Effective action approach to cosmological perturbations in dark energy and modified gravity

Effective action approach to cosmological perturbations in dark energy and modified gravity

Richard A. Battye a    and Jonathan A. Pearson richard.battye@manchester.ac.uk jp@jb.man.ac.uk Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, U.K
Abstract

In light of upcoming observations modelling perturbations in dark energy and modified gravity models has become an important topic of research. We develop an effective action to construct the components of the perturbed dark energy momentum tensor which appears in the perturbed generalized gravitational field equations, for linearized perturbations. Our method does not require knowledge of the Lagrangian density of the dark sector to be provided, only its field content. The method is based on the fact that it is only necessary to specify the perturbed Lagrangian to quadratic order and couples this with the assumption of global statistical isotropy of spatial sections to show that the model can be specified completely in terms of a finite number of background dependent functions. We present our formalism in a coordinate independent fashion and provide explicit formulae for the perturbed conservation equation and the components of for two explicit generic examples: (i) the dark sector does not contain extra fields, and (ii) the dark sector contains a scalar field and its first derivative . We discuss how the formalism can be applied to modified gravity models containing derivatives of the metric, curvature tensors, higher derivatives of the scalar fields and vector fields.

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August 30, 2019

1 Introduction

The standard model of cosmology uses General Relativity (GR) to describe gravitational interactions, an homogeneous/isotropic FRW metric to describe the geometry and matter content of cold dark matter (CDM)/photons/baryons to describe its constituents. Observations of the cosmic microwave background, supernovae, baryon acoustic oscillations, gravitational lensing and structure formation point to the existence of an additional component dubbed “dark energy”, or a modification to gravity, which needs to be introduced to explain the the observed acceleration 1933AcHPh…6..110Z ; Riess:1998cb ; Perlmutter:1998np ; Riess:1998dv ; 0067-0049-192-2-18 .

The simplest explanation is a cosmological constant, , and the standard paradigm is the CDM model. However, there is still considerable flexibility for the explanation to be something radically different. In general, we can model all possible theories as an extra “dark sector” component to the stress-energy-momentum tensor. The structure of the gravitational field equations means that this extra component can be used to model either “exotic matter” with an equation of state or a modification to GR (i.e. modifying exactly how gravity responds to the presence of matter). Constructing viable models of modified gravity has become an important task with the discovery of the acceleration of the Universe; some modified gravity models may also be able to account for observations which otherwise require dark matter.

One way to model the dark sector is “Lagrangian engineering”: write down ever more complicated new theories with a view of constraining their parameters and free functions to fit observation with the hope that self-accelerating solutions can be found. Theories where explicit forms of dark energy are written down also fall into this category. They include TeVeS PhysRevD.70.083509 ; Skordis:2009bf , Einstein-æther PhysRevD.75.044017 , Brans-Dicke PhysRev.124.925 , Horndeski springerlink:10.1007/BF01807638 ; Kobayashi:2011nu ; Charmousis:2011bf and gravities Capozziello:2003tk ; PhysRevD.70.043528 , quintessence PhysRevD.26.2580 ; Copeland:2006wr , -essence ArmendarizPicon:1999rj ; ArmendarizPicon:2000ah and Gallileons PhysRevD.79.064036 . This is by no means an exhaustive list, and we have made no mention of the plethora of higher dimensional theories. The reader is directed to the recent extensive review of modified gravity theories Clifton:2011jh .

Given this proliferation of modified gravity and dark energy models, it would be a good idea to construct a generic way of parameterising deviations from the GR+CDM picture and various suggestions have been made 0004-637X-506-2-485 ; Weller:2003hw ; PhysRevD.69.083503 ; PhysRevD.76.104043 ; PhysRevD.77.103524 ; 1475-7516-2008-04-013 ; PhysRevD.79.123527 ; PhysRevD.81.083534 ; PhysRevD.81.104023 ; Appleby:2010dx ; Hojjati:2011ix ; Baker:2011jy ; Baker:2011wt to do this for perturbations. This approach is called the “Parameterized-Post-Friedmannian” (PPF) framework, in analogy to the well established Parameterized-Post-Newtonian (PPN) framework which was invented for Solar System tests of General Relativity will_PPN . However, as we describe below, to date no generic approach has been proposed which has a physical basis.

