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Advances in Astronomy
Volume 2010 (2010), Article ID 576273, 26 pages
http://dx.doi.org/10.1155/2010/576273
Review Article

Second-Order Gauge-Invariant Cosmological Perturbation Theory: Current Status

Optical and Infrared Astronomy Division, National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo 181-8588, Japan

Received 19 April 2010; Accepted 12 July 2010

Academic Editor: Eiichiro Komatsu

Copyright © 2010 Kouji Nakamura. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

The general relativistic cosmological linear perturbation theory has been developed to a high degree of sophistication during the last 30 years [13]. One of the motivations of this development was to clarify the relation between the scenarios of the early universe and cosmological data, such as the cosmic microwave background (CMB) anisotropies. Recently, the first-order approximation 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 developments in observations were also supported by the theoretical sophistication of the linear-order cosmological perturbation theory.

The observational results of CMB also suggest that the fluctuations of our universe are adiabatic and Gaussian 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 theoretical sides [621] through the non-Gaussianity, the nonadiabaticity, 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 perturbations 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 general relativity is an old topic, a general framework of the gauge-invariant formulation of the general relativistic second-order perturbation has been proposed [22, 23]. This general formulation is an extension of the works of Bruni et al. [24] and has also been applied to cosmological perturbations: the derivation of the second-order Einstein equation in a gauge-invariant manner without any gauge fixing [25, 26]; applicability in more generic situations [27]; confirmation of the consistency between all components of the second-order Einstein equations and equations of motions [28]. We also note that the radiation case has recently been discussed by treating the Boltzmann equation up to second order [29, 30] along the gauge-invariant manner of the above series of papers by the present author.

In this paper, 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. Through this paper, 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 theory, 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.

The organization of this paper is as follows. In Section 2, we review the general framework of the second-order gauge-invariant perturbation theory developed in [22, 23, 25, 26, 31]. This review also includes additional explanation not given in those papers. In Section 3, we present 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 paper, we only consider a single scalar field as a matter content. The ingredients of Sections 2 and 3 will be applicable to perturbation theory in any theory with general covariance, if the decomposition formula (23) for the linear-order metric perturbation is correct. In Section 4, we summarize the Einstein equations in the case of a background homogeneous isotropic universe, which are used in the derivation of the first- and second-order Einstein equations. In Section 5, the first-order perturbation of the Einstein equations and the Klein-Gordon equations are summarized. The derivation of the second-order perturbations of the Einstein equations and the Klein-Gordon equations, and their consistency are reviewed in Section 6. The final section, Section 7, is devoted to a summary and discussions.

2. General Framework of the General Relativistic Gauge-Invariant Perturbation Theory

In this section, we review the general framework of the gauge-invariant perturbation theory developed in [2226, 3139]. To develop the general relativistic gauge-invariant perturbation theory, we first explain the general arguments of the Taylor expansion on a manifold without introducing an explicit coordinate system in Section 2.1. Further, we also have to clarify the notion of “gauge” in general relativity to develop the gauge-invariant perturbation theory from general point of view, which is explained in Section 2.2. After clarifying the notion of “gauge” in general relativistic perturbations, in Section 2.3, we explain the formulation of the general relativistic gauge-invariant perturbation theory from general point of view. Although our understanding of “gauge” in general relativistic perturbations essentially is different from “degree of freedom of coordinates” as in many literature, “a coordinate transformation” is induced by our understanding of “gauge.” This situation is explained in Section 2.4. To exclude “gauge degree of freedom” which is unphysical degree of freedom in perturbations, we construct “gauge-invariant variables” of perturbations as reviewed in Section 2.5. These “gauge-invariant variables” are regarded as physical quantities.

2.1. Taylor Expansion of Tensors on a Manifold

First, we briefly review the issues on the general form of the Taylor expansion of tensors on a manifold . The gauge issue of general relativistic perturbation theories which we will discuss is related to the coordinate transformation. Therefore, we have to discuss the general form of the Taylor expansion without the explicit introduction of coordinate systems. Although we only consider the Taylor expansion of a scalar function 𝑓, here, the resulting formula is extended to that for any tensor field on a manifold as in Appendix A. We have to emphasize that the general formula of the Taylor expansion shown here is the starting point of our gauge-invariant formulation of the second-order general relativistic perturbation theory.

The Taylor expansion of a function 𝑓 is an approximated form of 𝑓(𝑞) at 𝑞 in terms of the variables at 𝑝, where 𝑞 is in the neighborhood of 𝑝. To derive the formula for the Taylor expansion of 𝑓, we have to compare the values of 𝑓 at the different points on the manifold. To accomplish this, we introduce a one-parameter family of diffeomorphisms Φ𝜆, where Φ𝜆(𝑝)=𝑞 and Φ𝜆=0(𝑝)=𝑝. One example of a diffeomorphisms Φ𝜆 is an exponential map with a generator. However, we consider a more general class of diffeomorphisms.

The diffeomorphism Φ𝜆 induces the pull-back Φ𝜆 of the function 𝑓 and this pull-back enables us to compare the values of the function 𝑓 at different points. Further, the Taylor expansion of the function 𝑓(𝑞) is given by 𝑓Φ(𝑞)=𝑓𝜆Φ(𝑝)=𝜆𝑓𝜕(𝑝)=𝑓(𝑝)+𝜕𝜆(Φ𝜆|||𝑓)𝑝1𝜆+2𝜕2𝜕𝜆2(Φ𝜆||||𝑓)𝑝𝜆2𝜆+𝑂3.(1) Since this expression hold for an arbitrary smooth function 𝑓, the function 𝑓 in (1) can be regarded as a dummy. Therefore, we may regard the Taylor expansion (1) to be the expansion of the pull-back Φ𝜆 of the diffeomorphism Φ𝜆, rather than the expansion of the function 𝑓.

According to this point of view, Sonego and Bruni [36] showed that there exist vector fields 𝜉𝑎1 and 𝜉𝑎2 such that the expansion (1) is given by 𝑓Φ(𝑞)=𝜆𝑓£(𝑝)=𝑓(𝑝)+𝜉1𝑓||𝑝1𝜆+2£𝜉2+£2𝜉1𝑓|||𝑝𝜆2𝜆+𝑂3,(2) without loss of generality (see Appendix A). Equation (2) is not only the representation of the Taylor expansion of the function 𝑓, but also the definitions of the generators 𝜉𝑎1 and 𝜉𝑎2. These generators of the one-parameter family of diffeomorphisms Φ𝜆 represent the direction along which the Taylor expansion is carried out. The generator 𝜉𝑎1 is the first-order approximation of the flow of the diffeomorphism Φ𝜆, and the generator 𝜉𝑎2 is the second-order correction to this flow. We should regard the generators 𝜉𝑎1 and 𝜉𝑎2 to be independent. Further, as shown in Appendix A, the representation of the Taylor expansion of an arbitrary scalar function 𝑓 is extended to that for an arbitrary tensor field 𝑄 just through the replacement 𝑓𝑄.

We must note that, in general, the representation (2) of the Taylor expansion is different from an usual exponential map which is generated by a vector field. In general, Φ𝜎Φ𝜆Φ𝜎+𝜆,Φ𝜆1Φ𝜆.(3) As noted in [24], if the second-order generator 𝜉2 in (2) is proportional to the first-order generator 𝜉1 in (2), the diffeomorphism Φ𝜆 is reduced to an exponential map. Therefore, one may reasonably doubt that Φ𝜆 forms a group except under very special conditions. However, we have to note that the properties (3) do not directly mean that Φ𝜆 does not form a group. There will be possibilities that Φ𝜆 form a group in a different sense from exponential maps, in which the properties (3) will be maintained.

Now, we give an intuitive explanation of the representation (2) of the Taylor expansion through the case where the scalar function 𝑓 in (2) is a coordinate function. When two points 𝑝,𝑞 in (2) are in the neighborhood of each other, we can apply a coordinate system 𝑛 (𝑛=dim), which is denoted by {𝑥𝜇}, to an open set which includes these two points. Then, we can measure the relative position of these two points 𝑝 and 𝑞 in in terms of this coordinate system in 𝑛 through the Taylor expansion (2). In this case, we may regard that the scalar function 𝑓 in (2) is a coordinate function 𝑥𝜇 and (2) yields 𝑥𝜇Φ(𝑞)=𝜆𝑥𝜇(𝑝)=𝑥𝜇(𝑝)+𝜆𝜉11(𝑝)+2𝜆2𝜉2+𝜉𝜈1𝜕𝜈𝜉𝜇1||𝑝𝜆+𝑂3.(4) The second term 𝜆𝜉1(𝑝) in the right-hand side of (4) is familiar. This is regarded as the vector which point from the point 𝑥𝜇(𝑝) to the points 𝑥𝜇(𝑞) in the sense of the first-order correction as shown in Figure 1(a). However, in the sense of the second order, this vector 𝜆𝜉1(𝑝) may fail to point to 𝑥𝜇(𝑞). Therefore, it is necessary to add the second-order correction as shown in Figure 1(b). As a correction of the second order, we may add the term (1/2)𝜆2𝜉𝜈1(𝑝)𝜕𝜈𝜉𝜇1(𝑝). This second-order correction corresponds to that comeing from the exponential map which is generated by the vector field 𝜉𝜇1. However, this correction completely determined by the vector field 𝜉𝜇1. Even if we add this correction that comes from the exponential map, there is no guarantee that the corrected vector 𝜆𝜉1(𝑝)+(1/2)𝜆2𝜉𝜈1(𝑝)𝜕𝜈𝜉𝜇1(𝑝) does point to 𝑥𝜇(𝑞) in the sense of the second order. Thus, we have to add the new correction (1/2)𝜆2𝜉𝜈2(𝑝) of the second order, in general.

fig1
Figure 1: (a) The second term 𝜆𝜉1(𝑝) in (4) is the vector which points from the point 𝑥𝜇(𝑝) to the point 𝑥𝜇(𝑞) in the sense of the first-order correction. (b) If we look at the neighborhood of the point 𝑥𝜇(𝑞) in detail, the vector 𝜆𝜉1(𝑝) may fail to point to 𝑥𝜇(𝑞) in the sense of the second order. Therefore, it is necessary to add the second-order correction (1/2)𝜆2(𝜉𝜇2+𝜉𝜈1(𝑝)𝜕𝜈𝜉𝜇1(𝑝)).

Of course, without this correction (1/2)𝜆2𝜉𝜈2(𝑝), the vector which comes only from the exponential map generated by the vector field 𝜉1 might point to the point 𝑥𝜇(𝑞). Actually, this is possible if we carefully choose the vector field 𝜉𝜇1 taking into account the deviations at the second order. However, this means that we have to take care of the second-order correction when we determine the first-order correction. This contradicts to the philosophy of the Taylor expansion as a perturbative expansion, in which we can determine everything order-by-order. Therefore, we should regard that the correction (1/2)𝜆2𝜉𝜈2(𝑝) is necessary in general situations.

