Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature

TL;DR

This paper provides a rigorous proof for the existence of a decomposition theorem for symmetric tensor fields on spaces with constant curvature, which is crucial for cosmological perturbation analysis.

Contribution

It establishes the existence of a symmetric tensor decomposition on constant curvature spaces, filling a gap in the mathematical foundation of cosmological perturbation theory.

Findings

Proves the existence of tensor decomposition using elliptic PDE theory.

Ensures the decomposition is well-defined for smooth tensor fields.

Supports the mathematical rigor of cosmological perturbation methods.

Abstract

In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses -- beside the Hodge decomposition for one-forms -- an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss' theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations.