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Advances in Astronomy
Volume 2010 (2010), Article ID 576273, 26 pages
Second-Order Gauge-Invariant Cosmological Perturbation Theory: Current Status
Optical and Infrared Astronomy Division, National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo 181-8588, Japan
Received 19 April 2010; Accepted 12 July 2010
Academic Editor: Eiichiro Komatsu
Copyright © 2010 Kouji Nakamura. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The general relativistic cosmological linear 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  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.  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 ; confirmation of the consistency between all components of the second-order Einstein equations and equations of motions . We also note that the radiation case has recently been discussed by treating the Boltzmann equation up to second order [29, 30] along the gauge-invariant manner of the above series of papers by the present author.
In this paper, we summarize the current status of this development of the second-order gauge-invariant cosmological perturbation theory through the simple system of a scalar field. Through this paper, we point out the problems which should be clarified and directions of the further development of the theoretical sophistication of the general relativistic higher-order perturbation theory, especially in cosmological perturbations. We expect that this sophistication will be also useful to discuss the theoretical predictions of non-Gaussianity in CMB and comparison with observations.
The organization of this paper is as follows. In Section 2, we review the general framework of the second-order gauge-invariant perturbation theory developed in [22, 23, 25, 26, 31]. This review also includes additional explanation not given in those papers. In Section 3, we present also the derivations of the second-order perturbation of the Einstein equation and the energy-momentum tensor from general point of view. For simplicity, in this paper, we only consider a single scalar field as a matter content. The ingredients of Sections 2 and 3 will be applicable to perturbation theory in any theory with general covariance, if the decomposition formula (23) for the linear-order metric perturbation is correct. In Section 4, we summarize the Einstein equations in the case of a background homogeneous isotropic universe, which are used in the derivation of the first- and second-order Einstein equations. In Section 5, the first-order perturbation of the Einstein equations and the Klein-Gordon equations are summarized. The derivation of the second-order perturbations of the Einstein equations and the Klein-Gordon equations, and their consistency are reviewed in Section 6. The final section, Section 7, is devoted to a summary and discussions.
2. General Framework of the General Relativistic Gauge-Invariant Perturbation Theory
In this section, we review the general framework of the gauge-invariant perturbation theory developed in [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 manifold . The gauge issue of general relativistic perturbation theories which we will discuss is related to the coordinate transformation. Therefore, we have to discuss the general form of the Taylor expansion without the explicit introduction of coordinate systems. Although we only consider the Taylor expansion of a scalar function , here, the resulting formula is extended to that for any tensor field on a manifold as in Appendix A. We have to emphasize that the general formula of the Taylor expansion shown here is the starting point of our gauge-invariant formulation of the second-order general relativistic perturbation theory.
The Taylor expansion of a function is an approximated form of at in terms of the variables at , where is in the neighborhood of . To derive the formula for the Taylor expansion of , we have to compare the values of at the different points on the manifold. To accomplish this, we introduce a one-parameter family of diffeomorphisms , where and . One example of a diffeomorphisms is an exponential map with a generator. However, we consider a more general class of diffeomorphisms.
The diffeomorphism induces the pull-back of the function and this pull-back enables us to compare the values of the function at different points. Further, the Taylor expansion of the function is given by Since this expression hold for an arbitrary smooth function , the function in (1) can be regarded as a dummy. Therefore, we may regard the Taylor expansion (1) to be the expansion of the pull-back of the diffeomorphism , rather than the expansion of the function .
According to this point of view, Sonego and Bruni  showed that there exist vector fields and such that the expansion (1) is given by without loss of generality (see Appendix A). Equation (2) is not only the representation of the Taylor expansion of the function , but also the definitions of the generators and . These generators of the one-parameter family of diffeomorphisms represent the direction along which the Taylor expansion is carried out. The generator is the first-order approximation of the flow of the diffeomorphism , and the generator is the second-order correction to this flow. We should regard the generators and to be independent. Further, as shown in Appendix A, the representation of the Taylor expansion of an arbitrary scalar function is extended to that for an arbitrary tensor field just through the replacement .