2.2. Gauge Degree of Freedom in General Relativity

Since we want to explain the gauge-invariant perturbation theory in general relativity, first of all, we have to explain the notion of “gauge” in general relativity [31]. General relativity is a theory with general covariance, which intuitively states that there is no preferred coordinate system in nature. This general covariance also introduces the notion of “gauge” in the theory. In the theory with general covariance, these “gauges” give rise to the unphysical degree of freedom and we have to fix the “gauges” or to extract some invariant quantities to obtain physical result. Therefore, treatments of “gauges” are crucial in general relativity and this situation becomes more delicate in general relativistic perturbation theory as explained below.

In 1964, Sachs [32] pointed out that there are two kinds of “gauges” in general relativity. Sachs called these two “gauges” as the first- and the second-kind of gauges, respectively. Here, we review these concepts of “gauge.”

2.2.1. First-Kind Gauge

The first-kind gauge is a coordinate system on a single manifold . Although this first-kind gauge is not important in this paper, we explain this to emphasize the “gauge” discuss in this paper is different from this first-kind gauge.

In the standard text book of manifolds (e.g., see [40]), the following property of a manifold is written: on a manifold, we can always introduce a coordinate system as a diffeomorphism 𝜓𝛼 from an open set 𝑂𝛼 to an open set 𝜓𝛼(𝑂𝛼)𝑛 (𝑛=dim). This diffeomorphism 𝜓𝛼, that is, coordinate system of the open set 𝑂𝛼, is called gauge choice (of the first-kind). If we consider another open set in 𝑂𝛽, we have another gauge choice 𝜓𝛽𝑂𝛽𝜓𝛽(𝑂𝛽)𝑛 for 𝑂𝛽. If these two open sets 𝑂𝛼 and 𝑂𝛽 have the intersection 𝑂𝛼𝑂𝛽, we can consider the diffeomorphism 𝜓𝛽𝜓𝛼1. This diffeomorphism 𝜓𝛽𝜓𝛼1 is just a coordinate transformation: 𝜓𝛼(𝑂𝛼𝑂𝛽)𝑛𝜓𝛽(𝑂𝛼𝑂𝛽)𝑛, which is called gauge transformation (of the first-kind) in general relativity.

According to the theory of a manifold, coordinate system are not on a manifold itself but we can always introduce a coordinate system through a map from an open set in the manifold to an open set of 𝑛. For this reason, general covariance in general relativity is automatically included in the premise that our spacetime is regarded as a single manifold. The first-kind gauge does arise due to this general covariance. The gauge issue of the first-kind is represented by the question, which coordinate system is convenient? The answer to this question depends on the problem which we are addressing, that is, what we want to clarify. In some case, this gauge issue of the first-kind is an important. However, in many case, it becomes harmless if we apply a covariant theory on the manifold.

2.2.2. Second-Kind Gauge

The second-kind gauge appears in perturbation theories in a theory with general covariance. This notion of the second-kind “gauge” is the main issue of this paper. To explain this, we have to remind our selves what we are doing in perturbation theories.

First, in any perturbation theories, we always treat two spacetime manifolds. One is the physical spacetime . This physical spacetime is our nature itself and we want to describe the properties of this physical spacetime through perturbations. The other is the background spacetime 0. This background spacetime have nothing to do with our nature and this is a fictitious manifold which is prepared by us. This background spacetime is just a reference to carry out perturbative analyses. We emphasize that these two spacetime manifolds and 0 are distinct. Let us denote the physical spacetime by (,𝑔𝑎𝑏) and the background spacetime by (0,𝑔𝑎𝑏), where 𝑔𝑎𝑏 is the metric on the physical spacetime manifold, , and 𝑔𝑎𝑏 is the metric on the background spacetime manifold, 0. Further, we formally denote the spacetime metric and the other physical tensor fields on by 𝑄 and its background value on 0 by 𝑄0.

Second, in any perturbation theories, we always write equations for the perturbation of the physical variable 𝑄 in the form 𝑄("𝑝")=𝑄0(𝑝)+𝛿𝑄(𝑝).(5) Usually, this equation is simply regarded as a relation between the physical variable 𝑄 and its background value 𝑄0, or as the definition of the deviation 𝛿𝑄 of the physical variable 𝑄 from its background value 𝑄0. However, (5) has deeper implications. Keeping in our mind that we always treat two different spacetimes, and 0, in perturbation theory, (5) is a rather curious equation in the following sense: the variable on the left-hand side of (5) is a variable on , while the variables on the right-hand side of (5) are variables on 0. Hence, (5) gives a relation between variables on two different manifolds.

Further, through (5), we have implicitly identified points in these two different manifolds. More specifically, 𝑄("𝑝") on the left-hand side of (5) is a field on , and "𝑝". Similarly, we should regard the background value 𝑄0(𝑝) of 𝑄("𝑝") and its deviation 𝛿𝑄(𝑝) of 𝑄("𝑝") from 𝑄0(𝑝), which are on the right-hand side of (5), as fields on 0, and 𝑝0. Because (5) is regarded as an equation for field variables, it implicitly states that the points "𝑝" and 𝑝0 are same. This represents the implicit assumption of the existence of a map 0𝑝0"𝑝", which is usually called a gauge choice (of the second-kind) in perturbation theory [3335].

It is important to note that the second-kind gauge choice between points on 0 and , which is established by such a relation as (5), is not unique to the theory with general covariance. Rather, (5) involves the degree of freedom corresponding to the choice of the map 𝒳0. This is called the gauge degree of freedom (of the second-kind). Such a degree of freedom always exists in perturbations of a theory with general covariance. General covariance intuitively means that there is no preferred coordinate system in the theory as mentioned above. If general covariance is not imposed on the theory, there is a preferred coordinate system in the theory, and we naturally introduce this preferred coordinate system onto both 0 and . Then, we can choose the identification map 𝒳 using this preferred coordinate system. However, there is no such coordinate system in general relativity due to the general covariance, and we have no guiding principle to choose the identification map 𝒳. Indeed, we may identify "𝑝" with 𝑞0 (𝑞𝑝) instead of 𝑝0. In the above understanding of the concept of “gauge” (of the second-kind) in general relativistic perturbation theory, a gauge transformation is simply a change of the map 𝒳.

These are the basic ideas of gauge degree of freedom (of the second-kind) in the general relativistic perturbation theory which are pointed out by Sacks [32] and mathematically clarified by Stewart and Walker [3335]. Based on these ideas, higher-order perturbation theory has been developed in [2228, 31, 38, 39, 41].

2.3. Formulation of Perturbation Theory

To formulate the above understanding in more detail, we introduce an infinitesimal parameter 𝜆 for the perturbation. Further, we consider the 4+1-dimensional manifold 𝒩=×, where 4=dim and 𝜆. The background spacetime 0=𝒩|𝜆=0 and the physical spacetime =𝜆=𝒩|=𝜆 are also submanifolds embedded in the extended manifold 𝒩. Each point on 𝒩 is identified by a pair (𝑝,𝜆), where 𝑝𝜆, and each point in 0𝒩 is identified by 𝜆=0.

Through this construction, the manifold 𝒩 is foliated by four-dimensional submanifolds 𝜆 of each 𝜆, and these are diffeomorphic to and 0. The manifold 𝒩 has a natural differentiable structure consisting of the direct product of and . Further, the perturbed spacetimes 𝜆 for each 𝜆 must have the same differential structure with this construction. In other words, we require that perturbations be continuous in the sense that and 0 are connected by a continuous curve within the extended manifold 𝒩. Hence, the changes of the differential structure resulting from the perturbation, for example, the formation of singularities and singular perturbations in the sense of fluid mechanics, are excluded from consideration.

Let us consider the set of field equations 𝑄𝜆=0(6) on the physical spacetime 𝜆 for the physical variables 𝑄𝜆 on 𝜆. The field equation (6) formally represents the Einstein equation for the metric on 𝜆 and the equations for matter fields on 𝜆. If a tensor field 𝑄𝜆 is given on each 𝜆, 𝑄𝜆 is automatically extended to a tensor field on 𝒩 by 𝑄(𝑝,𝜆)=𝑄𝜆(𝑝), where 𝑝𝜆. In this extension, the field equation (6) is regarded as an equation on the extended manifold 𝒩. Thus, we have extended an arbitrary tensor field and the field equations (6) on each 𝜆 to those on the extended manifold 𝒩.

Tensor fields on 𝒩 obtained through the above construction are necessarily “tangent” to each 𝜆. To consider the basis of the tangent space of 𝒩, we introduce the normal form and its dual, which are normal to each 𝜆 in 𝒩. These are denoted by (𝑑𝜆)𝑎 and (𝜕/𝜕𝜆)𝑎, respectively, and they satisfy (𝑑𝜆)𝑎(𝜕/𝜕𝜆)𝑎=1. The form (𝑑𝜆)𝑎 and its dual, (𝜕/𝜕𝜆)𝑎, are normal to any tensor field extended from the tangent space on each 𝜆 through the above construction. The set consisting of (𝑑𝜆)𝑎, (𝜕/𝜕𝜆)𝑎, and the basis of the tangent space on each 𝜆 is regarded as the basis of the tangent space of 𝒩.

Now, we define the perturbation of an arbitrary tensor field 𝑄. We compare 𝑄 on 𝜆 with 𝑄0 on 0, and it is necessary to identify the points of 𝜆 with those of 0 as mentioned above. This point identification map is the gauge choice of the second-kind as mentioned above. The gauge choice is made by assigning a diffeomorphism 𝒳𝜆𝒩𝒩 such that 𝒳𝜆0𝜆. Following the paper of Bruni et al. [24], we introduce a gauge choice 𝒳𝜆 as an one-parameter groups of diffeomorphisms, that is, an exponential map, for simplicity. We denote the generator of this exponential map by 𝒳𝜂𝑎. This generator 𝒳𝜂𝑎 is decomposed by the basis on 𝒩 which are constructed above. Although the generator 𝒳𝜂𝑎 should satisfy some appropriate properties [22], the arbitrariness of the gauge choice 𝒳𝜆 is represented by the tangential component of the generator 𝒳𝜂𝑎 to 𝜆.

