Einstein--Gauss--Bonnet Inflationary Cosmology in Phase-$\theta$ Formalism

A central challenge in testing inflationary scenarios with cosmic microwave background (CMB) data is to derive predictions for cosmological observables that remain accurate beyond the slow-roll regime, where the standard consistency relations are no longer applicable. We address this issue in the context of Einstein–Gauss–Bonnet (EGB) gravity by introducing a phase-$\theta$ parametrization of the inflationary dynamics. Within this framework, the background evolution and the coefficients governing scalar and tensor perturbations are expressed entirely in terms of a single monotonic phase variable and the Hubble parameter. This construction provides a closed, analytical mapping between the background parameters of a given model and the associated inflationary observables. A notable advantage of the proposed parametrization is that it remains regular throughout the inflationary epoch, including the end-of-inflation regime, in which the conventional slow-roll consistency relations no longer hold. As an illustrative application, we consider a Starobinsky-type potential supplemented by a linear coupling between the inflaton and the Gauss–Bonnet invariant. Using the proposed formalism, we demonstrate that the resulting predictions of the Starobinsky model are consistent with the most recent constraints from ACT DR6 and BICEP/Keck, explicitly incorporating the impact of the non-minimal Gauss–Bonnet coupling on the expected values of the cosmological perturbation parameters. Finally, we compute the corresponding relic stochastic gravitational-wave background and evaluate its detectability with current and forthcoming gravitational-wave observatories.