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, As noted in , if the second-order generator in (2) is proportional to the first-order generator in (2), the diffeomorphism is reduced to an exponential map. Therefore, one may reasonably doubt that forms a group except under very special conditions. However, we have to note that the properties (3) do not directly mean that does not form a group. There will be possibilities that 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 in (2) is a coordinate function. When two points in (2) are in the neighborhood of each other, we can apply a coordinate system (), which is denoted by , to an open set which includes these two points. Then, we can measure the relative position of these two points and in in terms of this coordinate system in through the Taylor expansion (2). In this case, we may regard that the scalar function in (2) is a coordinate function and (2) yields The second term in the right-hand side of (4) is familiar. This is regarded as the vector which point from the point to the points in the sense of the first-order correction as shown in Figure 1(a). However, in the sense of the second order, this vector may fail to point to . 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 . This second-order correction corresponds to that comeing from the exponential map which is generated by the vector field . However, this correction completely determined by the vector field . Even if we add this correction that comes from the exponential map, there is no guarantee that the corrected vector does point to in the sense of the second order. Thus, we have to add the new correction of the second order, in general.
Of course, without this correction , the vector which comes only from the exponential map generated by the vector field might point to the point . Actually, this is possible if we carefully choose the vector field 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 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 . 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  pointed out that there are two kinds of “gauges” in general relativity. Sachs called these two “gauges” as the first- and the second-kind of gauges, respectively. Here, we review these concepts of “gauge.”
2.2.1. First-Kind Gauge
The first-kind gauge is a coordinate system on a single manifold . Although this first-kind gauge is not important in this paper, we explain this to emphasize the “gauge” discuss in this paper is different from this first-kind gauge.
In the standard text book of manifolds (e.g., see ), the following property of a manifold is written: on a manifold, we can always introduce a coordinate system as a diffeomorphism from an open set to an open set (). This diffeomorphism , that is, coordinate system of the open set , is called gauge choice (of the first-kind). If we consider another open set in , we have another gauge choice for . If these two open sets and have the intersection , we can consider the diffeomorphism . This diffeomorphism is just a coordinate transformation: , which is called gauge transformation (of the first-kind) in general relativity.
According to the theory of a manifold, coordinate system are not on a manifold itself but we can always introduce a coordinate system through a map from an open set in the manifold to an open set of . For this reason, general covariance in general relativity is automatically included in the premise that our spacetime is regarded as a single manifold. The first-kind gauge does arise due to this general covariance. The gauge issue of the first-kind is represented by the question, which coordinate system is convenient? The answer to this question depends on the problem which we are addressing, that is, what we want to clarify. In some case, this gauge issue of the first-kind is an important. However, in many case, it becomes harmless if we apply a covariant theory on the manifold.
2.2.2. Second-Kind Gauge
The second-kind gauge appears in perturbation theories in a theory with general covariance. This notion of the second-kind “gauge” is the main issue of this paper. To explain this, we have to remind our selves what we are doing in perturbation theories.
First, in any perturbation theories, we always treat two spacetime manifolds. One is the physical spacetime . This physical spacetime is our nature itself and we want to describe the properties of this physical spacetime through perturbations. The other is the background spacetime . This background spacetime have nothing to do with our nature and this is a fictitious manifold which is prepared by us. This background spacetime is just a reference to carry out perturbative analyses. We emphasize that these two spacetime manifolds and are distinct. Let us denote the physical spacetime by and the background spacetime by , where is the metric on the physical spacetime manifold, , and is the metric on the background spacetime manifold, . Further, we formally denote the spacetime metric and the other physical tensor fields on by and its background value on by .
Second, in any perturbation theories, we always write equations for the perturbation of the physical variable in the form Usually, this equation is simply regarded as a relation between the physical variable and its background value , or as the definition of the deviation of the physical variable from its background value . However, (5) has deeper implications. Keeping in our mind that we always treat two different spacetimes, and , in perturbation theory, (5) is a rather curious equation in the following sense: the variable on the left-hand side of (5) is a variable on , while the variables on the right-hand side of (5) are variables on . 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, on the left-hand side of (5) is a field on , and . Similarly, we should regard the background value of and its deviation of from , which are on the right-hand side of (5), as fields on , and . Because (5) is regarded as an equation for field variables, it implicitly states that the points and are same. This represents the implicit assumption of the existence of a map , 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 on and , which is established by such a relation as (5), is not unique to the theory with general covariance. Rather, (5) involves the degree of freedom corresponding to the choice of the map . 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 and . Then, we can choose the identification map 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 . Indeed, we may identify with () instead of . 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 .
