Cosmological perturbation in f(R,G) theories with a perfect fluid

TL;DR

This paper analyzes linear cosmological perturbations in f(R,G) modified gravity theories with a perfect fluid, identifying degrees of freedom, stability conditions, and propagation speeds, including potential super-luminal modes and ghost conditions.

Contribution

It provides a comprehensive classification of perturbation modes in f(R,G) theories and derives conditions to avoid instabilities like super-luminal propagation and ghosts.

Findings

Two modes obey de Broglie wave dispersion, indicating potential instability or super-luminal propagation.

Vector perturbations decay over time, indicating stability.

Tensor perturbations can be super-luminal or sub-luminal depending on the model.

Abstract

In order to classify modified gravity models according to their physical properties, we analyze the cosmological linear perturbations for f(R,G) theories (R being the Ricci scalar and G, the Gauss-Bonnet term) with a minimally coupled perfect fluid. For the scalar type perturbations, we identify in general six degrees of freedom. We find that two of these physical modes obey the same dispersion relation as the one for a non-relativistic de Broglie wave. This means that spacetime is either highly unstable or its fluctuations undergo a scale-dependent super-luminal propagation. Two other modes correspond to the degrees of freedom of the perfect fluid, and propagate with the sound speed of such a fluid. The remaining two modes correspond to the entropy and temperature perturbations of the perfect fluid, and completely decouple from the other modes for a barotropic equation of state. We then provide a concise condition on f(R,G) theories, that both f(R) and R+f(G) do fulfill, to avoid the de Broglie type dispersion relation. For the vector type perturbation, we find that the perturbations decay in time. For the tensor type perturbation, the perturbations can be either super-luminal or sub-luminal, depending on the model. No-ghost conditions are also obtained for each type of perturbation.