In this paper we describe a new way of parameterizing perturbations in the dark sector requiring as an assumption knowledge of the field content. We do not assume a specific Lagrangian density, but we are able to model the possible effects on observations by using an effective action to compute the possible perturbations to the gravitational field equations. This is done by limiting the action to terms which are quadratic in the perturbed field content which is sufficient to model linearized perturbations, and assuming that the spatial sections are isotropic. In this paper we only consider the case that the dark sector contains first order derivatives of scalar fields and the metric; we will discuss higher order derivatives and vector fields in a follow-up paper.

Our theories will be completely general allowing for all possible degrees of freedom. Initially we do not impose reparametrization, or gauge, invariance. This is something which we would expect of a fundamental theory of dark energy, but not necessarily one for which the field content is just a coarse grained description. We will find that this can lead to an phenomological vector degree of freedom, . In the elastic dark energy theory Battye:2005mm ; Battye:2006mb ; PhysRevD.76.023005 ; Battye:2009ze , which can be used to describe the effects of a dark energy component composed of a topological defect lattice, this represents a perturbation of the elastic medium from its equilibrium. We will see that the imposition of reparametrization invariance substantially reduces the number of free functions.

We note that many authors have consider possible dark energy theories which are effective Lagrangians in the traditional sense, that is, the terms in the Lagrangian represent an expansion of field operators which are suppressed at low energies Weinberg:2008hq ; Cheung:2007st ; Creminelli:2008wc . Our approach here is sufficiently similar to this approach to share the epitaph “effective action”, but it is completely different in many ways. It is completely classical and is in no sense an expansion energy scale. Moreover, it is just an effective action for the perturbations, and in no sense represents the full field theory of the dark energy.

2 Approaches to parameterizing dark sector perturbations

In this section we will provide a brief review of current approaches to studying generalized gravitational theories, concluding with a short discussion on the generalities of our approach.

2.1 Parameterized post-Friedmannian approach

A popular way to parameterize the dark sector takes an “observational” perspective. One can modify the equations governing the predictions of the Newtonian gravitational potential and shear by introducing extra functions space and time into the relevant equations and then parametrizing these extra functions in an ad hoc fashion. Since it is possible to explicitly observe and via the evolution structure and gravitational shear PhysRevD.69.083503 ; PhysRevD.76.023507 ; PhysRevD.81.083534 ; PhysRevD.81.104023 ; Hojjati:2011ix (see also the more recent papers Dossett:2011tn ; Kirk:2011sw ; Laszlo:2011sv ), one can then compare them with the predictions of particular ad hoc choice and determine constraints on the deviation of a particular parameter from its value in General Relativity.

One way of doing this is by modifying the Poisson and gravitational slip equations, introducing two scale- and time-dependent functions, and . The Poisson and gravitational-slip equations then become

(1)

where is the comoving density perturbation, the density contrast, the velocity divergence field, the equation of state and is the anisotropic stress. When these equations are derived in GR one finds that , and so if, by comparison to data, either of these parameters are shown to be inconsistent with unity, then deviations from GR can be established. In Baker:2011jy ; Zuntz:2011aq it was shown that the two functions are not necessarily independent: they can be linked by the perturbed Bianchi identity, depending on the structure of the underlying theory.

2.2 Generalized gravitational field equations

Another way to investigate the dark sector takes a more theoretical standpoint, and is based on a more consistent modification of the governing field equations. The method stems from the fact that any modified gravity theory or model of dark energy can be encapsulated by writing the generalized gravitational field equations

(2)

where is the Einstein tensor calculated from the spacetime metric, is the energy-momentum tensor of all known species (radiation, Baryons, CDM etc) and is a tensor which contains all unknown contributions to the gravitational field equations, which we call the dark energy-momentum tensor PhysRevD.76.104043 ; PhysRevD.77.103524 ; PhysRevD.79.123527 .

