content_image/381370.jpg content_image/381371.jpg content_image/381372.jpg content_image/381373.jpg
Finally, scalar part of Eqs. (6.21)–(6.22) are summarized as 

<img src='content_image/381370.jpg'>

<img src='content_image/381371.jpg'>

<img src='content_image/381372.jpg'>

<img src='content_image/381373.jpg'>

where Γ j i := γ $^{kj}$Γ$_{ik}$ and Γ k k = γ $^{ij}$Γ$_{ij}$ . Eq. (6.31) is the second-order extension of Eq. (5.19), which is the master equation of scalar mode of the second-order cosmological perturbation in a universe filled with a single scalar field. 

Thus, we have a set of ten equations for the second- order perturbations of a universe filled with a single scalar field, Eqs. (6.26)–(6.31). To solve this system of equa- tions of the second-order Einstein equation, first of all, we have to solve the linear-order system. This is ac- complished by solving Eq. (5.19) to obtain the potential 

(1) Φ , ϕ$_{1}$ is given through (5.14), and the tensor mode (1) χ ij is given by solving Eq. (5.17). Next, we evaluate the quadratic terms, Γ$_{0}$, Γ$_{i}$ and Γ$_{ij}$ of the linear-order pertur- bations, which are defined by Eqs. (6.23)–(6.25). Then, using the information of Eqs. (6.23)–(6.25), we estimate the source term in Eq. (6.31). If we know the two in- dependent solutions to the linear-order master equation (5.19), we can solve Eq. (6.31) through the method us- ing the Green functions. After constructing the solution 

(2) Φ to Eq. (6.31), we can obtain the second-order met- 

ric perturbation (2) Ψ through Eq. (6.29). Thus, we have obtained the second-order gauge invariant perturbation ϕ$_{2}$ of the scalar field through Eq. (6.28). Thus, the all 

scalar modes (2) Φ , (2) Ψ , ϕ$_{2}$ are obtained. Equation (6.30) is then used to check the consistency of the second-order perturbation of the Klein Gordon equation (6.18) as in Sec. VI E. 

For the vector-mode, (1) ν$_{i}$ of the first-order identically vanishes due to the momentum constraint (5.16) for the linear-order metric perturbations. On the other hand, in the second-order, we have evolution equation (6.27) 

of the vector mode (2) ν$_{i}$ with the initial value constraint. This evolution equation of the second-order vector mode should be consistent with the initial value constraint, which is confirmed in Sec. VI E. Equations (6.27) also im- 

ply that the second-order vector-mode perturbation may be generated by the mode couplings of the linear order perturbations. As the simple situations, the generation of the second-order vector mode due to the scalar-scalar mode coupling is discussed in Refs. [26]. 

The second-order tensor mode is also generated by the mode-coupling of the linear-order perturbations through the source term in Eq. (6.26). Note that Eq. (6.26) is al- most same as Eq. (5.17) for the linear-order tensor mode, except for the existence of the source term in Eq. (6.26). If we know the solution to the linear-order Einstein equa- tions (5.17) and (5.19), we can evaluate the source term in Eq. (6.26). Further, we can solve Eq. (6.26) through the Green function method. This leads the generation of the gravitational wave of the second order. Actually, in the simple situation where the first-order tensor mode ne- glected, the generation of the second-order gravitational waves discussed in some literature[27]. 

## E. Consistency of equations for second-order perturbations 

Now, we consider the consistency of the second-order perturbations of the Einstein equations (6.28)–(6.31) for the scalar modes, Eqs. (6.27) for vector mode, and the Klein-Gordon equation (6.18). The consistency check of these equations are necessary to guarantee that the de- rived equations are correct, since the second-order Ein- stein equations have complicated forms owing to the quadratic terms of the linear-order perturbations that arise from the nonlinear effects of the Einstein equations. Since the first equation in Eqs. (6.27) is the initial value 

constraint for the vector mode (2) ν$_{i}$ and it should be consis- tent with the evolution equation, i.e., the second equation of Eqs. (6.27). these equations should be consistent with each other from the general arguments of the Einstein equation. Explicitly, these equations are consistent with 

