For all degenerate f(R) models, the linearized spectrum on maximally symmetric backgrounds is empty: no graviton and no scalaron.
The quantum theory of scalar fields on the de Sitter expanding universe
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New quantum modes of the free scalar field are derived in a special time-evolution picture that may be introduced in moving charts of de Sitter backgrounds. The wave functions of these new modes are solutions of the Klein-Gordon equation and energy eigenfunctions, defining the energy basis. This completes the scalar quantum mechanics where the momentum basis is well-known from long time. In this enlarged framework the quantization of the scalar field can be done in canonical way obtaining the principal conserved one-particle operators and the Green functions.
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Propagating degrees of freedom on maximally-symmetric backgrounds in $f(R)$ theories of gravity
For all degenerate f(R) models, the linearized spectrum on maximally symmetric backgrounds is empty: no graviton and no scalaron.