Because the Bianchi identity automatically holds for the Einstein tensor, , in the standard case where the known and unknown sectors are decoupled (that is ) we have the conservation law

(3)

This represents a constraint equation on the extra parameters and functions that may appear in a parameterization of the dark sector at the level of the background. At perturbed order, the parameterization of is constrained by the perturbed conservation law

(4)

The shortcoming of this approach is that one must supply the components of . Skordis PhysRevD.79.123527 does this by expanding the components in terms of pseudo derivative operators acting upon gauge invariant combinations of metric perturbations, by imposing the principles that (a) the field equations remain at most second order and (b) the equations are gauge-form invariant. A particular form of these components were considered in PhysRevD.79.123527 :

(5a)
(5b)

where is a set of pseudo differential operators and are gauge invariant combinations of perturbed metric variables. The possible form that the elements of can take is constrained by the perturbed Bianchi identity. For instance, it was shown that is one of the sufficient consistency relations. A generalized version of this method can be found in Baker:2011jy ; Clifton:2011jh .

This scheme provides a way to compute and constrain observables without ever having to write down an explicit theory for the dark sector. There appears to be, however, a weakness in the current formulation of this strategy: there does not seem to be a physically obvious way to interpret the ; for example, if one were to find that is “required” for consistency with observational data, what does that impose physically upon the system? It is exactly this issue we address in this paper.

2.3 Effective action approach

The generalized gravitational field equations (2) can be constructed from an action

(6)

The matter Lagrangian density contains all known matter fields (e.g. baryons, photons) and is used to construct the known energy momentum tensor , and the dark sector Lagrangian density contains all “unknown” contributions to the gravitational sector, and will be used to construct the dark energy momentum tensor . One can define

(7)

The dark sector Lagrangian may contain known fields in an unknown configuration or extra fields, but of course we do not know a priori what the dark sector Lagrangian density is.

Two simple cases are (i) a slowly-rolling minimally coupled scalar field parameterized by a potential, , and (ii) a modified gravity model parameterized by a free function of the Ricci scalar, . There are restrictions on the form of both of these functions to achieve acceleration, but once they have been applied there is still considerable freedom in the choices of and and wide ranges of behaviour of the expansion history, , can be arranged for particular choices of the functions. One would expect this to be the case in any self consistent dark energy model compatible with FRW metric and therefore it might seen reasonable to make the assumption that the dark stress-energy-momentum tensor where is in 1-1 correspondence with . The important question, which we are concerned with, is how to parametrize the perturbations in a general way based on some general physical principle. In this way our approach is similar to that discussed in section 2.2

The overall ethos which we advocate is to write down an effective action, inspired by the approach that is taken in particle physics (see, e.g. PhysRevD.8.1226 ) where, for example, the most general modifications to the standard model are written down for a given field content that are compatible with some assumed symmetry/symmetries. Then all the free coefficients are constrained by experiment. In our case, we will specify the field content of the dark sector, for example, scalar or vector fields, and write down a general quadratic Lagrangian density for the perturbed field variables which is sufficient to generate equations of motion for linearized perturbations. We will also make the assumption that the spatial sections are isotropic which substantially reduces the number of free coefficients.

3 Formalism

3.1 Second order Lagrangian

The underlying principle behind our method is to write down a effective Lagrangian density for perturbed field variables. If our theory is constructed from a set of field variables , then we write each field variable as a linearized perturbation about some background value,

(8)

The action for the perturbed field variables is computed by integrating a Lagrangian density which is quadratic in the perturbed field variables. If there are “N” perturbed field variables, the effective Lagrangian density for the perturbed field variables is given by

(9)

where is a set of arbitrary functions only depending on the background field variables; clearly, . To obtain the equation of motion of the perturbed field variables we must induce some variation in the and subsequently demand that is independent of these variations. If we vary the perturbed field variables with a variational operator ,

(10)

then the effective Lagrangian will vary according to , where

(11)

The demand that the effective Lagrangian is independent of these variations is the statement that

(12)

that is,

(13)

These equations provide the equations of motion of the perturbed field variables. We will now show how to obtain the effective action for perturbations by directly perturbing the background action.

