  The striking GW170817 event indicated that the graviton is nearly massless,
since the gamma rays emitted from the two neutron stars merging arrived almost
simultaneously with the gravitational waves. Thus, the graviton must also be
massless during the inflationary and post-inflationary era, since there is no
obvious reason to believe otherwise. In this letter we shall investigate the
theoretical implications of the constraint that the graviton is massless to an
Einstein-Gauss-Bonnet theory with linear coupling of the scalar field to the
four dimensional Gauss-Bonnet invariant. As we show, the constraint of having
gravitational wave speed of the primordial gravitational waves equal to one,
severely restricts the dynamics of the scalar field, imposing a direct
constant-roll evolution on it. Also, as we show, the spectral index of the
primordial scalar perturbations for the GW170817-compatible
Einstein-Gauss-Bonnet theory with linear coupling is different in comparison to
the same theory with non-linear coupling. Thus the phenomenology of the model
is expected to be different, and we briefly discuss this issue too. In
addition, the constant-roll condition is always related to non-Gaussianities,
thus it is interesting that the imposition of a massless graviton in an
Einstein-Gauss-Bonnet theory with linear coupling may lead to
non-Gaussianities, so we briefly discuss this issue too.
