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 4 degrees of freedom. Two of these degrees will have a superluminal propagation-with k-dependent speed of propagation-whereas the other two will have 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.