content_image/437430.jpg content_image/437433.jpg content_image/437434.jpg content_image/437427.jpg content_image/437429.jpg content_image/437435.jpg content_image/437428.jpg content_image/437431.jpg content_image/437432.jpg
derstood the properties of the second-order perturbation of the Einstein tensor. The next task is to clarify the nature of the second-order perturbation of the energy- momentum tensor through the extension to multi-fluid or multi-field systems. Further, we also have to extend our arguments to the Einstein Boltzmann system to dis- cuss CMB physics, since we have to treat photon and neutrinos through the Boltzmann distribution functions. This issue is also discussed in some literature[7, 14]. If we accomplish these extension, we will be able to clarify the Non-linear effects in CMB physics. 

Finally, readers might think that the ingredients of this paper is too mathematical as Astronomy. However, we have to emphasize that a high degree of the theoretical sophistication leads unambiguous theoretical predictions in many case. As in the case of the linear-order cosmo- logical perturbation theory, the developments in observa- tions are also supported by the theoretical sophistication and the theoretical sophistication are accomplished mo- tivated by observations. In this sense, now, we have an opportunity to develop the general relativistic second- order perturbation theory to a high degree of sophistica- tion which is motivated by observations. We also expect that this theoretical sophistication will be also useful to discuss the theoretical predictions of Non-Gaussianity in CMB and comparison with observations. Therefore, I think that this opportunity is opened not only for obser- vational cosmologists but also for theoretical and math- ematical physicists. 

The author thanks participants in the GCOE/YITP workshop YITP-W-0901 on “Non-linear cosmological perturbations” which was held at YITP in Kyoto, Japan in April, 2009, for valuable discussions, in particular, Prof. M. Bruni, Prof. R. Maartens, Prof. M. Sasaki, Prof. T. Tanaka, and Prof. K. Tomita. This review is an extension of the contribution to this workshop by the author. 

## Appendix A: Derivation of the generic representation of the Taylor expansion of tensors on a manifold 

In this section, we derive the representation of the co- efficients of the formal Taylor expansion (2.2) of the pull- back of a diffeomorphism in terms of the suitable deriva- tive operators. The guide principle of our arguments is the following theorem[20, 23]. 

Theorem A.1. Let D be a derivative operator acting on the set of all the tensor fields defined on a differentiable manifold M and satisfying the following conditions: (i) it is linear and satisfies the Leibniz rule; (ii) it is tensor- type preserving; (iii) it commutes with every contraction 

## Acknowledgments 

of a tensor field; and (iv) it commutes with the exte- rior differentiation d . Then, D is equivalent to the Lie derivative operator with respect to some vector field ξ , i.e., D = £$_{ξ}$. 

The prove of the assertion of Theorem A.1 is given in Ref. [20] as follows. When acting on functions, the derivative operator D defines a vector field ξ through the relation 

The assertion of the Theorem for an arbitrary tensor field is hold iff the assertions for an arbitrary scalar function and for an arbitrary vector field V are hold. To do this, we consider the scalar function V ( f ) and we obtain 

through Eq. (A1). Through the conditions (i)-(iv) of D , D ( V ( f )) is also given by 

Then we obtain 

<img src='content_image/437430.jpg'>

for an arbitrary f , i.e., 

Through Eqs. (A1) and (A5), we can recursively show 

for an arbitrary tensor field Q [23]. 

Now, we consider the derivation of the Taylor expan- sion (2.1). As in the main text, we first consider the representation of the Taylor expansion of Φ ∗ $_{λ}$f for an ar- bitrary scalar function f ∈ F ( M ): 

<img src='content_image/437433.jpg'>

where F ( M ) denotes the algebra of C functions on M . Although the operator ∂/∂λ in the bracket {∗}$_{λ}$$_{=0}$ of Eq. (A7) are simply symbolic notation, we stipulate the properties 

<img src='content_image/437434.jpg'>

<img src='content_image/437427.jpg'>

<img src='content_image/437429.jpg'>

<img src='content_image/437435.jpg'>

<img src='content_image/437428.jpg'>

<img src='content_image/437431.jpg'>

<img src='content_image/437432.jpg'>

