LAUSR

We consider inflationary models with the inflaton coupled to the Gauss-Bonnet term assuming a special relation ${\ensuremath{\delta}}_{1}=2\ensuremath{\lambda}{\ensuremath{\epsilon}}_{1}$ between the two slow-roll parameters ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\epsilon}}_{1}$. For the slow-roll inflation, the assumed relation leads to the reciprocal relation between the Gauss-Bonnet coupling function $\ensuremath{\xi}(\ensuremath{\phi})$ and the potential $V(\ensuremath{\phi})$, and it leads to the relation $r=16(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\epsilon}}_{1}$ that reduces the tensor-to-scalar ratio $r$ by a factor of $1\ensuremath{-}\ensuremath{\lambda}$. 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 ${\ensuremath{\epsilon}}_{1}$ by using the method of Bessel function approximation. The tensor-to-scalar ratio is reduced by a factor of $1\ensuremath{-}\ensuremath{\lambda}+\ensuremath{\lambda}\stackrel{\texttildelow{}}{\ensuremath{\eta}}$. Comparing the derived ${n}_{s}\ensuremath{-}r$ with the observations, we obtain the constraints on the model parameters $\stackrel{\texttildelow{}}{\ensuremath{\eta}}$ and $\ensuremath{\lambda}$.