13 where the properties D i (1) ν i := γ $^{ij}$D$_{i}$ (1) ν j = (1) χ i i := γ ij (1) χ$_{ij}$ = D i (1) χ$_{ij}$ = 0 are satisfied as consequences of Eqs. (5.2). Thus, we may say that our assumption for the de- composition (2.23) in linear-order metric perturbation is correct in the case of cosmological perturbations. How- ever, we have to note that to accomplish Eq. (2.23), we assumed the existence of the Green functions Δ − $^{1}$, (Δ + 2 K ) − $^{1}$, and (Δ + 3 K ) − $^{1}$. As shown in Ref. [11], this assumption is necessary to guarantee the one to one correspondence between the variables { h$_{ηη}$, h$_{iη}$, h$_{ij}$ } and { h$_{ηη}$, h$_{(}$$_{V L}$$_{)}$, h$_{(}$$_{V}$$_{)}$$_{i}$, h$_{(}$$_{L}$$_{)}$, h$_{(}$$_{T L}$$_{)}$, h$_{(}$$_{T V}$$_{)}$$_{j}$, h$_{(}$$_{T T}$$_{)}$$_{ij}$ } , but excludes some perturbative modes of the metric pertur- bations which belong to the kernel of the operator Δ, (Δ + 2 K ), and (Δ + 3 K ) from our consideration. For example, homogeneous modes belong to the kernel of the operator Δ and are excluded from our consideration. If we have to treat these modes, the separate treatments are necessary. In this paper, we ignore these modes, for simplicity. We also note the fact that the definition (2.23) of the gauge-invariant variables is not unique. This comes from the fact that we can always construct new gauge- invariant quantities by the combination of the gauge- invariant variables. For example, using the gauge- invariant variables (1) Φ and (1) ν$_{i}$ of the first-order met- ric perturbation, we can define a vector field Z$_{a}$ by Z$_{a}$ := − a (1) Φ ( dη )$_{a}$+ a (1) ν$_{i}$ ( dx $^{i}$)$_{a}$ which is gauge-invariant. Then, we can rewrite the decomposition formula (2.23) for the linear-order metric perturbation as where we have defined new gauge-invariant variable K$_{ab}$ by K$_{ab}$ := H$_{ab}$ − £$_{Z}$ g$_{ab}$ . Clearly, K$_{ab}$ is gauge-invariant and the vector field X $^{a}$+ Z a satisfies Eq. (2.24). In spite of this non-uniqueness, we specify the components of the tensor H$_{ab}$ as Eq. (5.4), which is the gauge-invariant part of the linear-order metric perturbation associated with the longitudinal gauge. The non-uniqueness of the definitions of gauge- invariant variables is related to the “gauge-fixing” for the linear-order metric perturbations. Due to this non- uniqueness, we can consider the gauge-fixing in the first- order metric perturbation from two different points of view. The first point of view is that the gauge-fixing is to specify the gauge-variant part X $^{a}$. For example, the longitudinal gauge is realized by the gauge fixing X a = 0. Due to this gauge fixing X a = 0, we can regard the fact that perturbative variables in the longitudinal gauge are the completely gauge fixed variables. On the other hand, we may also regard that the gauge fixing is the specification of the gauge-invariant vector field Z a in Eq. (5.5). In this point of view, we do not specify the vector field X $^{a}$. Instead, we have to specify the gauge- invariant vector Z a or equivalently to specify the gauge- invariant metric perturbation K$_{ab}$ without specifying X a so that the first-order metric perturbation h$_{ab}$ coincides with the gauge-invariant variables K$_{ab}$ when we fix the gauge X a so that X a + Z a = 0. These two different point of view of “gauge fixing” is equivalent with each other due to the non-uniqueness of the definition (5.5) of the gauge-invariant variables. Here, we derive the linear-order Einstein equation (3.32). To derive the components of the gauge invariant part of the linearized Einstein tensor $^{(1)}$G b a [ H ], which is defined by Eqs. (3.8), we first derive the components of the tensor H c ab [ H ], which is defined in Eq. (3.11) with A$_{ab}$ = H$_{ab}$ and its component (5.4). These components are summarized in Ref. [11]. From Eq. (3.8), the component of $^{(1)}$G b a [ H ] are sum- marized as B. First-order Einstein equations h$_{ab}$ = H$_{ab}$ − £$_{Z}$ g$_{ab}$ + £$_{Z}$ g$_{ab}$ + £$_{X}$ g$_{ab}$, =: K$_{ab}$ + £$_{X}$$_{+}$$_{Z}$ g$_{ab}$, (5.5) $^{(1)}$G η η [ H ] = − 1 a 2 { ( − 6 H ∂$_{η}$ + 2Δ + 6 K ) (1) Ψ − 6 H 2 (1) Φ } , (5.6) $^{(1)}$G η i [ H ] = − 1 a 2 ( 2 ∂$_{η}$D$_{i}$ (1) Ψ +2 H D$_{i}$ (1) Φ − 1 2 (Δ + 2 K ) (1) ν$_{i}$ ) , (5.7) $^{(1)}$G i η [ H ] = 1 a 2 { 2 ∂$_{η}$D i (1) Ψ +2 H D i (1) Φ + 1 2 ( − Δ + 2 K + 4 H 2 − 4 ∂$_{η}$ H )$^{(1)}$ ν i } , (5.8) $^{(1)}$G j i [ H ] = 1 a 2 [ D$_{i}$D j ( (1) Ψ − (1) Φ ) + { ( − Δ + 2 ∂ 2 η + 4 H ∂$_{η}$ − 2 K ) (1) Ψ + ( 2 H ∂$_{η}$ + 4 ∂$_{η}$ H + 2 H 2 + Δ ) (1) Φ } γ j i − 1 2 a $_{2}$∂$_{η}$ { a 2 ( D$_{i}$ (1) ν j + D j (1) ν$_{i}$ )} + 1 2 ( ∂ 2 η + 2 H ∂$_{η}$ + 2 K − Δ )$^{(1)}$ χ j i ] . (5.9)