The pull-back 𝒳𝜆𝑄, which is induced by the exponential map 𝒳𝜆, maps a tensor field 𝑄 on the physical manifold 𝜆 to a tensor field 𝒳𝜆𝑄 on the background spacetime. In terms of this generator 𝒳𝜂𝑎, the pull-back 𝒳𝜆𝑄 is represented by the Taylor expansion 𝑄𝒳(𝑟)=𝑄𝜆(𝑝)=𝒳𝜆𝑄(𝑝)=𝑄(𝑝)+𝜆£𝒳𝜂𝑄||𝑝+12𝜆2£2𝒳𝜂𝑄||𝑝𝜆+𝑂3,(7) where 𝑟=𝒳𝜆(𝑝)𝜆. Because 𝑝0, we may regard the equation 𝒳𝜆𝑄(𝑝)=𝑄0(𝑝)+𝜆£𝒳𝜂𝑄||0+1(𝑝)2𝜆2£2𝒳𝜂𝑄||0(𝜆𝑝)+𝑂3(8) as an equation on the background spacetime 0, where 𝑄0=𝑄|0 is the background value of the physical variable of 𝑄. Once the definition of the pull-back of the gauge choice 𝒳𝜆 is given, the first- and the second-order perturbations 𝒳(1)𝑄 and 𝒳(2)𝑄 of a tensor field 𝑄 under the gauge choice 𝒳𝜆 are simply given by the expansion 𝒳𝜆𝑄𝜆||0=𝑄0+𝜆𝒳(1)1𝑄+2𝜆2𝒳(2)𝜆𝑄+𝑂3(9) with respect to the infinitesimal parameter 𝜆. Comparing (8) and (9), we define the first- and the second-order perturbations of a physical variable 𝑄𝜆 under the gauge choice 𝒳𝜆 by 𝒳(1)𝑄=£𝒳𝜂𝑄||0,𝒳(2)𝑄=£2𝒳𝜂𝑄||0.(10) We note that all variables in (9) are defined on 0.

Now, we consider two different gauge choices based on the above understanding of the second-kind gauge choice. Suppose that 𝒳𝜆 and 𝒴𝜆 are two exponential maps with the generators 𝒳𝜂𝑎 and 𝒴𝜂𝑎 on 𝒩, respectively. In other words, 𝒳𝜆 and 𝒴𝜆 are two gauge choices (see Figure 2). Then, the integral curves of each 𝒳𝜂𝑎 and 𝒴𝜂𝑎 in 𝒩 are the orbits of the actions of the gauge choices 𝒳𝜆 and 𝒴𝜆, respectively. Since we choose the generators 𝒳𝜂𝑎 and 𝒴𝜂𝑎 so that these are transverse to each 𝜆 everywhere on 𝒩, the integral curves of these vector fields intersect with each 𝜆. Therefore, points lying on the same integral curve of either of the two are to be regarded as the same point within the respective gauges. When these curves are not identical, that is, the tangential components to each 𝜆 of 𝒳𝜂𝑎 and 𝒴𝜂𝑎 are different, these point identification maps 𝒳𝜆 and 𝒴𝜆 are regarded as two different gauge choices.

576273.fig.002
Figure 2: The second-kind gauge is a point-identification between the physical spacetime 𝜆 and the background spacetime 0 on the extended manifold 𝒩. Through (5), we implicitly assume the existence of a point-identification map between 𝜆 and 0. However, this point-identification is not unique by virtue of the general covariance in the theory. We may chose the gauge of the second-kind so that 𝑝0 and “𝑝𝜆 is same (𝒳𝜆). We may also choose the gauge so that 𝑞0 and “𝑝𝜆 is same (𝒴𝜆). These are different gauge choices. The gauge transformation 𝒳𝜆𝒴𝜆 is given by the diffeomorphism Φ=𝒳𝜆1𝒴𝜆.

We next introduce the concept of gauge invariance. In particular, in this paper, we consider the concept of order-by-order gauge invariance [27]. Suppose that 𝒳𝜆 and 𝒴𝜆 are two different gauge choices which are generated by the vector fields 𝒳𝜂𝑎 and 𝒴𝜂𝑎, respectively. These gauge choices also pull back a generic tensor field 𝑄 on 𝒩 to two other tensor fields, 𝒳𝜆𝑄 and 𝒴𝜆𝑄, for any given value of 𝜆. In particular, on 0, we now have three tensor fields associated with a tensor field 𝑄; one is the background value 𝑄0 of 𝑄, and the other two are the pulled-back variables of 𝑄 from 𝜆 to 0 by the two different gauge choices 𝒳𝑄𝜆=𝒳𝜆𝑄||0=𝑄0+𝜆𝒳(1)1𝑄+2𝜆2𝒳(2)𝜆𝑄+𝑂3,(11)𝒴𝑄𝜆=𝒴𝜆𝑄||0=𝑄0+𝜆𝒴(1)1𝑄+2𝜆2𝒴(2)𝜆𝑄+𝑂3.(12) Here, we have used (9). Because 𝒳𝜆 and 𝒴𝜆 are gauge choices which map from 0 to 𝜆, 𝒳𝑄𝜆 and 𝒴𝑄𝜆 are the different representations on 0 in the two different gauges of the same perturbed tensor field 𝑄 on 𝜆. The quantities 𝒳(𝑘)𝑄 and 𝒴(𝑘)𝑄 in (11) and (12) are the perturbations of 𝑂(𝑘) in the gauges 𝒳𝜆 and 𝒴𝜆, respectively. We say that the 𝑘th-order perturbation 𝒳(𝑘)𝑄 of 𝑄 is order-by-order gauge-invariant if and only if for any two gauges 𝒳𝜆 and 𝒴𝜆 the following holds: 𝒳(𝑘)𝑄=𝒴(𝑘)𝑄.(13)

Now, we consider the gauge transformation rules between different gauge choices. In general, the representation 𝒳𝑄𝜆 on 0 of the perturbed variable 𝑄 on 𝜆 depends on the gauge choice 𝒳𝜆. If we employ a different gauge choice, the representation of 𝑄𝜆 on 0 may change. Suppose that 𝒳𝜆 and 𝒴𝜆 are different gauge choices, which are the point identification maps from 0 to 𝜆, and the generators of these gauge choices are given by 𝒳𝜂𝑎 and 𝒴𝜂𝑎, respectively. Then, the change of the gauge choice from 𝒳𝜆 to 𝒴𝜆 is represented by the diffeomorphism Φ𝜆𝒳=𝜆1𝒴𝜆.(14) This diffeomorphism Φ𝜆 is the map Φ𝜆00 for each value of 𝜆. The diffeomorphism Φ𝜆 does change the point identification, as expected from the understanding of the gauge choice discussed above. Therefore, the diffeomorphism Φ𝜆 is regarded as the gauge transformation Φ𝜆𝒳𝜆𝒴𝜆.

The gauge transformation Φ𝜆 induces a pull-back from the representation 𝒳𝑄𝜆 of the perturbed tensor field 𝑄 in the gauge choice 𝒳𝜆 to the representation 𝒴𝑄𝜆 in the gauge choice 𝒴𝜆. Actually, the tensor fields 𝒳𝑄𝜆 and 𝒴𝑄𝜆, which are defined on 0, are connected by the linear map Φ𝜆 as 𝒴𝑄𝜆=𝒴𝜆𝑄||0=𝒴𝜆𝒳𝜆𝒳𝜆1𝑄|||0=𝒳𝜆1𝒴𝜆𝒳𝜆𝑄||0=Φ𝜆𝒳𝑄𝜆.(15) According to generic arguments concerning the Taylor expansion of the pull-back of a tensor field on the same manifold, given in Section 2.1, it should be possible to express the gauge transformation Φ𝜆𝒳𝑄𝜆 in the form Φ𝜆𝒳𝑄=𝒳𝑄+𝜆£𝜉1𝒳𝜆𝑄+22£𝜉2+£2𝜉1𝒳𝑄𝜆+𝑂3,(16) where the vector fields 𝜉𝑎1 and 𝜉𝑎2 are the generators of the gauge transformation Φ𝜆 (see (2)).

Comparing the representation (16) of the Taylor expansion in terms of the generators 𝜉𝑎1 and 𝜉𝑎2 of the pull-back Φ𝜆𝒳𝑄 and that in terms of the generators 𝒳𝜂𝑎 and 𝒴𝜂𝑎 of the pull-back 𝒴𝜆(𝒳𝜆1)𝒳𝑄 (=Φ𝜆𝒳𝑄), we readily obtain explicit expressions for the generators 𝜉𝑎1 and 𝜉𝑎2 of the gauge transformation Φ=𝒳𝜆1𝒴𝜆 in terms of the generators 𝒳𝜂𝑎 and 𝒴𝜂𝑎 of each gauge choices as follows: 𝜉𝑎1=𝒴𝜂𝑎𝒳𝜂𝑎,𝜉𝑎2=𝒴𝜂,𝒳𝜂𝑎.(17) Further, because the gauge transformation Φ𝜆 is a map within the background spacetime 0, the generator should consist of vector fields on 0. This can be satisfied by imposing some appropriate conditions on the generators 𝒴𝜂𝑎 and 𝒳𝜂𝑎.

We can now derive the relation between the perturbations in the two different gauges. Up to second order, these relations are derived by substituting (11) and (12) into (16):𝒴(1)𝑄𝒳(1)𝑄=£𝜉1𝑄0,(18)𝒴(2)𝑄𝒳(2)𝑄=2£𝜉1𝒳(1)£𝑄+𝜉2+£2𝜉1𝑄0.(19)

Here, we should comment on the gauge choice in the above explanation. We have introduced an exponential map 𝒳𝜆 (or 𝒴𝜆) as the gauge choice, for simplicity. However, this simplified introduction of 𝒳𝜆 as an exponential map is not essential to the gauge transformation rules (18) and (19). Actually, we can generalize the diffeomorphism 𝒳𝜆 from an exponential map. For example, the diffeomorphism whose pull-back is represented by the Taylor expansion (2) is a candidate of the generalization. If we generalize the diffeomorphism 𝒳𝜆, the representation (8) of the pulled-back variable 𝒳𝜆𝑄(𝑝), the representations of the perturbations (10), and the relations (17) between generators of Φ𝜆, 𝒳𝜆, and 𝒴𝜆 will be changed. However, the gauge transformation rules (18) and (19) are direct consequences of the generic Taylor expansion (16) of Φ𝜆. Generality of the representation of the Taylor expansion (16) of Φ𝜆 implies that the gauge transformation rules (18) and (19) will not be changed, even if we generalize the each gauge choice 𝒳𝜆. Further, the relations (17) between generators also imply that, even if we employ simple exponential maps as gauge choices, both of the generators 𝜉𝑎1 and 𝜉𝑎2 are naturally induced by the generators of the original gauge choices. Hence, we conclude that the gauge transformation rules (18) and (19) are quite general and irreducible. In this paper, 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). Hence, the developments of the cosmological perturbation theory presented below will not be changed even if we generalize the gauge choice 𝒳𝜆 from a simple exponential map.

