Renormalizing a Viscous Fluid Model for Large Scale Structure Formation

TL;DR

This paper investigates the renormalization of the Stochastic Adhesion Model (SAM) for cosmic structure formation, demonstrating how to handle cutoff dependence and maintain Galilean invariance through non-local or local-in-time formulations.

Contribution

It introduces a renormalization approach for SAM, showing how to incorporate viscosity and noise terms consistently while preserving symmetries, and compares non-local and local-in-time methods.

Findings

Non-local-in-time corrections act as counter terms for vertex diagrams.

Galilean invariance constrains the form of viscosity and noise terms.

Local-in-time formulation simplifies renormalization with fewer parameters.

Abstract

Using the Stochastic Adhesion Model (SAM) as a simple toy model for cosmic structure formation, we study renormalization and the removal of the cutoff dependence from loop integrals in perturbative calculations. SAM shares the same symmetry with the full system of continuity+Euler equations and includes a viscosity term and a stochastic noise term, similar to the effective theories recently put forward to model CDM clustering. We show in this context that if the viscosity and noise terms are treated as perturbative corrections to the standard eulerian perturbation theory, they are necessarily non-local in time. To ensure Galilean Invariance higher order vertices related to the viscosity and the noise must then be added and we explicitly show at one-loop that these terms act as counter terms for vertex diagrams. The Ward Identities ensure that the non-local-in-time theory can be renormalized consistently. Another possibility is to include the viscosity in the linear propagator, resulting in exponential damping at high wavenumber. The resulting local-in-time theory is then renormalizable to one loop, requiring less free parameters for its renormalization.