$k$-Inflation-corrected Einstein-Gauss-Bonnet Gravity with Massless
  Primordial Gravitons
  In the present paper, we study the inflationary phenomenology of a
$k$-inflation corrected Einstein-Gauss-Bonnet theory. Non-canonical kinetic
terms are known for producing Jean instabilities or superluminal sound wave
velocities in the aforementioned era, but we demonstrate in this work that by
adding Gauss-Bonnet string corrections and assuming that the non-canonical
kinetic term $\omega X^\gamma$ is in quadratic, one can obtain a ghost free
description. Demanding compatibility with the recent GW170817 event forces one
to accept that the relation $\ddot\xi=H\dot\xi$ for the scalar coupling
function $\xi (\phi)$. As a result, the scalar functions of the theory are
revealed to be interconnected and by assuming a specific form for one of them,
specifies immediately the other. Here, we shall assume that the scalar
potential is directly derivable from the equations of motion, once the
Gauss-Bonnet coupling is appropriately chosen, but obviously the opposite is
feasible as well. As a result, each term entering the equations of motion, can
be written in terms of the scalar field and a relatively tractable
phenomenology is produced. For quadratic kinetic terms, the resulting scalar
potential is quite elegant functionally. Different exponents, which lead to
either a more perplexed solution for the scalar potential, are still a
possibility which was not further studied. We also discuss in brief the
non-Gaussianities issue under the slow-roll and constant-roll conditions
holding true, and we demonstrate that the predicted amount of non-Gaussianities
is significantly enhanced in comparison to the $k$-inflation free
Einstein-Gauss-Bonnet theory.
