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Inspired by a similar analysis for the vacuum conformal Einstein fieldequations by Paetz [Ann. H. Poincar\'e 16, 2059 (2015)], in this article weshow how to construct a system of quasilinear wave equations for the geometricfields associated to the conformal Einstein field equations coupled to mattermodels whose energy-momentum tensor has vanishing trace. In this case, theequation of conservation for the energy-momentum tensor is conformallyinvariant. Our analysis includes the construction of a subsidiary evolutionsystem which allows to prove the propagation of the constraints. We discuss howthe underlying structure behind these systems of equations is the integrabilityconditions satisfied by the conformal field equations. The main result of ouranalysis is that both the evolution and subsidiary equations for the geometricpart of the conformal Einstein-tracefree matter field equations close withoutthe need of any further assumption on the matter models other than thevanishing of the trace of the energy-momentum tensor. Our work is supplementedby an analysis of the evolution and subsidiary equations associated to threebasic tracefree matter models: the conformally invariant scalar field, theMaxwell field and the Yang-Mills field. As an application we provide a globalexistence and stability result for de Sitter-like spacetimes. In particular,the result for the conformally coupled scalar field is new in the literature.

Conformal wave equations for the Einstein-tracefree matter system