Abstract
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 paper, 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.
Highlights
The general relativistic cosmological linear perturbation theory has been developed to a high degree of sophistication during the last 30 years [1,2,3]
The first-order approximation of our universe from a homogeneous isotropic one was revealed through the observation of the cosmic microwave background (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
We summarize the current status of this development of the second-order gauge-invariant cosmological perturbation theory through the simple system of a scalar field
Summary
The general relativistic cosmological linear perturbation theory has been developed to a high degree of sophistication during the last 30 years [1,2,3]. Because (5) is regarded as an equation for field variables, it implicitly states that the points “p” ∈ M and p ∈ M0 are same This represents the implicit assumption of the existence of a map M0 → M : p ∈ M0 → “p” ∈ M, which is usually called a gauge choice (of the second-kind) in perturbation theory [33,34,35]. According to generic arguments concerning the Taylor expansion of the pull-back of a tensor field on the same manifold, given, it should be possible to express the gauge transformation Φ∗λ XQλ in the form. We review the development of a second-order gauge-invariant cosmological perturbation theory based on the above understanding of the gauge degree of freedom only through the gauge transformation rules (18) and (19). These decomposition formulae (36) and (37) are important ingredients in the general framework of the second-order general relativistic gauge-invariant perturbation theory
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