We also have to emphasize the physical implication of the gauge transformation rules (18) and (19). According to the above construction of the perturbation theory, gauge degree of freedom, which induces the transformation rules (18) and (19), is unphysical degree of freedom. As emphasized above, the physical spacetime 𝜆 is our nature itself, while there is no background spacetime 0 in our nature. The background spacetime 0 is a fictitious spacetime and it has nothing to do with our nature. Since the gauge choice 𝒳𝜆 just gives a relation between 𝜆 and 0, the gauge choice 𝒳𝜆 also has nothing to do with our nature. On the other hand, any observations and experiments are carried out only on the physical spacetime 𝜆 through the physical processes on the physical spacetime 𝜆. Therefore, any direct observables in any observations and experiments should be independent of the gauge choice 𝒳𝜆, that is, should be gauge-invariant. Keeping this fact in our mind, the gauge transformation rules (18) and (19) imply that the perturbations 𝒳(1)𝑄 and 𝒳(2)𝑄 include unphysical degree of freedom, that is, gauge degree of freedom, if these perturbations are transformed as (18) or (19) under the gauge transformation 𝒳𝜆𝒴𝜆. If the perturbations 𝒳(1)𝑄 and 𝒳(2)𝑄 are independent of the gauge choice, these variables are order-by-order gauge-invariant. Therefore, order-by-order gauge-invariant variables does not include unphysical degree of freedom and should be related to the physics on the physical spacetime 𝜆.

2.4. Coordinate Transformations Induced by the Second Kind Gauge Transformation

In many literature, gauge degree of freedom is regarded as the degree of freedom of the coordinate transformation. In the linear-order perturbation theory, these two degree of freedom are equivalent with each other. However, in the higher-order perturbations, we should regard that these two degree of freedom are different. Although the essential understanding of the gauge degree of freedom (of the second-kind) is as that explained above, the gauge transformation (of the second-kind) also induces the infinitesimal coordinate transformation on the physical spacetime 𝜆 as a result. In many cases, the understanding of “gauges” in perturbations based on coordinate transformations leads mistakes. Therefore, we did not use any ingredient of this subsection in our series of papers [22, 23, 2528] concerning about higher-order general relativistic gauge-invariant perturbation theory. However, we comment on the relations between the coordinate transformation, briefly. Details can be seen in [22, 37, 38].

To see that the gauge transformation of the second-kind induces the coordinate transformation, we introduce the coordinate system {𝑂𝛼,𝜓𝛼} on the “background spacetime” 0, where 𝑂𝛼 are open sets on the background spacetime and 𝜓𝛼 are diffeomorphisms from 𝑂𝛼 to 4 (4=dim0). The coordinate system {𝑂𝛼,𝜓𝛼} is the set of the collection of the pair of open sets 𝑂𝛼 and diffeomorphism 𝑂𝛼4. If we employ a gauge choice 𝒳𝜆, we have the correspondence of 𝜆 and 0. Together with the coordinate system 𝜓𝛼 on 0, this correspondence between 𝜆 and 0 induces the coordinate system on 𝜆. Actually, 𝑋𝜆(𝑂𝛼) for each 𝛼 is an open set of 𝜆. Then, 𝜓𝛼𝒳𝜆1 becomes a diffeomorphism from an open set 𝑋𝜆(𝑂𝛼)𝜆 to 4. This diffeomorphism 𝜓𝛼𝒳𝜆1 induces a coordinate system of an open set on 𝜆.

When we have two different gauge choices 𝒳𝜆 and 𝒴𝜆, 𝜓𝛼𝒳𝜆1 and 𝜓𝛼𝒴𝜆1 become different coordinate systems on 𝜆. We can also consider the coordinate transformation from the coordinate system 𝜓𝛼𝒳𝜆1 to another coordinate system 𝜓𝛼𝒴𝜆1. Since the gauge transformation 𝒳𝜆𝒴𝜆 is induced by the diffeomorphism Φ𝜆 defined by (14), the induced coordinate transformation is given by 𝑦𝜇(𝑞)=𝑥𝜇Φ(𝑝)=1𝑥𝜇(𝑞)(20) in the passive point of view [22, 37, 38]. If we represent this coordinate transformation in terms of the Taylor expansion in Section 2.1, up to third order, we have the coordinate transformation 𝑦𝜇(𝑞)=𝑥𝜇(𝑞)𝜆𝜉𝜇1𝜆(𝑞)+22𝜉𝜇2(𝑞)+𝜉𝜈1(𝑞)𝜕𝜈𝜉𝜇1𝜆(𝑞)+𝑂3.(21)

2.5. Gauge-Invariant Variables

Here, inspecting the gauge transformation rules (18) and (19), we define the gauge-invariant variables for a metric perturbation and for arbitrary matter fields (tensor fields). Employing the idea of order-by-order gauge invariance for perturbations [27], we proposed a procedure to construct gauge-invariant variables of higher-order perturbations [22]. This proposal is as follows. First, we decompose a linear-order metric perturbation into its gauge-invariant and variant parts. The procedure for decomposing linear-order metric perturbations is extended to second-order metric perturbations, and we can decompose the second-order metric perturbation into gauge-invariant and variant parts. Then, we can define the gauge-invariant variables for the first- and second-order perturbations of an arbitrary field other than the metric by using the gauge variant parts of the first- and second-order metric perturbations. Although the procedure for finding gauge-invariant variables for linear-order metric perturbations is highly nontrivial, once we know this procedure, we can easily define the gauge-invariant variables of a higher-order perturbation through a simple extension of the procedure for the linear-order perturbations.

Now, we review the above strategy to construct gauge-invariant variables. To consider a metric perturbation, we expand the metric on the physical spacetime 𝜆, which is pulled back to the background spacetime 0 using a gauge choice in the form given in (9): 𝒳𝜆𝑔𝑎𝑏=𝑔𝑎𝑏+𝜆𝒳𝑎𝑏+𝜆22𝒳𝑙𝑎𝑏+𝑂3(𝜆),(22) where 𝑔𝑎𝑏 is the metric on 0. Of course, the expansion (22) of the metric depends entirely on the gauge choice 𝒳𝜆. Nevertheless, henceforth, we do not explicitly express the index of the gauge choice 𝒳𝜆 in an expression if there is no possibility of confusion.

Our starting point to construct gauge-invariant variables is the assumption that we already know the procedure for finding gauge-invariant variables for the linear metric perturbations. Then, a linear metric perturbation 𝑎𝑏 is decomposed as 𝑎𝑏=𝑎𝑏+£𝑋𝑔𝑎𝑏,(23) where 𝑎𝑏 and 𝑋𝑎 are the gauge-invariant and variant parts of the linear-order metric perturbations, that is, under the gauge transformation (18), these are transformed as 𝒴𝑎𝑏𝒳𝑎𝑏=0,𝒴𝑋𝑎𝒳𝑋𝑎=𝜉𝑎1.(24) The first-order metric perturbation (23) together with the gauge transformation rules (24) does satisfy the gauge transformation rule (18) for the first-order metric perturbation, that is, 𝒴(1)𝑎𝑏𝒳(1)𝑎𝑏=£𝜉1𝑔𝑎𝑏.(25)

As emphasized in our series of papers [22, 23, 2528], the above assumption is quite non-trivial and it is not simple to carry out the systematic decomposition (23) on an arbitrary background spacetime, since this procedure depends completely on the background spacetime (0,𝑔𝑎𝑏). However, as we will show below, this procedure exists at least in the case of cosmological perturbations of a homogeneous and isotropic universe in Section 5.1.

Once we accept this assumption for linear-order metric perturbations, we can always find gauge-invariant variables for higher-order perturbations [22]. According to the gauge transformation rule (19), the second-order metric perturbation 𝑙𝑎𝑏 is transformed as 𝒴(2)𝑙𝑎𝑏𝒳(2)𝑙𝑎𝑏=2£𝜉1𝒳𝑎𝑏+£𝜉2+£2𝜉1𝑔𝑎𝑏(26) under the gauge transformation Φ𝜆=(𝒳𝜆)1𝒴𝜆𝒳𝜆𝒴𝜆. Although this gauge transformation rule is slightly complicated, inspecting this gauge transformation rule, we first introduce the variable 𝐿𝑎𝑏 defined by 𝐿𝑎𝑏=𝑙𝑎𝑏2£𝑋𝑎𝑏+£2𝑋𝑔𝑎𝑏.(27) Under the gauge transformation Φ𝜆=(𝒳𝜆)1𝒴𝜆𝒳𝜆𝒴𝜆, the variable 𝐿𝑎𝑏 is transformed as 𝒴𝐿𝑎𝑏𝒳𝐿𝑎𝑏=£𝜎𝑔𝑎𝑏,𝜎(28)𝑎=𝜉𝑎2+𝜉1,𝑋𝑎.(29) The gauge transformation rule (28) is identical to that for a linear metric perturbation. Therefore, we may apply the above procedure to decompose 𝑎𝑏 into 𝑎𝑏 and 𝑋𝑎 when we decompose of the components of the variable 𝐿𝑎𝑏. Then, 𝐿𝑎𝑏 can be decomposed as 𝐿𝑎𝑏=𝑎𝑏+£𝑌𝑔𝑎𝑏,(30) where 𝑎𝑏 is the gauge-invariant part of the variable 𝐿𝑎𝑏, or equivalently, of the second-order metric perturbation 𝑙𝑎𝑏, and 𝑌𝑎 is the gauge variant part of 𝐿𝑎𝑏, that is, the gauge variant part of 𝑙𝑎𝑏. Under the gauge transformation Φ𝜆=(𝒳𝜆)1𝒴𝜆, the variables 𝑎𝑏 and 𝑌𝑎 are transformed as 𝒴𝑎𝑏𝒳𝑎𝑏=0,𝒴𝑌𝑎𝒴𝑌𝑎=𝜎𝑎,(31) respectively. Thus, once we accept the assumption (23), the second-order metric perturbations are decomposed as 𝑙𝑎𝑏=𝑎𝑏+2£𝑋𝑎𝑏+£𝑌£2𝑋𝑔𝑎𝑏,(32) where 𝑎𝑏 and 𝑌𝑎 are the gauge-invariant and variant parts of the second order metric perturbations, that is, 𝒴𝑎𝑏𝒳𝑎𝑏=0,𝒴𝑌𝑎𝒳𝑌𝑎=𝜉𝑎2+𝜉1,𝑋𝑎.(33)

Furthermore, as shown in [22], using the first- and second-order gauge variant parts, 𝑋𝑎 and 𝑌𝑎, of the metric perturbations, the gauge-invariant variables for an arbitrary field 𝑄 other than the metric are given by (1)𝒬=(1)𝑄£𝑋𝑄0,(34)(2)𝒬=(2)𝑄2£𝑋(1)£𝑄𝑌£2𝑋𝑄0.(35) It is straightforward to confirm that the variables (𝑝)𝒬 defined by (34) and (35) are gauge-invariant under the gauge transformation rules (18) and (19), respectively.

Equations (34) and (35) have very important implications. To see this, we represent these equations as (1)𝑄=(1)𝒬+£𝑋𝑄0,(36)(2)𝑄=(2)𝒬+2£𝑋(1)£𝑄+𝑌£2𝑋𝑄0.(37) These equations imply that any perturbation of first and second order can always be decomposed into gauge-invariant and gauge-variant parts as (36) and (37), respectively. These decomposition formulae (36) and (37) are important ingredients in the general framework of the second-order general relativistic gauge-invariant perturbation theory.

