in [203, 204] $^{9}$: Gravitational waves are linearized tensor perturbations of the metric and do not couple to the energy momentum tensor. Therefore, they do not give rise a gravitational instability, but carry the underlying geometric structure of the space-time. The first calculation of the gravitational wave production was made in [14], and the topic has been considered by many authors [15–17]. For reviews on gravitational waves, see [12, 209]. The gravitational wave perturbations are described by a line element ds 2 + δds $^{2}$, where Transient violation of slow-roll conditions were studied in hybrid inflation [206], for com- putation of the power spectrum, see [207, 208]. Models of inflation with large η were also considered in [151]. 4. Gravitational waves It gives for the slow-roll parameter ϵ$_{1}$ = − ˙ H/H $^{2}$, and inflation takes place only when ¨ a > 0 which is equivalent to ϵ$_{1}$ < 1. Slow-roll inflation takes place when ∀ n, ϵ$_{n}$ ≪ 1. In the slow-roll limit, these parameters can be related to the slow-roll parameters, ϵ$_{0}$ ≡ H , ϵ$_{n}$$_{+1}$ ≡ ln | ϵ$_{n}$ | N . (41) ϵ$_{1}$ ≃ ϵ + O ( ϵ $^{2}$, η$^{2}$, ξ$^{2}$) , ϵ$_{2}$ ≃ 4 ϵ − 2 η + O ( ϵ $^{2}$, η$^{2}$, ξ$^{2}$) . (42) ds 2 = a $^{2}$( τ )(d τ 2 − d x $^{i}$d x$_{i}$ ) , δds 2 = − a $^{2}$( τ ) h$_{ij}$ d x $^{i}$d x j . (43) The gauge invariant and conformally invariant 3-tensor h$_{ij}$ is symmetric, traceless δ $^{ij}$h$_{ij}$ = 0, and divergenceless ∇ $_{i}$h$_{ij}$ = 0 ( ∇ i is a covariant derivative). Massless spin 2 gravitons have two degrees of freedom and as a result are also transverse. This means that in a Fourier domain the gravitational wave has a form For the Einstein gravity, the gravitational wave equation of motion follows that of a massless Klein Gordon equation [12]. Especially, for a flat universe 9 It is possible to extend the calculation of metric perturbation beyond the slow-roll approximations based on a formalism similar to that developed in Refs. [179, 182, 184, 205]. 20 h$_{ij}$ = h$_{+}$e + ij + $^{h}$×$_{e}$× ij . (44) ¨ h i j + 3 H ˙ h i j + ( k 2 a 2 ) h i j = 0 , (45)