tion defined by Eq. (3.28) is given by Eq. (2.28) under the gauge transformation and we may regard the generator σ$_{a}$ defined by Eq. (2.29) as an ar- bitrary vector field on M$_{0}$ from the fact that the gen- erator ξ a 2 in Eq. (2.29) is arbitrary. We can apply the procedure to find gauge invariant variables for the first- order metric perturbations (5.4) in Sec. V A. Then, we can accomplish the decomposition (2.30). Following to the same argument as in the linear case, we may choose the components of the gauge invariant variables L$_{ab}$ in Eq. (2.32) as Through the background Einstein equations (4.6), (4.7), and the first-order perturbations (5.14) and (5.19) of the Einstein equation, we can easily derive the first-order perturbation of the Klein-Gordon equation (5.20)[13]. Hence, the first-order perturbation of the Klein-Gordon equation is not independent of the back- ground and the first-order perturbation of the Einstein equation. Therefore, from the viewpoint of the Cauchy problem, any information obtained from the first-order perturbation of the Klein-Gordon equation should also be obtained from the set of the background and the first- order the Einstein equation, in principle. Now, we develop the second-order perturbation the- ory on the cosmological background spacetime in Sec. IV within the general framework of the gauge-invariant per- turbation theory reviewed in Sec. II. Since we have al- ready confirm the important step of our general frame- work, i.e., the assumption for the decomposition (2.23) of the linear-order metric perturbation is correct. Hence, the general framework reviewed in Sec. II is applica- ble. Applying this framework, we define the second-order gauge invariant variables of the metric perturbation in Sec. VI A. In Sec. VI B, we summarize the explicit com- ponents of the gauge invariant parts of the second-order perturbation of the Einstein tensor. In Sec. VI C, we summarize the explicit components of the second-order perturbation of the energy-momentum tensor and the Klein-Gordon equations. Then, in Sec. VI D, we derive the second-order Einstein equations in terms of gauge- invariant variables. The resulting equations have the source terms which constitute of the quadratic terms of the linear-order perturbations. Although these source terms have complicated forms, we give identities which comes from the consistency of all the second-order per- turbations of the Einstein equation and the Klein-Gordon equation in Sec. VI E. The gauge invariant variables (2) Φ and (2) Ψ are the scalar mode perturbations of second order, and (2) ν$_{i}$ and (2) χ$_{ij}$ are the second-order vector and tensor modes of the metric perturbations, respectively. Here, we also note the fact that the decomposition (2.32) is not unique. This situation is similar to the case of the linear-order, but more complicated. In the defini- tion of the gauge invariant variables of the second-order metric perturbation, we may replace where Z $^{′}$a is gauge invariant and X $^{′}$a is transformed as under the gauge transformation X$_{λ}$ → Y$_{λ}$ . This Z $^{′}$a may be different from the vector Z a in Eq. (5.5). By the replacement (6.3), the second-order metric perturbation (2.32) is given in the form where we defined Here, the vector field W a in Eq. (6.7) constitute of some components of gauge invariant second-order metric per- turbation L$_{ab}$ like Z a in Eq. (5.5). The tensor field J$_{ab}$ is manifestly gauge invariant. The gauge transformation rule of the new gauge-variant part Y $^{′}$a of the second-order metric perturbation is given by First, we consider the components of the gauge invari- ant variables for the metric perturbation of second order. The variable ˆ L$_{ab}$ defined by Eq. (2.27) is transformed as A. Gauge-invariant metric perturbations VI. EQUATIONS FOR THE SECOND-ORDER COSMOLOGICAL PERTURBATIONS − a 2 (1) C$_{(}$$_{K}$$_{)}$ = ∂ 2 $_{η}$ϕ$_{1}$ + 2 H ∂$_{η}$ϕ$_{1}$ − Δ ϕ$_{1}$ − ( ∂$_{η}$ (1) Φ +3 ∂$_{η}$ (1) Ψ ) ∂$_{η}$ϕ +2 a 2 (1) Φ ∂V ∂ ¯ ϕ ( ϕ ) + a $^{2}$ϕ$_{1}$ ∂ $^{2}$V ∂ ¯ ϕ 2 ( ϕ ) = 0 . (5.20) L$_{ab}$ = − 2 a 2 (2) Φ ( dη )$_{a}$( dη )$_{b}$ + 2 a 2 (2) ν$_{i}$ ( dη )$_{(}$$_{a}$( dx $^{i}$)$_{b}$$_{)}$ + a 2 ( − 2 (2) Ψ γ$_{ij}$ + (2) χ$_{ij}$ ) ( dx $^{i}$)$_{a}$ ( dx $^{j}$)$_{b}$ , (6.1) D i (2) ν$_{i}$ = 0 , (2) χ i i = 0 , D i (2) χ$_{ij}$ = 0 . (6.2) where (2) ν i and (2) χ$_{ij}$ satisfy the equations X a = X $^{′}$a − Z $^{$^{′}$a}$, (6.3) Y X $^{′}$a − X $^{′}$a = ξ a 1 (6.4) J$_{ab}$ := L$_{ab}$ − £$_{W}$ g$_{ab}$ − 2 £$_{Z}$ ′ K$_{ab}$ − 2 £$_{Z}$ ′ £$_{Z}$ g$_{ab}$ + £ 2 Z ′ g$_{ab}$, (6.6) Y $^{′}$a := Y a + W a + [ X $^{′}$, Z $^{′}$] $^{a}$. (6.7) Y $^{′}$a − X Y $^{′}$a = ξ a (2) + [ ξ$_{(1)}$, X $^{′}$] $^{a}$. (6.8) 15 l$_{ab}$ =: J$_{ab}$ + 2 £$_{X}$ ′ h$_{ab}$ + ( £$_{Y}$ ′ − £ 2 X ′ ) g$_{ab}$, (6.5)