Modified propagation of gravitational waves from the early radiation era

We study the propagation of cosmological gravitational wave (GW) backgrounds from the early radiation era until the present day in modified theories of gravity. Comparing to general relativity (GR), we study the effects that Horndeski parameters, such as the run rate of the effective Planck mass αM and the tensor speed excess αT, have on the present-day GW spectrum. We use both the WKB estimate, which provides an analytical description but fails at superhorizon scales, and numerical simulations that allow us to go beyond the WKB approximation. We show that αT makes relatively insignificant changes to the GR solution, especially taking into account the constraints on its value from GW observations by the LIGO-Virgo collaboration, while αM can introduce modifications to the spectral slopes of the GW energy spectrum in the low-frequency regime depending on the considered time evolution of αM. The latter effect is additional to the damping or growth occurring equally at all scales that can be predicted by the WKB approximation. In light of the recent observations by pulsar timing array collaborations and future detectors such as SKA, LISA, DECIGO, BBO, or ET, we show that, in most of the cases, constraints can not be placed on the effects of αM and the initial GW energy density separately, but only on the combined effects of the two.