3. Perturbations of the Field Equations

In terms of the gauge-invariant variables defined last section, we derive the field equations, that is, Einstein equations and the equation for a matter field. To derive the perturbation of the Einstein equations and the equation for a matter field (Klein-Gordon equation), first of all, we have to derive the perturbative expressions of the Einstein tensor [23]. This is reviewed in Section 3.1. We also derive the first and the second order perturbations of the energy momentum tensor for a scalar field and the Klein-Gordon equation [27] in Section 3.2. Finally, we consider the first- and the second-order the Einstein equations in Section 3.3.

3.1. Perturbations of the Einstein Curvature

The relation between the curvatures associated with the metrics on the physical spacetime 𝜆 and the background spacetime 0 is given by the relation between the pulled-back operator 𝒳𝜆𝑎(𝒳𝜆1) of the covariant derivative 𝑎 associated with the metric 𝑔𝑎𝑏 on 𝜆 and the covariant derivative 𝑎 associated with the metric 𝑔𝑎𝑏 on 0. The pulled-back covariant derivative 𝒳𝜆𝑎(𝒳𝜆1) depends on the gauge choice 𝒳𝜆. The property of the derivative operator 𝒳𝜆𝑎(𝒳𝜆1) as the covariant derivative on 𝜆 is given by 𝒳𝜆𝑎𝒳𝜆1𝒳𝜆𝑔𝑎𝑏=0,(38) where 𝒳𝜆𝑔𝑎𝑏 is the pull-back of the metric on 𝜆, which is expanded as (22). In spite of the gauge dependence of the operator 𝒳𝜆𝑎(𝒳𝜆1), we simply denote this operator by 𝑎, because our calculations are carried out only on 0 in the same gauge choice 𝒳𝜆. Further, we denote the pulled-back metric 𝒳𝜆𝑔𝑎𝑏 on 𝜆 by 𝑔𝑎𝑏, as mentioned above.

Since the derivative operator 𝑎 (=𝒳𝑎(𝒳1)) may be regarded as a derivative operator on 0 that satisfies the property (38), there exists a tensor field 𝐶𝑐𝑎𝑏 on 0 such that 𝑎𝜔𝑏=𝑎𝜔𝑏𝐶𝑐𝑎𝑏𝜔𝑐,(39) where 𝜔𝑎 is an arbitrary one-form on 0. From the property (38) of the covariant derivative operator 𝑎 on 𝜆, the tensor field 𝐶𝑐𝑎𝑏 on 0 is given by 𝐶𝑐𝑎𝑏=12𝑔𝑐𝑑𝑎𝑔𝑑𝑏+𝑏𝑔𝑑𝑎𝑑𝑔𝑎𝑏,(40) where 𝑔𝑎𝑏 is the inverse of 𝑔𝑎𝑏 (see Appendix B). We note that the gauge dependence of the covariant derivative 𝑎 appears only through 𝐶𝑐𝑎𝑏. The Riemann curvature 𝑅𝑑𝑎𝑏𝑐 on 𝜆, which is also pulled back to 0, is given by [42] 𝑅𝑑𝑎𝑏𝑐=𝑅𝑑𝑎𝑏𝑐2[𝑎𝐶𝑑𝑏]𝑐+2𝐶𝑒𝑐[𝑎𝐶𝑑𝑏]𝑒,(41) where 𝑅𝑑𝑎𝑏𝑐 is the Riemann curvature on 0. The perturbative expression for the curvatures are obtained from the expansion of (41) through the expansion of 𝐶𝑐𝑎𝑏.

The first- and the-second order perturbations of the Riemann, the Ricci, the scalar, the Weyl curvatures, and the Einstein tensors on the general background spacetime are summarized in [23]. We also derived the perturbative form of the divergence of an arbitrary tensor field of second rank to check the perturbative Bianchi identities in [23]. In this paper, we only present the perturbative expression for the Einstein tensor, and its derivations in Appendix B.

We expand the Einstein tensor 𝐺𝑎𝑏=𝑅𝑏𝑎(1/2)𝛿𝑏𝑎𝑅 on 𝜆 as 𝐺𝑎𝑏=𝐺𝑎𝑏+𝜆(1)𝐺𝑎𝑏+12𝜆2(2)𝐺𝑎𝑏𝜆+𝑂3.(42) As shown in Appendix B, each order perturbation of the Einstein tensor is given by(1)𝐺𝑎𝑏=(1)𝒢𝑎𝑏[]+£𝒳𝐺𝑎𝑏,(43)(2)𝐺𝑎𝑏=(1)𝒢𝑏𝑎[]+(2)𝒢𝑏𝑎[],+2£𝑋(1)𝐺𝑎𝑏+£𝑌£2𝑋𝐺𝑎𝑏,(44) where (1)𝒢𝑎𝑏[𝐴]=(1)Σ𝑎𝑏[𝐴]12𝛿𝑎𝑏(1)Σ𝑐𝑐[𝐴],(1)Σ𝑎𝑏[𝐴]=2[𝑎𝐻𝑑]𝑏𝑑[𝐴]𝐴𝑐𝑏𝑅𝑎𝑐,(45)(2)𝒢𝑎𝑏[]𝐴,𝐵=(2)Σ𝑎𝑏[]1𝐴,𝐵2𝛿𝑎𝑏(2)Σ𝑐𝑐[]𝐴,𝐵,(46)(2)Σ𝑎𝑏[]𝐴,𝐵=2𝑅𝑎𝑑𝐵𝑐(𝑏𝐴𝑑)𝑐+2𝐻[𝑎𝑑𝑒[𝐴]𝐻𝑏𝑒𝑑][𝐵]+2𝐻[𝑎𝑑𝑒[𝐵]𝐻𝑏𝑒𝑑][𝐴]+2𝐴𝑒𝑑[𝑎𝐻𝑑]𝑏𝑒[𝐵]+2𝐵𝑒𝑑[𝑎𝐻𝑑]𝑏𝑒[𝐴]+2𝐴𝑐𝑏[𝑎𝐻𝑑]𝑐𝑑[𝐵]+2𝐵𝑐𝑏[𝑎𝐻𝑑]𝑐𝑑[𝐴],𝐻(47)𝑐𝑎𝑏[𝐴]=(𝑎𝐴𝑐𝑏)12𝑐𝐴𝑎𝑏𝐻,(48)𝑎𝑏𝑐[𝐴]=𝑔𝑐𝑑𝐻𝑑𝑎𝑏[𝐴],𝐻𝑎𝑏𝑐[𝐴]=𝑔𝑏𝑑𝐻𝑐𝑎𝑑[𝐴],𝐻𝑎𝑏𝑐[𝐴]=𝑔𝑐𝑑𝐻𝑎𝑏𝑑[𝐴].(49) We note that (1)𝒢𝑎𝑏[] and (2)𝒢𝑎𝑏[,] in (43) and (44) are the gauge-invariant parts of the perturbative Einstein tensors, and (43) and (44) have the same forms as (34) and (37), respectively. The expression of (2)𝒢𝑎𝑏[𝐴,𝐵] in (46) with (47) is derived by the consideration of the general relativistic gauge-invariant perturbation theory with two infinitesimal parameters in [22, 23].

We also note that (1)𝒢𝑎𝑏[] and (2)𝒢𝑎𝑏[,] defined by (45)–(47) satisfy the identities𝑎(1)𝒢𝑎𝑏[𝐴]=𝐻𝑎𝑐𝑎[𝐴]𝐺𝑏𝑐+𝐻𝑐𝑏𝑎[𝐴]𝐺𝑎𝑐,(50)𝑎(2)𝒢𝑏𝑎[]𝐴,𝐵=𝐻𝑎𝑐𝑎[𝐴](1)𝒢𝑏𝑐[𝐵]𝐻𝑎𝑐𝑎[𝐵](1)𝒢𝑏𝑐[𝐴]+𝐻𝑒𝑏𝑎[𝐴](1)𝒢𝑒𝑎[𝐵]+𝐻𝑒𝑏𝑎[𝐵](1)𝒢𝑒𝑎[𝐴]𝐻𝑏𝑎𝑑[𝐵]𝐴𝑑𝑐+𝐻𝑏𝑎𝑑[𝐴]𝐵𝑑𝑐𝐺𝑐𝑎+𝐻𝑐𝑎𝑑[𝐵]𝐴𝑎𝑑+𝐻𝑐𝑎𝑑[𝐴]𝐵𝑎𝑑𝐺𝑏𝑐,(51) for arbitrary tensor fields 𝐴𝑎𝑏 and 𝐵𝑎𝑏, respectively. We can directly confirm these identities without specifying arbitrary tensors 𝐴𝑎𝑏 and 𝐵𝑎𝑏 of the second rank, respectively. This implies that our general framework of the second-order gauge-invariant perturbation theory discussed here gives a self-consistent formulation of the second-order perturbation theory. These identities (50) and (51) guarantee the first- and second-order perturbations of the Bianchi identity 𝑏𝐺𝑏𝑎=0 and are also useful when we check whether the derived components of (45) and (46) are correct.

3.2. Perturbations of the Energy Momentum Tensor and Klein-Gordon Equation

Here, we consider the perturbations of the energy momentum tensor of the equation of motion. As a model of the matter field, we only consider the scalar field, for simplicity. Then, equation of motion for a scalar field is the Klein-Gordon equation.

The energy momentum tensor for a scalar field 𝜑 is given by 𝑇𝑎𝑏=𝑎𝜑𝑏1𝜑2𝛿𝑎𝑏𝑐𝜑𝑐𝜑+2𝑉𝜑,(52) where 𝑉(𝜑) is the potential of the scalar field 𝜑. We expand the scalar field 𝜑 as 𝜑=𝜑+𝜆𝜑1+12𝜆2𝜑2𝜆+𝑂3,(53) where 𝜑 is the background value of the scalar field 𝜑. Further, following to the decomposition formulae (34) and (35), each order perturbation of the scalar field 𝜑 is decomposed as 𝜑1=𝜑1+£𝑋𝜑,𝜑2=𝜑2+2£𝑋𝜑1+£𝑌£2𝑋𝜑,(54) where 𝜑1 and 𝜑2 are the first- and the second-order gauge-invariant perturbations of the scalar field, respectively.