We will consider an action of the form

(14)

where is the determinant of the spacetime metric, , and is the Lagrangian density, which contains all fields in the theory. It will be useful to write the first and second variations of the action as

(15)

where “” is a useful measure-weighted pseudo-operator introduced in 0264-9381-11-11-010 ; Battye:1998zk and is defined by

(16)

We will only consider first perturbations of the field content of a theory. For the action (14) we can use the well known result

(17)

to show that to quadratic order in the perturbations that the integrands in (15) are given by

(18a)
(18b)

We treat the integrand of the second variation of the action, i.e. , as the effective Lagrangian, , for linearized perturbations, and it is called the second order Lagrangian. The final term of (18b) is an effective mass-term for the gravitational fluctuations which is always present even when the field which constitutes the dark sector does not vary, i.e. when .

Although we will be providing various explicit examples later on in the paper, we will briefly discuss how to write down once the field content has been specified. If the field content is , then we write , and then is written down by writing all quadratic interactions of the perturbed fields with appropriate coefficients,

(19)

Notice that we have moved from having complete ignorance of how the fields combine to construct the Lagrangian density to only requiring 3 “background” functions, to be able to write down. Typically, we would expect these functions to be specified in terms of the scale factor .

The theories we consider contribute to the gravitational field equations via the dark energy-momentum tensor, , which we define in the usual way, (7). The indices on the dark energy momentum tensor are symmetric by construction,

(20)

where tensor indices are symmetrised as . The dark energy-momentum tensor above can be directly perturbed to give

(21)

where are the perturbed field variables. This can be written in a more succinct way by using the second order Lagrangian,

(22)

Therefore, to obtain the gravitational contribution at perturbed order, due to our effective Lagrangian for perturbed field variables, one must compute the derivative of the second order Lagrangian with respect to the perturbed metric.

The equations of motion for a field and its perturbation are found by regarding and as the relevant Lagrangian densities. Explicitly, the equations of motion for the field and its perturbation, , are respectively given by

(23)

The equations of motion governing the perturbation to the metric, , are given by the perturbed gravitational field equations,

(24)

The perturbed conservation law for the dark energy-momentum tensor is

(25)

which can be written as

(26)

3.2 Isotropic (3+1) decomposition

We will impose isotropy of spatial sections on the background spacetime. The motivation for doing this is that our goal is to study perturbations about an FRW background. After imposing isotropy we are able to use an isotropic (3+1) decomposition to significantly simplify expressions. It is also possible to include anisotropic backgrounds as described in Battye:2006mb .

We will foliate the 4D spacetime by 3D surfaces orthogonal to a time-like vector , which is normalized via

(27)

This induces an embedding of a 3D surface in a 4D space. The 4D metric is and the 3D metric is , and they are related by

(28)

The foliation implies that the time-like vector is orthogonal to the 3D metric,

(29)

The foliation induces a symmetric extrinsic curvature, , which is entirely spatial, . We can use this to deduce that .

A common application of the (3+1) decomposition is to write down the only energy-momentum tensor compatible with the globally isotropic FRW metric,

(30)

There are only two “coefficients” used in the decomposition of the energy-momentum tensor: the energy-density and pressure ,

(31)

Writing a tensor as a sum over combinations of and defines the isotropic (3+1) decomposition. We will now show how to decompose tensors of higher rank. For example, an isotropic vector is completely decomposed as

(32)

where . Notice that before we imposed isotropy upon we would need 4 functions to specify all “free” components of ; by imposing isotropy we have reduced the number of “free” functions from . A symmetric rank-2 isotropic tensor is completely decomposed as

(33)

where . The time-like part of is and the space-like part is . A rank-3 tensor symmetric in its second two indices is completely decomposed as

(34)

This formalism can also be used to construct tensors which are entirely spatial. For example, a rank-4 tensor defined as

(35)

is entirely spatial, a fact which is manifested by , after one notes the symmetries in the indices .

