  We investigate the propagation of primordial gravitational waves within the
context of the Horndeski theories, for this, we present a generalized transfer
function quantifying the sub-horizon evolution of gravitational waves modes
after they enter the horizon. We compare the theoretical prediction of the
modified primordial gravitational waves spectral density with the aLIGO,
Einstein telescope, LISA, gLISA and DECIGO sensitivity curves. Assuming
reasonable and different values for the free parameters of the theory (in
agreement with the event GW170817 and stability conditions of the theory), we
note that the gravitational waves amplitude can vary significantly in
comparison with general relativity. We find that in some cases the
gravitational primordial spectrum can cross the sensitivity curves for DECIGO
detector with the maximum frequency sensitivity to the theoretical predictions
around 0.05 - 0.30 Hz. From our results, it is clear that the future
generations of space based interferometers can bring new perspectives to
probing modifications in general relativity.