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  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 parameter for the perturbation. Further, we consider the -dimensional manifold , where and . The background spacetime and the physical spacetime are also submanifolds embedded in the extended manifold . Each point on is identified by a pair , where , and each point in is identified by .
Through this construction, the manifold is foliated by four-dimensional submanifolds of each , and these are diffeomorphic to and . The manifold has a natural differentiable structure consisting of the direct product of and . Further, the perturbed spacetimes for each must have the same differential structure with this construction. In other words, we require that perturbations be continuous in the sense that and are connected by a continuous curve within the extended manifold . 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 on the physical spacetime for the physical variables on . The field equation (6) formally represents the Einstein equation for the metric on and the equations for matter fields on . If a tensor field is given on each , is automatically extended to a tensor field on by , where . In this extension, the field equation (6) is regarded as an equation on the extended manifold . Thus, we have extended an arbitrary tensor field and the field equations (6) on each to those on the extended manifold .
Tensor fields on obtained through the above construction are necessarily “tangent” to each . To consider the basis of the tangent space of , we introduce the normal form and its dual, which are normal to each in . These are denoted by and , respectively, and they satisfy . The form and its dual, , are normal to any tensor field extended from the tangent space on each through the above construction. The set consisting of , , and the basis of the tangent space on each is regarded as the basis of the tangent space of .
Now, we define the perturbation of an arbitrary tensor field . We compare on with on , and it is necessary to identify the points of with those of 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 such that . Following the paper of Bruni et al. , we introduce a gauge choice as an one-parameter groups of diffeomorphisms, that is, an exponential map, for simplicity. We denote the generator of this exponential map by . This generator is decomposed by the basis on which are constructed above. Although the generator should satisfy some appropriate properties , the arbitrariness of the gauge choice is represented by the tangential component of the generator to .
The pull-back , which is induced by the exponential map , maps a tensor field on the physical manifold to a tensor field on the background spacetime. In terms of this generator , the pull-back is represented by the Taylor expansion where . Because , we may regard the equation as an equation on the background spacetime , where is the background value of the physical variable of . Once the definition of the pull-back of the gauge choice is given, the first- and the second-order perturbations and of a tensor field under the gauge choice are simply given by the expansion with respect to the infinitesimal parameter . Comparing (8) and (9), we define the first- and the second-order perturbations of a physical variable under the gauge choice by We note that all variables in (9) are defined on .
Now, we consider two different gauge choices based on the above understanding of the second-kind gauge choice. Suppose that and are two exponential maps with the generators and on , respectively. In other words, and are two gauge choices (see Figure 2). Then, the integral curves of each and in are the orbits of the actions of the gauge choices and , respectively. Since we choose the generators and so that these are transverse to each everywhere on , the integral curves of these vector fields intersect with each . 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 of and are different, these point identification maps and are regarded as two different gauge choices.
We next introduce the concept of gauge invariance. In particular, in this paper, we consider the concept of order-by-order gauge invariance . Suppose that and are two different gauge choices which are generated by the vector fields and , respectively. These gauge choices also pull back a generic tensor field on to two other tensor fields, and , for any given value of . In particular, on , we now have three tensor fields associated with a tensor field ; one is the background value of , and the other two are the pulled-back variables of from to by the two different gauge choices Here, we have used (9). Because and are gauge choices which map from to , and are the different representations on in the two different gauges of the same perturbed tensor field on . The quantities and in (11) and (12) are the perturbations of in the gauges and , respectively. We say that the th-order perturbation of is order-by-order gauge-invariant if and only if for any two gauges and the following holds:
Now, we consider the gauge transformation rules between different gauge choices. In general, the representation on of the perturbed variable on depends on the gauge choice . If we employ a different gauge choice, the representation of on may change. Suppose that and are different gauge choices, which are the point identification maps from to , and the generators of these gauge choices are given by and , respectively. Then, the change of the gauge choice from to is represented by the diffeomorphism This diffeomorphism is the map for each value of . The diffeomorphism does change the point identification, as expected from the understanding of the gauge choice discussed above. Therefore, the diffeomorphism is regarded as the gauge transformation .
