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

Authors: Kouji Nakamura
Journal: Advances in Astronomy
 (2010)

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## 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.


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## Body

## 1. Introduction

The general relativistic cosmologicallinear perturbation theory has been developed to a high degree of sophistication during the last 30 years [1–3]. 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 [6–21] 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 Section2, 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 [22–26, 31–39]. 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 manifoldℳ. 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:ℳ↦ℝ, 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 functionf is an approximated form of f(q) at q∈ℳ in terms of the variables at p∈ℳ, 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:ℳ↦ℳ, 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 diffeomorphismΦλ 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 (1)f(q)=f(Φλ(p))=:(Φλ*f)(p)=f(p)+∂∂λ(Φλ*f)|pλ+12∂2∂λ2(Φλ*f)|pλ2+O(λ3).
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 ξ1a and ξ2a such that the expansion (1) is given by (2)f(q)=(Φλ*f)(p)=f(p)+(£ξ1f)|pλ+12(£ξ2+£ξ12)f|pλ2+O(λ3),
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 ξ1a and ξ2a. These generators of the one-parameter family of diffeomorphisms Φλ represent the direction along which the Taylor expansion is carried out. The generator ξ1a is the first-order approximation of the flow of the diffeomorphism Φλ, and the generator ξ2a is the second-order correction to this flow. We should regard the generators ξ1a and ξ2a 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 f→Q.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, (3)Φσ∘Φλ≠Φσ+λ,Φλ-1≠Φ-λ.
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∈ℳ in (2) are in the neighborhood of each other, we can apply a coordinate system ℳ↦ℝn (n=dimℳ), 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 ℳ in terms of this coordinate system in ℝn 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 (4)xμ(q)=(Φλ*xμ)(p)=xμ(p)+λξ1(p)+12λ2(ξ2+ξ1ν∂νξ1μ)|p+O(λ3).
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)λ2ξ1ν(p)∂νξ1μ(p). This second-order correction corresponds to that comeing from the exponential map which is generated by the vector field ξ1μ. However, this correction completely determined by the vector field ξ1μ. Even if we add this correction that comes from the exponential map, there is no guarantee that the corrected vector λξ1(p)+(1/2)λ2ξ1ν(p)∂νξ1μ(p) does point to xμ(q) in the sense of the second order. Thus, we have to add the new correction (1/2)λ2ξ2ν(p) of the second order, in general.(a) The second termλξ1(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(ξ2μ+ξ1ν(p)∂νξ1μ(p)).
(a)(b)Of course, without this correction(1/2)λ2ξ2ν(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 ξ1μ 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)λ2ξ2ν(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 ℳ. 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⊂ℳ to an open set ψα(Oα)⊂ℝn (n=dimℳ). 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⊂ℳ, we have another gauge choice ψβ:Oβ↦ψβ(Oβ)⊂ℝn for Oβ. If these two open sets Oα and Oβ have the intersection Oα∩Oβ≠∅, we can consider the diffeomorphism ψβ∘ψα-1. This diffeomorphism ψβ∘ψα-1 is just a coordinate transformation: ψα(Oα∩Oβ)⊂ℝn↦ψβ(Oα∩Oβ)⊂ℝn, 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 manifoldℳ to an open set of ℝn. 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 spacetimeℳ. This physical spacetime ℳ is our nature itself and we want to describe the properties of this physical spacetime ℳ through perturbations. The other is the background spacetime ℳ0. 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 ℳ and ℳ0 are distinct. Let us denote the physical spacetime by (ℳ,g̅ab) and the background spacetime by (ℳ0,gab), where g̅ab is the metric on the physical spacetime manifold, ℳ, and gab is the metric on the background spacetime manifold, ℳ0. Further, we formally denote the spacetime metric and the other physical tensor fields on ℳ by Q and its background value on ℳ0 by Q0.Second, in any perturbation theories, we always write equations for the perturbation of the physical variableQ in the form (5)Q(“p”)=Q0(p)+δQ(p).
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, ℳ and ℳ0, 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 ℳ, while the variables on the right-hand side of (5) are variables on ℳ0. 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 ℳ, and “p”∈ℳ. 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 ℳ0, and p∈ℳ0. 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 [33–35].It is important to note that the second-kind gauge choice between points onℳ0 and ℳ, 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 𝒳:ℳ0↦ℳ. 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 ℳ0 and ℳ. 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”∈ℳ with q∈ℳ0 (q≠p) instead of p∈ℳ0. 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 [33–35]. Based on these ideas, higher-order perturbation theory has been developed in [22–28, 31, 38, 39, 41].


### 2.3. Formulation of Perturbation Theory

To formulate the above understanding in more detail, we introduce an infinitesimal parameterλ for the perturbation. Further, we consider the 4+1-dimensional manifold 𝒩=ℳ×ℝ, where 4=dimℳ and λ∈ℝ. The background spacetime ℳ0=𝒩|λ=0 and the physical spacetime ℳ=ℳλ=𝒩|ℝ=λ are also submanifolds embedded in the extended manifold 𝒩. Each point on 𝒩 is identified by a pair (p,λ), where p∈ℳλ, and each point in ℳ0⊂𝒩 is identified by λ=0.Through this construction, the manifold𝒩 is foliated by four-dimensional submanifolds ℳλ of each λ, and these are diffeomorphic to ℳ and ℳ0. The manifold 𝒩 has a natural differentiable structure consisting of the direct product of ℳ and ℝ. Further, the perturbed spacetimes ℳλ 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 ℳ and ℳ0 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(6)ℰ[Qλ]=0
on the physical spacetime ℳλ for the physical variables Qλ on ℳλ. The field equation (6) formally represents the Einstein equation for the metric on ℳλ and the equations for matter fields on ℳλ. If a tensor field Qλ is given on each ℳλ, Qλ is automatically extended to a tensor field on 𝒩 by Q(p,λ):=Qλ(p), where p∈ℳλ. 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 ℳλ to those on the extended manifold 𝒩.Tensor fields on𝒩 obtained through the above construction are necessarily “tangent” to each ℳλ. To consider the basis of the tangent space of 𝒩, we introduce the normal form and its dual, which are normal to each ℳλ 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 ℳλ through the above construction. The set consisting of (dλ)a, (∂/∂λ)a, and the basis of the tangent space on each ℳλ is regarded as the basis of the tangent space of 𝒩.Now, we define the perturbation of an arbitrary tensor fieldQ. We compare Q on ℳλ with Q0 on ℳ0, and it is necessary to identify the points of ℳλ with those of ℳ0 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 𝒳λ:𝒩→𝒩 such that 𝒳λ:ℳ0→ℳλ. 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 ηa𝒳. This generator ηa𝒳 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 ℳλ.The pull-back𝒳λ*Q, which is induced by the exponential map 𝒳λ, maps a tensor field Q on the physical manifold ℳλ to a tensor field 𝒳λ*Q on the background spacetime. In terms of this generator 𝒳ηa, the pull-back 𝒳λ*Q is represented by the Taylor expansion (7)Q(r)=Q(𝒳λ(p))=𝒳λ*Q(p)=Q(p)+λ£𝒳ηQ|p+12λ2£𝒳η2Q|p+O(λ3),
where r=𝒳λ(p)∈ℳλ. Because p∈ℳ0, we may regard the equation (8)𝒳λ*Q(p)=Q0(p)+λ£𝒳ηQ|ℳ0(p)+12λ2£𝒳η2Q|ℳ0(p)+O(λ3)
as an equation on the background spacetime ℳ0, where Q0=Q|ℳ0 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 (9)𝒳λ*Qλ|ℳ0=Q0+λ𝒳(1)Q+12λ2𝒳(2)Q+O(λ3)
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 (10)𝒳(1)Q:=£𝒳ηQ|ℳ0,𝒳(2)Q:=£𝒳η2Q|ℳ0.
We note that all variables in (9) are defined on ℳ0.Now, we consider twodifferent 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 ℳλ everywhere on 𝒩, the integral curves of these vector fields intersect with each ℳλ. 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 ℳλ of 𝒳ηa and 𝒴ηa are different, these point identification maps 𝒳λ and 𝒴λ are regarded as two different gauge choices.Figure 2
The second-kind gauge is a point-identification between the physical spacetimeℳλ and the background spacetime ℳ0 on the extended manifold 𝒩. Through (5), we implicitly assume the existence of a point-identification map between ℳλ and ℳ0. 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 p∈ℳ0 and “p” ∈ℳλ is same (𝒳λ). We may also choose the gauge so that q∈ℳ0 and “p” ∈ℳλ is same (𝒴λ). These are different gauge choices. The gauge transformation 𝒳λ→𝒴λ is given by the diffeomorphism Φ=𝒳λ-1∘𝒴λ.We next introduce the concept ofgauge 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, 𝒳λ*Q and 𝒴λ*Q, for any given value of λ. In particular, on ℳ0, 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 ℳλ to ℳ0 by the two different gauge choices (11)𝒳Qλ:=𝒳λ*Q|ℳ0=Q0+λ𝒳(1)Q+12λ2𝒳(2)Q+O(λ3),(12)𝒴Qλ:=𝒴λ*Q|ℳ0=Q0+λ𝒴(1)Q+12λ2𝒴(2)Q+O(λ3).
Here, we have used (9). Because 𝒳λ and 𝒴λ are gauge choices which map from ℳ0 to ℳλ, 𝒳Qλ and 𝒴Qλ are the different representations on ℳ0 in the two different gauges of the same perturbed tensor field Q on ℳλ. 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: (13)𝒳(k)Q=𝒴(k)Q.Now, we consider thegauge transformation rules between different gauge choices. In general, the representation 𝒳Qλ on ℳ0 of the perturbed variable Q on ℳλ depends on the gauge choice 𝒳λ. If we employ a different gauge choice, the representation of Qλ on ℳ0 may change. Suppose that 𝒳λ and 𝒴λ are different gauge choices, which are the point identification maps from ℳ0 to ℳλ, 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 (14)Φλ:=(𝒳λ)-1∘𝒴λ.
This diffeomorphism Φλ is the map Φλ:ℳ0→ℳ0 for each value of λ∈ℝ. 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 Φλ:𝒳λ→𝒴λ.The gauge transformationΦλ 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 ℳ0, are connected by the linear map Φλ* as (15)𝒴Qλ=𝒴λ*Q|ℳ0=(𝒴λ*(𝒳λ𝒳λ-1)*Q)|ℳ0=(𝒳λ-1𝒴λ)*(𝒳λ*Q)|ℳ0=Φλ*𝒳Qλ.
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 Φλ*𝒳Qλ in the form (16)Φλ*𝒳Q=𝒳Q+λ£ξ1𝒳Q+λ22{£ξ2+£ξ12}Q𝒳+O(λ3),
where the vector fields ξ1a and ξ2a are the generators of the gauge transformation Φλ (see (2)).Comparing the representation (16) of the Taylor expansion in terms of the generators ξ1a and ξ2a of the pull-back Φλ*𝒳Q and that in terms of the generators 𝒳ηa and 𝒴ηa of the pull-back 𝒴λ*∘(𝒳λ-1)*Q𝒳 (=Φλ*𝒳Q), we readily obtain explicit expressions for the generators ξ1a and ξ2a of the gauge transformation Φ=𝒳λ-1∘𝒴λ in terms of the generators 𝒳ηa and 𝒴ηa of each gauge choices as follows: (17)ξ1a=𝒴ηa-𝒳ηa,ξ2a=[𝒴η,𝒳η]a.
Further, because the gauge transformation Φλ is a map within the background spacetime ℳ0, the generator should consist of vector fields on ℳ0. 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):(18)𝒴(1)Q-𝒳(1)Q=£ξ1Q0,(19)𝒴(2)Q-𝒳(2)Q=2£ξ1𝒳(1)Q+{£ξ2+£ξ12}Q0.Here, we should comment on the gauge choice in the above explanation. We have introduced an exponential map𝒳λ (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 𝒳λ*Q(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 ξ1a and ξ2a 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 ℳλ is our nature itself, while there is no background spacetime ℳ0 in our nature. The background spacetime ℳ0 is a fictitious spacetime and it has nothing to do with our nature. Since the gauge choice 𝒳λ just gives a relation between ℳλ and ℳ0, 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 ℳλ through the physical processes on the physical spacetime ℳλ. 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 𝒳λ→𝒴λ. 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 ℳλ.


### 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 spacetimeℳλ 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, 25–28] 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” ℳ0, where Oα are open sets on the background spacetime and ψα are diffeomorphisms from Oα to ℝ4 (4=dimℳ0). The coordinate system {Oα,ψα} is the set of the collection of the pair of open sets Oα and diffeomorphism Oα↦ℝ4. If we employ a gauge choice 𝒳λ, we have the correspondence of ℳλ and ℳ0. Together with the coordinate system ψα on ℳ0, this correspondence between ℳλ and ℳ0 induces the coordinate system on ℳλ. Actually, Xλ(Oα) for each α is an open set of ℳλ. Then, ψα∘𝒳λ-1 becomes a diffeomorphism from an open set Xλ(Oα)⊂ℳλ to ℝ4. This diffeomorphism ψα∘𝒳λ-1 induces a coordinate system of an open set on ℳλ.When we have two different gauge choices𝒳λ and 𝒴λ, ψα∘𝒳λ-1 and ψα∘𝒴λ-1 become different coordinate systems on ℳλ. We can also consider the coordinate transformation from the coordinate system ψα∘𝒳λ-1 to another coordinate system ψα∘𝒴λ-1. Since the gauge transformation 𝒳λ→𝒴λ is induced by the diffeomorphism Φλ defined by (14), the induced coordinate transformation is given by (20)yμ(q):=xμ(p)=((Φ-1)*xμ)(q)
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 (21)yμ(q)=xμ(q)-λξ1μ(q)+λ22{-ξ2μ(q)+ξ1ν(q)∂νξ1μ(q)}+O(λ3).


