  Classical gravity coupled to a CFT$_4$ (matter) is considered. The effect of
the quantum dynamics of matter on gravity is studied around maximally symmetric
spaces (flat, de Sitter and Anti de Sitter). The structure of the graviton
propagator is modified and non-trivial poles appear due to matter quantum
effects. The position and residues of such poles are mapped as a function of
the relevant parameters, the central charge of the CFT$_4$, the two $R^2$
couplings of gravity as well as the curvature of the background space-time. The
instabilities induced are determined. Such instabilities can be important in
cosmology as they trigger the departure from de Sitter space and in some
regions of parameters are more important than the well-known scalar
instabilities. It is also determined when the presence of such instabilities is
unreliable if the associated scales are larger than the ``species" cutoff of
the gravitational theory.
