Pinching estimates for solutions of the linearized Ricci flow system in higher dimensions

We prove pinching estimates for solutions of the linearized Ricci flow system on a closed manifold of dimension $n\geq 4$ with positive scalar curvature and vanishing Weyl tensor. If the vanishing Weyl tensor condition is removed, we only give a rough pinching estimate controlled by some blow-up function in a short time. These results extend the $3$-dimensional case due to Anderson and Chow (2005) \cite{[AC]}.