### 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 spacetimeℳλ, which is pulled back to the background spacetime ℳ0 using a gauge choice in the form given in (9): (22)𝒳λ*g̅ab=gab+λ𝒳hab+λ22lab𝒳+O3(λ),
where gab is the metric on ℳ0. 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 thatwe already know the procedure for finding gauge-invariant variables for the linear metric perturbations. Then, a linear metric perturbation hab is decomposed as (23)hab=:ℋab+£Xgab,
where ℋab 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 (24)𝒴ℋab-𝒳ℋab=0,𝒴Xa-𝒳Xa=ξ1a.
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, (25)𝒴(1)hab-𝒳(1)hab=£ξ1gab.As emphasized in our series of papers [22, 23, 25–28], 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 (ℳ0,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 (26)𝒴(2)lab-𝒳(2)lab=2£ξ1𝒳hab+{£ξ2+£ξ12}gab
under the gauge transformation Φλ=(𝒳λ)-1∘𝒴λ:𝒳λ→𝒴λ. Although this gauge transformation rule is slightly complicated, inspecting this gauge transformation rule, we first introduce the variable L̂ab defined by (27)L̂ab:=lab-2£Xhab+£X2gab.
Under the gauge transformation Φλ=(𝒳λ)-1∘𝒴λ:𝒳λ→𝒴λ, the variable L̂ab is transformed as (28)𝒴L̂ab-𝒳L̂ab=£σgab,(29)σa:=ξ2a+[ξ1,X]a.
The gauge transformation rule (28) is identical to that for a linear metric perturbation. Therefore, we may apply the above procedure to decompose hab into ℋab and Xa when we decompose of the components of the variable L̂ab. Then, L̂ab can be decomposed as (30)L̂ab=ℒab+£Ygab,
where ℒab is the gauge-invariant part of the variable L̂ab, or equivalently, of the second-order metric perturbation lab, and Ya is the gauge variant part of L̂ab, that is, the gauge variant part of lab. Under the gauge transformation Φλ=(𝒳λ)-1∘𝒴λ, the variables ℒab and Ya are transformed as (31)𝒴ℒab-𝒳ℒab=0,𝒴Ya-𝒴Ya=σa,
respectively. Thus, once we accept the assumption (23), the second-order metric perturbations are decomposed as (32)lab=:ℒab+2£Xhab+(£Y-£X2)gab,
where ℒab and Ya are the gauge-invariant and variant parts of the second order metric perturbations, that is, (33)𝒴ℒab-𝒳ℒab=0,𝒴Ya-𝒳Ya=ξ2a+[ξ1,X]a.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 (34)(1)𝒬:=(1)Q-£XQ0,(35)(2)𝒬:=(2)Q-2£X(1)Q-{£Y-£X2}Q0.
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 (36)Q(1)=𝒬(1)+£XQ0,(37)(2)Q=𝒬(2)+2£X(1)Q+{£Y-£X2}Q0.
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.


## 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 manifoldℳ. 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:ℳ↦ℝ, 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 functionf is an approximated form of f(q) at q∈ℳ in terms of the variables at p∈ℳ, 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:ℳ↦ℳ, 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 diffeomorphismΦλ 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 (1)f(q)=f(Φλ(p))=:(Φλ*f)(p)=f(p)+∂∂λ(Φλ*f)|pλ+12∂2∂λ2(Φλ*f)|pλ2+O(λ3).
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 ξ1a and ξ2a such that the expansion (1) is given by (2)f(q)=(Φλ*f)(p)=f(p)+(£ξ1f)|pλ+12(£ξ2+£ξ12)f|pλ2+O(λ3),
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 ξ1a and ξ2a. These generators of the one-parameter family of diffeomorphisms Φλ represent the direction along which the Taylor expansion is carried out. The generator ξ1a is the first-order approximation of the flow of the diffeomorphism Φλ, and the generator ξ2a is the second-order correction to this flow. We should regard the generators ξ1a and ξ2a 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 f→Q.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, (3)Φσ∘Φλ≠Φσ+λ,Φλ-1≠Φ-λ.
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∈ℳ in (2) are in the neighborhood of each other, we can apply a coordinate system ℳ↦ℝn (n=dimℳ), 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 ℳ in terms of this coordinate system in ℝn 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 (4)xμ(q)=(Φλ*xμ)(p)=xμ(p)+λξ1(p)+12λ2(ξ2+ξ1ν∂νξ1μ)|p+O(λ3).
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)λ2ξ1ν(p)∂νξ1μ(p). This second-order correction corresponds to that comeing from the exponential map which is generated by the vector field ξ1μ. However, this correction completely determined by the vector field ξ1μ. Even if we add this correction that comes from the exponential map, there is no guarantee that the corrected vector λξ1(p)+(1/2)λ2ξ1ν(p)∂νξ1μ(p) does point to xμ(q) in the sense of the second order. Thus, we have to add the new correction (1/2)λ2ξ2ν(p) of the second order, in general.(a) The second termλξ1(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(ξ2μ+ξ1ν(p)∂νξ1μ(p)).
(a)(b)Of course, without this correction(1/2)λ2ξ2ν(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 ξ1μ 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)λ2ξ2ν(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 ℳ. 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⊂ℳ to an open set ψα(Oα)⊂ℝn (n=dimℳ). 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⊂ℳ, we have another gauge choice ψβ:Oβ↦ψβ(Oβ)⊂ℝn for Oβ. If these two open sets Oα and Oβ have the intersection Oα∩Oβ≠∅, we can consider the diffeomorphism ψβ∘ψα-1. This diffeomorphism ψβ∘ψα-1 is just a coordinate transformation: ψα(Oα∩Oβ)⊂ℝn↦ψβ(Oα∩Oβ)⊂ℝn, 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 manifoldℳ to an open set of ℝn. 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 spacetimeℳ. This physical spacetime ℳ is our nature itself and we want to describe the properties of this physical spacetime ℳ through perturbations. The other is the background spacetime ℳ0. 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 ℳ and ℳ0 are distinct. Let us denote the physical spacetime by (ℳ,g̅ab) and the background spacetime by (ℳ0,gab), where g̅ab is the metric on the physical spacetime manifold, ℳ, and gab is the metric on the background spacetime manifold, ℳ0. Further, we formally denote the spacetime metric and the other physical tensor fields on ℳ by Q and its background value on ℳ0 by Q0.Second, in any perturbation theories, we always write equations for the perturbation of the physical variableQ in the form (5)Q(“p”)=Q0(p)+δQ(p).
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, ℳ and ℳ0, 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 ℳ, while the variables on the right-hand side of (5) are variables on ℳ0. 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 ℳ, and “p”∈ℳ. 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 ℳ0, and p∈ℳ0. 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 [33–35].It is important to note that the second-kind gauge choice between points onℳ0 and ℳ, 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 𝒳:ℳ0↦ℳ. 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 ℳ0 and ℳ. 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”∈ℳ with q∈ℳ0 (q≠p) instead of p∈ℳ0. 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 [33–35]. Based on these ideas, higher-order perturbation theory has been developed in [22–28, 31, 38, 39, 41].


## 2.2.1. First-Kind Gauge

The first-kind gauge is a coordinate system on a single manifold ℳ. 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⊂ℳ to an open set ψα(Oα)⊂ℝn (n=dimℳ). 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⊂ℳ, we have another gauge choice ψβ:Oβ↦ψβ(Oβ)⊂ℝn for Oβ. If these two open sets Oα and Oβ have the intersection Oα∩Oβ≠∅, we can consider the diffeomorphism ψβ∘ψα-1. This diffeomorphism ψβ∘ψα-1 is just a coordinate transformation: ψα(Oα∩Oβ)⊂ℝn↦ψβ(Oα∩Oβ)⊂ℝn, 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 manifoldℳ to an open set of ℝn. 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 spacetimeℳ. This physical spacetime ℳ is our nature itself and we want to describe the properties of this physical spacetime ℳ through perturbations. The other is the background spacetime ℳ0. 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 ℳ and ℳ0 are distinct. Let us denote the physical spacetime by (ℳ,g̅ab) and the background spacetime by (ℳ0,gab), where g̅ab is the metric on the physical spacetime manifold, ℳ, and gab is the metric on the background spacetime manifold, ℳ0. Further, we formally denote the spacetime metric and the other physical tensor fields on ℳ by Q and its background value on ℳ0 by Q0.Second, in any perturbation theories, we always write equations for the perturbation of the physical variableQ in the form (5)Q(“p”)=Q0(p)+δQ(p).
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, ℳ and ℳ0, 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 ℳ, while the variables on the right-hand side of (5) are variables on ℳ0. 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 ℳ, and “p”∈ℳ. 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 ℳ0, and p∈ℳ0. 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 [33–35].It is important to note that the second-kind gauge choice between points onℳ0 and ℳ, 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 𝒳:ℳ0↦ℳ. 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 ℳ0 and ℳ. 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”∈ℳ with q∈ℳ0 (q≠p) instead of p∈ℳ0. 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 [33–35]. Based on these ideas, higher-order perturbation theory has been developed in [22–28, 31, 38, 39, 41].


## 2.3. Formulation of Perturbation Theory

To formulate the above understanding in more detail, we introduce an infinitesimal parameterλ for the perturbation. Further, we consider the 4+1-dimensional manifold 𝒩=ℳ×ℝ, where 4=dimℳ and λ∈ℝ. The background spacetime ℳ0=𝒩|λ=0 and the physical spacetime ℳ=ℳλ=𝒩|ℝ=λ are also submanifolds embedded in the extended manifold 𝒩. Each point on 𝒩 is identified by a pair (p,λ), where p∈ℳλ, and each point in ℳ0⊂𝒩 is identified by λ=0.Through this construction, the manifold𝒩 is foliated by four-dimensional submanifolds ℳλ of each λ, and these are diffeomorphic to ℳ and ℳ0. The manifold 𝒩 has a natural differentiable structure consisting of the direct product of ℳ and ℝ. Further, the perturbed spacetimes ℳλ 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 ℳ and ℳ0 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(6)ℰ[Qλ]=0
on the physical spacetime ℳλ for the physical variables Qλ on ℳλ. The field equation (6) formally represents the Einstein equation for the metric on ℳλ and the equations for matter fields on ℳλ. If a tensor field Qλ is given on each ℳλ, Qλ is automatically extended to a tensor field on 𝒩 by Q(p,λ):=Qλ(p), where p∈ℳλ. 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 ℳλ to those on the extended manifold 𝒩.Tensor fields on𝒩 obtained through the above construction are necessarily “tangent” to each ℳλ. To consider the basis of the tangent space of 𝒩, we introduce the normal form and its dual, which are normal to each ℳλ 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 ℳλ through the above construction. The set consisting of (dλ)a, (∂/∂λ)a, and the basis of the tangent space on each ℳλ is regarded as the basis of the tangent space of 𝒩.Now, we define the perturbation of an arbitrary tensor fieldQ. We compare Q on ℳλ with Q0 on ℳ0, and it is necessary to identify the points of ℳλ with those of ℳ0 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 𝒳λ:𝒩→𝒩 such that 𝒳λ:ℳ0→ℳλ. 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 ηa𝒳. This generator ηa𝒳 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 ℳλ.The pull-back𝒳λ*Q, which is induced by the exponential map 𝒳λ, maps a tensor field Q on the physical manifold ℳλ to a tensor field 𝒳λ*Q on the background spacetime. In terms of this generator 𝒳ηa, the pull-back 𝒳λ*Q is represented by the Taylor expansion (7)Q(r)=Q(𝒳λ(p))=𝒳λ*Q(p)=Q(p)+λ£𝒳ηQ|p+12λ2£𝒳η2Q|p+O(λ3),
where r=𝒳λ(p)∈ℳλ. Because p∈ℳ0, we may regard the equation (8)𝒳λ*Q(p)=Q0(p)+λ£𝒳ηQ|ℳ0(p)+12λ2£𝒳η2Q|ℳ0(p)+O(λ3)
as an equation on the background spacetime ℳ0, where Q0=Q|ℳ0 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 (9)𝒳λ*Qλ|ℳ0=Q0+λ𝒳(1)Q+12λ2𝒳(2)Q+O(λ3)
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 (10)𝒳(1)Q:=£𝒳ηQ|ℳ0,𝒳(2)Q:=£𝒳η2Q|ℳ0.
We note that all variables in (9) are defined on ℳ0.Now, we consider twodifferent 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 ℳλ everywhere on 𝒩, the integral curves of these vector fields intersect with each ℳλ. 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 ℳλ of 𝒳ηa and 𝒴ηa are different, these point identification maps 𝒳λ and 𝒴λ are regarded as two different gauge choices.Figure 2
The second-kind gauge is a point-identification between the physical spacetimeℳλ and the background spacetime ℳ0 on the extended manifold 𝒩. Through (5), we implicitly assume the existence of a point-identification map between ℳλ and ℳ0. 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 p∈ℳ0 and “p” ∈ℳλ is same (𝒳λ). We may also choose the gauge so that q∈ℳ0 and “p” ∈ℳλ is same (𝒴λ). These are different gauge choices. The gauge transformation 𝒳λ→𝒴λ is given by the diffeomorphism Φ=𝒳λ-1∘𝒴λ.We next introduce the concept ofgauge 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, 𝒳λ*Q and 𝒴λ*Q, for any given value of λ. In particular, on ℳ0, 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 ℳλ to ℳ0 by the two different gauge choices (11)𝒳Qλ:=𝒳λ*Q|ℳ0=Q0+λ𝒳(1)Q+12λ2𝒳(2)Q+O(λ3),(12)𝒴Qλ:=𝒴λ*Q|ℳ0=Q0+λ𝒴(1)Q+12λ2𝒴(2)Q+O(λ3).
Here, we have used (9). Because 𝒳λ and 𝒴λ are gauge choices which map from ℳ0 to ℳλ, 𝒳Qλ and 𝒴Qλ are the different representations on ℳ0 in the two different gauges of the same perturbed tensor field Q on ℳλ. 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: (13)𝒳(k)Q=𝒴(k)Q.Now, we consider thegauge transformation rules between different gauge choices. In general, the representation 𝒳Qλ on ℳ0 of the perturbed variable Q on ℳλ depends on the gauge choice 𝒳λ. If we employ a different gauge choice, the representation of Qλ on ℳ0 may change. Suppose that 𝒳λ and 𝒴λ are different gauge choices, which are the point identification maps from ℳ0 to ℳλ, 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 (14)Φλ:=(𝒳λ)-1∘𝒴λ.
This diffeomorphism Φλ is the map Φλ:ℳ0→ℳ0 for each value of λ∈ℝ. 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 Φλ:𝒳λ→𝒴λ.The gauge transformationΦλ 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 ℳ0, are connected by the linear map Φλ* as (15)𝒴Qλ=𝒴λ*Q|ℳ0=(𝒴λ*(𝒳λ𝒳λ-1)*Q)|ℳ0=(𝒳λ-1𝒴λ)*(𝒳λ*Q)|ℳ0=Φλ*𝒳Qλ.
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 Φλ*𝒳Qλ in the form (16)Φλ*𝒳Q=𝒳Q+λ£ξ1𝒳Q+λ22{£ξ2+£ξ12}Q𝒳+O(λ3),
where the vector fields ξ1a and ξ2a are the generators of the gauge transformation Φλ (see (2)).Comparing the representation (16) of the Taylor expansion in terms of the generators ξ1a and ξ2a of the pull-back Φλ*𝒳Q and that in terms of the generators 𝒳ηa and 𝒴ηa of the pull-back 𝒴λ*∘(𝒳λ-1)*Q𝒳 (=Φλ*𝒳Q), we readily obtain explicit expressions for the generators ξ1a and ξ2a of the gauge transformation Φ=𝒳λ-1∘𝒴λ in terms of the generators 𝒳ηa and 𝒴ηa of each gauge choices as follows: (17)ξ1a=𝒴ηa-𝒳ηa,ξ2a=[𝒴η,𝒳η]a.
Further, because the gauge transformation Φλ is a map within the background spacetime ℳ0, the generator should consist of vector fields on ℳ0. 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):(18)𝒴(1)Q-𝒳(1)Q=£ξ1Q0,(19)𝒴(2)Q-𝒳(2)Q=2£ξ1𝒳(1)Q+{£ξ2+£ξ12}Q0.Here, we should comment on the gauge choice in the above explanation. We have introduced an exponential map𝒳λ (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 𝒳λ*Q(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 ξ1a and ξ2a 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 ℳλ is our nature itself, while there is no background spacetime ℳ0 in our nature. The background spacetime ℳ0 is a fictitious spacetime and it has nothing to do with our nature. Since the gauge choice 𝒳λ just gives a relation between ℳλ and ℳ0, 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 ℳλ through the physical processes on the physical spacetime ℳλ. 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 𝒳λ→𝒴λ. 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 ℳλ.


