  We prove an estimate for solutions to the linearized Ricci flow system on
closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is
preserved estimate for the Ricci curvatures of solutions to the Ricci flow on
3-manifolds with positive Ricci curvature. In our estimate we make no
assumption on the curvature of the initial metric. We show that the norm of the
solution of the Lichnerowicz Laplacian heat equation (coupled to the Ricci
flow) is bounded by a constant (depending on time) times the scalar curvature
plus a constant. This relies on a Bochner type formula and establishing the
nonnegativity of a degree 4 homogeneous polynomial in 6 variables.