The gauge transformation induces a pull-back from the representation of the perturbed tensor field in the gauge choice to the representation in the gauge choice . Actually, the tensor fields and , which are defined on , are connected by the linear map as 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 in the form where the vector fields and are the generators of the gauge transformation (see (2)).
Comparing the representation (16) of the Taylor expansion in terms of the generators and of the pull-back and that in terms of the generators and of the pull-back (), we readily obtain explicit expressions for the generators and of the gauge transformation in terms of the generators and of each gauge choices as follows: Further, because the gauge transformation is a map within the background spacetime , the generator should consist of vector fields on . This can be satisfied by imposing some appropriate conditions on the generators and .
Here, we should comment on the gauge choice in the above explanation. We have introduced an exponential map (or ) as the gauge choice, for simplicity. However, this simplified introduction of as an exponential map is not essential to the gauge transformation rules (18) and (19). Actually, we can generalize the diffeomorphism 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 , the representation (8) of the pulled-back variable , the representations of the perturbations (10), and the relations (17) between generators of , , and will be changed. However, the gauge transformation rules (18) and (19) are direct consequences of the generic Taylor expansion (16) of . Generality of the representation of the Taylor expansion (16) of implies that the gauge transformation rules (18) and (19) will not be changed, even if we generalize the each gauge choice . Further, the relations (17) between generators also imply that, even if we employ simple exponential maps as gauge choices, both of the generators and 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 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 is our nature itself, while there is no background spacetime in our nature. The background spacetime is a fictitious spacetime and it has nothing to do with our nature. Since the gauge choice just gives a relation between and , the gauge choice also has nothing to do with our nature. On the other hand, any observations and experiments are carried out only on the physical spacetime through the physical processes on the physical spacetime . Therefore, any direct observables in any observations and experiments should be independent of the gauge choice , that is, should be gauge-invariant. Keeping this fact in our mind, the gauge transformation rules (18) and (19) imply that the perturbations and include unphysical degree of freedom, that is, gauge degree of freedom, if these perturbations are transformed as (18) or (19) under the gauge transformation . If the perturbations and 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 .
2.4. Coordinate Transformations Induced by the Second Kind Gauge Transformation
In many literature, gauge degree of freedom is regarded as the degree of freedom of the coordinate transformation. In the linear-order perturbation theory, these two degree of freedom are equivalent with each other. However, in the higher-order perturbations, we should regard that these two degree of freedom are different. Although the essential understanding of the gauge degree of freedom (of the second-kind) is as that explained above, the gauge transformation (of the second-kind) also induces the infinitesimal coordinate transformation on the physical spacetime 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 on the “background spacetime” , where are open sets on the background spacetime and are diffeomorphisms from to (). The coordinate system is the set of the collection of the pair of open sets and diffeomorphism . If we employ a gauge choice , we have the correspondence of and . Together with the coordinate system on , this correspondence between and induces the coordinate system on . Actually, for each is an open set of . Then, becomes a diffeomorphism from an open set to . This diffeomorphism induces a coordinate system of an open set on .
When we have two different gauge choices and , and become different coordinate systems on . We can also consider the coordinate transformation from the coordinate system to another coordinate system . Since the gauge transformation is induced by the diffeomorphism defined by (14), the induced coordinate transformation is given by 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
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 , we proposed a procedure to construct gauge-invariant variables of higher-order perturbations . This proposal is as follows. First, we decompose a linear-order metric perturbation into its gauge-invariant and variant parts. The procedure for decomposing linear-order metric perturbations is extended to second-order metric perturbations, and we can decompose the second-order metric perturbation into gauge-invariant and variant parts. Then, we can define the gauge-invariant variables for the first- and second-order perturbations of an arbitrary field other than the metric by using the gauge variant parts of the first- and second-order metric perturbations. Although the procedure for finding gauge-invariant variables for linear-order metric perturbations is highly nontrivial, once we know this procedure, we can easily define the gauge-invariant variables of a higher-order perturbation through a simple extension of the procedure for the linear-order perturbations.
