  Gravitational waves (GWs) have opened a new window on fundamental physics in
a number of important ways. The next generation of GW detectors may reveal more
information about the polarization structure of GWs. Additionally, there is
growing interest in theories of gravity beyond GR. One such theory which
remains viable within the context of recent measurements of the speed of
propagation of GWs is the teleparallel analogue of Horndeski gravity. In this
work, we explore the polarization structure of this newly proposed formulation
of Horndeski theory. In curvature-based gravity, Horndeski theory is almost
synonymous with extensions to GR since it spans a large portion of these
possible extensions. We perform this calculation by taking perturbations about
a Minkowski background and consider which mode propagates. The result is that
the polarization structure depends on the choice of model parameters in the
teleparallel Horndeski Lagrangian with a maximum of seven propagating degrees
of freedom. While the curvature-based Horndeski results follows as a particular
limit within this setup, we find a much richer structure of both massive and
massless cases which produce scalar--vector--tensor propagating degrees of
freedom. We also find that the GW polarization that emerges from the
teleparallel analogue of Horndeski gravity results in analogous massive and
massless modes which take on at most four polarizations in the massless sector
and two scalar ones in the massive sector. In none of the cases do we find
vector polarizations.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory