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Reviving non-Minimal Horndeski-like Theories after GW170817: Kinetic Coupling Corrected Einstein-Gauss-Bonnet Inflation

After the recent GW170817 event of the two neutron stars merging, many string corrected cosmological theories confronted the non-viability peril. This was due to the fact that most of these theories produce massive gravitons primordially. Among these theories were the ones containing a non-minimal kinetic coupling correction term in the Lagrangian, which belong to a subclass of Horndeski theories. In this work we demonstrate how these theories may be revived and we show how these theories can produce primordial gravitational waves with speed $c_T^2=1$ in natural units, thus complying with the GW170817 event. As we show, if the gravitational action of an Einstein-Gauss-Bonnet theory also contains a kinetic coupling of the form $\sim ξ(ϕ) G^{μν}\partial_μϕ\partial_νϕ$, the condition of having primordial massless gravitons, or equivalently primordial gravitational waves with speed $c_T^2=1$ in natural units, results to certain conditions on the scalar field dependent coupling function of the Gauss-Bonnet term, which is also the non-minimal coupling of the kinetic coupling. We extensively study the phenomenological implications of such a theory focusing on the inflationary era, by only assuming slow-roll dynamics for the scalar field. Accordingly, we briefly study the case that the scalar field evolves in a constant-roll way. By using some illustrative examples, we demonstrate that the viability of the theoretical framework at hand may easily be achieved. Also, theories containing terms of the form $\sim ξ(ϕ)\Boxϕg^{μν}\partial_μϕ\partial_νϕ$ and $\sim ξ(ϕ)\left(g^{μν}\partial_μϕ\partial_νϕ\right)^2$ also lead to the same gravitational wave speed as the theory we shall study in this paper, so this covers a larger class of Horndeski theories.