## 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 spacetimeℳλ 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, 25–28] 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” ℳ0, where Oα are open sets on the background spacetime and ψα are diffeomorphisms from Oα to ℝ4 (4=dimℳ0). The coordinate system {Oα,ψα} is the set of the collection of the pair of open sets Oα and diffeomorphism Oα↦ℝ4. If we employ a gauge choice 𝒳λ, we have the correspondence of ℳλ and ℳ0. Together with the coordinate system ψα on ℳ0, this correspondence between ℳλ and ℳ0 induces the coordinate system on ℳλ. Actually, Xλ(Oα) for each α is an open set of ℳλ. Then, ψα∘𝒳λ-1 becomes a diffeomorphism from an open set Xλ(Oα)⊂ℳλ to ℝ4. This diffeomorphism ψα∘𝒳λ-1 induces a coordinate system of an open set on ℳλ.When we have two different gauge choices𝒳λ and 𝒴λ, ψα∘𝒳λ-1 and ψα∘𝒴λ-1 become different coordinate systems on ℳλ. We can also consider the coordinate transformation from the coordinate system ψα∘𝒳λ-1 to another coordinate system ψα∘𝒴λ-1. Since the gauge transformation 𝒳λ→𝒴λ is induced by the diffeomorphism Φλ defined by (14), the induced coordinate transformation is given by (20)yμ(q):=xμ(p)=((Φ-1)*xμ)(q)
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 (21)yμ(q)=xμ(q)-λξ1μ(q)+λ22{-ξ2μ(q)+ξ1ν(q)∂νξ1μ(q)}+O(λ3).


## 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 spacetimeℳλ, which is pulled back to the background spacetime ℳ0 using a gauge choice in the form given in (9): (22)𝒳λ*g̅ab=gab+λ𝒳hab+λ22lab𝒳+O3(λ),
where gab is the metric on ℳ0. 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 thatwe already know the procedure for finding gauge-invariant variables for the linear metric perturbations. Then, a linear metric perturbation hab is decomposed as (23)hab=:ℋab+£Xgab,
where ℋab 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 (24)𝒴ℋab-𝒳ℋab=0,𝒴Xa-𝒳Xa=ξ1a.
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, (25)𝒴(1)hab-𝒳(1)hab=£ξ1gab.As emphasized in our series of papers [22, 23, 25–28], 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 (ℳ0,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 (26)𝒴(2)lab-𝒳(2)lab=2£ξ1𝒳hab+{£ξ2+£ξ12}gab
under the gauge transformation Φλ=(𝒳λ)-1∘𝒴λ:𝒳λ→𝒴λ. Although this gauge transformation rule is slightly complicated, inspecting this gauge transformation rule, we first introduce the variable L̂ab defined by (27)L̂ab:=lab-2£Xhab+£X2gab.
Under the gauge transformation Φλ=(𝒳λ)-1∘𝒴λ:𝒳λ→𝒴λ, the variable L̂ab is transformed as (28)𝒴L̂ab-𝒳L̂ab=£σgab,(29)σa:=ξ2a+[ξ1,X]a.
The gauge transformation rule (28) is identical to that for a linear metric perturbation. Therefore, we may apply the above procedure to decompose hab into ℋab and Xa when we decompose of the components of the variable L̂ab. Then, L̂ab can be decomposed as (30)L̂ab=ℒab+£Ygab,
where ℒab is the gauge-invariant part of the variable L̂ab, or equivalently, of the second-order metric perturbation lab, and Ya is the gauge variant part of L̂ab, that is, the gauge variant part of lab. Under the gauge transformation Φλ=(𝒳λ)-1∘𝒴λ, the variables ℒab and Ya are transformed as (31)𝒴ℒab-𝒳ℒab=0,𝒴Ya-𝒴Ya=σa,
respectively. Thus, once we accept the assumption (23), the second-order metric perturbations are decomposed as (32)lab=:ℒab+2£Xhab+(£Y-£X2)gab,
where ℒab and Ya are the gauge-invariant and variant parts of the second order metric perturbations, that is, (33)𝒴ℒab-𝒳ℒab=0,𝒴Ya-𝒳Ya=ξ2a+[ξ1,X]a.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 (34)(1)𝒬:=(1)Q-£XQ0,(35)(2)𝒬:=(2)Q-2£X(1)Q-{£Y-£X2}Q0.
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 (36)Q(1)=𝒬(1)+£XQ0,(37)(2)Q=𝒬(2)+2£X(1)Q+{£Y-£X2}Q0.
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 spacetimeℳλ and the background spacetime ℳ0 is given by the relation between the pulled-back operator 𝒳λ*∇̅a(𝒳λ-1)* of the covariant derivative ∇̅a associated with the metric g̅ab on ℳλ and the covariant derivative ∇a associated with the metric gab on ℳ0. The pulled-back covariant derivative 𝒳λ*∇̅a(𝒳λ-1)* depends on the gauge choice 𝒳λ. The property of the derivative operator 𝒳λ*∇̅a(𝒳λ-1)* as the covariant derivative on ℳλ is given by (38)𝒳λ*∇̅a((𝒳λ-1)*𝒳λ*g̅ab)=0,
where 𝒳λ*g̅ab is the pull-back of the metric on ℳλ, which is expanded as (22). In spite of the gauge dependence of the operator 𝒳λ*∇̅a(𝒳λ-1)*, we simply denote this operator by ∇̅a, because our calculations are carried out only on ℳ0 in the same gauge choice 𝒳λ. Further, we denote the pulled-back metric 𝒳λ*g̅ab on ℳλ by g̅ab, as mentioned above.Since the derivative operator∇̅a (=𝒳*∇̅a(𝒳-1)*) may be regarded as a derivative operator on ℳ0 that satisfies the property (38), there exists a tensor field Ccab on ℳ0 such that (39)∇̅aωb=∇aωb-Ccabωc,
where ωa is an arbitrary one-form on ℳ0. From the property (38) of the covariant derivative operator ∇̅a on ℳλ, the tensor field Ccab on ℳ0 is given by (40)Ccab=12g̅cd(∇ag̅db+∇bg̅da-∇dg̅ab),
where g̅ab is the inverse of g̅ab (see Appendix B). We note that the gauge dependence of the covariant derivative ∇̅a appears only through Ccab. The Riemann curvature R̅abcd on ℳλ, which is also pulled back to ℳ0, is given by [42] (41)R̅abcd=Rabcd-2∇[aCb]cd+2Cc[aeCb]ed,
where Rabcd is the Riemann curvature on ℳ0. 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 tensorG̅ab:=R̅ab-(1/2)δabR̅ on ℳλ as (42)G̅ab=Gab+λ(1)G̅ab+12λ2(2)Gab+O(λ3).
As shown in Appendix B, each order perturbation of the Einstein tensor is given by(43)(1)Gab=(1)𝒢ab[ℋ]+£𝒳Gab,(44)(2)Gab=(1)𝒢ab[ℒ]+(2)𝒢ab[ℋ,ℋ]+2£X(1)G̅ab+{£Y-£X2}Gab,
where  (45)(1)𝒢ab[A]:=Σ(1)ab[A]-12δab(1)Σcc[A],Σ(1)ab[A]:=-2∇Hd][abd[A]-AcbRac,(46)(2)𝒢ab[A,B]:=Σ(2)ab[A,B]-12δab(2)Σcc[A,B],(47)(2)Σab[A,B]:=2RadBc(bAd)c+2H[ade[A]Hd]b[B]e+2H[ade[B]Hd]b[A]e+2Aed∇[aHd]be[B]+2Bed∇[aHd]be[A]+2Acb∇[aHd]cd[B]+2Bcb∇[aHd]cd[A],(48)Habc[A]:=∇(aAb)c-12∇cAab,(49)Habc[A]:=gcdHabd[A],Habc[A]:=gbdHadc[A],Hab[A]c:=gcdHabd[A].
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 identities(50)∇a(1)𝒢ab[A]=-Hcaa[A]Gbc+Hbac[A]Gca,(51)∇a(2)𝒢ba[A,B]=-Hcaa[A](1)𝒢bc[B]-Hcaa[B](1)𝒢bc[A]+Hbae[A](1)𝒢ea[B]+Hbae[B](1)𝒢ea[A]-(Hbad[B]Adc+Hbad[A]Bdc)Gca+(Hcad[B]Aad+Hcad[A]Bad)Gbc,
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 ∇̅bG̅ab=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 fieldφ̅ is given by (52)T̅ab=∇̅aφ̅∇̅bφ̅-12δab(∇̅cφ̅∇̅cφ̅+2V(φ̅)),
where V(φ̅) is the potential of the scalar field φ̅. We expand the scalar field φ̅ as (53)φ̅=φ+λφ̂1+12λ2φ̂2+O(λ3),
where φ is the background value of the scalar field φ̅. Further, following to the decomposition formulae (34) and (35), each order perturbation of the scalar field φ̅ is decomposed as (54)φ̂1=:φ1+£Xφ,φ̂2=:φ2+2£Xφ̂1+(£Y-£X2)φ,
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 φ̅ and the inverse metric, the energy momentum tensor (52) is also expanded as (55)T̅ab=Tab+λ(1)(Tab)+12λ2(2)(Tab)+O(λ3).
The background energy momentum tensor Tab is given by the replacement φ̅→φ 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(56)(1)(Tab)=:(1)𝒯ab+£XTab,(57)(2)(Tab)=:(2)𝒯ab+2£X(1)(Tab)+(£Y-£X2)Tab,
where the gauge-invariant parts (1)𝒯ab and (2)𝒯ab of the first and the second order are given by  (58)(1)𝒯ab:=∇aφ∇bφ1-∇aφℋbc∇cφ+∇aφ1∇bφ-δab(∇cφ∇cφ1-12∇cφℋdc∇dφ+φ1∂V∂φ),(59)(2)𝒯ab:=∇aφ∇bφ2+∇aφ2∇bφ-∇aφgbdℒdc∇cφ-2∇aφℋbc∇cφ1+2∇aφℋbdℋdc∇cφ+2∇aφ1∇bφ1-2∇aφ1ℋbc∇cφ-δab(∇cφ∇cφ2-12∇cφℒdc∇dφ+∇cφℋdeℋec∇dφ-2∇cφℋdc∇dφ1+∇cφ1∇cφ1+φ2∂V∂φ+φ12∂2V∂φ2).
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(60)C̅(K):=∇̅a∇̅aφ̅-∂V∂φ̅(φ̅)=0.
Through the perturbative expansions (53) and (22), the Klein-Gordon equation (60) is expanded as (61)C̅(K)=:C(K)+λC(K)(1)+12λ2C(K)(2)+O(λ3).C(K) is the background Klein-Gordon equation (62)C(K):=∇a∇aφ-∂V∂φ̅(φ)=0.
The first- and the second-order perturbations C(K)(1) and C(K)(2) are also decomposed into the gauge-invariant and the gauge-variant parts as  (63)C(K)(1)=:𝒞(K)(1)+£XC(K),C(K)(2)=:𝒞(K)(2)+2£XC(K)(1)+(£Y-£X2)C(K),
where
(64)𝒞(K)(1):=∇a∇aφ1-Haac[ℋ]∇cφ-ℋab∇a∇bφ-φ1∂2V∂φ̅2(φ),(65)𝒞(K)(2):=∇a∇aφ2-Haac[ℒ]∇cφ+2Haad[ℋ]ℋcd∇cφ-2Haac[ℋ]∇cφ1+2ℋabHabc[ℋ]∇cφ-ℒab∇a∇bφ+2ℋdaℋdb∇a∇bφ-2ℋab∇a∇bφ1-φ2∂2V∂φ̅2(φ)-(φ1)2∂3V∂φ̅3(φ).
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(66)𝒞(K)(1)=0,𝒞(K)(2)=0.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,(67)(1)Gab=8πGT(1)ab,(2)Gab=8πGT(2)ab.
Then, the perturbative Einstein equation is given by (68)(1)𝒢ab[ℋ]=8πG(1)𝒯ab
at linear order and (69)(1)𝒢ab[ℒ]+(2)𝒢ab[ℋ,ℋ]=8πG𝒯(2)ab
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 metricgab, 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.


