# Conformal wave equations for the Einstein-tracefree matter system

## TL;DR

This paper develops a system of quasilinear wave equations for the conformal Einstein field equations coupled with tracefree matter models, enabling analysis of global existence and stability of de Sitter-like spacetimes.

## Contribution

It constructs a closed system of wave equations for conformal Einstein-tracefree matter systems, including new results for the conformally coupled scalar field.

## Key findings

- Established a framework for the propagation of constraints in these systems.
- Proved global existence and stability for de Sitter-like spacetimes with tracefree matter.
- Provided new results for the conformally coupled scalar field.

## Abstract

Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz [Ann. H. Poincar\'e 16, 2059 (2015)], in this article we show how to construct a system of quasilinear wave equations for the geometric fields associated to the conformal Einstein field equations coupled to matter models whose energy-momentum tensor has vanishing trace. In this case, the equation of conservation for the energy-momentum tensor is conformally invariant. Our analysis includes the construction of a subsidiary evolution system which allows to prove the propagation of the constraints. We discuss how the underlying structure behind these systems of equations is the integrability conditions satisfied by the conformal field equations. The main result of our analysis is that both the evolution and subsidiary equations for the geometric part of the conformal Einstein-tracefree matter field equations close without the need of any further assumption on the matter models other than the vanishing of the trace of the energy-momentum tensor. Our work is supplemented by an analysis of the evolution and subsidiary equations associated to three basic tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang-Mills field. As an application we provide a global existence and stability result for de Sitter-like spacetimes. In particular, the result for the conformally coupled scalar field is new in the literature.