Through the perturbative expansions (53) and (B.2) of the scalar field 𝜑 and the inverse metric, the energy momentum tensor (52) is also expanded as 𝑇𝑎𝑏=𝑇𝑎𝑏+𝜆(1)𝑇𝑎𝑏+12𝜆2(2)𝑇𝑎𝑏𝜆+𝑂3.(55) The background energy momentum tensor 𝑇𝑎𝑏 is given by the replacement 𝜑𝜑 in (52). Further, through the decompositions (23), (32), (54), the perturbations of the energy momentum tensor (1)(𝑇𝑎𝑏) and (2)(𝑇𝑎𝑏) are also decomposed as(1)𝑇𝑎𝑏=(1)𝒯𝑎𝑏+£𝑋𝑇𝑎𝑏,(56)(2)𝑇𝑎𝑏=(2)𝒯𝑎𝑏+2£𝑋(1)𝑇𝑎𝑏+£𝑌£2𝑋𝑇𝑎𝑏,(57) where the gauge-invariant parts (1)𝒯𝑎𝑏 and (2)𝒯𝑎𝑏 of the first and the second order are given by (1)𝒯𝑎𝑏=𝑎𝜑𝑏𝜑1𝑎𝜑𝑏𝑐𝑐𝜑+𝑎𝜑1𝑏𝜑𝛿𝑏𝑎𝑐𝜑𝑐𝜑112𝑐𝜑𝑑𝑐𝑑𝜑+𝜑1𝜕𝑉,𝜕𝜑(58)(2)𝒯𝑎𝑏=𝑎𝜑𝑏𝜑2+𝑎𝜑2𝑏𝜑𝑎𝜑𝑔𝑏𝑑𝑑𝑐𝑐𝜑2𝑎𝜑𝑏𝑐𝑐𝜑1+2𝑎𝜑𝑏𝑑𝑑𝑐𝑐𝜑+2𝑎𝜑1𝑏𝜑12𝑎𝜑1𝑏𝑐𝑐𝜑𝛿𝑏𝑎𝑐𝜑𝑐𝜑212𝑐𝜑𝑑𝑐𝑑𝜑+𝑐𝜑𝑑𝑒𝑒𝑐𝑑𝜑2𝑐𝜑𝑑𝑐𝑑𝜑1+𝑐𝜑1𝑐𝜑1+𝜑2𝜕𝑉𝜕𝜑+𝜑21𝜕2𝑉𝜕𝜑2.(59) We note that (56) and (57) have the same form as (36) and (37), respectively.

Next, we consider the perturbation of the Klein-Gordon equation 𝐶(𝐾)=𝑎𝑎𝜑𝜕𝑉𝜕𝜑𝜑=0.(60) Through the perturbative expansions (53) and (22), the Klein-Gordon equation (60) is expanded as 𝐶(𝐾)=𝐶(𝐾)+𝜆(1)𝐶(𝐾)+12𝜆2(2)𝐶(𝐾)𝜆+𝑂3.(61)𝐶(𝐾) is the background Klein-Gordon equation 𝐶(𝐾)=𝑎𝑎𝜑𝜕𝑉𝜕𝜑(𝜑)=0.(62) The first- and the second-order perturbations (1)𝐶(𝐾) and (2)𝐶(𝐾) are also decomposed into the gauge-invariant and the gauge-variant parts as (1)𝐶(𝐾)=(1)𝒞(𝐾)+£𝑋𝐶(𝐾),(2)𝐶(𝐾)=(2)𝒞(𝐾)+2£𝑋(1)𝐶(𝐾)+£𝑌£2𝑋𝐶(𝐾),(63) where (1)𝒞(𝐾)=𝑎𝑎𝜑1𝐻𝑎𝑎𝑐[]𝑐𝜑𝑎𝑏𝑎𝑏𝜑𝜑1𝜕2𝑉𝜕𝜑2(𝜑),(64)(2)𝒞(𝐾)=𝑎𝑎𝜑2𝐻𝑎𝑎𝑐[]𝑐𝜑+2𝐻𝑎𝑎𝑑[]𝑐𝑑𝑐𝜑2𝐻𝑎𝑎𝑐[]𝑐𝜑1+2𝑎𝑏𝐻𝑐𝑎𝑏[]𝑐𝜑𝑎𝑏𝑎𝑏𝜑+2𝑎𝑑𝑑𝑏𝑎𝑏𝜑2𝑎𝑏𝑎𝑏𝜑1𝜑2𝜕2𝑉𝜕𝜑2𝜑(𝜑)12𝜕3𝑉𝜕𝜑3(𝜑).(65) Here, we note that (63) have the same form as (36) and (37).

By virtue of the order-by-order evaluations of the Klein-Gordon equation, the first- and the second-order perturbation of the Klein-Gordon equation are necessarily given in gauge-invariant form as (1)𝒞(𝐾)=0,(2)𝒞(𝐾)=0.(66)

We should note that, in [27], we summarized the formulae of the energy momentum tensors for an perfect fluid, an imperfect fluid, and a scalar field. Further, we also summarized the equations of motion of these three matter fields, that is, the energy continuity equation and the Euler equation for a perfect fluid; the energy continuity equation and the Navier-Stokes equation for an imperfect fluid; the Klein-Gordon equation for a scalar field. All these formulae also have the same form as the decomposition formulae (36) and (37). In this sense, we may say that the decomposition formulae (36) and (37) are universal.

3.3. Perturbations of the Einstein Equation

Finally, we impose the perturbed Einstein equation of each order, (1)𝐺𝑎𝑏=8𝜋𝐺(1)𝑇𝑎𝑏,(2)𝐺𝑎𝑏=8𝜋𝐺(2)𝑇𝑎𝑏.(67) Then, the perturbative Einstein equation is given by (1)𝒢𝑎𝑏[]=8𝜋𝐺(1)𝒯𝑎𝑏(68) at linear order and (1)𝒢𝑎𝑏[]+(2)𝒢𝑎𝑏[],=8𝜋𝐺(2)𝒯𝑎𝑏(69) at second order. These explicitly show that, order-by-order, the Einstein equations are necessarily given in terms of gauge-invariant variables only.

Together with (66), we have seen that the first- and the second-order perturbations of the Einstein equations and the Klein-Gordon equation are necessarily given in gauge-invariant form. This implies that we do not have to consider the gauge degree of freedom, at least in the level where we concentrate only on the equations of the system.

We have reviewed the general outline of the second-order gauge-invariant perturbation theory. We also note that the ingredients of this section are independent of the explicit form of the background metric 𝑔𝑎𝑏, except for the decomposition assumption (23) for the linear-order metric perturbations and are valid not only in cosmological perturbation case but also the other generic situations if (23) is correct. Within this general framework, we develop a second-order cosmological perturbation theory in terms of the gauge-invariant variables.

4. Cosmological Background Spacetime and Equations

The background spacetime 0 considered in cosmological perturbation theory is a homogeneous, isotropic universe that is foliated by the three-dimensional hypersurface Σ(𝜂), which is parametrized by 𝜂. Each hypersurface of Σ(𝜂) is a maximally symmetric three-space [43], and the spacetime metric of this universe is given by 𝑔𝑎𝑏=𝑎2(𝜂)(𝑑𝜂)𝑎(𝑑𝜂)𝑏+𝛾𝑖𝑗𝑑𝑥𝑖𝑎𝑑𝑥𝑗𝑏,(70) where 𝑎=𝑎(𝜂) is the scale factor, 𝛾𝑖𝑗 is the metric on the maximally symmetric 3-space with curvature constant 𝐾, and the indices 𝑖,𝑗,𝑘, for the spatial components run from 1 to 3.

To study the Einstein equation for this background spacetime, we introduce the energy-momentum tensor for a scalar field, which is given by 𝑇𝑎𝑏=𝑎𝜑𝑏1𝜑2𝛿𝑏𝑎𝑐𝜑𝑐1𝜑+2𝑉(𝜑)=2𝑎2𝜕𝜂𝜑2+𝑉(𝜑)(𝑑𝜂)𝑎𝜕𝜕𝜂𝑏+12𝑎2𝜕𝜂𝜑2𝛾𝑉(𝜑)𝑏𝑎,(71) where we assumed that the scalar field 𝜑 is homogeneous 𝜑=𝜑(𝜂)(72) and 𝛾𝑎𝑏 are defined as 𝛾𝑎𝑏=𝛾𝑖𝑗𝑑𝑥𝑖𝑎𝑑𝑥𝑗𝑏,𝛾𝑎𝑏=𝛾𝑗𝑖𝑑𝑥𝑖𝑎𝜕𝜕𝑥𝑗𝑏.(73)

The background Einstein equations 𝐺𝑎𝑏=8𝜋𝐺𝑇𝑎𝑏 for this background spacetime filled with the single scalar field are given by2+𝐾=8𝜋𝐺3𝑎212𝑎2𝜕𝜂𝜑2,+𝑉(𝜑)2𝜕𝜂+21+𝐾=8𝜋𝐺2𝜕𝜂𝜑2+𝑎2.𝑉(𝜑)(74) We also note that (74) lead to 2+𝐾𝜕𝜂𝜕=4𝜋𝐺𝜂𝜑2.(75) Equation (75) is also useful when we derive the perturbative Einstein equations.

Next, we consider the background Klein-Gordon equation which is the equation of motion 𝑎𝑇𝑎𝑏=0 for the scalar field 𝜕2𝜂𝜑+2𝜕𝜂𝜑+𝑎2𝜕𝑉𝜕𝜑=0.(76) The Klein-Gordon equation (76) is also derived from the Einstein equations (74). This is a well known fact and is just due to the Bianchi identity of the background spacetime. However, these types of relation are useful to check whether the derived system of equations is consistent.

5. Equations for the First-Order Cosmological Perturbations

On the cosmological background spacetime in the last section, we develop the perturbation theory in the gauge-invariant manner. In this section, we summarize the first-order perturbation of the Einstein equation and the Klein-Gordon equations. In Section 5.1, we show that the assumption on the decomposition (23) of the linear-order metric perturbation is correct. In Section 5.2, we summarize the first-order perturbation of the Einstein equation. Finally, in Section 5.3, we show the first-order perturbation of the Klein-Gordon equation.

