Cosmological teleparallel perturbations

There has been growing interest in $f(Q)$ gravity, which has led to significant advancements in the field. However, it is important to note that most studies in this area were based on the coincident gauge, thus overlooking the impact of the connection degrees of freedom. In this work, we pay special attention to the connection when studying perturbations in general teleparallel, metric teleparallel, and symmetric teleparallel theories of gravity. We do not just examine perturbations in the metric, but also in the affine connection. To illustrate this, we investigate cosmological perturbations in $f(G)$, $f(T)$, and $f(Q)$ gravity with and without matter in form of an additional scalar field for spatially flat and curved FLRW geometries. Our perturbative analysis reveals that for general $f(Q)$ backgrounds, there are up to seven degrees of freedom, depending on the background connection. This is in perfect agreement with the upper bound on degrees of freedom established for the first time in $\href{https://doi.org/10.1002/prop.202300185}{Fortschr. Phys. 2023, 2300185}$. In $f(G)$ and $f(T)$ gravity theories, only two tensor modes propagate in the gravity sector on generic curved cosmological backgrounds, indicating strong coupling problems. In the context of $f(Q)$ cosmology, we find that for a particular background connection, where all seven modes propagate, there is at least one ghost degree of freedom. For all other choices of the connection the ghost can be avoided at the cost of strong coupling problem, where only four degrees of freedom propagate. Hence, all of the cosmologies within the teleparallel families of theories in form of $f(G)$, $f(T)$, and $f(Q)$ suffer either from strong coupling or from ghost instabilities. A direct coupling of the matter field to the connection or non-minimal couplings might alter these results.