Inclusion of the first-order vector- and tensor-modes in the second-order gauge-invariant cosmological perturbation theory

Gauge-invariant treatments of the second-order cosmological perturbation in a four dimensional homogeneous isotropic universe are formulated without any gauge fixing. We have derived the Einstein equations in the case of the single perfect fluid without ignoring any modes. These equations imply that any types of mode-coupling arise due to the second-order effects of the Einstein equations. The second-order general relativistic cosmological perturbation theory has very wide physical motivation. In particular, the first order approximation of our universe from a homogeneous isotropic one is revealed by the recent observations of Cosmic Microwave Background (CMB) by Wilkinson Microwave Anisotropy Probe[1], which suggests that the fluctuations of our universe are adiabatic and Gaussian at least in the first order approximation. One of the next theoretical researches is to clarify the accuracy of this result through the non-Gaussianity, or non-adiabaticity, and so on. To carry out this, it is necessary to discuss the second-order cosmological perturbations. However, general relativistic perturbation theory requires delicate treatments of “gauges” and this situation becomes clearer by the general arguments of perturbation theories. Therefore, it is worthwhile to formulate the higher-order gauge-invariant perturbation theory from general point of view. According to this motivation, we proposed the general framework of the second-order gauge-invariant perturbation theory on a generic background spacetime[2]. This general framework was applied to cosmological perturbation theory[3] and all components of the second-order perturbation of the Einstein equation were derived in gauge invariant manner. The derived second-order Einstein equations are quite similar to the equations for the first-order one but there are source terms which consist of the quadratic terms of the linear-order perturbations. In this article, we show the extension of the formulation in Refs. [3] to include the first-order vectorand tensor-modes in the source terms of the second-order Einstein equation, which were ignored in Refs. [3]. As emphasized in Refs.[2, 3], in any perturbation theory, we always treat two spacetime manifolds. One is a physical spacetime Mλ and the other is the background spacetime M0. In this article, the background spacetime M0 is the Friedmann-Robertson-Walker universe filled with a perfect fluid whose metric is given by