In this work we introduce a new theoretical framework for Einstein-Gauss-Bonnet theories of gravity, which results to particularly elegant, functionally simple and transparent gravitational equations of motion, slow-roll indices and the corresponding observational indices. The main requirement is that the Einstein-Gauss-Bonnet theory has to be compatible with the GW170817 event, so the gravitational wave speed c 2T is required to be c 2 T ≃ 1 in natural units. This assumption was also made in a previous work of ours, but in this work we express all the related quantities as functions of the scalar field. The constraint c 2 T ≃ 1 restricts the functional form of the scalar Gauss-Bonnet coupling function ξ(φ) and of the scalar potential V (φ), which must satisfy a differential equation. However, by also assuming that the slow-roll conditions hold true, the resulting equations of motion and the slow-roll indices acquire particularly simple forms, and also the relation that yields the e-foldings number is N = φ f φ i ξ ′′ /ξ ′ dφ, a fact that enables us to perform particularly simple calculations in order to study the inflationary phenomenological implications of several models. As it proves, the models we presented are compatible with the observational data, and also satisfy all the assumptions made during the process of extracting the gravitational equations of motion. More interestingly, we also investigated the phenomenological implications of an additional condition ξ ′ /ξ ′′ ≪ 1, which is motivated by the slow-roll conditions that are imposed on the scalar field evolution and on the Hubble rate. As we shall show, the resulting constraint differential equation that constrains the functional form of the scalar Gauss-Bonnet coupling function ξ(φ) and of the scalar potential V (φ), is simpler in this case, and in effect the whole study becomes somewhat easier. As we also show, compatibility with the observational data can also be achieved in this case too, in a much simpler and less constrained way. Our approach opens a new window in viable Einstein-Gauss-Bonnet theories of gravity.

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