Rectifying Einstein-Gauss-Bonnet Inflation in View of GW170817
  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 $c_T^2$ is required to be
$c_T^2\simeq 1$ 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 $c_T^2\simeq 1$ restricts the
functional form of the scalar Gauss-Bonnet coupling function $\xi(\phi)$ and of
the scalar potential $V(\phi)$, 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
$N=\int_{\phi_i}^{\phi_f}\xi''/\xi'd \phi$, 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 $\xi'/\xi''\ll 1$, which is motivated
by the slow-roll conditions that are imposed on the scalar field evolution and
on the Hubble rate, in which case the study is easier. Our approach opens a new
window in viable Einstein-Gauss-Bonnet theories of gravity.
