7 choices. Hence, we conclude that the gauge transforma- tion rules (2.18) and (2.19) are quite general and irre- ducible. In this paper, we review the development of a second-order gauge-invariant cosmological perturbation theory based on the above understanding of the gauge degree of freedom only through the gauge transforma- tion rules (2.18) and (2.19). Hence, the developments of the cosmological perturbation theory presented below will not be changed even if we generalize the gauge choice X$_{λ}$ from a simple exponential map. We also have to emphasize the physical implication of the gauge transformation rules (2.18) and (2.19). Ac- cording to the above construction of the perturbation theory, gauge degree of freedom, which induces the trans- formation rules (2.18) and (2.19), is unphysical degree of freedom. As emphasized above, the physical spacetime M$_{λ}$ is our nature itself, while there is no background spacetime M$_{0}$ in our nature. The background spacetime M$_{0}$ is a fictitious spacetime and it have nothing to do with our nature. Since the gauge choice X$_{λ}$ just gives a relation between M$_{λ}$ and M$_{0}$ , the gauge choice X$_{λ}$ also have nothing to do with our nature. On the other hand, any observations and experiments are carried out only on the physical spacetime M$_{λ}$ through the physical processes on the physical spacetime M$_{λ}$ . Therefore, any direct observables in any observations and experiments should be independent of the gauge choice X$_{λ}$ , i.e., should be gauge invariant. Keeping this fact in our mind, the gauge transformation rules (2.18) and (2.19) imply that the perturbations (1) $_{X}$Q and (2) $_{X}$Q include unphysical de- gree of freedom, i.e., gauge degree of freedom, if these perturbations are transformed as (2.18) or (2.19) under the gauge transformation X$_{λ}$ → Y$_{λ}$ . If the perturbations (1) $_{X}$Q and (2) $_{X}$Q are independent of the gauge choice, these variables are order by order gauge invariant. Therefore, order by order gauge-invariant variables does not include unphysical degree of freedom and should be related to the physics on the physical spacetime M$_{λ}$ . In many literature, gauge degree of freedom is regarded as the degree of freedom of the coordinate transforma- tion. In the linear-order perturbation theory, these two degree of freedom are equivalent with each other. How- ever, in the higher order perturbations, we should regard that these two degree of freedom are different. Although the essential understanding of the gauge degree of free- dom (of the second kind) is as that explained above, the gauge transformation (of the second kind) also induces the infinitesimal coordinate transformation on the phys- ical spacetime M$_{λ}$ as a result. In many case, the un- derstanding of “gauges” in perturbations based on co- ordinate transformations leads mistakes. Therefore, we did not use any ingredient of this subsection in our se- ries of papers[8, 9, 11–13] concerning about higher-order Here, inspecting the gauge transformation rules (2.18) and (2.19), we define the gauge invariant variables for a metric perturbation and for arbitrary matter fields (ten- sor fields). Employing the idea of order by order gauge invariance for perturbations[12], we proposed a proce- dure to construct gauge invariant variables of higher- order perturbations[8]. This proposal is as follows. First, we decompose a linear-order metric perturbation into its gauge invariant and variant parts. The procedure for de- composing linear-order metric perturbations is extended to second-order metric perturbations, and we can decom- pose the second-order metric perturbation into gauge in- variant and variant parts. Then, we can define the gauge invariant variables for the first- and second-order pertur- bations of an arbitrary field other than the metric by us- ing the gauge variant parts of the first- and second-order general relativistic gauge-invariant perturbation theory. However, we comment on the relations between the co- ordinate transformation, briefly. Details can be seen in Refs. [8, 19, 20]. To see that the gauge transformation of the second kind induces the coordinate transformation, we intro- duce the coordinate system { O$_{α}$, ψ$_{α}$ } on the “background spacetime” M$_{0}$ , where O$_{α}$ are open sets on the back- ground spacetime and ψ$_{α}$ are diffeomorphisms from O$_{α}$ to R 4 (4 = dim M$_{0}$ ). The coordinate system { O$_{α}$, ψ$_{α}$ } is the set of the collection of the pair of open sets O$_{α}$ and diffeomorphism O$_{α}$ ↦→ R $^{4}$. If we employ a gauge choice X$_{λ}$ , we have the correspondence of M$_{λ}$ and M$_{0}$ . To- gether with the coordinate system ψ$_{α}$ on M$_{0}$ , this corre- spondence between M$_{λ}$ and M$_{0}$ induces the coordinate system on M$_{λ}$ . Actually, X$_{λ}$ ( O$_{α}$ ) for each α is an open set of M$_{λ}$ . Then, ψ$_{α}$ ◦ X − 1 λ becomes a diffeomorphism from an open set X$_{λ}$ ( O$_{α}$ ) ⊂ M$_{λ}$ to R $^{4}$. This diffeomor- phism ψ$_{α}$ ◦ X − 1 λ induces a coordinate system of an open set on M$_{λ}$ . When we have two different gauge choices X$_{λ}$ and Y$_{λ}$ , ψ$_{α}$ ◦ X − 1 λ and ψ$_{α}$ ◦ Y − 1 λ become different coordinate sys- tems on M$_{λ}$ . We can also consider the coordinate trans- formation from the coordinate system ψ$_{α}$ ◦ X − 1 λ to an- other coordinate system ψ$_{α}$ ◦Y − 1 λ . Since the gauge trans- formation X$_{λ}$ → Y$_{λ}$ is induced by the diffeomorphism Φ$_{λ}$ defined by Eq. (2.14), the induced coordinate transforma- tion is given by D. Coordinate transformations induced by the second kind gauge transformation E. Gauge-invariant variables y $^{µ}$( q ) := x $^{µ}$( p ) = (( Φ − $^{1}$) ∗ x µ ) ( q ) (2.20) in the passive point of view[8, 19, 20]. If we represent this coordinate transformation in terms of the Taylor expan- sion in Sec. II A, up to third order, we have the coordinate transformation y $^{µ}$( q ) = x $^{µ}$( q ) − λξ µ $_{1}$( q ) + λ 2 2 {− ξ µ $_{2}$( q ) + ξ ν $_{1}$( q ) ∂$_{ν}$ξ µ $_{1}$( q ) } + O ( λ $^{3}$) . (2.21)