of density, divergence and vorticity of the velocity in the Effective Field Theory of Large Scale Structures. Because of momentum and mass conservation, the corrections from short scales to the large-scale power spectra of density, divergence and vorticity must start at order $k^{4}$. For the vorticity this constitutes one of the two leading terms. Exact (approximated) self-similarity of an Einstein-de Sitter ($\Lambda$CDM) background fixes the time dependence so that the vorticity power spectrum at leading order is determined by the symmetries of the problem and the power spectrum around the non-linear scale. We show that to cancel all divergences in the velocity correlators one needs new counterterms. These fix the definition of velocity and do not represent new properties of the system. For an Einstein-de Sitter universe, we show that all three renormalized cross- and auto-correlation functions have the same structure but different numerical coefficients, which we compute. We elucidate the differences between using momentum and velocity.