as emphasized above, the direct observables in observa- tions should be gauge invariant and the gauge-variant term in Eq. (2.37) should be disappear in the direct ob- servables. Therefore, we have to clarify the how gauge- invariant variables are related to the observed tempera- ture fluctuations and the gauge-variant term disappear in the observable. 

Although there are some rooms to accomplish the com- plete formulation of the second-order cosmological per- turbation theory, we derived all the components of the second-order perturbation of the Einstein equation with- out ignoring any types modes (scalar-, vector-, tensor- types) of perturbations in the case of a scalar field system. In our formulation, any gauge fixing is not necessary and we can obtain all equations in the gauge-invariant form, which are equivalent to the complete gauge fixing. In other words, our formulation gives complete gauge-fixed equations without any gauge fixing. Therefore, equations obtained in a gauge-invariant manner cannot be reduced without physical restrictions any more. In this sense, the equations shown here are irreducible. This is one of the advantages of the gauge-invariant perturbation theory. 

The resulting Einstein equations of the second order show that any type of mode-coupling appears as the quadratic terms of the linear-order perturbations owing to the nonlinear effect of the Einstein equations, in prin- ciple. Perturbations in cosmological situations are clas- sified into three types: scalar, vector, and tensor. In the second-order perturbations, we also have these three types of perturbations as in the case of the first-order perturbations. Furthermore, in the equations for the second-order perturbations, there are many quadratic terms of linear-order perturbations owing to the non- linear effects of the system. Owing to these nonlinear effects, the above three types of perturbations couple with each other. In the scalar field system shown in this paper, the first-order vector mode does not appear due to the momentum constraint of the first-order perturba- tion of the Einstein equation. Therefore, we have seen that three types of mode-coupling appear in the second- order Einstein equations, i.e., scalar-scalar, scalar-tensor, and tensor-tensor type of mode coupling. In general, all types of mode-coupling may appear in the second-order Einstein equations. Actually, in Ref. [13], we also de- rived the all components of the Einstein equations for a perfect fluid system and we can see all types of mode- coupling, i.e., scalar-scalar, scalar-vector, scalar-tensor, vector-vector, vector-tensor, tensor-tensor types mode- coupling, appear in the second-order Einstein equation, in general. Of course, in the some realistic situations of cosmology, we may neglect some modes. In this case, we may neglect some mode-coupling. However, even in this case, we should keep in mind the fact that all types of mode-couplings may appear in principle when we discuss the realistic situations of cosmology. We cannot deny the possibility that the mode-couplings of any type pro- duces observable effects when the quite high accuracy of observations is accomplished. 

Even in the case of the single scalar field discussed in this paper, the source terms of the second-order Ein- stein equation show the mode-coupling of scalar-scalar, scalar-tensor, and the tensor-tensor types as mentioned above. Since the tensor mode of the linear order is also generated due to quantum fluctuations during the infla- tionary phase, the mode-couplings of the scalar-tensor and tensor-tensor types may appear in the inflation. If these mode-couplings occur during the inflationary phase, these effects will depend on the scalar-tensor ratio r . If so, there is a possibility that the accurate observa- tions of the second-order effects in the fluctuations of the scalar type in our universe also restrict the scalar-tensor ratio r or give some consistency relations between the other observations such as the measurements of the B- mode of the polarization of CMB. This will be a new effect that gives some information on the scalar-tensor ratio r . 

Furthermore, we have also checked the consistency be- tween the second-order perturbations of the equations of motion of matter field and the Einstein equations. In the case of a scalar field, we checked the consistency between the second-order perturbations of the Klein- Gordon equation and the Einstein equations. Due to this consistency check, we have obtained the consistency re- lations between the source terms in these equations Γ$_{0}$, Γ$_{i}$, Γ$_{ij}$, and Ξ$_{(}$$_{K}$$_{)}$, which are given by Eqs. (6.32) and (6.34). We note that the relation (6.32) comes from the consistency in the Einstein equations of the second order by itself, while the relation (6.34) comes from the con- sistency between the second-order perturbation of the Klein-Gordon equation and the Einstein equation. We also showed that these relations between the source terms are satisfied through the background and the first-order perturbation of the Einstein equations in Ref. [13]. This implies that the set of all equations are self-consistent and the derived source terms Γ$_{0}$, Γ$_{i}$, Γ$_{ij}$, and Ξ$_{(}$$_{K}$$_{)}$ are correct. We also note that these relations are indepen- dent of the details of the potential of the scalar field. 

Thus, we have derived the self-consistent set of equa- tions of the second-order perturbation of the Einstein equations and the evolution equations of matter fields in terms of gauge-invariant variables. As the current status of the second-order gauge-invariant cosmological pertur- bation theory, we may say that the curvature terms in the second-order Einstein tensor (3.33), i.e., the second- order perturbations of the Einstein tensor, are almost completely derived although there remains the problem of homogeneous modes as mentioned above. After com- plete the problem of homogeneous modes, we have to clarify the physical behaviors of the second-order cosmo- logical perturbation in the single scalar field system in the context of the inflationary scenario. This is the pre- liminary step to clarify the quantum behaviors of second- order perturbations in the inflationary universe. Further, we also have to carry out the comparison with the re- sult by long-wavelength approximations. If these issues are completed, we may say that we have completely un- 

