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In this work we study the GW170817-compatible Einstein-Gauss-Bonnet theoriesduring the reheating and the end of inflationary era. Given the scalar fieldpotential $V(\phi)$ which can have some intrinsic importance for the theory,determining the scalar coupling function $\xi(\phi)$ can be cumbersome due tolack of analyticity. The GW170817 observation constrains the scalar couplingfunction and the scalar field potential to have some interdependence, thusduring the slow-roll era one can calculate the scalar coupling function.However, when the slow-roll era ends, it is expected that the scalar couplingfunction should have a different form and the same applies for the reheatingera, assuming that the scalar potential of the theory does not change. In thiswork we exactly aim to highlight this feature of Einstein-Gauss-Bonnettheories, as the Universe evolves through distinct sequential evolution eras,and we focus on how to determine the scalar coupling function during thevarious evolutionary eras, from inflation to the reheating era. Regarding boththe end of the inflationary era and the reheating era, it is found that theHubble rate obeys a constant-roll-like condition of the form $\dot{H}=\deltaH^2$, thus the determination of the scalar Gauss-Bonnet function $\xi(\phi)$ isreduced to solving a differential equation. A mentionable feature of the eraexactly at the end of inflation is that the Klein-Gordon equation is decoupledfrom the field equations, because the Gauss-Bonnet invariant is zero. Weprovide several examples of interest to support our arguments.

Einstein-Gauss-Bonnet Cosmological Theories at Reheating and at the End of the Inflationary Era