  In this work we shall calculate in detail the effect of an
GW170817-compatible Einstein-Gauss-Bonnet theory on the energy spectrum of the
primordial gravitational waves. The spectrum is affected by two
characteristics, the overall amplification/damping factor caused by the
GW170817-compatible Einstein-Gauss-Bonnet theory and by the tensor spectral
index and the tensor-to-scalar ratio. We shall present the formalism for
studying the inflationary dynamics and post-inflationary dynamics of
GW170817-compatible Einstein-Gauss-Bonnet theories for all redshifts starting
from the radiation era up to the dark energy era. We exemplify our formalism by
using two characteristic models, which produce viable inflationary and dark
energy eras. As we demonstrate, remarkably the overall damping/amplification
factor is of the order of unity, thus the GW170817-compatible
Einstein-Gauss-Bonnet models affect the primordial gravitational waves energy
spectrum only via their tensor spectral index and the tensor-to-scalar ratio.
Both models have a blue tilted tensor spectrum, and therefore the predicted
energy spectrum of the primordial gravity waves can be detectable by most of
the future gravitational waves experiments, for various reheating temperatures.
