Scalar mode propagation in modified gravity with a scalar field
  We study the propagation of the scalar modes around a
Friedmann-Lemaitre-Robertson-Walker universe for general modifications of
gravity in the presence of a real scalar field. In general, there will be two
propagating scalar perturbation fields, which will have in total four degrees
of freedom. Two of these degrees will have a superluminal propagation--with
k-dependent speed of propagation--whereas the other two will travel with the
speed of light. Therefore, the scalar degrees of freedom do not modify the
general feature of modified gravity models: the appearance of modes whose
frequency depends on the second power of the modulus of the wave vector.
Constraints are given and special cases are discussed.
