21 mulation of the perturbation theory. This general class of diffeomorphism is represented in terms of the Tay- lor expansion (2.2) of its pull-back. As commented in Sec. II A, this general class of diffeomorphism does not form a one-parameter group of diffeomorphism as shown through Eq. (2.3). However, the properties (2.3) do not directly mean that this general class of diffeomorphism does not form a group. One of the key points of the prop- erties of this diffeomorphism is the non-commutativity of generators ξ a 1 and ξ a 2 of each order. Although the expres- sion of the n -th order Taylor expansion of the pull-back of this general class is discussed in Ref. [22], when we con- sider the situation of the n -th order perturbation, this non-commutativity becomes important[8]. Therefore, to clarify the properties of this general class of diffeomor- phism, we have to take care of this non-commutativity of generators. Thus, there is a room to clarify the properties of this general class of diffeomorphism. Further, in Sec. II C, we introduced a gauge choice X$_{λ}$ as an exponential map, for simplicity. On the other hand, we have the concept of the general class of dif- feomorphism which is wider than the class of the ex- ponential map. Therefore, we may introduce a gauge choice as one of the element of this general class of dif- feomorphism. However, the gauge-transformation rules (2.18) and (2.19) will not be changed even if we general- ize the definition of a each gauge choice as emphasized in Sec. II C. Although there is a room to sophisticate in log- ical arguments to derive the gauge-transformation rules (2.18) and (2.19), these are harmless to the development of the general framework of the gauge-invariant pertur- bation theory shown in Secs. II C, II E, III, and their application to cosmological perturbations in Sec. IV. As emphasize in Sec. II E, our starting point to con- struct gauge invariant variables is the assumption that we already know the procedure for finding gauge in- variant variables for the linear metric perturbations as Eq. (2.23) . This is highly nontrivial assumption on a generic background spacetime. The procedure to accom- plish the decomposition (2.23) completely depends on the details of the background spacetime. In spite of this non-triviality, this assumption is almost correct in some background spacetime[28]. Further, once we accept this assumption, we can develop the higher-order perturba- tion theory in an independent manner of the details of the background spacetime. We also expect that this gen- eral framework of the gauge-invariant perturbation the- ory is extensible to n -th order perturbation theory, since our procedure to construct gauge-invariant variables can be extended to the third-order perturbation theory with two-parameter[8]. Due to this situation, in Ref. [12], we propose the conjecture which state that the above as- sumption for the decomposition of the linear-order met- ric perturbation is correct for any background spacetime. We may also say that the most nontrivial part of our gen- eral framework of higher-order gauge-invariant perturba- tion theory is in the above assumption. Further, as em- phasized in Sec. V A, we assumed the existence of some Green functions to accomplish the decomposition (2.23) and this assumption exclude some perturbative modes of the metric perturbations from our consideration, even in the case of cosmological perturbations. For example, ho- mogeneous modes of perturbations are excluded in our current arguments of the cosmological perturbation the- ory. These homogeneous modes will be necessary to dis- cuss the comparison with the arguments based on the long-wavelength approximation. Therefore, we have to say that there is a room to clarify even in the cosmolog- ical perturbation theory. Even if the assumption is correct on any background spacetime, the other problem is in the interpretations of the gauge-invariant variables. We have commented on the non-uniqueness in the definitions of the gauge- invariant variables through Eqs. (5.5) and (6.5). This non-uniqueness in the definition of gauge-invariant vari- ables also leads some ambiguities in the interpretations of gauge-invariant variables. On the other hand, as empha- size in Sec. II C, any observations and experiments are carried out only on the physical spacetime through the physical processes on the physical spacetime. For this reason, any direct observables in any observations and experiments should be independent of the gauge choice. Further, the non-uniqueness in the definitions the gauge- invariant variables expressed by Eqs. (5.5) and (6.5) have the same form as the decomposition formulae (2.36) and (2.37). Therefore, if the statement that any direct ob- servables in any observations and experiments is indepen- dent of the gauge choice is really true, this also confirm that the non-uniqueness of the definition of the gauge- invariant variables also have nothing to do with the direct observables in observations and experiments. These will be confirmed by the clarification of the relations between gauge-invariant variables and observables in experiments and observations. To accomplish this, we have to spec- ify the concrete process of experiments and observations and clarify the problem what are the direct observables in the experiments and observations and derive the rela- tions between the gauge-invariant variables and observ- ables in concrete observations and experiments. If these arguments are completed, we will be able to show that the gauge degree of freedom is just unphysical degree of freedom and the non-uniqueness of the gauge-invariant variables have nothing to do with the direct observables in the concrete observation or experiment, and then, we will be able to clarify the precise physical interpretation of the gauge-invariant variables. For example, in the case of the CMB physics, we can easily see that the first-order perturbation of the CMB temperature is automatically gauge invariant from Eq. (2.36), because the background temperature of CMB is homogeneous. On the other hand, the decomposi- tion formula (2.37) of the second order yields that the theoretical prediction of the second-order perturbation of the CMB temperature may depend on gauge choice, since we do know the existence of the first-order fluctu- ations as the temperature anisotropy in CMB. However,