## 3.1. Perturbations of the Einstein Curvature

The relation between the curvatures associated with the metrics on the physical spacetimeℳλ and the background spacetime ℳ0 is given by the relation between the pulled-back operator 𝒳λ*∇̅a(𝒳λ-1)* of the covariant derivative ∇̅a associated with the metric g̅ab on ℳλ and the covariant derivative ∇a associated with the metric gab on ℳ0. The pulled-back covariant derivative 𝒳λ*∇̅a(𝒳λ-1)* depends on the gauge choice 𝒳λ. The property of the derivative operator 𝒳λ*∇̅a(𝒳λ-1)* as the covariant derivative on ℳλ is given by (38)𝒳λ*∇̅a((𝒳λ-1)*𝒳λ*g̅ab)=0,
where 𝒳λ*g̅ab is the pull-back of the metric on ℳλ, which is expanded as (22). In spite of the gauge dependence of the operator 𝒳λ*∇̅a(𝒳λ-1)*, we simply denote this operator by ∇̅a, because our calculations are carried out only on ℳ0 in the same gauge choice 𝒳λ. Further, we denote the pulled-back metric 𝒳λ*g̅ab on ℳλ by g̅ab, as mentioned above.Since the derivative operator∇̅a (=𝒳*∇̅a(𝒳-1)*) may be regarded as a derivative operator on ℳ0 that satisfies the property (38), there exists a tensor field Ccab on ℳ0 such that (39)∇̅aωb=∇aωb-Ccabωc,
where ωa is an arbitrary one-form on ℳ0. From the property (38) of the covariant derivative operator ∇̅a on ℳλ, the tensor field Ccab on ℳ0 is given by (40)Ccab=12g̅cd(∇ag̅db+∇bg̅da-∇dg̅ab),
where g̅ab is the inverse of g̅ab (see Appendix B). We note that the gauge dependence of the covariant derivative ∇̅a appears only through Ccab. The Riemann curvature R̅abcd on ℳλ, which is also pulled back to ℳ0, is given by [42] (41)R̅abcd=Rabcd-2∇[aCb]cd+2Cc[aeCb]ed,
where Rabcd is the Riemann curvature on ℳ0. 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 tensorG̅ab:=R̅ab-(1/2)δabR̅ on ℳλ as (42)G̅ab=Gab+λ(1)G̅ab+12λ2(2)Gab+O(λ3).
As shown in Appendix B, each order perturbation of the Einstein tensor is given by(43)(1)Gab=(1)𝒢ab[ℋ]+£𝒳Gab,(44)(2)Gab=(1)𝒢ab[ℒ]+(2)𝒢ab[ℋ,ℋ]+2£X(1)G̅ab+{£Y-£X2}Gab,
where  (45)(1)𝒢ab[A]:=Σ(1)ab[A]-12δab(1)Σcc[A],Σ(1)ab[A]:=-2∇Hd][abd[A]-AcbRac,(46)(2)𝒢ab[A,B]:=Σ(2)ab[A,B]-12δab(2)Σcc[A,B],(47)(2)Σab[A,B]:=2RadBc(bAd)c+2H[ade[A]Hd]b[B]e+2H[ade[B]Hd]b[A]e+2Aed∇[aHd]be[B]+2Bed∇[aHd]be[A]+2Acb∇[aHd]cd[B]+2Bcb∇[aHd]cd[A],(48)Habc[A]:=∇(aAb)c-12∇cAab,(49)Habc[A]:=gcdHabd[A],Habc[A]:=gbdHadc[A],Hab[A]c:=gcdHabd[A].
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 identities(50)∇a(1)𝒢ab[A]=-Hcaa[A]Gbc+Hbac[A]Gca,(51)∇a(2)𝒢ba[A,B]=-Hcaa[A](1)𝒢bc[B]-Hcaa[B](1)𝒢bc[A]+Hbae[A](1)𝒢ea[B]+Hbae[B](1)𝒢ea[A]-(Hbad[B]Adc+Hbad[A]Bdc)Gca+(Hcad[B]Aad+Hcad[A]Bad)Gbc,
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 ∇̅bG̅ab=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 fieldφ̅ is given by (52)T̅ab=∇̅aφ̅∇̅bφ̅-12δab(∇̅cφ̅∇̅cφ̅+2V(φ̅)),
where V(φ̅) is the potential of the scalar field φ̅. We expand the scalar field φ̅ as (53)φ̅=φ+λφ̂1+12λ2φ̂2+O(λ3),
where φ is the background value of the scalar field φ̅. Further, following to the decomposition formulae (34) and (35), each order perturbation of the scalar field φ̅ is decomposed as (54)φ̂1=:φ1+£Xφ,φ̂2=:φ2+2£Xφ̂1+(£Y-£X2)φ,
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 φ̅ and the inverse metric, the energy momentum tensor (52) is also expanded as (55)T̅ab=Tab+λ(1)(Tab)+12λ2(2)(Tab)+O(λ3).
The background energy momentum tensor Tab is given by the replacement φ̅→φ 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(56)(1)(Tab)=:(1)𝒯ab+£XTab,(57)(2)(Tab)=:(2)𝒯ab+2£X(1)(Tab)+(£Y-£X2)Tab,
where the gauge-invariant parts (1)𝒯ab and (2)𝒯ab of the first and the second order are given by  (58)(1)𝒯ab:=∇aφ∇bφ1-∇aφℋbc∇cφ+∇aφ1∇bφ-δab(∇cφ∇cφ1-12∇cφℋdc∇dφ+φ1∂V∂φ),(59)(2)𝒯ab:=∇aφ∇bφ2+∇aφ2∇bφ-∇aφgbdℒdc∇cφ-2∇aφℋbc∇cφ1+2∇aφℋbdℋdc∇cφ+2∇aφ1∇bφ1-2∇aφ1ℋbc∇cφ-δab(∇cφ∇cφ2-12∇cφℒdc∇dφ+∇cφℋdeℋec∇dφ-2∇cφℋdc∇dφ1+∇cφ1∇cφ1+φ2∂V∂φ+φ12∂2V∂φ2).
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(60)C̅(K):=∇̅a∇̅aφ̅-∂V∂φ̅(φ̅)=0.
Through the perturbative expansions (53) and (22), the Klein-Gordon equation (60) is expanded as (61)C̅(K)=:C(K)+λC(K)(1)+12λ2C(K)(2)+O(λ3).C(K) is the background Klein-Gordon equation (62)C(K):=∇a∇aφ-∂V∂φ̅(φ)=0.
The first- and the second-order perturbations C(K)(1) and C(K)(2) are also decomposed into the gauge-invariant and the gauge-variant parts as  (63)C(K)(1)=:𝒞(K)(1)+£XC(K),C(K)(2)=:𝒞(K)(2)+2£XC(K)(1)+(£Y-£X2)C(K),
where
(64)𝒞(K)(1):=∇a∇aφ1-Haac[ℋ]∇cφ-ℋab∇a∇bφ-φ1∂2V∂φ̅2(φ),(65)𝒞(K)(2):=∇a∇aφ2-Haac[ℒ]∇cφ+2Haad[ℋ]ℋcd∇cφ-2Haac[ℋ]∇cφ1+2ℋabHabc[ℋ]∇cφ-ℒab∇a∇bφ+2ℋdaℋdb∇a∇bφ-2ℋab∇a∇bφ1-φ2∂2V∂φ̅2(φ)-(φ1)2∂3V∂φ̅3(φ).
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(66)𝒞(K)(1)=0,𝒞(K)(2)=0.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,(67)(1)Gab=8πGT(1)ab,(2)Gab=8πGT(2)ab.
Then, the perturbative Einstein equation is given by (68)(1)𝒢ab[ℋ]=8πG(1)𝒯ab
at linear order and (69)(1)𝒢ab[ℒ]+(2)𝒢ab[ℋ,ℋ]=8πG𝒯(2)ab
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 metricgab, 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 spacetimeℳ0 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 (70)gab=a2(η)(-(dη)a(dη)b+γij(dxi)a(dxj)b),
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(71)Tab=∇aφ∇bφ-12δab(∇cφ∇cφ+2V(φ))=-(12a2(∂ηφ)2+V(φ))(dη)a(∂∂η)b+(12a2(∂ηφ)2-V(φ))γab,
where we assumed that the scalar field φ is homogeneous (72)φ=φ(η)
and γab are defined as (73)γab:=γij(dxi)a(dxj)b,γab:=γij(dxi)a(∂∂xj)b.The background Einstein equationsGab=8πGTab for this background spacetime filled with the single scalar field are given by(74)ℋ2+K=8πG3a2(12a2(∂ηφ)2+V(φ)),2∂ηℋ+ℋ2+K=8πG(-12(∂ηφ)2+a2V(φ)).
We also note that (74) lead to (75)ℋ2+K-∂ηℋ=4πG(∂ηφ)2.
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∇aTba=0 for the scalar field (76)∂η2φ+2ℋ∂ηφ+a2∂V∂φ=0.
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 Section5.1, 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 perturbationhab 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 Δ-1:=(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 hab as(77)hab=hηη(dη)a(dη)b+2(Dih(VL)+h(V)i)(dη)(a(dxi)b)+a2{h(L)γij+(DiDj-13γijΔ)h(TL)+2D(ih(TV)j)+h(TT)ij(DiDj-13γijΔ)}(dxi)a(dxj)b,
where h(V)i, h(TV)j, and h(TT)ij satisfy the properties (78)Dih(V)i=0,Dih(TV)i=0,h(TT)ij=h(TT)ji,h(T)ii:=γijh(T)ij=0,Dih(TT)ij=0.
The gauge-transformation rules for the variables hηη, h(VL), h(V)i, h(L), h(TL), h(TV)j and h(TT)ij are derived from (25). Inspecting these gauge-transformation rules, we define the gauge-variant part Xa in (23): (79)Xa:=(h(VL)-12a2∂ηh(TL))(dη)a+a2(h(TV)i+12Dih(TL))(dxi)a.
We can easily check this vector field Xa satisfies (24). Subtracting gauge variant-part £Xgab from hab, we have the gauge-invariant part ℋab in (23): (80)ℋab=a2{-2Φ(1)(dη)a(dη)b+2ν(1)i(dη)(dxi)(ab)+(-2Ψ(1)γij+χ(1)ij)(dxi)a(dxj)b},
where the properties Diν(1)i:=γijDiν(1)j=χii(1):=γijχij(1)=Diχ(1)ij=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 Δ-1, (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 {hηη,hiη,hij} and {hηη,h(VL),h(V)i,h(L),h(TL),h(TV)j,h(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 Φ(1) and νi(1) of the first-order metric perturbation, we can define a vector field Za by Za:=-aΦ(1)(dη)a+aνi(1)(dxi)a which is gauge invariant. Then, we can rewrite the decomposition formula (23) for the linear-order metric perturbation as (81)hab=ℋab-£Zgab+£Zgab+£Xgab,=:𝒦ab+£X+Zgab,
where we have defined new gauge invariant variable Kab by 𝒦ab:=ℋab-£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 ℋab 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 partXa. 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 hab 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)𝒢ab[ℋ], which is defined by (45), we first derive the components of the tensor ℋabc[ℋ], which is defined in (48) with Aab=ℋab and its component (80). These components are summarized in [25, 26].From (45), the component of (1)𝒢ab[ℋ] are summarized as(82)(1)𝒢ηη[ℋ]=-1a2{(-6ℋ∂η+2Δ+6K)Ψ(1)-6ℋ2Φ(1)},(1)𝒢iη[ℋ]=-1a2(2∂ηDiΨ(1)+2ℋDiΦ(1)-12(Δ+2K)νi(1)),(1)𝒢ηi[ℋ]=1a2{2∂ηDiΨ(1)+2ℋDiΦ(1)+12(-Δ+2K+4ℋ2-4∂ηℋ)νi(1)},(1)𝒢ij[ℋ]=1a2[DiDj(Ψ(1)-Φ(1))+{(-Δ+2∂η2+4ℋ∂η-2K)Ψ(1)+(2ℋ∂η+4∂ηℋ+2ℋ2+Δ)Φ(1)}γij-12a2∂η{a2(Diνj(1)+Djνi(1))}+12(∂η2+2ℋ∂η+2K-Δ)χij(1)].
Straightforward calculations show that these components of the first-order gauge-invariant perturbation (1)𝒢ab[ℋ] of the Einstein tensor satisfies the identity (50). Although this confirmation is also possible without specification of the tensor ℋab, the confirmation of (50) through the explicit components (82) implies that we have derived the components of (1)𝒢ab[ℋ] 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 fieldφ is homogeneous as (72), the components of the gauge-invariant part of the first-order energy-momentum tensor (1)𝒯ab are given by(83)(1)𝒯ηη=-1a2(∂ηφ1∂ηφ-Φ(1)(∂ηφ)2+a2dVdφφ1),(1)𝒯iη=-1a2Diφ1∂ηφ,(84)(1)𝒯ηi=1a2∂ηφ(Diφ1+(∂ηφ)νi(1)),(1)𝒯ij=1a2γij(∂ηφ1∂ηφ-Φ(1)(∂ηφ)2-a2dVdφφ1).
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(85)Φ(1)=Ψ(1).
From (82)–(84) and (85), the components of scalar parts of the linearized Einstein equation (68) are given as [3] (86)(Δ-3ℋ∂η+4K-∂ηℋ-2ℋ2)Φ(1)=4πG(∂ηφ1∂ηφ+a2dVdφφ1),(87)∂ηΦ(1)+ℋΦ(1)=4πGφ1∂ηφ,(88)(∂η2+3ℋ∂η+∂ηℋ+2ℋ2)Φ(1)=4πG(∂ηφ1∂ηφ-a2dVdφφ1).
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[ℋ]=8πG(1)𝒯iη shows that (89)ν(1)i=0.
The equation for the tensor mode χij(1) is given by (90)(∂η2+2ℋ∂η+2K-Δ)χij(1)=0.Combining (86) and (88), we eliminate the potential term of the scalar field and thereby obtain that(91)(∂η2+Δ+4K)Φ(1)=8πG∂ηφ1∂ηφ.
Further, using (87) to express ∂ηφ1 in terms of ∂ηΦ(1) and Φ(1), we also eliminate ∂ηφ1 in (91). Hence, we have  (92){∂η2+2(ℋ-2∂η2φ∂ηφ)∂η-Δ-4K+2(∂ηℋ-ℋ∂η2φ∂ηφ)}Φ(1)=0.
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 𝒞(K)(1) of the first-order Klein-Gordon equation defined by (64) is given by(93)-a2𝒞(K)(1)=∂η2φ1+2ℋ∂ηφ1-Δφ1-(∂ηΦ(1)+3∂ηΨ(1))∂ηφ+2a2Φ(1)∂V∂φ̅(φ)+a2φ1∂2V∂φ̅2(φ)=0.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.