Now, we review the above strategy to construct gauge-invariant variables. To consider a metric perturbation, we expand the metric on the physical spacetime , which is pulled back to the background spacetime using a gauge choice in the form given in (9): where is the metric on . Of course, the expansion (22) of the metric depends entirely on the gauge choice . Nevertheless, henceforth, we do not explicitly express the index of the gauge choice in an expression if there is no possibility of confusion.
Our starting point to construct gauge-invariant variables is the assumption that we already know the procedure for finding gauge-invariant variables for the linear metric perturbations. Then, a linear metric perturbation is decomposed as where and are the gauge-invariant and variant parts of the linear-order metric perturbations, that is, under the gauge transformation (18), these are transformed as 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,
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 . 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 . According to the gauge transformation rule (19), the second-order metric perturbation is transformed as under the gauge transformation . Although this gauge transformation rule is slightly complicated, inspecting this gauge transformation rule, we first introduce the variable defined by Under the gauge transformation , the variable is transformed as The gauge transformation rule (28) is identical to that for a linear metric perturbation. Therefore, we may apply the above procedure to decompose into and when we decompose of the components of the variable . Then, can be decomposed as where is the gauge-invariant part of the variable , or equivalently, of the second-order metric perturbation , and is the gauge variant part of , that is, the gauge variant part of . Under the gauge transformation , the variables and are transformed as respectively. Thus, once we accept the assumption (23), the second-order metric perturbations are decomposed as where and are the gauge-invariant and variant parts of the second order metric perturbations, that is,
Furthermore, as shown in , using the first- and second-order gauge variant parts, and , of the metric perturbations, the gauge-invariant variables for an arbitrary field other than the metric are given by It is straightforward to confirm that the variables 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 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 . 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  in Section 3.2. Finally, we consider the first- and the second-order the Einstein equations in Section 3.3.
3.1. Perturbations of the Einstein Curvature
The relation between the curvatures associated with the metrics on the physical spacetime and the background spacetime is given by the relation between the pulled-back operator of the covariant derivative associated with the metric on and the covariant derivative associated with the metric on . The pulled-back covariant derivative depends on the gauge choice . The property of the derivative operator as the covariant derivative on is given by where is the pull-back of the metric on , which is expanded as (22). In spite of the gauge dependence of the operator , we simply denote this operator by , because our calculations are carried out only on in the same gauge choice . Further, we denote the pulled-back metric on by , as mentioned above.
Since the derivative operator () may be regarded as a derivative operator on that satisfies the property (38), there exists a tensor field on such that where is an arbitrary one-form on . From the property (38) of the covariant derivative operator on , the tensor field on is given by where is the inverse of (see Appendix B). We note that the gauge dependence of the covariant derivative appears only through . The Riemann curvature on , which is also pulled back to , is given by  where is the Riemann curvature on . The perturbative expression for the curvatures are obtained from the expansion of (41) through the expansion of .
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 . We also derived the perturbative form of the divergence of an arbitrary tensor field of second rank to check the perturbative Bianchi identities in . In this paper, we only present the perturbative expression for the Einstein tensor, and its derivations in Appendix B.
We expand the Einstein tensor on as As shown in Appendix B, each order perturbation of the Einstein tensor is given by where We note that and 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 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 and defined by (45)–(47) satisfy the identities for arbitrary tensor fields and , respectively. We can directly confirm these identities without specifying arbitrary tensors and 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 and are also useful when we check whether the derived components of (45) and (46) are correct.
3.2. Perturbations of the Energy Momentum Tensor and Klein-Gordon Equation
Here, we consider the perturbations of the energy momentum tensor of the equation of motion. As a model of the matter field, we only consider the scalar field, for simplicity. Then, equation of motion for a scalar field is the Klein-Gordon equation.