5.1. Gauge-Invariant Metric Perturbations

Here, we consider the first-order metric perturbation 𝑎𝑏 and show the assumption on the decomposition (23) is correct in the background metric (70). To accomplish the decomposition (23), first, we assume the existence of the Green functions Δ1=(𝐷𝑖𝐷𝑖)1, (Δ+2𝐾)1, and (Δ+3𝐾)1, where 𝐷𝑖 is the covariant derivative associated with the metric 𝛾𝑖𝑗 and 𝐾 is the curvature constant of the maximally symmetric three space. Next, we consider the decomposition of the linear-order metric perturbation 𝑎𝑏 as𝑎𝑏=𝜂𝜂(𝑑𝜂)𝑎(𝑑𝜂)𝑏𝐷+2𝑖(𝑉𝐿)+(𝑉)𝑖(𝑑𝜂)(𝑎𝑑𝑥𝑖𝑏)+𝑎2(𝐿)𝛾𝑖𝑗+𝐷𝑖𝐷𝑗13𝛾𝑖𝑗Δ(𝑇𝐿)+2𝐷(𝑖(𝑇𝑉)𝑗)+(𝑇𝑇)𝑖𝑗𝑑𝑥𝑖𝑎𝑑𝑥𝑗𝑏,(77) where (𝑉)𝑖, (𝑇𝑉)𝑗, and (𝑇𝑇)𝑖𝑗 satisfy the properties 𝐷𝑖(𝑉)𝑖=0,𝐷𝑖(𝑇𝑉)𝑖=0,(𝑇𝑇)𝑖𝑗=(𝑇𝑇)𝑗𝑖,𝑖𝑖(𝑇)=𝛾𝑖𝑗(𝑇)𝑖𝑗𝐷=0,𝑖(𝑇𝑇)𝑖𝑗=0.(78) The gauge-transformation rules for the variables 𝜂𝜂, (𝑉𝐿), (𝑉)𝑖, (𝐿), (𝑇𝐿), (𝑇𝑉)𝑗 and (𝑇𝑇)𝑖𝑗 are derived from (25). Inspecting these gauge-transformation rules, we define the gauge-variant part 𝑋𝑎 in (23): 𝑋𝑎=(𝑉𝐿)12𝑎2𝜕𝜂(𝑇𝐿)(𝑑𝜂)𝑎+𝑎2(𝑇𝑉)𝑖+12𝐷𝑖(𝑇𝐿)𝑑𝑥𝑖𝑎.(79) We can easily check this vector field 𝑋𝑎 satisfies (24). Subtracting gauge variant-part £𝑋𝑔𝑎𝑏 from 𝑎𝑏, we have the gauge-invariant part 𝑎𝑏 in (23): 𝑎𝑏=𝑎22(1)Φ(𝑑𝜂)𝑎(𝑑𝜂)𝑏+2(1)𝜈𝑖(𝑑𝜂)(𝑎𝑑𝑥𝑖𝑏)+2(1)Ψ𝛾𝑖𝑗+(1)𝜒𝑖𝑗𝑑𝑥𝑖𝑎𝑑𝑥𝑗𝑏,(80) where the properties 𝐷𝑖(1)𝜈𝑖=𝛾𝑖𝑗𝐷𝑖(1)𝜈𝑗=(1)𝜒𝑖𝑖=𝛾𝑖𝑗(1)𝜒𝑖𝑗=𝐷𝑖(1)𝜒𝑖𝑗=0 are satisfied as consequences of (78).

Thus, we may say that our assumption for the decomposition (23) in linear-order metric perturbation is correct in the case of cosmological perturbations. However, we have to note that to accomplish (23), we assumed the existence of the Green functions Δ1, (Δ+2𝐾)1, and (Δ+3𝐾)1. As shown in [25, 26], this assumption is necessary to guarantee the one to one correspondence between the variables {𝜂𝜂,𝑖𝜂,𝑖𝑗} and {𝜂𝜂,(𝑉𝐿),(𝑉)𝑖,(𝐿),(𝑇𝐿),(𝑇𝑉)𝑗,(𝑇𝑇)𝑖𝑗}, but excludes some perturbative modes of the metric perturbations which belong to the kernel of the operator Δ, (Δ+2𝐾), and (Δ+3𝐾) from our consideration. For example, homogeneous modes belong to the kernel of the operator Δ and are excluded from our consideration. If we have to treat these modes, the separate treatments are necessary. In this paper, we ignore these modes, for simplicity.

We also note the fact that the definition (23) of the gauge-invariant variables is not unique. This comes from the fact that we can always construct new gauge-invariant quantities by the combination of the gauge-invariant variables. For example, using the gauge-invariant variables (1)Φ and (1)𝜈𝑖 of the first-order metric perturbation, we can define a vector field 𝑍𝑎 by 𝑍𝑎=𝑎(1)Φ(𝑑𝜂)𝑎+𝑎(1)𝜈𝑖(𝑑𝑥𝑖)𝑎 which is gauge invariant. Then, we can rewrite the decomposition formula (23) for the linear-order metric perturbation as 𝑎𝑏=𝑎𝑏£𝑍𝑔𝑎𝑏+£𝑍𝑔𝑎𝑏+£𝑋𝑔𝑎𝑏,=𝒦𝑎𝑏+£𝑋+𝑍𝑔𝑎𝑏,(81) where we have defined new gauge invariant variable 𝐾𝑎𝑏 by 𝒦𝑎𝑏=𝑎𝑏£𝑍𝑔𝑎𝑏. Clearly, 𝒦𝑎𝑏 is gauge-invariant and the vector field 𝑋𝑎+𝑍𝑎 satisfies (24). In spite of this nonuniqueness, we specify the components of the tensor 𝑎𝑏 as (80), which is the gauge-invariant part of the linear-order metric perturbation associated with the longitudinal gauge.

The non-uniqueness of the definitions of gauge-invariant variables is related to the “gauge-fixing” for the linear-order metric perturbations. Due to this non-uniqueness, we can consider the gauge-fixing in the first-order metric perturbation from two different points of view. The first point of view is that the gauge-fixing is to specify the gauge-variant part 𝑋𝑎. For example, the longitudinal gauge is realized by the gauge fixing 𝑋𝑎=0. Due to this gauge fixing 𝑋𝑎=0, we can regard the fact that perturbative variables in the longitudinal gauge are the completely gauge fixed variables. On the other hand, we may also regard that the gauge fixing is the specification of the gauge-invariant vector field 𝑍𝑎 in (81). In this point of view, we do not specify the vector field 𝑋𝑎. Instead, we have to specify the gauge-invariant vector 𝑍𝑎 or equivalently to specify the gauge-invariant metric perturbation 𝒦𝑎𝑏 without specifying 𝑋𝑎 so that the first-order metric perturbation 𝑎𝑏 coincides with the gauge-invariant variables 𝒦𝑎𝑏 when we fix the gauge 𝑋𝑎 so that 𝑋𝑎+𝑍𝑎=0. These two different point of view of “gauge fixing” is equivalent with each other due to the non-uniqueness of the definition (81) of the gauge-invariant variables.

5.2. First-Order Einstein Equations

Here, we derive the linear-order Einstein equation (68). To derive the components of the gauge-invariant part of the linearized Einstein tensor (1)𝒢𝑏𝑎[], which is defined by (45), we first derive the components of the tensor 𝑐𝑎𝑏[], which is defined in (48) with 𝐴𝑎𝑏=𝑎𝑏 and its component (80). These components are summarized in [25, 26].

From (45), the component of (1)𝒢𝑏𝑎[] are summarized as(1)𝒢𝜂𝜂[]1=𝑎26𝜕𝜂+2Δ+6𝐾(1)Ψ62(1)Φ,(1)𝒢𝑖𝜂[]1=𝑎22𝜕𝜂𝐷𝑖(1)Ψ+2𝐷𝑖(1)1Φ2(Δ+2𝐾)(1)𝜈𝑖,(1)𝒢𝜂𝑖[]=1𝑎22𝜕𝜂𝐷𝑖(1)Ψ+2𝐷𝑖(1)Φ+12Δ+2𝐾+424𝜕𝜂(1)𝜈𝑖,(1)𝒢𝑖𝑗[]=1𝑎2𝐷𝑖𝐷𝑗(1)Ψ(1)Φ+Δ+2𝜕2𝜂+4𝜕𝜂2𝐾(1)Ψ+2𝜕𝜂+4𝜕𝜂+22+Δ(1)Φ𝛾𝑗𝑖12𝑎2𝜕𝜂𝑎2𝐷𝑖(1)𝜈𝑗+𝐷𝑗(1)𝜈𝑖+12𝜕2𝜂+2𝜕𝜂+2𝐾Δ(1)𝜒𝑗𝑖.(82) Straightforward calculations show that these components of the first-order gauge-invariant perturbation (1)𝒢𝑎𝑏[] of the Einstein tensor satisfies the identity (50). Although this confirmation is also possible without specification of the tensor 𝑎𝑏, the confirmation of (50) through the explicit components (82) implies that we have derived the components of (1)𝒢𝑎𝑏[] consistently.

Next, we summarize the first-order perturbation of the energy momentum tensor for a scalar field. Since, at the background level, we assume that the scalar field 𝜑 is homogeneous as (72), the components of the gauge-invariant part of the first-order energy-momentum tensor (1)𝒯𝑎𝑏 are given by(1)𝒯𝜂𝜂1=𝑎2𝜕𝜂𝜑1𝜕𝜂𝜑(1)Φ𝜕𝜂𝜑2+𝑎2𝑑𝑉𝜑𝑑𝜑1,(1)𝒯𝑖𝜂1=𝑎2𝐷𝑖𝜑1𝜕𝜂𝜑,(83)(1)𝒯𝜂𝑖=1𝑎2𝜕𝜂𝜑𝐷𝑖𝜑1+𝜕𝜂𝜑(1)𝜈𝑖,(1)𝒯𝑖𝑗=1𝑎2𝛾𝑖𝑗𝜕𝜂𝜑1𝜕𝜂𝜑(1)Φ𝜕𝜂𝜑2𝑎2𝑑𝑉𝜑𝑑𝜑1.(84) The second equation in (84) shows that there is no anisotropic stress in the energy-momentum tensor of the single scalar field. Then, we obtain that(1)Φ=(1)Ψ.(85) From (82)–(84) and (85), the components of scalar parts of the linearized Einstein equation (68) are given as [3] Δ3𝜕𝜂+4𝐾𝜕𝜂22(1)Φ𝜕=4𝜋𝐺𝜂𝜑1𝜕𝜂𝜑+𝑎2𝑑𝑉𝜑𝑑𝜑1,𝜕(86)𝜂(1)Φ+(1)Φ=4𝜋𝐺𝜑1𝜕𝜂𝜕𝜑,(87)2𝜂+3𝜕𝜂+𝜕𝜂+22(1)Φ𝜕=4𝜋𝐺𝜂𝜑1𝜕𝜂𝜑𝑎2𝑑𝑉𝜑𝑑𝜑1.(88) In the derivation of (86)–(88), we have used (75). We also note that only two of these equations are independent. Further, the vector part of the component (1)𝒢𝑖𝜂[]=8𝜋𝐺(1)𝒯𝑖𝜂 shows that (1)𝜈𝑖=0.(89) The equation for the tensor mode (1)𝜒𝑖𝑗 is given by 𝜕2𝜂+2𝜕𝜂+2𝐾Δ(1)𝜒𝑖𝑗=0.(90)

Combining (86) and (88), we eliminate the potential term of the scalar field and thereby obtain that𝜕2𝜂+Δ+4𝐾(1)Φ=8𝜋𝐺𝜕𝜂𝜑1𝜕𝜂𝜑.(91) Further, using (87) to express 𝜕𝜂𝜑1 in terms of 𝜕𝜂(1)Φ and (1)Φ, we also eliminate 𝜕𝜂𝜑1 in (91). Hence, we have 𝜕2𝜂+22𝜕2𝜂𝜑𝜕𝜂𝜑𝜕𝜂𝜕Δ4𝐾+2𝜂𝜕2𝜂𝜑𝜕𝜂𝜑(1)Φ=0.(92) This is the master equation for the scalar mode perturbation of the cosmological perturbation in universe filled with a single scalar field. It is also known that (92) reduces to a simple equation through a change of variables [3].

