arXiv:1001.2621v2 [gr-qc] 10 Dec 2010 Second-order Gauge-invariant Cosmological Perturbation Theory: Current Status Kouji Nakamura Optical and Infrared Astronomy Division, National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo 181-8588, Japan. (Dated: November 1, 2018) The current status of the recent developments of the second-order gauge-invariant cosmological perturbation theory is reviewed. To show the essence of this perturbation theory, we concentrate only on the universe filled with a single scalar field. Through this review, we point out the problems which should be clarified for the further theoretical sophistication of this perturbation theory. 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. The general relativistic cosmological linear perturba- tion theory has been developed to a high degree of so- phistication during the last 30 years[1–3]. One of the mo- tivations of this development was to clarify the relation between the scenarios of the early universe and cosmo- logical data, such as the cosmic microwave background (CMB) anisotropies. Recently, the first-order approxi- mation of our universe from a homogeneous isotropic one was revealed through the observation of the CMB by the Wilkinson Microwave Anisotropy Probe (WMAP)[4, 5], the cosmological parameters are accurately measured, we have obtained the standard cosmological model, and the so-called “precision cosmology” has begun. These de- velopments in observations were also supported by the theoretical sophistication of the linear order cosmologi- cal perturbation theory. The observational results of CMB also suggest that the fluctuations of our universe are adiabatic and Gaus- sian at least in the first-order approximation. We are now on the stage to discuss the deviation from this first- order approximation from the observational[5] and theo- retical sides[6, 7] through the non-Gaussianity, the non- adiabaticity, and so on. These will be goals of future satellite missions. With the increase of precision of the CMB data, the study of relativistic cosmological pertur- bations beyond linear order is a topical subject. The second-order cosmological perturbation theory is one of such perturbation theories beyond linear order. Although the second-order perturbation theory in gen- eral relativity is an old topic, a general framework of the gauge-invariant formulation of the general relativistic second-order perturbation has been proposed[8, 9]. This general formulation is an extension of the works of Bruni et al.[10] and has also been applied to cosmological per- turbations: The derivation of the second-order Einstein equation in a gauge-invariant manner without any gauge fixing[11]; Applicability in more generic situations[12]; Confirmation of the consistency between all components of the second-order Einstein equations and equations of motions[13]. We also note that the radiation case has recently been discussed by treating the Boltzmann equa- tion up to second order[14] along the gauge-invariant In this section, we review the general framework of the gauge invariant perturbation theory developed in Refs. [8–11, 15–21]. To develop the general relativistic The organization of this paper is as follows. In Sec. II, we review the general framework of the second- order gauge invariant perturbation theory developed in Refs. [8, 9, 11, 15]. This review also includes additional explanation not given in those papers. In Sec. III, we also the derivations of the second-order perturbation of the Einstein equation and the energy-momentum tensor from general point of view. For simplicity, in this pa- per, we only consider a single scalar field as a matter content. The ingredients of Sec. II and III will be appli- cable to perturbation theory in any theory with general covariance, if the decomposition formula (2.23) for the linear-order metric perturbation is correct. In Sec. IV, we summarize the Einstein equations in the case of a back- ground homogeneous isotropic universe, which are used in the derivation of the first- and second-order Einstein equations. In Sec. V, the first-order perturbation of the Einstein equations and the Klein-Gordon equations are summarized. The derivation of the second-order pertur- bations of the Einstein equations and the Klein-Gordon equations, and their consistency are reviewed in Sec. VI. The final section, Sec. VII, is devoted to a summary and discussions. II. GENERAL FRAMEWORK OF THE GENERAL RELATIVISTIC GAUGE-INVARIANT PERTURBATION THEORY I. INTRODUCTION manner of the above series of papers by the present au- thor. In this review article, we summarize the current status of this development of the second-order gauge-invariant cosmological perturbation theory through the simple sys- tem of a sclar field. Through this review, we point out the problems which should be clarified and directions of the further development of the theoretical sophistication of the general relativistic higher-order perturbation the- ory, especially in cosmological perturbations. We expect that this sophistication will be also useful to discuss the theoretical predictions of Non-Gaussianity in CMB and comparison with observations.