16 Although Eq. (6.5) is similar to Eq. (2.32), the ten- sor fields L$_{ab}$ and J$_{ab}$ are different from each other. Thus, the definition of the gauge invariant variables for the second-order metric perturbation is not unique in a more complicated way than the linear order. This non-uniqueness of gauge-invariant variables for the met- ric perturbations propagates to the definition (2.34) and (2.35) of the gauge invariant variables for matter fields. In spite of the existence of infinitely many definitions of the gauge invariant variables, in this paper, we con- sider the components of L$_{ab}$ given by Eq. (6.1). Eq. (6.1) corresponds to the second-order extension of the longitu- dinal gauge, which is called Poisson gauge X a = Y a = 0. (4.1). We evaluate the term $^{(1)}$G b a [ L ] and $^{(2)}$G b a [ H , H ] in the Einstein equation (3.33). First, we evaluate the term $^{(1)}$G b a [ L ] in the Einstein equation (3.33). Because the components (6.1) of L$_{ab}$ are obtained through the replacements in the components (5.4) of H$_{ab}$ , we easily obtain the com- ponents of $^{(1)}$G b a [ L ] through the replacements (6.9) in Eqs. (5.6)–(5.9). From Eq. (5.4), we can derive the components of $^{(2)}$G b a = $^{(2)}$G b a [ H , H ] defined by Eqs. (3.9)–(3.12) in a straightforward manner. Here, we use the results (5.12) and (5.16) of the first-order Einstein equations, for simplicity. Then the explicit components $^{(2)}$G b a = $^{(2)}$G b a [ H , H ] are summarized as Here, we evaluate the second-order perturbation of the Einstein tensor (3.7) with the cosmological background B. Einstein tensor (1) Φ → (2) Φ , (1) ν$_{i}$ → (2) ν $_{i}$, (1) Ψ → (2) Ψ , (1) χ$_{ij}$ → (2) χ$_{ij}$ (6.9) $^{(2)}$G η η = 2 a 2 [ − 3 D$_{k}$ (1) Φ D k (1) Φ − 8 (1) Φ Δ (1) Φ − 3 ( ∂$_{η}$ (1) Φ ) 2 − 12 ( H 2 + K ) ( (1) Φ ) 2 + D$_{l}$D$_{k}$ (1) Φ (1) χ lk + 1 8 ∂$_{η}$ (1) χ kl ( ∂$_{η}$ + 8 H ) (1) χ$_{kl}$ + 1 2 D$_{k}$ (1) χ$_{lm}$ D [ l (1) χ k ] m − 1 8 D$_{k}$ (1) χ$_{lm}$ D k (1) χ ml − 1 2 (1) χ lm (Δ − K ) (1) χ$_{lm}$ ] , (6.10) $^{(2)}$G i η = 2 a 2 [ 8 (1) Φ ∂$_{η}$D i (1) Φ − D$_{j}$ (1) Φ ∂$_{η}$ (1) χ ji − ( ∂$_{η}$D$_{j}$ (1) Φ +2 H D$_{j}$ (1) Φ ) (1) χ ij + 1 4 ∂$_{η}$ (1) χ$_{jk}$ D i (1) χ kj + (1) χ$_{kl}$ ∂$_{η}$D [ i (1) χ k ] l ] , (6.11) $^{(2)}$G η i = 2 a 2 [ 8 H (1) Φ D$_{i}$ (1) Φ − 2 D$_{i}$ (1) Φ ∂$_{η}$ (1) Φ + D j (1) Φ ∂$_{η}$ (1) χ$_{ij}$ − ∂$_{η}$D j (1) Φ (1) χ$_{ij}$ − 1 4 ∂$_{η}$ (1) χ kj D$_{i}$ (1) χ$_{kj}$ + (1) χ kj ∂$_{η}$D$_{[}$$_{j}$ (1) χ$_{i}$$_{]}$$_{k}$ ] , (6.12) $^{(2)}$G j i = 2 a 2 [{ − 3 D$_{k}$ (1) Φ D k (1) Φ − 4 (1) Φ (Δ + K ) (1) Φ − ∂$_{η}$ (1) Φ ∂$_{η}$ (1) Φ − 8 H (1) Φ ∂$_{η}$ (1) Φ − 4 ( 2 ∂$_{η}$ H + H $^{2}$) ( (1) Φ ) $^{2}$} γ j i +2 D$_{i}$ (1) Φ D j (1) Φ +4 (1) Φ D$_{i}$D j (1) Φ + (1) χ j i ( ∂ 2 η + 2 H ∂$_{η}$ ) (1) Φ + D$_{k}$ (1) Φ ( D$_{i}$ (1) χ jk + D j (1) χ$_{ik}$ ) − 2 D k (1) Φ D$_{k}$ (1) χ j i − 2 (1) Φ (Δ − 2 K ) (1) χ j i − Δ (1) Φ (1) χ j i + D$_{k}$D$_{i}$ (1) Φ (1) χ jk + D $^{m}$Dj (1) Φ (1) χ$_{im}$ − D$_{l}$D$_{k}$ (1) Φ (1) χ lk γ j i − 1 2 ∂$_{η}$ (1) χ$_{ik}$ ∂$_{η}$ (1) χ kj + D$_{k}$ (1) χ$_{il}$ D [ k (1) χ l ] j + 1 4 D j (1) χ$_{lk}$ D$_{i}$ (1) χ lk + 1 2 (1) χ$_{lm}$ D$_{i}$D j (1) χ ml − 1 2 (1) χ$_{lm}$ D $^{l}$D$_{i}$ (1) χ mj − 1 2 (1) χ lm D$_{l}$D j (1) χ$_{mi}$ + 1 2 (1) χ lm D$_{m}$D$_{l}$ (1) χ j i − 1 2 (1) χ jk ( ∂ 2 η + 2 H ∂$_{η}$ − Δ + 2 K ) (1) χ$_{ik}$ + 1 2 { 3 4 ∂$_{η}$ (1) χ$_{lk}$ ∂$_{η}$ (1) χ kl + (1) χ$_{kl}$ ( ∂ 2 η + 2 H ∂$_{η}$ − Δ + K ) (1) χ lk − 1 4 D$_{k}$ (1) χ$_{lm}$ D k (1) χ ml + D$_{k}$ (1) χ$_{lm}$ D [ l (1) χ k ] m } γ j i ] . (6.13) (6.13), Then, we may say that the expressions (6.10)– We have checked the identity (3.14) through Eqs. (6.10)–