The coefficients which appear in an isotropic decomposition can only have time-like derivatives. For the coefficients in (33) we have

(36)

where an overdot is used to denote differentiation in the direction of the time-like vector: .

3.3 Perturbation theory

We will be making substantial use of perturbation theory in this paper, and so here we will take the time to concrete the notation and terminology we use. A large portion of the technology we are about to discuss was developed, amongst other things, to model relativistic elastic materials Carter21111972 ; BF01645505 ; PhysRevD.7.1590 ; 1978ApJ2221119F ; carterqunt_1977 ; Carter26081980 ; ll_elast ; PhysRevD.60.043505 ; Battye:2006mb ; PhysRevD.76.023005 ; Carter:1982xm ; Azeyanagi:2009zd ; Fukuma:2011pr ; we will recapitulate the ideas and bring the technology into the language of perturbation theory to be used with a gravitational theory.

A quantity is perturbed about a background value, , as . For example, the metric perturbed about a background is written as

(37)

It is important to realize that the operation of index raising and lowering does not commute with the variation. For example, for the metric and for the derivative of a scalar field .

Consider a quantity which is perturbed about some background value, . We can then employ two classes of coordinate system to follow the perturbation through evolution; time evolution can be thought of as Lie-dragging a quantity along a time-like vector, , to “carve out” the world-line of the perturbation, i.e. operating on a quantity with . The first is where the density of the perturbations remains fixed (i.e. the coordinate system evolves to comove with the perturbations); this is a Lagrangian system. In the second, the coordinate system is fixed by some means (such as knowledge of the background geometry) and the density of the perturbations changes; this is an Eulerian system. We write perturbations in the Lagrangian system as and perturbations in the Eulerian system as . Evidently, a coordinate transformation can be used to transfer between the two systems, . The Eulerian and Lagrangian variations are linked by

(38)

where is the Lie derivative along the gauge field . This setup is schematically depicted in Figure 1.

Figure 1: Schematic view of the Eulerian and Lagrangian coordinate systems. The Lagrangian system can be said to be comoving, and the Eulerian system as being fixed. The Lagrangian system retains the density of a field, whereas the Eulerian system does not. This is schematically depicted by the “grid square” becoming deformed in the Lagrangian system on the left, to accommodate the movement of “particles” upon evolution in time. The grid square in the Eulerian system has remained fixed, meaning that the number of particles in a given square changes upon evolution. In cosmology we are perturbing against a fixed background: the FRW background, however calculations are often easier to perform in a comoving system. This means that physical relevance is taken from equations perturbed according to a Eulerian scheme.

Without loss of generality we can set the gauge field and time-like vector to be mutually orthogonal,

(39)

This is because the time-like transformations which the component could induce are world-line preserving, and are redundant when is present (which is inherently a world-line preserving evolution). See Figure 2 for a schematic view illustrating this point.

Figure 2: Schematic view of the foliation and evolution, with three example world-lines drawn on, each piercing two 3D surfaces; is a time-like vector satisfying . A quantity on a surface with spacetime location can be transformed into a quantity on the same surface but at a different location by transforming the coordinate on the surface, . This is a diffeomorphism which drags one world-line into another. If the time coordinate is transformed then the quantity is evaluated on a different 3d surface, but on the same world-line. Thus, if we were to have a transformation , where is an arbitrary scalar field, the time-like part of is redundant. Hence, we are free to set , fixing the time-like part of the diffeomorphism field to be zero. So, we should have the interpretation that moves between world-lines and moves along world-lines.

There is an important question which arises: which perturbation scheme should we use to derive cosmologically relevant results, i.e. which should we use: or ? In cosmological perturbation theory a quantity is perturbed from its value in a fixed (or known) background (such as its value in an FRW background). Therefore, equations should be perturbed relative to a fixed background, and so we should employ the Eulerian scheme.