## 5.1. Gauge-Invariant Metric Perturbations

Here, we consider the first-order metric perturbationhab 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 Δ-1:=(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 hab as(77)hab=hηη(dη)a(dη)b+2(Dih(VL)+h(V)i)(dη)(a(dxi)b)+a2{h(L)γij+(DiDj-13γijΔ)h(TL)+2D(ih(TV)j)+h(TT)ij(DiDj-13γijΔ)}(dxi)a(dxj)b,
where h(V)i, h(TV)j, and h(TT)ij satisfy the properties (78)Dih(V)i=0,Dih(TV)i=0,h(TT)ij=h(TT)ji,h(T)ii:=γijh(T)ij=0,Dih(TT)ij=0.
The gauge-transformation rules for the variables hηη, h(VL), h(V)i, h(L), h(TL), h(TV)j and h(TT)ij are derived from (25). Inspecting these gauge-transformation rules, we define the gauge-variant part Xa in (23): (79)Xa:=(h(VL)-12a2∂ηh(TL))(dη)a+a2(h(TV)i+12Dih(TL))(dxi)a.
We can easily check this vector field Xa satisfies (24). Subtracting gauge variant-part £Xgab from hab, we have the gauge-invariant part ℋab in (23): (80)ℋab=a2{-2Φ(1)(dη)a(dη)b+2ν(1)i(dη)(dxi)(ab)+(-2Ψ(1)γij+χ(1)ij)(dxi)a(dxj)b},
where the properties Diν(1)i:=γijDiν(1)j=χii(1):=γijχij(1)=Diχ(1)ij=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 Δ-1, (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 {hηη,hiη,hij} and {hηη,h(VL),h(V)i,h(L),h(TL),h(TV)j,h(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 Φ(1) and νi(1) of the first-order metric perturbation, we can define a vector field Za by Za:=-aΦ(1)(dη)a+aνi(1)(dxi)a which is gauge invariant. Then, we can rewrite the decomposition formula (23) for the linear-order metric perturbation as (81)hab=ℋab-£Zgab+£Zgab+£Xgab,=:𝒦ab+£X+Zgab,
where we have defined new gauge invariant variable Kab by 𝒦ab:=ℋab-£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 ℋab 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 partXa. 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 hab 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)𝒢ab[ℋ], which is defined by (45), we first derive the components of the tensor ℋabc[ℋ], which is defined in (48) with Aab=ℋab and its component (80). These components are summarized in [25, 26].From (45), the component of (1)𝒢ab[ℋ] are summarized as(82)(1)𝒢ηη[ℋ]=-1a2{(-6ℋ∂η+2Δ+6K)Ψ(1)-6ℋ2Φ(1)},(1)𝒢iη[ℋ]=-1a2(2∂ηDiΨ(1)+2ℋDiΦ(1)-12(Δ+2K)νi(1)),(1)𝒢ηi[ℋ]=1a2{2∂ηDiΨ(1)+2ℋDiΦ(1)+12(-Δ+2K+4ℋ2-4∂ηℋ)νi(1)},(1)𝒢ij[ℋ]=1a2[DiDj(Ψ(1)-Φ(1))+{(-Δ+2∂η2+4ℋ∂η-2K)Ψ(1)+(2ℋ∂η+4∂ηℋ+2ℋ2+Δ)Φ(1)}γij-12a2∂η{a2(Diνj(1)+Djνi(1))}+12(∂η2+2ℋ∂η+2K-Δ)χij(1)].
Straightforward calculations show that these components of the first-order gauge-invariant perturbation (1)𝒢ab[ℋ] of the Einstein tensor satisfies the identity (50). Although this confirmation is also possible without specification of the tensor ℋab, the confirmation of (50) through the explicit components (82) implies that we have derived the components of (1)𝒢ab[ℋ] 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 fieldφ is homogeneous as (72), the components of the gauge-invariant part of the first-order energy-momentum tensor (1)𝒯ab are given by(83)(1)𝒯ηη=-1a2(∂ηφ1∂ηφ-Φ(1)(∂ηφ)2+a2dVdφφ1),(1)𝒯iη=-1a2Diφ1∂ηφ,(84)(1)𝒯ηi=1a2∂ηφ(Diφ1+(∂ηφ)νi(1)),(1)𝒯ij=1a2γij(∂ηφ1∂ηφ-Φ(1)(∂ηφ)2-a2dVdφφ1).
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(85)Φ(1)=Ψ(1).
From (82)–(84) and (85), the components of scalar parts of the linearized Einstein equation (68) are given as [3] (86)(Δ-3ℋ∂η+4K-∂ηℋ-2ℋ2)Φ(1)=4πG(∂ηφ1∂ηφ+a2dVdφφ1),(87)∂ηΦ(1)+ℋΦ(1)=4πGφ1∂ηφ,(88)(∂η2+3ℋ∂η+∂ηℋ+2ℋ2)Φ(1)=4πG(∂ηφ1∂ηφ-a2dVdφφ1).
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[ℋ]=8πG(1)𝒯iη shows that (89)ν(1)i=0.
The equation for the tensor mode χij(1) is given by (90)(∂η2+2ℋ∂η+2K-Δ)χij(1)=0.Combining (86) and (88), we eliminate the potential term of the scalar field and thereby obtain that(91)(∂η2+Δ+4K)Φ(1)=8πG∂ηφ1∂ηφ.
Further, using (87) to express ∂ηφ1 in terms of ∂ηΦ(1) and Φ(1), we also eliminate ∂ηφ1 in (91). Hence, we have  (92){∂η2+2(ℋ-2∂η2φ∂ηφ)∂η-Δ-4K+2(∂ηℋ-ℋ∂η2φ∂ηφ)}Φ(1)=0.
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 𝒞(K)(1) of the first-order Klein-Gordon equation defined by (64) is given by(93)-a2𝒞(K)(1)=∂η2φ1+2ℋ∂ηφ1-Δφ1-(∂ηΦ(1)+3∂ηΨ(1))∂ηφ+2a2Φ(1)∂V∂φ̅(φ)+a2φ1∂2V∂φ̅2(φ)=0.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 Section4 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 variableL̂ab 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 ℳ0 from the fact that the generator ξ2a 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 ℒab in (32) as(94)ℒab=-2a2Φ(2)(dη)a(dη)b+2a2νi(2)(dη)(a(dxi)b)+a2(-2Ψ(2)γij+χij(2))(dxi)a(dxj)b,
where ν(2)i and χij(2) satisfy the equations(95)Diνi(2)=0,χii(2)=0,Diχij(2)=0.
The gauge-invariant variables Φ(2) and Ψ(2) are the scalar mode perturbations of second order, and νi(2) and χij(2) 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 (96)Xa=X'a-Z'a,
where Z'a is gauge invariant and X'a is transformed as (97)𝒴X'a-𝒳X'a=ξ1a
under the gauge transformation 𝒳λ→𝒴λ. This Z'a may be different from the vector Za in (81). By the replacement (96), the second-order metric perturbation (32) is given in the form (98)lab=:𝒥ab+2£X′hab+(£Y'-£X′2)gab,
where we defined (99)𝒥ab:=ℒab-£Wgab-2£Z'𝒦ab-2£Z′£Zgab+£Z′2gab,(100)Y'a:=Ya+Wa+[X′,Z′]a.
Here, the vector field Wa in (100) constitute of some components of gauge-invariant second-order metric perturbation ℒab like Za in (81). The tensor field 𝒥ab is manifestly gauge invariant. The gauge transformation rule of the new gauge-variant part Y'a of the second-order metric perturbation is given by (101)𝒴Y'a-𝒳Y'a=ξ(2)a+[ξ(1),X′]a.
Although (98) is similar to (32), the tensor fields ℒab 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 ofℒab 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[ℒ] and (2)𝒢ab[ℋ,ℋ] in the Einstein equation (69).First, we evaluate the term(1)𝒢ab[ℒ] in the Einstein equation (69). Because the components (94) of ℒab are obtained through the replacements (102)Φ(1)→Φ(2),νi(1)→ν(2)i,Ψ(1)→Ψ(2),χij(1)→χij(2)
in the components (80) of ℋab, we easily obtain the components of (1)𝒢ab[ℒ] through the replacements (102) in (82).From (80), we can derive the components of (2)𝒢ab=(2)𝒢ab[ℋ,ℋ] 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[ℋ,ℋ] are summarized as(103)(2)𝒢ηη=2a2[-3DkΦ(1)DkΦ(1)-8Φ(1)ΔΦ(1)-3(∂ηΦ(1))2-12(ℋ2+K)(Φ(1))2+DlDkΦ(1)χlk(1)+18∂ηχkl(1)(∂η+8ℋ)χkl(1)+12Dkχlm(1)D[lχk]m(1)-18Dkχlm(1)Dkχml(1)-12χlm(1)(Δ-K)χlm(1)],(2)𝒢ηi=2a2[8Φ(1)∂ηDiΦ(1)-DjΦ(1)∂ηχji(1)-(∂ηDjΦ(1)+2ℋDjΦ(1))χij(1)+14∂ηχjk(1)Diχkj(1)+χkl(1)∂ηD[iχk]l(1)],(2)𝒢iη=2a2[8ℋΦ(1)DiΦ(1)-2DiΦ(1)∂ηΦ(1)+DjΦ(1)∂ηχij(1)-∂ηDjΦ(1)χij(1)-14∂ηχkj(1)Diχkj(1)+χkj(1)∂ηD[jχi]k(1)],(2)𝒢ij=2a2[{-3DkΦ(1)DkΦ(1)-4Φ(1)(Δ+K)Φ(1)-∂ηΦ(1)∂ηΦ(1)-8ℋΦ(1)∂ηΦ(1)-4(2∂ηℋ+ℋ2)(Φ(1))2}γij+2DiΦ(1)DjΦ(1)+4Φ(1)DiDjΦ(1)+χij(1)(∂η2+2ℋ∂η)Φ(1)+DkΦ(1)(Diχjk(1)+Djχik(1))-2DkΦ(1)Dkχij(1)-2Φ(1)(Δ-2K)χij(1)-ΔΦ(1)χij(1)+DkDiΦ(1)χjk(1)+DmDjΦ(1)χim(1)-DlDkΦ(1)χlk(1)γij-12∂ηχik(1)∂ηχkj(1)+Dkχil(1)D[kχl]j(1)+14Djχlk(1)Diχlk(1)+12χlm(1)DiDjχml(1)-12χlm(1)DlDiχmj(1)-12χlm(1)DlDjχmi(1)+12χlm(1)DmDlχij(1)-12χjk(1)(∂η2+2ℋ∂η-Δ+2K)χik(1)+12{34∂ηχlk(1)∂ηχkl(1)+χkl(1)(∂η2+2ℋ∂η-Δ+K)χlk(1)-14Dkχlm(1)Dkχml(1)+Dkχlm(1)D[lχk]m(1)}γij].
We have checked the identity (51) through (103), Then, we may say that the expressions (103) are self-consistent.


### 6.3. Energy-Momentum Tensor and Klein-Gordon Equation

Here, we summarize the explicit components of the gauge-invariant parts (59) of the second-order perturbation of energy momentum tensor for a single scalar field in terms of gauge-invariant variables. Through (72), (80), and (94), the components of (59) are derived by the straightforward calculations. In this paper, we just summarize the components of (2)𝒯ab in the situation where the first-order Einstein equations (85) and (89) are satisfied:(104)a2(2)𝒯ηη=-∂ηφ∂ηφ2+(∂ηφ)2Φ(2)-a2φ2∂V∂φ+4∂ηφΦ(1)∂ηφ1-4(∂ηφ)2(Φ(1))2-(∂ηφ1)2-Diφ1Diφ1-a2(φ1)2∂2V∂φ2,a2(2)𝒯iη=-∂ηφDiφ2+4∂ηφDiφ1Φ(1)-2Diφ1∂ηφ1,a2(2)𝒯ηi=∂ηφDiφ2+2∂ηφ1Diφ1+4∂ηφΦ(1)Diφ1-2∂ηφχil(1)Dlφ1,a2(2)𝒯ij=Diφ1Djφ1+12γij{∂ηφ∂ηφ2-4∂ηφΦ(1)∂ηφ1+4(∂ηφ)2(Φ(1))2-(∂ηφ)2Φ(2)+(∂ηφ1)2-Dlφ1Dlφ1-a2φ2∂V∂φ-a2(φ1)2∂2V∂φ2}.
More generic formulae for the components of (2)𝒯ab are given in [27].Next, we show the gauge-invariant second-order the Klein-Gordon equation. We only consider the simple situation where (85) and (89) are satisfied. The formulae for more generic situation is given in [27]. Through (80), (94), and (72), the second-order perturbation of the Klein-Gordon equation (65) is given by (105)-a2𝒞(K)(2)=∂η2φ2+2ℋ∂ηφ2-Δφ2-(∂ηΦ(2)+3∂ηΨ(2))∂ηφ+2a2Φ(2)∂V∂φ̅(φ)+a2φ2∂2V∂φ̅2(φ)-Ξ(K)=0,
where we defined (106)Ξ(K):=8∂ηΦ(1)∂ηφ1+8Φ(1)Δφ1-4a2Φ(1)φ1∂2V∂φ̅2(φ)-a2(φ1)2∂3V∂φ̅3(φ)+8Φ(1)∂ηΦ(1)∂ηφ-2χij(1)DjDiφ1+∂ηφχij(1)∂ηχij(1).In (105), Ξ(K) is the source term which is the collection of the quadratic terms of the linear-order perturbations in the second-order perturbation of the Klein-Gordon equation. If we ignore this source term, (105) coincides with the first-order perturbation of the Klein-Gordon equation. From this source term (106) of the Klein-Gordon equation, we can see that the mode-mode coupling due to the nonlinear effects appear in the second-order Klein-Gordon equation.We cannot discuss solutions to (105) only through this equation, since this includes metric perturbations. To determine the behavior of the metric perturbations, we have to treat the Einstein equations simultaneously. The second-order Einstein equation is shown in Section 6.4.


