Gauge-invariant cosmological perturbations in general teleparallel gravity

We study linear cosmological perturbations in the most general teleparallel gravity setting, where gravity is mediated by the torsion and nonmetricity of a flat connection alongside the metric. For a general linear perturbation of this geometry around a homogeneous and isotropic background geometry, we derive the irreducible decomposition of the perturbation variables, as well as their behavior under gauge transformations, i.e., infinitesimal diffeomorphisms generated by a vector field. In addition, we also study these properties for the most general set of matter variables and gravitational field equations. We then make use of these result to construct gauge-invariant perturbation variables, using a general approach based on gauge conditions. We further calculate these quantities also in the metric and symmetric teleparallel geometries, where nonmetricity or torsion is imposed to vanish. To illustrate our results, we derive the energy-momentum-hypermomentum conservation equations for both the cosmological background and the linear perturbations. As another example, we study the propagation of tensor perturbations in the $f(G)$, $f(T)$ and $f(Q)$ class of theories.