The equation of motion governing the metric perturbations is

(40)

and the perturbed conservation law that should be solved is the one evaluated in a Eulerian system,

(41)

which can be written as

(42)

If the Lagrangian variation of the dark energy-momentum tensor is the quantity that is supplied, (i.e. is given), then one must be careful to use (38), to obtain the Eulerian perturbed quantity,

(43)

Furthermore, to obtain the components of the mixed Eulerian perturbed dark energy-momentum tensor, one must use

(44)

The Lagrangian and Eulerian perturbations of the metric are linked by

(45)

For a vector field one finds that

(46)

As final explicit example, the Eulerian and Lagrangian variations of a scalar field are linked via

(47)

An interesting lemma is that if then by (39) we find that the Eulerian and Lagrangian variations of a scalar field are identical, . This means that a diffeomorphism does not change the perturbations of the scalar field; this is a consequence of the background field being homogeneous.

4 No extra fields:

Our first and simplest example is where the dark sector does not contain any extra fields: only the metric is present, albeit in an arbitrary combination. This class of theories contains the cosmological constant and elastic dark energy PhysRevD.60.043505 ; PhysRevD.76.023005 , and will also include more general theories that have not been previously considered. In this section we do not allow the dark sector to contain derivatives of the metric – this is discussed in a subsequent section. One of the aims is to build an intuition for understanding how to write down perturbative quantities and how to decompose tensors which arise in the perturbative equations.

The Lagrangian density we will consider in this section is of the form

(48)

so that the second order Lagrangian is given by

(49)

The rank-4 tensor is only a function of background quantities, and is therefore manifestly gauge invariant. We can use (22) and (49) to show that the perturbations to the dark energy momentum tensor are given by

(50)

By inspecting (49) it follows that the tensor enjoys the following symmetries,

(51)

This shows us how to construct the Lagrangian perturbations to the generalized gravitational field equations, under the assumption that the dark sector Lagrangian is a function of the metric only. Because it is the Lagrangian variation which appears above we must convert to Eulerian variations to obtain cosmologically relevant perturbations. By using (45) and (43) in (50) we obtain

To find the equation of motion of the vector field, , we must compute the Eulerian perturbed Bianchi identity. Substituting (4) into (42) we obtain

(53a)
where, for convenience, we have defined
(53b)
and where the perturbed source term, , is given by
(53c)

Here we observe that the metric perturbations and the diffeomorphism field are intimately linked: one cannot consistently set either to zero. The equation (53) is the constraint equation for any parameters/functions that appear in a parameterization of the dark sector, under the rather general assumption that ; the only freedom that remains is how to construct out of background quantities. In Section 6 we will provide the components of the equation of motion for a perturbed FRW spacetime.

The only way to write the tensors with an isotropic (3+1) decomposition which respects the symmetries (51) is

(54a)
(54b)
where respects the same symmetries as , satisfies (i.e. is entirely spatial) and is given by
(54c)

A concrete example of a theory which only contains the metric is the elastic dark energy theory PhysRevD.60.043505 ; PhysRevD.76.023005 where one can find that the coefficients in terms of physical quantities such as energy density , pressure , bulk and shear moduli are given by

(55a)
(55b)

where the bulk modulus is defined via , and the pressure and shear modulus are functions of the density (e.g. one way to choose these functional dependancies is with an “equation of state”, and , so that ).

In this section we have identified that just five functions are required to specify the perturbations in the dark sector when no extra fields are present. These five functions are

(56)

and each function only depends on background quantities and are governed by the background evolution.

5 Scalar fields:

The second example is when the dark sector contains an arbitrary combination of scalar field , the first derivative of the field , and the metric . This encompasses scalar field theories such as quintessence and -essence, but we could also encompass a range of other possible theories.

For a Lagrangian density given by

(57)

the second order Lagrangian is given by

(58)

The coefficients above comprise: one scalar , one vector , two rank-2 tensors , one rank-3 tensor, and one rank-4 tensor , all of which are only functions of background quantities and are therefore gauge invariant. At first sight these are all independent quantities, but we will show later that the conservation and Euler-Lagrange equations can be used to link the quantities. By inspecting (58) these tensors enjoy the following symmetries,

(59a)
(59b)

In what follows we will assume (alternatively this can be stated as ), so that . This is the covariant statement that is entirely time-like, while using the fact that the diffeomorphism is entirely space-like. Therefore, because the Eulerian and Lagrangian perturbations of a scalar field are identical we will not distinguish between them and we will write .