The energy momentum tensor for a scalar field is given by where is the potential of the scalar field . We expand the scalar field as where is the background value of the scalar field . Further, following to the decomposition formulae (34) and (35), each order perturbation of the scalar field is decomposed as where and 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 and the inverse metric, the energy momentum tensor (52) is also expanded as The background energy momentum tensor is given by the replacement in (52). Further, through the decompositions (23), (32), (54), the perturbations of the energy momentum tensor and are also decomposed as where the gauge-invariant parts and of the first and the second order are given by 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 Through the perturbative expansions (53) and (22), the Klein-Gordon equation (60) is expanded as is the background Klein-Gordon equation The first- and the second-order perturbations and are also decomposed into the gauge-invariant and the gauge-variant parts as where 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
We should note that, in , 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, Then, the perturbative Einstein equation is given by at linear order and at second order. These explicitly show that, order-by-order, the Einstein equations are necessarily given in terms of gauge-invariant variables only.
Together with (66), we have seen that the first- and the second-order perturbations of the Einstein equations and the Klein-Gordon equation are necessarily given in gauge-invariant form. This implies that we do not have to consider the gauge degree of freedom, at least in the level where we concentrate only on the equations of the system.
We have reviewed the general outline of the second-order gauge-invariant perturbation theory. We also note that the ingredients of this section are independent of the explicit form of the background metric , except for the decomposition assumption (23) for the linear-order metric perturbations and are valid not only in cosmological perturbation case but also the other generic situations if (23) is correct. Within this general framework, we develop a second-order cosmological perturbation theory in terms of the gauge-invariant variables.
4. Cosmological Background Spacetime and Equations
The background spacetime considered in cosmological perturbation theory is a homogeneous, isotropic universe that is foliated by the three-dimensional hypersurface , which is parametrized by . Each hypersurface of is a maximally symmetric three-space , and the spacetime metric of this universe is given by where is the scale factor, is the metric on the maximally symmetric 3-space with curvature constant , and the indices 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 where we assumed that the scalar field is homogeneous and are defined as
The background Einstein equations for this background spacetime filled with the single scalar field are given by We also note that (74) lead to 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 for the scalar field The Klein-Gordon equation (76) is also derived from the Einstein equations (74). This is a well known fact and is just due to the Bianchi identity of the background spacetime. However, these types of relation are useful to check whether the derived system of equations is consistent.
5. Equations for the First-Order Cosmological Perturbations
On the cosmological background spacetime in the last section, we develop the perturbation theory in the gauge-invariant manner. In this section, we summarize the first-order perturbation of the Einstein equation and the Klein-Gordon equations. In Section 5.1, we show that the assumption on the decomposition (23) of the linear-order metric perturbation is correct. In Section 5.2, we summarize the first-order perturbation of the Einstein equation. Finally, in Section 5.3, we show the first-order perturbation of the Klein-Gordon equation.
5.1. Gauge-Invariant Metric Perturbations
Here, we consider the first-order metric perturbation 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 , , and , where is the covariant derivative associated with the metric and is the curvature constant of the maximally symmetric three space. Next, we consider the decomposition of the linear-order metric perturbation as where , , and satisfy the properties The gauge-transformation rules for the variables , , , , , and are derived from (25). Inspecting these gauge-transformation rules, we define the gauge-variant part in (23): We can easily check this vector field satisfies (24). Subtracting gauge variant-part from , we have the gauge-invariant part in (23): where the properties 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 , , and . As shown in [25, 26], this assumption is necessary to guarantee the one to one correspondence between the variables and , but excludes some perturbative modes of the metric perturbations which belong to the kernel of the operator , , and from our consideration. For example, homogeneous modes belong to the kernel of the operator 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 and of the first-order metric perturbation, we can define a vector field by which is gauge invariant. Then, we can rewrite the decomposition formula (23) for the linear-order metric perturbation as where we have defined new gauge invariant variable by . Clearly, is gauge-invariant and the vector field satisfies (24). In spite of this nonuniqueness, we specify the components of the tensor as (80), which is the gauge-invariant part of the linear-order metric perturbation associated with the longitudinal gauge.
The non-uniqueness of the definitions of gauge-invariant variables is related to the “gauge-fixing” for the linear-order metric perturbations. Due to this non-uniqueness, we can consider the gauge-fixing in the first-order metric perturbation from two different points of view. The first point of view is that the gauge-fixing is to specify the gauge-variant part . For example, the longitudinal gauge is realized by the gauge fixing . Due to this gauge fixing , 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 in (81). In this point of view, we do not specify the vector field . Instead, we have to specify the gauge-invariant vector or equivalently to specify the gauge-invariant metric perturbation without specifying so that the first-order metric perturbation coincides with the gauge-invariant variables when we fix the gauge so that . 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 , which is defined by (45), we first derive the components of the tensor , which is defined in (48) with and its component (80). These components are summarized in [25, 26].
