Gravitational waves in modified teleparallel theories

We investigate the gravitational waves and their properties in various modified teleparallel theories, such as f(T), f(T,B), and f(T,TG) gravities. We perform the perturbation analysis both around a Minkowski background and in the case where a cosmological constant is present, and for clarity we use both the metric and the tetrad languages. For f(T) gravity we verify the result that no further polarization modes comparing to general relativity are present at first-order perturbation level, and we show that in order to see extra modes one should look at third-order perturbations. For nontrivial f(T,B) gravity, by examining the geodesic deviation equations, we show that extra polarization models, namely the longitudinal and breathing modes, do appear at first-order perturbation level, and the reason for this behavior is the fact that although the first-order perturbation does not have any effect on T, it does affect the boundary term B. Finally, for f(T,TG) gravity we show that at first-order perturbations the gravitational waves exhibit the same behavior as those of f(T) gravity. Since different modified teleparallel theories exhibit different gravitational wave properties, the advancing gravitational-wave astronomy would help to alleviate the degeneracy not only between curvature and torsional modified gravity but also between different subclasses of modified teleparallel gravities.