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

TL;DR

This paper develops the EFTofLSS framework for velocity-related observables, showing that short-scale physics renormalizes large-scale density, velocity divergence, and vorticity through a controlled set of counterterms. Due to mass and momentum conservation, leading UV corrections scale as $k^{4}$, with vorticity receiving one of the leading contributions; self-similarity in EdS fixes the time dependence of the leading vorticity term and constrains the form of renormalized correlators. The authors demonstrate that additional counterterms in the continuity equation are necessary to cancel all divergences in velocity correlators, deriving explicit structures for the renormalized two-point functions and identifying the finite parameters that must be fit to simulations or observations. They compare the velocity and momentum formulations, showing how they are related and how their IR/UV properties differ, and discuss the implications for extracting velocity fields from simulations. The results provide a coherent, renormalized picture of large-scale velocity statistics, including the leading $P_{\omega\omega}$ contribution and the shared structure of $P_{\delta\delta}$, $P_{\delta\theta}$, and $P_{\theta\theta}$, with clear guidance for simulation-based calibration and future extension to $\,\Lambda$CDM cosmologies.

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 ($Λ$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.

Figures (1)

• Figure 1: Scaling dependence of the invariant correlator $\mathcal{I}$. The solid, dashed, dot-dashed and dotted lines are the the tree-level, one-loop, LEC and noise contributions to $\mathcal{I}$. In addition, we plot the scaling of the two-loop and the higher order derivative terms which are represented by the thin dashed and dot-dashed lines, respectively. The gray shaded regions delimit the regions of dominance of one of the contributions. For $n\approx-3/2$, which is relevant for our universe, we note that the LEC and one-loop contributions are much more important than the noise term.