The equation of motion of the perturbed scalar field, , is given by the Euler-Lagrange equation (23). Using (58) one finds

(60)

where the “perturbed source” piece, , is given by

(61)

where . We note that plays the role of an “effective metric”, due to its resemblance to the corresponding term in the perturbed Klein-Gordon equation, namely , and there is also an effective mass of the -field, .

The isotropic (3+1) decomposition of the coefficients , whilst respecting the symmetries (59), is

(62a)
(62b)
(62c)
(62d)
(62e)

The decompositions of are identical to those given in eq.(54).

Only the terms in (58) which involve are relevant for writing down the perturbations to the dark energy-momentum tensor. We obtain

(63)

The Eulerian perturbed conservation law (42) can be computed using (63). We obtain

(64)

where is given by (53c), and represents the wave equation for and is given by

(65)

Equation (64) is an evolution equation for the scalar field perturbation , sourced by the metric perturbations, , and the vector field, . The scalar field perturbation sources the equation of motion for ,(40), via the components . In general one cannot consistently solve the evolution equations for independently from those for the vector field ; we will soon show how these two fields might decouple, but the decoupling only occurs in special cases.

The perturbed Euler-Lagrange equation (60) and perturbed conservation law (64) are both evolution equations for , and both have apparently different coefficients, resulting in an over-determined system. We can choose to remove this apparent over-determination by “forcing” the time-like (i.e. scalar) part of the perturbed conservation law to be identical to the perturbed Euler-Lagrange equation. It is important to realize that the perturbed conservation law is a vector equation, and only one of its components can be set equal to the perturbed Euler-Lagrange equation; the other components of the vector equation will introduce a set of constraint equations.

When we contract the perturbed Bianchi identity with a time-like vector (where ), we can read off a set of conditions that link the coefficients appearing in the Euler-Lagrange equation (60) and the perturbed Bianchi identity (64). Doing this we obtain the linking conditions

(66a)
(66b)
(66c)
(66d)

We see, therefore, that the coefficients and that appear in (58) are not independent, which is now obvious from (66). By differentiating (66a) and comparing with (66b) one finds that

(67)

where is the induced extrinsic curvature and an overdot is used to denote differentiation along the time-like vector.

The (3+1) decomposition introduces some interesting structure and can be used to explicitly evaluate the linking conditions (66). From (66a) we find that

(68)

After combining (66a) and (66b) to yield (67) we find that

(69)

In a similar fashion, it follows from (66c) that

(70)

where .

One can think of as being an “artificial” vector field whose role was to restore reparameterization invariance, and it would therefore be desirable to have a theory that does not require to be present and reparameterization invariance is manifest. We will derive conditions that the tensors in the Lagrangian must satisfy in order for reparameterization invariance to be manifest.

We can rewrite the Lagrangian with the vector field explicitly present to show how the three fields interact and how the parameters can be arranged so that they ultimately decouple. To ease our calculation we will write , We use (45) to replace with in the Lagrangian (58). Rearranging, whilst keeping track of total derivatives yields

(71)

To enable us to identify the “free” and “interaction” Lagrangians, we note that (71) can be written schematically as

(72)

where of a single field variable represents the self-interaction of that field and of two fields represents the interaction between the two fields. The final line of (71) is a pure surface term, and will not contribute to the dynamics, and thus does not require consideration in what we are about to discuss. However, if we note the definition of and compare to the perturbed EMT (63), we find that .

Notice that the perturbed scalar field, , and vector field are coupled in the Lagrangian, and only decouple when their interaction Lagrangian, , vanishes. This will remove the direct coupling but they may remain indirectly coupled if the interaction Lagrangian for the perturbed metric and vector field remains non-zero (i.e. if ), since the perturbed metric and scalar field will remain coupled, . So, the interaction Lagrangian between the vector field and perturbed scalar field vanishes, i.e. , when