From (45), the component of are summarized as Straightforward calculations show that these components of the first-order gauge-invariant perturbation of the Einstein tensor satisfies the identity (50). Although this confirmation is also possible without specification of the tensor , the confirmation of (50) through the explicit components (82) implies that we have derived the components of consistently.
Next, we summarize the first-order perturbation of the energy momentum tensor for a scalar field. Since, at the background level, we assume that the scalar field is homogeneous as (72), the components of the gauge-invariant part of the first-order energy-momentum tensor are given by 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 From (82)–(84) and (85), the components of scalar parts of the linearized Einstein equation (68) are given as  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 shows that The equation for the tensor mode is given by
Combining (86) and (88), we eliminate the potential term of the scalar field and thereby obtain that Further, using (87) to express in terms of and , we also eliminate in (91). Hence, we have 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 .
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 summarized in [25, 26], the gauge-invariant part of the first-order Klein-Gordon equation defined by (64) is given by
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) . Hence, the first-order perturbation of the Klein-Gordon equation is not independent of the background and the first-order perturbation of the Einstein equation. Therefore, from the viewpoint of the Cauchy problem, any information obtained from the first-order perturbation of the Klein-Gordon equation should also be obtained from the set of the background and the first-order the Einstein equation, in principle.
6. Equations for the Second-Order Cosmological Perturbations
Now, we develop the second-order perturbation theory on the cosmological background spacetime in Section 4 within the general framework of the gauge-invariant perturbation theory reviewed in Section 2. Since we have already confirmed the important step of our general framework, that is, the assumption for the decomposition (23) of the linear-order metric perturbation is correct. Hence, the general framework reviewed in Section 2 is applicable. Applying this framework, we define the second-order gauge-invariant variables of the metric perturbation in Section 6.1. In Section 6.2, we summarize the explicit components of the gauge-invariant parts of the second-order perturbation of the Einstein tensor. In Section 6.3, we summarize the explicit components of the second-order perturbation of the energy-momentum tensor and the Klein-Gordon equations. Then, in Section 6.4, we derive the second-order Einstein equations in terms of gauge-invariant variables. The resulting equations have the source terms which constitute of the quadratic terms of the linear-order perturbations. Although these source terms have complicated forms, we give identities which comes from the consistency of all the second-order perturbations of the Einstein equation and the Klein-Gordon equation in Section 6.5.
6.1. Gauge-Invariant Metric Perturbations
First, we consider the components of the gauge-invariant variables for the metric perturbation of second order. The variable defined by (27) is transformed as (28) under the gauge transformation and we may regard the generator defined by (29) as an arbitrary vector field on from the fact that the generator 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 in (32) as where and satisfy the equations The gauge-invariant variables and are the scalar mode perturbations of second order, and and 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 where is gauge invariant and is transformed as under the gauge transformation . This may be different from the vector in (81). By the replacement (96), the second-order metric perturbation (32) is given in the form where we defined Here, the vector field in (100) constitute of some components of gauge-invariant second-order metric perturbation like in (81). The tensor field is manifestly gauge invariant. The gauge transformation rule of the new gauge-variant part of the second-order metric perturbation is given by Although (98) is similar to (32), the tensor fields and are different from each other. Thus, the definition of the gauge-invariant variables for the second-order metric perturbation is not unique in a more complicated way than the linear order. This nonuniqueness of gauge-invariant variables for the metric perturbations propagates to the definition (34) and (35) of the gauge-invariant variables for matter fields.
In spite of the existence of infinitely many definitions of the gauge-invariant variables, in this paper, we consider the components of given by (94). Equation (94) corresponds to the second-order extension of the longitudinal gauge, which is called Poisson gauge .
6.2. Einstein Tensor
First, we evaluate the term in the Einstein equation (69). Because the components (94) of are obtained through the replacements in the components (80) of , we easily obtain the components of through the replacements (102) in (82).
From (80), we can derive the components of 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 are summarized as