Short note on second-order gauge-invariant cosmological perturbation theory

The second order perturbations in Friedmann-Robertson-Walker universe filled with a perfect fluid are completely formulated in the gauge invariant manner without any gauge fixing. All components of the Einstein equations are derived neglecting the first order vector and tensor modes. These equations imply that the tensor and the vector mode of the second order metric perturbations may be generated by the non-linear effects of the Einstein equations from the first order density perturbations. Recently, the first order approximation of the early universe from a homogeneous isotropic one is revealed by the observation of the CMB by Wilkinson Microwave Anisotropy Probe (WMAP)[1] and is suggested that fluctuations in the early universe are adiabatic and Gaussian at least in the first order approximation. One of the next theoretical tasks is to clarify the accuracy of these results, for example, through the non-Gaussianity. To do this, the second order cosmological perturbation theory is necessary. In this article, we show the gauge invariant formulation of the general relativistic second order cosmological perturbations on the background Friedmann-Robertson-Walker (FRW) universe M0 filled with the perfect fluid whose metric is given by gab = a (η) ( −(dη)a(dη)b + γij(dx )a(dx )b ) , (1) where γij is the metric on maximally symmetric three space. The details of our formulation is given in Refs.[2]. The gauge transformation rules for the variable Q, which is expanded as Qλ = Q0 + λ Q+ 12λ Q, are given by (1) Y Q− (1) X Q = £ξ(1)Q0, (2) Y Q− (2) X Q = 2£ξ(1) (1) X Q+ {