### 6.4. Einstein Equations

Here, we show all the components of the second-order Einstein equation (69). All components of (69) are summarized as(107)(-3ℋ∂η+Δ+3K)Ψ(2)+(-∂ηℋ-2ℋ2+K)Φ(2)-4πG(∂ηφ∂ηφ2+a2φ2∂V∂φ)=Γ0,(108)2∂ηDiΨ(2)+2ℋDiΦ(2)-12(Δ+2K)νi(2)-8πGDiφ2∂ηφ=Γi,(109)DiDj(Ψ(2)-Φ(2))+{(-Δ+2∂η2+4ℋ∂η-2K)Ψ(2)+(2ℋ∂η+2∂ηℋ+4ℋ2+Δ+2K)Φ(2)}γij-1a2∂η(a2D(iνj)(2))+12(∂η2+2ℋ∂η+2K-Δ)χ(2)ij-8πG(∂ηφ∂ηφ2-a2φ2∂V∂φ(φ))γij=Γij,
where Γ0, Γi and Γij are the collection of the quadratic term of the first-order perturbations as follows: (110)Γ0:=4πG((∂ηφ1)2+Diφ1Diφ1+a2(φ1)2∂2V∂φ2)-4∂ηℋ(Φ(1))2-2Φ(1)∂η2Φ(1)-3DkΦ(1)DkΦ(1)-10Φ(1)ΔΦ(1)-3(∂ηΦ(1))2-16K(Φ(1))2-8ℋ2(Φ(1))2+DlDkΦ(1)χlk(1)+18∂ηχlk(1)∂ηχkl(1)+ℋχkl(1)∂ηχlk(1)-38Dkχlm(1)Dkχml(1)+14Dkχlm(1)Dlχmk(1)-12χlm(1)Δχlm(1)+12Kχlm(1)χlm(1),(111)Γi:=16πG∂ηφ1Diφ1-4∂ηΦ(1)DiΦ(1)+8ℋΦ(1)DiΦ(1)-8Φ(1)∂ηDiΦ(1)+2DjΦ(1)∂ηχji(1)-2∂ηDjΦ(1)χij(1)-12∂ηχjk(1)Diχkj(1)-χkl(1)∂ηDiχlk(1)+χkl(1)∂ηDkχil(1),(112)Γij:=16πGDiφ1Djφ1+8πG{(∂ηφ1)2-Dlφ1Dlφ1-a2(φ1)2∂2V∂φ2}γij-4DiΦ(1)DjΦ(1)-8Φ(1)DiDjΦ(1)+(6DkΦ(1)DkΦ(1)+4Φ(1)ΔΦ(1)+2(∂ηΦ(1))2+8∂ηℋ(Φ(1))2+16ℋ2(Φ(1))2+16ℋΦ(1)∂ηΦ(1)-4Φ(1)∂η2Φ(1))γij-4ℋ∂ηΦ(1)χij(1)-2∂η2Φ(1)χij(1)-4DkΦ(1)D(iχj)k(1)+4DkΦ(1)Dkχij(1)-8KΦ(1)χij(1)+4Φ(1)Δχij(1)-4DkD(iΦ(1)χj)k(1)+2ΔΦ(1)χij(1)+2DlDkΦ(1)χlk(1)γij+∂ηχik(1)∂ηχjk(1)-Dkχil(1)Dkχjl(1)+Dkχil(1)Dlχjk(1)-12Diχlk(1)Djχlk(1)-χlm(1)DiDjχml(1)+2χlm(1)DlD(iχj)m(1)-χlm(1)DmDlχij(1)-14(3∂ηχlk(1)∂ηχkl(1)-3Dkχlm(1)Dkχml(1)+2Dkχlm(1)Dlχmk(1)-4Kχlm(1)χlm(1))γij.
Here, we used (75), (85), (87), (89), and (91).The tensor part of (109) is given by (113)(∂η2+2ℋ∂η+2K-Δ)χij(2)=2Γij-23γijΓkk-3(DiDj-13γijΔ)(Δ+3K)-1×(Δ-1DkDlΓkl-13Γkk)+4{D(i(Δ+2K)-1Dj)Δ-1DlDkΓlk-D(i(Δ+2K)-1DkΓj)k}.
This tensor mode is also called the second-order gravitational waves.Further, the vector part of (108) yields the initial value constraint and the evolution equation of the vector mode νj(2): (114)νi(2)=2Δ+2K{DiΔ-1DkΓk-Γi},∂η(a2νi(2))=2a2Δ+2K{DiΔ-1DkDlΓkl-DkΓik}.
Finally, scalar part of (108) are summarized as (115)2∂ηΨ(2)+2ℋΦ(2)-8πGφ2∂ηφ=Δ-1DkΓk,(116)Ψ(2)-Φ(2)=32(Δ+3K)-1{Δ-1DiDjΓij-13Γkk},(117)(-∂η2-5ℋ∂η+43Δ+4K)Ψ(2)-(2∂ηℋ+ℋ∂η+4ℋ2+13Δ)Φ(2)-8πGa2φ2∂V∂φ=Γ0-16Γkk,(118){∂η2+2(ℋ-∂η2φ∂ηφ)∂η-Δ-4K+2(∂ηℋ-∂η2φ∂ηφℋ)}Φ(2)=-Γ0-12Γkk+Δ-1DiDjΓij+(∂η-∂η2φ∂ηφ)Δ-1DkΓk-32{∂η2-(2∂η2φ∂ηφ-ℋ)∂η}(Δ+3K)-1×{Δ-1DiDjΓij-13Γkk},
where Γij:=γkjΓik and Γkk=γijΓij. Equation (118) is the second-order extension of (92), which is the master equation of scalar mode of the second-order cosmological perturbation in a universe filled with a single scalar field.Thus, we have a set of ten equations for the second-order perturbations of a universe filled with a single scalar field, (113)–(118). To solve this system of equations of the second-order Einstein equation, first of all, we have to solve the linear-order system. This is accomplished by solving (92) to obtain the potential Φ(1), φ1 is given through (87), and the tensor mode χij(1) is given by solving (90). Next, we evaluate the quadratic terms Γ0, Γi, and Γij of the linear-order perturbations, which are defined by (110)–(112). Then, using the information of (110)–(112), we estimate the source term in (118). If we know the two independent solutions to the linear-order master equation (92), we can solve (118) through the method using the Green functions. After constructing the solution Φ(2) to (118), we can obtain the second-order metric perturbation Ψ(2) through (116). Thus, we have obtained the second-order gauge-invariant perturbation φ2 of the scalar field through (115). Thus, the all scalar modes Φ(2), Ψ(2), and φ2 are obtained. Equation (117) is then used to check the consistency of the second-order perturbation of the Klein Gordon equation (105) as in Section 6.5.For the vector-mode,νi(1) of the first-order identically vanishes due to the momentum constraint (89) for the linear-order metric perturbations. On the other hand, in the second-order, we have evolution equation (114) of the vector mode νi(2) with the initial value constraint. This evolution equation of the second-order vector mode should be consistent with the initial value constraint, which is confirmed in Section 6.5. Equations (114) also imply that the second-order vector-mode perturbation may be generated by the mode couplings of the linear-order perturbations. As the simple situations, the generation of the second-order vector mode due to the scalar-scalar mode coupling is discussed in [44–47].The second-order tensor mode is also generated by the mode-coupling of the linear-order perturbations through the source term in (113). Note that (113) is almost same as (90) for the linear-order tensor mode, except for the existence of the source term in (113). If we know the solution to the linear-order Einstein equations (90) and (92), we can evaluate the source term in (113). Further, we can solve (113) through the Green function method. This leads the generation of the gravitational wave of the second order. Actually, in the simple situation where the first-order tensor mode neglected, the generation of the second-order gravitational waves discussed in some literature [48–54].


### 6.5. Consistency of Equations for Second-Order Perturbations

Now, we consider the consistency of the second-order perturbations of the Einstein equations (115)–(118) for the scalar modes, (114) for vector mode, and the Klein-Gordon equation (105). The consistency check of these equations are necessary to guarantee that the derived equations are correct, since the second-order Einstein equations have complicated forms owing to the quadratic terms of the linear-order perturbations that arise from the nonlinear effects of the Einstein equations.Since the first equation in (114) is the initial value constraint for the vector mode νi(2) and it should be consistent with the evolution equation, that is, the second equation of (114). these equations should be consistent with each other from the general arguments of the Einstein equation. Explicitly, these equations are consistent with each other if the equation (119)∂ηΓk+2ℋΓk-DlΓlk=0
is satisfied. Actually, through the first-order perturbative Einstein equations (87), (92), and (90), we can confirm (119). This is a trivial result from a general viewpoint, because the Einstein equation is the first class constrained system. However, this trivial result implies that we have derived the source terms Γi and Γij of the second-order Einstein equations consistently.Next, we consider (117). Through the second-order Einstein equations (115), (116), (118), and the background Klein-Gordon equation (76), we can confirm that (117) is consistent with the set of the background, first-order and other second-order Einstein equation if the equation (120)(∂η+2ℋ)DkΓk-DjDiΓij=0
is satisfied under the background and first-order Einstein equations. Actually, we have already seen that (119) is satisfied under the background and first-order Einstein equations. Taking the divergence of (119), we can immediately confirm (120). Then, (117) gives no information.Thus, we have seen that the derived Einstein equations of the second-order (114)–(118) are consistent with each other through (119). This fact implies that the derived source terms Γi and Γij of the second-order perturbations of the Einstein equations, which are defined by (111) and (112), are correct source terms of the second-order Einstein equations. On the other hand, for Γ0, we have to consider the consistency between the perturbative Einstein equations and the perturbative Klein-Gordon equation as seen below.Now, we consider the consistency of the second-order perturbation of the Klein-Gordon equation and the Einstein equations. The second-order perturbation of the Klein-Gordon equation is given by (105) with the source term (106). Since the vector mode νi(2) and tensor mode χij(2) of the second-order do not appear in the expressions (105) of the second-order perturbation of the Klein-Gordon equation, we may concentrate on the Einstein equations for scalar mode of the second order, that is, (115), (116), and (118) with the definitions (110)–(112) of the source terms. As in the linear case, the second-order perturbation of the Klein-Gordon equation should also be derived from the set of equations consisting of the second-order perturbations of the Einstein equations (115), (116), (118), the first-order perturbations of the Einstein equations (85), (87), (92), and the background Einstein equations (74). Actually, from these equations, we can show that the second-order perturbation of the Klein-Gordon equation is consistent with the background and the second-order Einstein equations if the equation (121)2(∂η+H)Γ0-DkΓk+ℋΓkk+8πG∂ηφΞ(K)=0
is satisfied under the background and the first-order Einstein equations. Further, we can also confirm (121) through the background Einstein equations (74), the scalar part of the first-order perturbation of the momentum constraint (87), and the evolution equations (92) and (90) for the first-order scalar and tensor modes in the Einstein equation.As shown in [28], the first-order perturbation of the Klein-Gordon equation is derived from the background and the first-order perturbations of the Einstein equation. In the case of the second-order perturbation, the Klein-Gordon equation (105) can be also derived from the background, the first-order, and the second-order Einstein equations. The second-order perturbations of the Einstein equation and the Klein-Gordon equation include the source terms Γ0, Γi, Γij, and Ξ(K) due to the mode-coupling of the linear-order perturbations. The second-order perturbation of the Klein-Gordon equation gives the relation (121) between the source terms Γ0, Γi, Γij, Ξ(K), and we have also confirmed that (121) is satisfied due to the background, the first-order perturbation of the Einstein equations, and the Klein-Gordon equation. Thus, the second-order perturbation of the Klein-Gordon equation is not independent of the set of the background, the first-order, and the second-order Einstein equations if we impose on the Einstein equation at any conformal time η. This also implies that the derived formulae of the source terms Γ0, Γi, Γij, and Ξ(K) are consistent with each other. In this sense, we may say that the formulae (110)–(112) and (106) for these source terms are correct.


## 6.1. Gauge-Invariant Metric Perturbations

First, we consider the components of the gauge-invariant variables for the metric perturbation of second order. The variableL̂ab 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 ℳ0 from the fact that the generator ξ2a 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 ℒab in (32) as(94)ℒab=-2a2Φ(2)(dη)a(dη)b+2a2νi(2)(dη)(a(dxi)b)+a2(-2Ψ(2)γij+χij(2))(dxi)a(dxj)b,
where ν(2)i and χij(2) satisfy the equations(95)Diνi(2)=0,χii(2)=0,Diχij(2)=0.
The gauge-invariant variables Φ(2) and Ψ(2) are the scalar mode perturbations of second order, and νi(2) and χij(2) 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 (96)Xa=X'a-Z'a,
where Z'a is gauge invariant and X'a is transformed as (97)𝒴X'a-𝒳X'a=ξ1a
under the gauge transformation 𝒳λ→𝒴λ. This Z'a may be different from the vector Za in (81). By the replacement (96), the second-order metric perturbation (32) is given in the form (98)lab=:𝒥ab+2£X′hab+(£Y'-£X′2)gab,
where we defined (99)𝒥ab:=ℒab-£Wgab-2£Z'𝒦ab-2£Z′£Zgab+£Z′2gab,(100)Y'a:=Ya+Wa+[X′,Z′]a.
Here, the vector field Wa in (100) constitute of some components of gauge-invariant second-order metric perturbation ℒab like Za in (81). The tensor field 𝒥ab is manifestly gauge invariant. The gauge transformation rule of the new gauge-variant part Y'a of the second-order metric perturbation is given by (101)𝒴Y'a-𝒳Y'a=ξ(2)a+[ξ(1),X′]a.
Although (98) is similar to (32), the tensor fields ℒab 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 ofℒab 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[ℒ] and (2)𝒢ab[ℋ,ℋ] in the Einstein equation (69).First, we evaluate the term(1)𝒢ab[ℒ] in the Einstein equation (69). Because the components (94) of ℒab are obtained through the replacements (102)Φ(1)→Φ(2),νi(1)→ν(2)i,Ψ(1)→Ψ(2),χij(1)→χij(2)
in the components (80) of ℋab, we easily obtain the components of (1)𝒢ab[ℒ] through the replacements (102) in (82).From (80), we can derive the components of (2)𝒢ab=(2)𝒢ab[ℋ,ℋ] 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[ℋ,ℋ] are summarized as(103)(2)𝒢ηη=2a2[-3DkΦ(1)DkΦ(1)-8Φ(1)ΔΦ(1)-3(∂ηΦ(1))2-12(ℋ2+K)(Φ(1))2+DlDkΦ(1)χlk(1)+18∂ηχkl(1)(∂η+8ℋ)χkl(1)+12Dkχlm(1)D[lχk]m(1)-18Dkχlm(1)Dkχml(1)-12χlm(1)(Δ-K)χlm(1)],(2)𝒢ηi=2a2[8Φ(1)∂ηDiΦ(1)-DjΦ(1)∂ηχji(1)-(∂ηDjΦ(1)+2ℋDjΦ(1))χij(1)+14∂ηχjk(1)Diχkj(1)+χkl(1)∂ηD[iχk]l(1)],(2)𝒢iη=2a2[8ℋΦ(1)DiΦ(1)-2DiΦ(1)∂ηΦ(1)+DjΦ(1)∂ηχij(1)-∂ηDjΦ(1)χij(1)-14∂ηχkj(1)Diχkj(1)+χkj(1)∂ηD[jχi]k(1)],(2)𝒢ij=2a2[{-3DkΦ(1)DkΦ(1)-4Φ(1)(Δ+K)Φ(1)-∂ηΦ(1)∂ηΦ(1)-8ℋΦ(1)∂ηΦ(1)-4(2∂ηℋ+ℋ2)(Φ(1))2}γij+2DiΦ(1)DjΦ(1)+4Φ(1)DiDjΦ(1)+χij(1)(∂η2+2ℋ∂η)Φ(1)+DkΦ(1)(Diχjk(1)+Djχik(1))-2DkΦ(1)Dkχij(1)-2Φ(1)(Δ-2K)χij(1)-ΔΦ(1)χij(1)+DkDiΦ(1)χjk(1)+DmDjΦ(1)χim(1)-DlDkΦ(1)χlk(1)γij-12∂ηχik(1)∂ηχkj(1)+Dkχil(1)D[kχl]j(1)+14Djχlk(1)Diχlk(1)+12χlm(1)DiDjχml(1)-12χlm(1)DlDiχmj(1)-12χlm(1)DlDjχmi(1)+12χlm(1)DmDlχij(1)-12χjk(1)(∂η2+2ℋ∂η-Δ+2K)χik(1)+12{34∂ηχlk(1)∂ηχkl(1)+χkl(1)(∂η2+2ℋ∂η-Δ+K)χlk(1)-14Dkχlm(1)Dkχml(1)+Dkχlm(1)D[lχk]m(1)}γij].
We have checked the identity (51) through (103), Then, we may say that the expressions (103) are self-consistent.