5.3. First-Order Klein-Gordon Equations

Next, we consider the first-order perturbation of the Klein-Gordon equation (64). By the straightforward calculations using (70), (80), (72), (76), and the components 𝐻𝑎𝑎𝑐 summarized in [25, 26], the gauge-invariant part (1)𝒞(𝐾) of the first-order Klein-Gordon equation defined by (64) is given by𝑎2(1)𝒞(𝐾)=𝜕2𝜂𝜑1+2𝜕𝜂𝜑1Δ𝜑1𝜕𝜂(1)Φ+3𝜕𝜂(1)Ψ𝜕𝜂𝜑+2𝑎2(1)Φ𝜕𝑉𝜕𝜑(𝜑)+𝑎2𝜑1𝜕2𝑉𝜕𝜑2(𝜑)=0.(93)

Through the background Einstein equations (74) and the first-order perturbations (87) and (92) of the Einstein equation, we can easily derive the first-order perturbation of the Klein-Gordon equation (93) [28]. Hence, the first-order perturbation of the Klein-Gordon equation is not independent of the background and the first-order perturbation of the Einstein equation. Therefore, from the viewpoint of the Cauchy problem, any information obtained from the first-order perturbation of the Klein-Gordon equation should also be obtained from the set of the background and the first-order the Einstein equation, in principle.

6. Equations for the Second-Order Cosmological Perturbations

Now, we develop the second-order perturbation theory on the cosmological background spacetime in Section 4 within the general framework of the gauge-invariant perturbation theory reviewed in Section 2. Since we have already confirmed the important step of our general framework, that is, the assumption for the decomposition (23) of the linear-order metric perturbation is correct. Hence, the general framework reviewed in Section 2 is applicable. Applying this framework, we define the second-order gauge-invariant variables of the metric perturbation in Section 6.1. In Section 6.2, we summarize the explicit components of the gauge-invariant parts of the second-order perturbation of the Einstein tensor. In Section 6.3, we summarize the explicit components of the second-order perturbation of the energy-momentum tensor and the Klein-Gordon equations. Then, in Section 6.4, we derive the second-order Einstein equations in terms of gauge-invariant variables. The resulting equations have the source terms which constitute of the quadratic terms of the linear-order perturbations. Although these source terms have complicated forms, we give identities which comes from the consistency of all the second-order perturbations of the Einstein equation and the Klein-Gordon equation in Section 6.5.

6.1. Gauge-Invariant Metric Perturbations

First, we consider the components of the gauge-invariant variables for the metric perturbation of second order. The variable 𝐿𝑎𝑏 defined by (27) is transformed as (28) under the gauge transformation and we may regard the generator 𝜎𝑎 defined by (29) as an arbitrary vector field on 0 from the fact that the generator 𝜉𝑎2 in (29) is arbitrary. We can apply the procedure to find gauge-invariant variables for the first-order metric perturbations (80) in Section 5.1. Then, we can accomplish the decomposition (30). Following to the same argument as in the linear case, we may choose the components of the gauge-invariant variables 𝑎𝑏 in (32) as𝑎𝑏=2𝑎2(2)Φ(𝑑𝜂)𝑎(𝑑𝜂)𝑏+2𝑎2(2)𝜈𝑖(𝑑𝜂)(𝑎𝑑𝑥𝑖𝑏)+𝑎22(2)Ψ𝛾𝑖𝑗+(2)𝜒𝑖𝑗𝑑𝑥𝑖𝑎𝑑𝑥𝑗𝑏,(94) where (2)𝜈𝑖 and (2)𝜒𝑖𝑗 satisfy the equations𝐷𝑖(2)𝜈𝑖=0,(2)𝜒𝑖𝑖=0,𝐷𝑖(2)𝜒𝑖𝑗=0.(95) The gauge-invariant variables (2)Φ and (2)Ψ are the scalar mode perturbations of second order, and (2)𝜈𝑖 and (2)𝜒𝑖𝑗 are the second-order vector and tensor modes of the metric perturbations, respectively.

Here, we also note the fact that the decomposition (32) is not unique. This situation is similar to the case of the linear order, but more complicated. In the definition of the gauge-invariant variables of the second-order metric perturbation, we may replace 𝑋𝑎=𝑋𝑎𝑍𝑎,(96) where 𝑍𝑎 is gauge invariant and 𝑋𝑎 is transformed as 𝒴𝑋𝑎𝒳𝑋𝑎=𝜉𝑎1(97) under the gauge transformation 𝒳𝜆𝒴𝜆. This 𝑍𝑎 may be different from the vector 𝑍𝑎 in (81). By the replacement (96), the second-order metric perturbation (32) is given in the form 𝑙𝑎𝑏=𝒥𝑎𝑏+2£𝑋𝑎𝑏+£𝑌£2𝑋𝑔𝑎𝑏,(98) where we defined 𝒥𝑎𝑏=𝑎𝑏£𝑊𝑔𝑎𝑏2£𝑍𝒦𝑎𝑏2£𝑍£𝑍𝑔𝑎𝑏+£2𝑍𝑔𝑎𝑏,𝑌(99)𝑎=𝑌𝑎+𝑊𝑎+𝑋,𝑍𝑎.(100) Here, the vector field 𝑊𝑎 in (100) constitute of some components of gauge-invariant second-order metric perturbation 𝑎𝑏 like 𝑍𝑎 in (81). The tensor field 𝒥𝑎𝑏 is manifestly gauge invariant. The gauge transformation rule of the new gauge-variant part 𝑌𝑎 of the second-order metric perturbation is given by 𝒴𝑌𝑎𝒳𝑌𝑎=𝜉𝑎(2)+𝜉(1),𝑋𝑎.(101) Although (98) is similar to (32), the tensor fields 𝑎𝑏 and 𝒥𝑎𝑏 are different from each other. Thus, the definition of the gauge-invariant variables for the second-order metric perturbation is not unique in a more complicated way than the linear order. This nonuniqueness of gauge-invariant variables for the metric perturbations propagates to the definition (34) and (35) of the gauge-invariant variables for matter fields.

In spite of the existence of infinitely many definitions of the gauge-invariant variables, in this paper, we consider the components of 𝑎𝑏 given by (94). Equation (94) corresponds to the second-order extension of the longitudinal gauge, which is called Poisson gauge 𝑋𝑎=𝑌𝑎=0.

6.2. Einstein Tensor

Here, we evaluate the second-order perturbation of the Einstein tensor (44) with the cosmological background (70). We evaluate the term (1)𝒢𝑎𝑏[] and (2)𝒢𝑎𝑏[,] in the Einstein equation (69).

First, we evaluate the term (1)𝒢𝑎𝑏[] in the Einstein equation (69). Because the components (94) of 𝑎𝑏 are obtained through the replacements (1)Φ(2)Φ,(1)𝜈𝑖(2)𝜈𝑖,(1)Ψ(2)Ψ,(1)𝜒𝑖𝑗(2)𝜒𝑖𝑗(102) in the components (80) of 𝑎𝑏, we easily obtain the components of (1)𝒢𝑎𝑏[] through the replacements (102) in (82).

From (80), we can derive the components of (2)𝒢𝑎𝑏=(2)𝒢𝑎𝑏[,] defined by (46)–(49) in a straightforward manner. Here, we use the results (85) and (89) of the first-order Einstein equations, for simplicity. Then the explicit components (2)𝒢𝑎𝑏=(2)𝒢𝑎𝑏[,] are summarized as(2)𝒢𝜂𝜂=2𝑎23𝐷𝑘(1)Φ𝐷𝑘(1)Φ8(1)ΦΔ(1)𝜕Φ3𝜂(1)Φ2122+𝐾(1)Φ2+𝐷𝑙𝐷𝑘(1)Φ(1)𝜒𝑙𝑘+18𝜕𝜂(1)𝜒𝑘𝑙𝜕𝜂+8(1)𝜒𝑘𝑙+12𝐷𝑘(1)𝜒𝑙𝑚𝐷[𝑙(1)𝜒𝑘]𝑚18𝐷𝑘(1)𝜒𝑙𝑚𝐷𝑘(1)𝜒𝑚𝑙12(1)𝜒𝑙𝑚(Δ𝐾)(1)𝜒𝑙𝑚,(2)𝒢𝜂𝑖=2𝑎28(1)Φ𝜕𝜂𝐷𝑖(1)Φ𝐷𝑗(1)Φ𝜕𝜂(1)𝜒𝑗𝑖𝜕𝜂𝐷𝑗(1)Φ+2𝐷𝑗(1)Φ(1)𝜒𝑖𝑗+14𝜕𝜂(1)𝜒𝑗𝑘𝐷𝑖(1)𝜒𝑘𝑗+(1)𝜒𝑘𝑙𝜕𝜂𝐷[𝑖(1)𝜒𝑘]𝑙,(2)𝒢𝑖𝜂=2𝑎28(1)Φ𝐷𝑖(1)Φ2𝐷𝑖(1)Φ𝜕𝜂(1)Φ+𝐷𝑗(1)Φ𝜕𝜂(1)𝜒𝑖𝑗𝜕𝜂𝐷𝑗(1)Φ(1)𝜒𝑖𝑗14𝜕𝜂(1)𝜒𝑘𝑗𝐷𝑖(1)𝜒𝑘𝑗+(1)𝜒𝑘𝑗𝜕𝜂𝐷[𝑗(1)𝜒𝑖]𝑘,(2)𝒢𝑖𝑗=2𝑎23𝐷𝑘(1)Φ𝐷𝑘(1)Φ4(1)Φ(Δ+𝐾)(1)Φ𝜕𝜂(1)Φ𝜕𝜂(1)Φ8(1)Φ𝜕𝜂(1)Φ42𝜕𝜂+2(1)Φ2𝛾𝑖𝑗+2𝐷𝑖(1)Φ𝐷𝑗(1)Φ+4(1)Φ𝐷𝑖𝐷𝑗(1)Φ+(1)𝜒𝑗𝑖𝜕2𝜂+2𝜕𝜂(1)Φ+𝐷𝑘(1)Φ𝐷𝑖(1)𝜒𝑗𝑘+𝐷𝑗(1)𝜒𝑖𝑘2𝐷𝑘(1)Φ𝐷𝑘(1)𝜒𝑗𝑖2(1)Φ(Δ2𝐾)(1)𝜒𝑗𝑖Δ(1)Φ(1)𝜒𝑗𝑖+𝐷𝑘𝐷𝑖(1)Φ(1)𝜒𝑗𝑘+𝐷𝑚𝐷𝑗(1)Φ(1)𝜒𝑖𝑚𝐷𝑙𝐷𝑘(1)Φ(1)𝜒𝑙𝑘𝛾𝑖𝑗1