11 Next, we consider the perturbation of the Klein- Gordon equation Through the perturbative expansions (3.16) and (2.22), the Klein-Gordon equation (3.24) is expanded as C$_{(}$$_{K}$$_{)}$ is the background Klein-Gordon equation C$_{(}$$_{K}$$_{)}$ := ∇$_{a}$∇ $^{a}$ϕ − ∂V ∂ ¯ ϕ ( ϕ ) = 0 . (3.26) ¯ C$_{(}$$_{K}$$_{)}$ := ¯ ∇ a ¯ ∇$_{a}$ ¯ ϕ − ∂V ∂ ¯ ϕ ( ¯ ϕ ) = 0 . (3.24) ¯ C$_{(}$$_{K}$$_{)}$ =: C$_{(}$$_{K}$$_{)}$ + λ (1) C$_{(}$$_{K}$$_{)}$ + 1 2 λ 2 (2) C$_{(}$$_{K}$$_{)}$ + O ( λ $^{3}$) . (3.25) The first- and the second-order perturbations (1) C$_{(}$$_{K}$$_{)}$ and (2) C$_{(}$$_{K}$$_{)}$ are also decomposed into the gauge-invariant and the gauge-variant parts as where Here, we note that Eqs. (3.27) have the same form as Eqs. (2.36) and (2.37). By virtue of the order by order evaluations of the Klein- Gordon equation, the first- and the second-order pertur- bation of the Klein-Gordon equation are necessarily given in gauge-invariant form as Then, the perturbative Einstein equation is given by at linear order and We should note that, in Ref. [12], we summarized the formulae of the energy momentum tensors for an perfect fluid, an imperfect fluid, and a scalar field. Further, we also summarized the equations of motion of these three matter fields: i.e., the energy continuity equation and the Euler equation for a perfect fluid; the energy con- tinuity equation and the Navier-Stokes equation for an imperfect fluid; the Klein-Gordon equation for a scalar field. All these formulae also have the same form as the decomposition formulae (2.36) and (2.37). In this sense, we may say that the decomposition formulae (2.36) and (2.37) are universal. at second order. These explicitly show that, order by or- der, the Einstein equations are necessarily given in terms of gauge invariant variables only. Together with Eqs. (3.30), we have seen that the first- and the second-order perturbations of the Einstein equations and the Klein-Gordon equation are necessarily given in gauge-invariant form. This implies that we do not have to consider the gauge degree of freedom, at least in the level where we concentrate only on the equations of the system. We have reviewed the general outline of the second- order gauge invariant perturbation theory. We also note that the ingredients of this section are independent of the explicit form of the background metric g$_{ab}$ , except for the decomposition assumption (2.23) for the linear-order metric perturbations and are valid not only in cosmologi- cal perturbation case but also the other generic situations if Eq. (2.23) is correct. Within this general framework, we develop a second-order cosmological perturbation the- ory in terms of the gauge invariant variables. Finally, we impose the perturbed Einstein equation of each order, C. Perturbations of the Einstein equation (1) C$_{(}$$_{K}$$_{)}$ =: (1) C$_{(}$$_{K}$$_{)}$ + £$_{X}$ C$_{(}$$_{K}$$_{)}$, (2) C$_{(}$$_{K}$$_{)}$ =: (2) C$_{(}$$_{K}$$_{)}$ +2 £$_{X}$ (1) C$_{(}$$_{K}$$_{)}$ + ( £$_{Y}$ − £ 2 X ) C$_{(}$$_{K}$$_{)}$, (3.27) (1) C$_{(}$$_{K}$$_{)}$ := ∇ $^{a}$∇$_{a}$ ϕ$_{1}$ − H ac a [ H ] ∇$_{c}$ ϕ − H $^{ab}$∇$_{a}$ ∇$_{b}$ ϕ − ϕ$_{1}$ ∂ $^{2}$V ∂ ¯ ϕ 2 ( ϕ ) , (3.28) (2) C$_{(}$$_{K}$$_{)}$ := ∇ $^{a}$∇$_{a}$ ϕ$_{2}$ − H ac a [ L ] ∇$_{c}$ ϕ + 2 H ad a [ H ] H$_{cd}$∇ $^{c}$ϕ − 2 H ac a [ H ] ∇$_{c}$ ϕ$_{1}$ + 2 H $^{ab}$H c ab [ H ] ∇$_{c}$ ϕ −L $^{ab}$∇$_{a}$ ∇$_{b}$ ϕ + 2 H $^{a}$$_{d}$H$^{db}$∇$_{a}$ ∇$_{b}$ ϕ − 2 H $^{ab}$∇$_{a}$ ∇$_{b}$ ϕ$_{1}$ − ϕ$_{2}$ ∂ $^{2}$V ∂ ¯ ϕ 2 ( ϕ ) − ( ϕ$_{1}$ ) $_{2}$∂$^{3}$V ∂ ¯ ϕ 3 ( ϕ ) . (3.29) $^{(1)}$G b a [ H ] = 8 πG $^{(1)}$T b a (3.32) (1) C$_{(}$$_{K}$$_{)}$ = 0 , (2) C$_{(}$$_{K}$$_{)}$ = 0 . (3.30) $^{(1)}$G b a [ L ] + $^{(2)}$G b a [ H , H ] = 8 πG $^{(2)}$T b a (3.33) $^{(1)}$G b a = 8 πG $^{(1)}$T b a , $^{(2)}$G b a = 8 πG $^{(2)}$T b a . (3.31)