## 6.3. Energy-Momentum Tensor and Klein-Gordon Equation

Here, we summarize the explicit components of the gauge-invariant parts (59) of the second-order perturbation of energy momentum tensor for a single scalar field in terms of gauge-invariant variables. Through (72), (80), and (94), the components of (59) are derived by the straightforward calculations. In this paper, we just summarize the components of (2)𝒯ab in the situation where the first-order Einstein equations (85) and (89) are satisfied:(104)a2(2)𝒯ηη=-∂ηφ∂ηφ2+(∂ηφ)2Φ(2)-a2φ2∂V∂φ+4∂ηφΦ(1)∂ηφ1-4(∂ηφ)2(Φ(1))2-(∂ηφ1)2-Diφ1Diφ1-a2(φ1)2∂2V∂φ2,a2(2)𝒯iη=-∂ηφDiφ2+4∂ηφDiφ1Φ(1)-2Diφ1∂ηφ1,a2(2)𝒯ηi=∂ηφDiφ2+2∂ηφ1Diφ1+4∂ηφΦ(1)Diφ1-2∂ηφχil(1)Dlφ1,a2(2)𝒯ij=Diφ1Djφ1+12γij{∂ηφ∂ηφ2-4∂ηφΦ(1)∂ηφ1+4(∂ηφ)2(Φ(1))2-(∂ηφ)2Φ(2)+(∂ηφ1)2-Dlφ1Dlφ1-a2φ2∂V∂φ-a2(φ1)2∂2V∂φ2}.
More generic formulae for the components of (2)𝒯ab are given in [27].Next, we show the gauge-invariant second-order the Klein-Gordon equation. We only consider the simple situation where (85) and (89) are satisfied. The formulae for more generic situation is given in [27]. Through (80), (94), and (72), the second-order perturbation of the Klein-Gordon equation (65) is given by (105)-a2𝒞(K)(2)=∂η2φ2+2ℋ∂ηφ2-Δφ2-(∂ηΦ(2)+3∂ηΨ(2))∂ηφ+2a2Φ(2)∂V∂φ̅(φ)+a2φ2∂2V∂φ̅2(φ)-Ξ(K)=0,
where we defined (106)Ξ(K):=8∂ηΦ(1)∂ηφ1+8Φ(1)Δφ1-4a2Φ(1)φ1∂2V∂φ̅2(φ)-a2(φ1)2∂3V∂φ̅3(φ)+8Φ(1)∂ηΦ(1)∂ηφ-2χij(1)DjDiφ1+∂ηφχij(1)∂ηχij(1).In (105), Ξ(K) is the source term which is the collection of the quadratic terms of the linear-order perturbations in the second-order perturbation of the Klein-Gordon equation. If we ignore this source term, (105) coincides with the first-order perturbation of the Klein-Gordon equation. From this source term (106) of the Klein-Gordon equation, we can see that the mode-mode coupling due to the nonlinear effects appear in the second-order Klein-Gordon equation.We cannot discuss solutions to (105) only through this equation, since this includes metric perturbations. To determine the behavior of the metric perturbations, we have to treat the Einstein equations simultaneously. The second-order Einstein equation is shown in Section 6.4.


## 6.4. Einstein Equations

Here, we show all the components of the second-order Einstein equation (69). All components of (69) are summarized as(107)(-3ℋ∂η+Δ+3K)Ψ(2)+(-∂ηℋ-2ℋ2+K)Φ(2)-4πG(∂ηφ∂ηφ2+a2φ2∂V∂φ)=Γ0,(108)2∂ηDiΨ(2)+2ℋDiΦ(2)-12(Δ+2K)νi(2)-8πGDiφ2∂ηφ=Γi,(109)DiDj(Ψ(2)-Φ(2))+{(-Δ+2∂η2+4ℋ∂η-2K)Ψ(2)+(2ℋ∂η+2∂ηℋ+4ℋ2+Δ+2K)Φ(2)}γij-1a2∂η(a2D(iνj)(2))+12(∂η2+2ℋ∂η+2K-Δ)χ(2)ij-8πG(∂ηφ∂ηφ2-a2φ2∂V∂φ(φ))γij=Γij,
where Γ0, Γi and Γij are the collection of the quadratic term of the first-order perturbations as follows: (110)Γ0:=4πG((∂ηφ1)2+Diφ1Diφ1+a2(φ1)2∂2V∂φ2)-4∂ηℋ(Φ(1))2-2Φ(1)∂η2Φ(1)-3DkΦ(1)DkΦ(1)-10Φ(1)ΔΦ(1)-3(∂ηΦ(1))2-16K(Φ(1))2-8ℋ2(Φ(1))2+DlDkΦ(1)χlk(1)+18∂ηχlk(1)∂ηχkl(1)+ℋχkl(1)∂ηχlk(1)-38Dkχlm(1)Dkχml(1)+14Dkχlm(1)Dlχmk(1)-12χlm(1)Δχlm(1)+12Kχlm(1)χlm(1),(111)Γi:=16πG∂ηφ1Diφ1-4∂ηΦ(1)DiΦ(1)+8ℋΦ(1)DiΦ(1)-8Φ(1)∂ηDiΦ(1)+2DjΦ(1)∂ηχji(1)-2∂ηDjΦ(1)χij(1)-12∂ηχjk(1)Diχkj(1)-χkl(1)∂ηDiχlk(1)+χkl(1)∂ηDkχil(1),(112)Γij:=16πGDiφ1Djφ1+8πG{(∂ηφ1)2-Dlφ1Dlφ1-a2(φ1)2∂2V∂φ2}γij-4DiΦ(1)DjΦ(1)-8Φ(1)DiDjΦ(1)+(6DkΦ(1)DkΦ(1)+4Φ(1)ΔΦ(1)+2(∂ηΦ(1))2+8∂ηℋ(Φ(1))2+16ℋ2(Φ(1))2+16ℋΦ(1)∂ηΦ(1)-4Φ(1)∂η2Φ(1))γij-4ℋ∂ηΦ(1)χij(1)-2∂η2Φ(1)χij(1)-4DkΦ(1)D(iχj)k(1)+4DkΦ(1)Dkχij(1)-8KΦ(1)χij(1)+4Φ(1)Δχij(1)-4DkD(iΦ(1)χj)k(1)+2ΔΦ(1)χij(1)+2DlDkΦ(1)χlk(1)γij+∂ηχik(1)∂ηχjk(1)-Dkχil(1)Dkχjl(1)+Dkχil(1)Dlχjk(1)-12Diχlk(1)Djχlk(1)-χlm(1)DiDjχml(1)+2χlm(1)DlD(iχj)m(1)-χlm(1)DmDlχij(1)-14(3∂ηχlk(1)∂ηχkl(1)-3Dkχlm(1)Dkχml(1)+2Dkχlm(1)Dlχmk(1)-4Kχlm(1)χlm(1))γij.
Here, we used (75), (85), (87), (89), and (91).The tensor part of (109) is given by (113)(∂η2+2ℋ∂η+2K-Δ)χij(2)=2Γij-23γijΓkk-3(DiDj-13γijΔ)(Δ+3K)-1×(Δ-1DkDlΓkl-13Γkk)+4{D(i(Δ+2K)-1Dj)Δ-1DlDkΓlk-D(i(Δ+2K)-1DkΓj)k}.
This tensor mode is also called the second-order gravitational waves.Further, the vector part of (108) yields the initial value constraint and the evolution equation of the vector mode νj(2): (114)νi(2)=2Δ+2K{DiΔ-1DkΓk-Γi},∂η(a2νi(2))=2a2Δ+2K{DiΔ-1DkDlΓkl-DkΓik}.
Finally, scalar part of (108) are summarized as (115)2∂ηΨ(2)+2ℋΦ(2)-8πGφ2∂ηφ=Δ-1DkΓk,(116)Ψ(2)-Φ(2)=32(Δ+3K)-1{Δ-1DiDjΓij-13Γkk},(117)(-∂η2-5ℋ∂η+43Δ+4K)Ψ(2)-(2∂ηℋ+ℋ∂η+4ℋ2+13Δ)Φ(2)-8πGa2φ2∂V∂φ=Γ0-16Γkk,(118){∂η2+2(ℋ-∂η2φ∂ηφ)∂η-Δ-4K+2(∂ηℋ-∂η2φ∂ηφℋ)}Φ(2)=-Γ0-12Γkk+Δ-1DiDjΓij+(∂η-∂η2φ∂ηφ)Δ-1DkΓk-32{∂η2-(2∂η2φ∂ηφ-ℋ)∂η}(Δ+3K)-1×{Δ-1DiDjΓij-13Γkk},
where Γij:=γkjΓik and Γkk=γijΓij. Equation (118) is the second-order extension of (92), which is the master equation of scalar mode of the second-order cosmological perturbation in a universe filled with a single scalar field.Thus, we have a set of ten equations for the second-order perturbations of a universe filled with a single scalar field, (113)–(118). To solve this system of equations of the second-order Einstein equation, first of all, we have to solve the linear-order system. This is accomplished by solving (92) to obtain the potential Φ(1), φ1 is given through (87), and the tensor mode χij(1) is given by solving (90). Next, we evaluate the quadratic terms Γ0, Γi, and Γij of the linear-order perturbations, which are defined by (110)–(112). Then, using the information of (110)–(112), we estimate the source term in (118). If we know the two independent solutions to the linear-order master equation (92), we can solve (118) through the method using the Green functions. After constructing the solution Φ(2) to (118), we can obtain the second-order metric perturbation Ψ(2) through (116). Thus, we have obtained the second-order gauge-invariant perturbation φ2 of the scalar field through (115). Thus, the all scalar modes Φ(2), Ψ(2), and φ2 are obtained. Equation (117) is then used to check the consistency of the second-order perturbation of the Klein Gordon equation (105) as in Section 6.5.For the vector-mode,νi(1) of the first-order identically vanishes due to the momentum constraint (89) for the linear-order metric perturbations. On the other hand, in the second-order, we have evolution equation (114) of the vector mode νi(2) with the initial value constraint. This evolution equation of the second-order vector mode should be consistent with the initial value constraint, which is confirmed in Section 6.5. Equations (114) also imply that the second-order vector-mode perturbation may be generated by the mode couplings of the linear-order perturbations. As the simple situations, the generation of the second-order vector mode due to the scalar-scalar mode coupling is discussed in [44–47].The second-order tensor mode is also generated by the mode-coupling of the linear-order perturbations through the source term in (113). Note that (113) is almost same as (90) for the linear-order tensor mode, except for the existence of the source term in (113). If we know the solution to the linear-order Einstein equations (90) and (92), we can evaluate the source term in (113). Further, we can solve (113) through the Green function method. This leads the generation of the gravitational wave of the second order. Actually, in the simple situation where the first-order tensor mode neglected, the generation of the second-order gravitational waves discussed in some literature [48–54].


