| Summary: | In this work we shall introduce a theoretical framework comprised by a non-minimal coupled canonical scalar field, a non-minimal coupling to the Gauss-Bonnet invariant and a non-minimal kinetic coupling. This theoretical framework is basically a non-minimally coupled Einstein-Gauss-Bonnet theory with extra corrections of the form of a non-minimal kinetic coupling. In order to comply with the GW170817 event, we shall impose a constraint on the propagation speed of the primordial tensor perturbations that it is equal to that of light's in vacuum, and this constraint basically specifies the way that the scalar potential and the non-minimal couplings of the theory can be chosen. The whole theoretical framework, which belongs to the larger class of Horndeski theories, cannot yield viable results, due to the fact that the primordial gravitational wave speed is not equal to that of light's. Thus we study this theory by also imposing the constraint of having gravity wave speed equal to the light speed. We directly examine the inflationary phenomenology of our theoretical framework and by assuming the slow-roll conditions, we derived the equations of motion in such a way so that analytical results may be extracted. By using several well motivated models we demonstrate the framework leads to a viable phenomenology.
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