Inflation with Gauss-Bonnet coupling

We consider inflationary models with the inflaton coupled to the Gauss-Bonnet term assuming a special relation ?1=2??1 between the two slow-roll parameters ?1 and ?1. For the slow-roll inflation, the assumed relation leads to the reciprocal relation between the Gauss-Bonnet coupling function ?(?) and the potential V(?), and it leads to the relation r=16(1??)?1 that reduces the tensor-to-scalar ratio r by a factor of 1??. For the constant-roll inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor-to-scalar ratio to the first order of ?1 by using the method of Bessel function approximation. The tensor-to-scalar ratio is reduced by a factor of 1??+??˜. Comparing the derived ns?r with the observations, we obtain the constraints on the model parameters ?˜ and ?.