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We choose c$_{1}$ = 1 and c$_{2}$ = 0, so that the usual Minkowski vacuum state is recovered in the asymptotic past ( c$_{R}$k | τ | → ∞ ). The power spectrum of curvature perturbations P R = k $^{3}$| v/z$_{R}$ | $^{2}$/ 2 π 2 on the large scales ( c$_{R}$k ≪ aH ) is − 3 2 3 − 2 ν R 

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with spectral index 

As in the case of scalar perturbations, the power spec- trum of tensor perturbations P T = 2 k $^{3}$| 2 u/z$_{T}$ | $^{2}$/ 2 π 2 is given by 

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with spectral index 

where we have defined ν 2 T ≡ τ $^{2}$z ′′ $_{T}$/z$_{T}$ + 1 / 4. All back- ground quantities above are evaluated at the moment such that c$_{T}$ k = aH . This is not exactly the same time as the horizon-crossing time in Eq. (33) for scalar modes, but to lowest order in the slow-roll parameters this differ- ence is unimportant. We can use the slow-roll approxi- mation to estimate the amount of e-folds between horizon crossing of the scalar mode and the tensor mode with a reference scale k , Δ N ∼ ln( c$_{T}$ /c$_{R}$ ) ∼ δ$_{1}$/ 2. An impor- tant observational quantity is the tensor-to-scalar ratio which is defined as 

To first order in the slow-roll approximation, we have 

The spectral indices of scalar and tensor perturbations read 

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which show that the spectral index of scalar perturbation contains not only the Hubble flow parameters but also the GB flow parameters. Even for a solution very close to de Sitter inflation (i.e., ϵ$_{i}$ ≈ 0), the GB term can lead to a red ( δ$_{2}$ > 0) or blue ( δ$_{2}$ < 0) power spectrum of scalar perturbation. If | δ$_{1}$ | ≪ ϵ$_{1}$ , the spectral indices are the same as for a potential-driven slow-roll inflation. 

The tensor-to-scalar ratio (37) is approximately 

which is the modified consistency relation. The degen- eracy of standard consistency relation is broken in the slow-roll inflation with the Gauss-Bonnet correction. For this reason, the future experimental checking of this re- lation is usually regarded as an important test of the simplest forms of inflation. 

The Hubble and GB flow parameters can be expressed in terms of the potential and the GB coupling 

where Q ≡ ω ( V$_{,φ}$/V + 4 ξ$_{,φ}$V/ 3). 

The key result of our paper is the general slow-roll ex- pression for GB inflation, Eqs. (42), (43) and (44), which is new and follows from a nontrivial calculation. 

Let us consider a specific inflation model 

This potential has been widely studied. The specific choice of GB coupling allows us to find an analytic re- lation between the spectral index of curvature perturba- tions and the tensor-to-scalar ratio. If α ≡ 4 V$_{0}$ξ$_{0}$/ 3 = 1, all flow parameters vanish. The motion of the inflaton is frozen because the force due to the slope of the potential is exactly balanced by one, the slope of the GB coupling. In this case, exact de Sitter inflation can be realized for the monomial potential and the inverse monomial GB coupling. If α < 1, choosing ω = 1 is required for a positive ϵ$_{1}$ . In this case the contribution of the positive GB term increases the Hubble expansion rate during in- flation, which makes the evolution of the inflaton slower than in the case of standard slow-roll inflation, while the contribution of the negative GB term decreases the Hub- ble expansion rate. If α > 1, we choose ω = − 1 to guar- antee ϵ$_{1}$ > 0. The potential force drives the inflaton to climb up the potential while the GB force drives the field 

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## IV. AN EXAMPLE MODEL 

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