In this work we shall study Einstein Gauss-Bonnet theories and we investigate when these can have their gravitational wave speed equal to the speed of light , which is unity in natural units , thus becoming compatible with the striking event GW170817 .
We demonstrate how this is possible and we show that if the scalar coupling to the Gauss-Bonnet invariant is constrained to satisfy a differential equation , the gravitational wave speed becomes equal to one .
Accordingly , we investigate the inflationary phenomenology of the resulting restricted Einstein Gauss-Bonnet model , by assuming that the slow-roll conditions hold true .
As we demonstrate , the compatibility with the observational data coming from the Planck 2018 collaboration , can be achieved , even for a power-law potential .
We restricted ourselves to the study of the power-law potential , due to the lack of analyticity , however more realistic potentials can be used , in this case though the calculations are not easy to be performed analytically .
We also pointed out that a string-corrected extension of the Einstein Gauss-Bonnet model we studied , containing terms of the form \sim \xi ( \phi ) G ^ { ab } \partial _ { a } \phi \partial _ { b } \phi can also provide a theory with gravity waves speed c _ { T } ^ { 2 } = 1 in natural units , if the function \xi ( \phi ) is appropriately constrained , however in the absence of the Gauss-Bonnet term \sim \xi ( \phi ) \mathcal { G } the gravity waves speed can never be c _ { T } ^ { 2 } = 1 .
Finally , we discuss which extensions of the above models can provide interesting cosmologies , since any combination of f ( R,X, \phi ) gravities with the above string-corrected Einstein Gauss-Bonnet models can yield c _ { T } ^ { 2 } = 1 , with X = \frac { 1 } { 2 } \partial _ { \mu } \phi \partial ^ { \mu } \phi .