Inevitable ghost and the degrees of freedom in f(R,G) gravity

The study of linear perturbation theory for general functions of the Ricci and Gauss-Bonnet scalars is done over an empty anisotropic universe, i.e. the Kasner-type background, in order to show that an anisotropic background in general has ghost degrees of freedom, which are absent on Friedmann-Lemaitre-Robertson-Walker (FLRW) backgrounds. The study of the scalar perturbation reveals that on this background the number of independent propagating degrees of freedom is four and reduces to three on FLRW backgrounds, as one mode becomes highly massive to decouple from the physical spectrum. When this mode remains physical, there is inevitably a ghost mode.