  Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547],
we study in this paper the deformation of the weighted scalar curvature. By
studying the kernel of the formal $L_\phi^2$-adjoint for the linearization of
the weighted scalar curvature, we prove several geometric results. In
particular, we define a weighted vacuum static space, and study locally
conformally flat weighted vacuum static spaces. We then prove some stability
results of the weighted scalar curvature on flat spaces. Finally, we consider
the prescribed weighted scalar curvature problem on closed smooth metric
measure spaces.
