  This paper investigates the propagation of cosmological gravitational waves
interacting with free-streaming neutrinos within the context of Horndeski
theories of gravity constrained by the detection of GW170817. We apply the
theory of cosmological perturbations to explicitly derive the
Einstein-Boltzmann equation for the damped propagation of first-order
transverse traceless gravitational waves. In contrast to general relativity, we
argue that modified gravity can give rise to non-vanishing free-streaming
damping effects during the cosmological matter dominated era. We also provide
an analytic formula for the main multipole order with which modified gravity
and free-streaming neutrinos damp the variety of tensor correlation functions
of the cosmic microwave background.
