content_image/394208.jpg content_image/394206.jpg content_image/394207.jpg
each other if the equation 

is satisfied. Actually, through the first-order perturbative Einstein equations (5.14), (5.19), (5.17), we can confirm the equation (6.32). This is a trivial result from a general viewpoint, because the Einstein equation is the first class constrained system. However, this trivial result implies that we have derived the source terms Γ$_{i}$ and Γ$_{ij}$ of the second-order Einstein equations consistently. 

Next, we consider Eq. (6.30). Through the second- order Einstein equations (6.28), (6.29), (6.31), and the background Klein-Gordon equation (4.9), we can con- firm that Eq. (6.30) is consistent with the set of the background, first-order and other second-order Einstein equation if the equation 

is satisfied under the background and first-order Einstein equations. Actually, we have already seen that Eq. (6.32) is satisfied under the background and first-order Einstein equations. Taking the divergence of Eq. (6.32), we can immediately confirm Eq. (6.33). Then, Eq. (6.30) gives no information. 

Thus, we have seen that the derived Einstein equations of the second order (6.27)–(6.31) are consistent with each other through Eq. (6.32). This fact implies that the de- rived source terms Γ$_{i}$ and Γ$_{ij}$ of the second-order per- turbations of the Einstein equations, which are defined by Eqs. (6.24) and (6.25), are correct source terms of the second-order Einstein equations. On the other hand, for Γ$_{0}$, we have to consider the consistency between the per- turbative Einstein equations and the perturbative Klein- Gordon equation as seen below. 

Now, we consider the consistency of the second-order perturbation of the Klein-Gordon equation and the Ein- stein equations. The second-order perturbation of the Klein-Gordon equation is given by Eq. (6.18) with the source term (6.19). Since the vector mode (2) ν$_{i}$ and tensor mode (2) χ ij of the second-order do not appear in the ex- pressions (6.18) of the second-order perturbation of the Klein-Gordon equation, we may concentrate on the Ein- stein equations for scalar mode of the second order, i.e., Eqs. (6.28), (6.29), and (6.31) with the definitions (6.23)– (6.25) of the source terms. As in the linear case, the second-order perturbation of the Klein-Gordon equation should also be derived from the set of equations consisting of the second-order perturbations of the Einstein equa- tions (6.28), (6.29), (6.31), the first-order perturbations of the Einstein equations (5.12), (5.14), (5.19), and the background Einstein equations (4.6) and (4.7). Actually, from these equation, we can show that the second-order perturbation of the Klein-Gordon equation is consistent with the background and the second-order Einstein equa- tions if the equation 

<img src='content_image/394208.jpg'>

<img src='content_image/394206.jpg'>

<img src='content_image/394207.jpg'>

is satisfied under the background and the first-order Ein- stein equations. Further, we can also confirm Eq. (6.34) through the background Einstein equations (4.6) and (4.7), the scalar part of the first-order perturbation of the momentum constraint (5.14), the evolution equations (5.19) and (5.17) for the first order scalar and tensor modes in the Einstein equation. 

As shown in Ref. [13], the first-order perturbation of the Klein-Gordon equation is derived from the back- ground and the first-order perturbations of the Einstein equation. In the case of the second-order perturbation, the Klein-Gordon equation (6.18) can be also derived from the background, the first-order, and the second- order Einstein equations. The second-order perturba- tions of the Einstein equation and the Klein-Gordon equation include the source terms Γ$_{0}$, Γ$_{i}$, Γ$_{ij}$, and Ξ$_{(}$$_{K}$$_{)}$ due to the mode-coupling of the linear-order pertur- bations. The second-order perturbation of the Klein- Gordon equation gives the relation (6.34) between the source terms Γ$_{0}$, Γ$_{i}$, Γ$_{ij}$, Ξ$_{(}$$_{K}$$_{)}$ and we have also confirmed that Eq. (6.34) is satisfied due to the background, the first-order perturbation of the Einstein equations, and the Klein-Gordon equation. Thus, the second-order per- turbation of the Klein-Gordon equation is not indepen- dent of the set of the background, the first-order, and the second-order Einstein equations if we impose on the Ein- stein equation at any conformal time η . This also implies that the derived formulae of the source terms Γ$_{0}$, Γ$_{i}$, Γ$_{ij}$, and Ξ$_{(}$$_{K}$$_{)}$ are consistent with each other. In this sense, we may say that the formulae (6.23)–(6.25) and (6.19) for these source terms are correct. 

## VII. SUMMARY AND DISCUSSIONS 

In this paper, we summarized the current status of the formulation of the gauge-invariant second-order cosmo- logical perturbations. Although the presentation in this paper is restricted to the case of the universe filled by a single scalar field, the essence of the general framework of the gauge-invariant perturbation theory is transpar- ent through this simple case. The general framework of the general relativistic higher-order gauge-invariant perturbation theory can be separated into three parts. First one is the general formulation to derive the gauge- transformation rules (2.18) and (2.19). Second one is the construction of the gauge-invariant variables for the per- turbations on the generic background spacetime inspect- ing gauge-transformation rules (2.18) and (2.19) and the decomposition formula (2.36) and (2.37) for perturba- tions of any tensor field. Third one is the application of the above general framework of the gauge-invariant perturbation theory to the cosmological situations. 

To derive the gauge-transformation rules (2.18) and (2.19), we considered the general arguments on the Tay- lor expansion of an arbitrary tensor field on a manifold, the general class of the diffeomorphism which is wider than the usual exponential map, and the general for- 