## 6.5. Consistency of Equations for Second-Order Perturbations

Now, we consider the consistency of the second-order perturbations of the Einstein equations (115)–(118) for the scalar modes, (114) for vector mode, and the Klein-Gordon equation (105). The consistency check of these equations are necessary to guarantee that the derived equations are correct, since the second-order Einstein equations have complicated forms owing to the quadratic terms of the linear-order perturbations that arise from the nonlinear effects of the Einstein equations.Since the first equation in (114) is the initial value constraint for the vector mode νi(2) and it should be consistent with the evolution equation, that is, the second equation of (114). these equations should be consistent with each other from the general arguments of the Einstein equation. Explicitly, these equations are consistent with each other if the equation (119)∂ηΓk+2ℋΓk-DlΓlk=0
is satisfied. Actually, through the first-order perturbative Einstein equations (87), (92), and (90), we can confirm (119). This is a trivial result from a general viewpoint, because the Einstein equation is the first class constrained system. However, this trivial result implies that we have derived the source terms Γi and Γij of the second-order Einstein equations consistently.Next, we consider (117). Through the second-order Einstein equations (115), (116), (118), and the background Klein-Gordon equation (76), we can confirm that (117) is consistent with the set of the background, first-order and other second-order Einstein equation if the equation (120)(∂η+2ℋ)DkΓk-DjDiΓij=0
is satisfied under the background and first-order Einstein equations. Actually, we have already seen that (119) is satisfied under the background and first-order Einstein equations. Taking the divergence of (119), we can immediately confirm (120). Then, (117) gives no information.Thus, we have seen that the derived Einstein equations of the second-order (114)–(118) are consistent with each other through (119). This fact implies that the derived source terms Γi and Γij of the second-order perturbations of the Einstein equations, which are defined by (111) and (112), are correct source terms of the second-order Einstein equations. On the other hand, for Γ0, we have to consider the consistency between the perturbative Einstein equations and the perturbative Klein-Gordon equation as seen below.Now, we consider the consistency of the second-order perturbation of the Klein-Gordon equation and the Einstein equations. The second-order perturbation of the Klein-Gordon equation is given by (105) with the source term (106). Since the vector mode νi(2) and tensor mode χij(2) of the second-order do not appear in the expressions (105) of the second-order perturbation of the Klein-Gordon equation, we may concentrate on the Einstein equations for scalar mode of the second order, that is, (115), (116), and (118) with the definitions (110)–(112) of the source terms. As in the linear case, the second-order perturbation of the Klein-Gordon equation should also be derived from the set of equations consisting of the second-order perturbations of the Einstein equations (115), (116), (118), the first-order perturbations of the Einstein equations (85), (87), (92), and the background Einstein equations (74). Actually, from these equations, we can show that the second-order perturbation of the Klein-Gordon equation is consistent with the background and the second-order Einstein equations if the equation (121)2(∂η+H)Γ0-DkΓk+ℋΓkk+8πG∂ηφΞ(K)=0
is satisfied under the background and the first-order Einstein equations. Further, we can also confirm (121) through the background Einstein equations (74), the scalar part of the first-order perturbation of the momentum constraint (87), and the evolution equations (92) and (90) for the first-order scalar and tensor modes in the Einstein equation.As shown in [28], the first-order perturbation of the Klein-Gordon equation is derived from the background and the first-order perturbations of the Einstein equation. In the case of the second-order perturbation, the Klein-Gordon equation (105) can be also derived from the background, the first-order, and the second-order Einstein equations. The second-order perturbations of the Einstein equation and the Klein-Gordon equation include the source terms Γ0, Γi, Γij, and Ξ(K) due to the mode-coupling of the linear-order perturbations. The second-order perturbation of the Klein-Gordon equation gives the relation (121) between the source terms Γ0, Γi, Γij, Ξ(K), and we have also confirmed that (121) is satisfied due to the background, the first-order perturbation of the Einstein equations, and the Klein-Gordon equation. Thus, the second-order perturbation of the Klein-Gordon equation is not independent of the set of the background, the first-order, and the second-order Einstein equations if we impose on the Einstein equation at any conformal time η. This also implies that the derived formulae of the source terms Γ0, Γi, Γij, and Ξ(K) are consistent with each other. In this sense, we may say that the formulae (110)–(112) and (106) for these source terms are correct.


## 7. Summary and Discussions

In this paper, we summarized the current status of the formulation of the gauge-invariant second-order cosmological perturbations. Although the presentation in this paper is restricted to the case of the universe filled by a single scalar field, the essence of the general framework of the gauge-invariant perturbation theory is transparent through this simple case. The general framework of the general relativistic higher-order gauge-invariant perturbation theory can be separated into three parts. First one is the general formulation to derive the gauge-transformation rules (18) and (19). Second one is the construction of the gauge-invariant variables for the perturbations on the generic background spacetime inspecting gauge-transformation rules (18) and (19) and the decomposition formula (36) and (37) for perturbations of any tensor field. Third one is the application of the above general framework of the gauge-invariant perturbation theory to the cosmological situations.To derive the gauge-transformation rules (18) and (19), we considered the general arguments on the Taylor expansion of an arbitrary tensor field on a manifold, the general class of the diffeomorphism which is wider than the usual exponential map, and the general formulation of the perturbation theory. This general class of diffeomorphism is represented in terms of the Taylor expansion (2) of its pull-back. As commented in Section 2.1, this general class of diffeomorphism does not form a one-parameter group of diffeomorphism as shown through (3). However, the properties (3) do not directly mean that this general class of diffeomorphism does not form a group. One of the key points of the properties of this diffeomorphism is the noncommutativity of generators ξ1a and ξ2a of each order. Although the expression of the nth-order Taylor expansion of the pull-back of this general class is discussed in [41], when we consider the situation of the nth-order perturbation, this noncommutativity becomes important [22]. Therefore, to clarify the properties of this general class of diffeomorphism, we have to take care of this noncommutativity of generators. Thus, there is a room to clarify the properties of this general class of diffeomorphism.Further, in Section2.3, we introduced a gauge choice 𝒳λ as an exponential map, for simplicity. On the other hand, we have the concept of the general class of diffeomorphism which is wider than the class of the exponential map. Therefore, we may introduce a gauge choice as one of the element of this general class of diffeomorphism. However, the gauge-transformation rules (18) and (19) will not be changed even if we generalize the definition of a each gauge choice as emphasized in Section 2.3. Although there is a room to sophisticate in logical arguments to derive the gauge-transformation rules (18) and (19), these are harmless to the development of the general framework of the gauge-invariant perturbation theory shown in Sections 2.3, 2.5, 3, and their application to cosmological perturbations in Section 4.As emphasized in Section2.5, 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 as (23). This is highly nontrivial assumption on a generic background spacetime. The procedure to accomplish the decomposition (23) completely depends on the details of the background spacetime. In spite of this nontriviality, this assumption is almost correct in some background spacetime [55–59]. Further, once we accept this assumption, we can develop the higher-order perturbation theory in an independent manner of the details of the background spacetime. We also expect that this general framework of the gauge-invariant perturbation theory is extensible to nth-order perturbation theory, since our procedure to construct gauge-invariant variables can be extended to the third-order perturbation theory with two-parameter [22]. Due to this situation, in [27], we propose the conjecture which states that the above assumption for the decomposition of the linear-order metric perturbation is correct for any background spacetime. We may also say that the most nontrivial part of our general framework of higher-order gauge-invariant perturbation theory is in the above assumption. Further, as emphasized in Section 5.1, we assumed the existence of some Green functions to accomplish the decomposition (23) and this assumption exclude some perturbative modes of the metric perturbations from our consideration, even in the case of cosmological perturbations. For example, homogeneous modes of perturbations are excluded in our current arguments of the cosmological perturbation theory. These homogeneous modes will be necessary to discuss the comparison with the arguments based on the long-wavelength approximation. Therefore, we have to say that there is a room to clarify even in the cosmological perturbation theory.Even if the assumption is correct on any background spacetime, the other problem is in the interpretations of the gauge-invariant variables. We have commented on the nonuniqueness in the definitions of the gauge-invariant variables through (81) and (98). This nonuniqueness in the definition of gauge-invariant variables also leads some ambiguities in the interpretations of gauge-invariant variables. On the other hand, as emphasized in Section 2.3, any observations and experiments are carried out only on the physical spacetime through the physical processes on the physical spacetime. For this reason, any direct observables in any observations and experiments should be independent of the gauge choice. Further, the nonuniqueness in the definitions the gauge-invariant variables expressed by (81) and (98) have the same form as the decomposition formulae (36) and (37). Therefore, if the statement that any direct observables in any observations and experiments is independent of the gauge choice is really true, this also confirms that the nonuniqueness of the definition of the gauge-invariant variables also have nothing to do with the direct observables in observations and experiments. These will be confirmed by the clarification of the relations between gauge-invariant variables and observables in experiments and observations. To accomplish this, we have to specify the concrete process of experiments and observations and clarify the problem of what are the direct observables in the experiments and observations and derive the relations between the gauge-invariant variables and observables in concrete observations and experiments. If these arguments are completed, we will be able to show that the gauge degree of freedom is just an unphysical degree of freedom and the nonuniqueness of the gauge-invariant variables has nothing to do with the direct observables in the concrete observation or experiment, and then, we will be able to clarify the precise physical interpretation of the gauge-invariant variables.For example, in the case of the CMB physics, we can easily see that the first-order perturbation of the CMB temperature is automatically gauge invariant from (36), because the background temperature of CMB is homogeneous. On the other hand, the decomposition formula (37) of the second order yields that the theoretical prediction of the second-order perturbation of the CMB temperature may depend on gauge choice, since we do know the existence of the first-order fluctuations as the temperature anisotropy in CMB. However, as emphasized above, the direct observables in observations should be gauge invariant and the gauge-variant term in (37) should be disappear in the direct observables. Therefore, we have to clarify how the gauge-invariant variables are related to the observed temperature fluctuations and the gauge-variant term disappeares in the observable.Although there are some rooms to accomplish the complete formulation of the second-order cosmological perturbation theory, we derived all the components of the second-order perturbation of the Einstein equation without ignoring any types modes (scalar-, vector-, tensor-types) of perturbations in the case of a scalar field system. In our formulation, any gauge fixing is not necessary and we can obtain all equations in the gauge-invariant form, which are equivalent to the complete gauge fixing. In other words, our formulation gives complete gauge-fixed equations without any gauge fixing. Therefore, equations obtained in a gauge-invariant manner cannot be reduced without physical restrictions any more. In this sense, the equations shown here are irreducible. This is one of the advantages of the gauge-invariant perturbation theory.The resulting Einstein equations of the second order show that any type of mode-coupling appears as the quadratic terms of the linear-order perturbations owing to the nonlinear effect of the Einstein equations, in principle. Perturbations in cosmological situations are classified into three types: scalar, vector, and tensor. In the second-order perturbations, we also have these three types of perturbations as in the case of the first-order perturbations. Furthermore, in the equations for the second-order perturbations, there are many quadratic terms of linear-order perturbations owing to the nonlinear effects of the system. Owing to these nonlinear effects, the above three types of perturbations couple with each other. In the scalar field system shown in this paper, the first-order vector mode does not appear due to the momentum constraint of the first-order perturbation of the Einstein equation. Therefore, we have seen that three types of mode-coupling appear in the second-order Einstein equations, that is, scalar-scalar, scalar-tensor, and tensor-tensor type of mode coupling. In general, all types of mode-coupling may appear in the second-order Einstein equations. Actually, in [28], we also derived all the components of the Einstein equations for a perfect fluid system and we can see all types of mode-coupling, that is, scalar-scalar, scalar-vector, scalar-tensor, vector-vector, vector-tensor, and tensor-tensor types mode-coupling, appear in the second-order Einstein equation, in general. Of course, in the some realistic situations of cosmology, we may neglect some modes. In this case, we may neglect some mode-coupling. However, even in this case, we should keep in mind the fact that all types of mode-couplings may appear in principle when we discuss the realistic situations of cosmology. We cannot deny the possibility that the mode-couplings of any type produces observable effects when the quite high accuracy of observations is accomplished.Even in the case of the single scalar field discussed in this paper, the source terms of the second-order Einstein equation show the mode-coupling of scalar-scalar, scalar-tensor, and the tensor-tensor types as mentioned above. Since the tensor mode of the linear order is also generated due to quantum fluctuations during the inflationary phase, the mode-couplings of the scalar-tensor and tensor-tensor types may appear in the inflation. If these mode-couplings occur during the inflationary phase, these effects will depend on the scalar-tensor ratior. If so, there is a possibility that the accurate observations of the second-order effects in the fluctuations of the scalar type in our universe also restrict the scalar-tensor ratio r or give some consistency relations between the other observations such as the measurements of the B-mode of the polarization of CMB. This will be a new effect that gives some information on the scalar-tensor ratio r.Furthermore, we have also checked the consistency between the second-order perturbations of the equations of motion of matter field and the Einstein equations. In the case of a scalar field, we checked the consistency between the second-order perturbations of the Klein-Gordon equation and the Einstein equations. Due to this consistency check, we have obtained the consistency relations between the source terms in these equationsΓ0, Γi, Γij, and Ξ(K), which are given by (119) and (121). We note that the relation (119) comes from the consistency in the Einstein equations of the second order by itself, while the relation (121) comes from the consistency between the second-order perturbation of the Klein-Gordon equation and the Einstein equation. We also showed that these relations between the source terms are satisfied through the background and the first-order perturbation of the Einstein equations in [28]. This implies that the set of all equations are self-consistent and the derived source terms Γ0, Γi, Γij, and Ξ(K) are correct. We also note that these relations are independent of the details of the potential of the scalar field.Thus, we have derived the self-consistent set of equations of the second-order perturbation of the Einstein equations and the evolution equations of matter fields in terms of gauge-invariant variables. As the current status of the second-order gauge-invariant cosmological perturbation theory, we may say that the curvature terms in the second-order Einstein tensor (69), that is, the second-order perturbations of the Einstein tensor, are almost completely derived although there remains the problem of homogeneous modes as mentioned above. After complete the problem of homogeneous modes, we have to clarify the physical behaviors of the second-order cosmological perturbation in the single scalar field system in the context of the inflationary scenario. This is the preliminary step to clarify the quantum behaviors of second-order perturbations in the inflationary universe. Further, we also have to carry out the comparison with the result by long-wavelength approximations. If these issues are completed, we may say that we have completely understood the properties of the second-order perturbation of the Einstein tensor. The next task is to clarify the nature of the second-order perturbation of the energy-momentum tensor through the extension to multi-fluid or multi-field systems. Further, we also have to extend our arguments to the Einstein Boltzmann system to discuss CMB physics, since we have to treat photon and neutrinos through the Boltzmann distribution functions. This issue is also discussed in some literature [13–21, 29, 30]. If we accomplish these extension, we will be able to clarify the Non-linear effects in CMB physics.Finally, readers might think that the ingredients of this paper is too mathematical as Astronomy. However, we have to emphasize that a high degree of the theoretical sophistication leads unambiguous theoretical predictions in many cases. As in the case of the linear-order cosmological perturbation theory, the developments in observations are also supported by the theoretical sophistication and the theoretical sophistication are accomplished motivated by observations. In this sense, now, we have an opportunity to develop the general relativistic second-order perturbation theory to a high degree of sophistication which is motivated by observations. 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. Therefore, I think that this opportunity is opened not only for observational cosmologists but also for theoretical and mathematical physicists.



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