Contents


A Class of Effective Field Theory Models of Cosmic Acceleration

Jolyon K. Bloomfield, Éanna É. Flanagan,

Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853.

Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853.

jkb84@cornell.edu

eef3@cornell.edu


January 2011

  Abstract
We explore a class of effective field theory models of cosmic acceleration involving a metric and a single scalar field. These models can be obtained by starting with a set of ultralight pseudo-Nambu-Goldstone bosons whose couplings to matter satisfy the weak equivalence principle, assuming that one boson is lighter than all the others, and integrating out the heavier fields. The result is a quintessence model with matter coupling, together with a series of correction terms in the action in a covariant derivative expansion, with specific scalings for the coefficients. After eliminating higher derivative terms and exploiting the field redefinition freedom, we show that the resulting theory contains nine independent free functions of the scalar field when truncated at four derivatives. This is in contrast to the four free functions found in similar theories of single-field inflation, where matter is not present. We discuss several different representations of the theory that can be obtained using the field redefinition freedom. For perturbations to the quintessence field today on subhorizon lengthscales larger than the Compton wavelength of the heavy fields, the theory is weakly coupled and natural in the sense of t’Hooft. The theory admits a regime where the perturbations become modestly nonlinear, but very strong nonlinearities lie outside its domain of validity.  

1 Introduction and Summary

1.1 Background and Motivation

The recent discovery of the accelerating expansion of the Universe [1, 2] has prompted many theoretical speculations about the underlying mechanism. The most likely mechanism is a cosmological constant, which is the simplest model and is in good agreement with observational data [3]. More complicated models involve new dynamical sources of gravity that act as dark energy, and/or modifications to general relativity on large scales. A plethora of models have been postulated and explored in recent years, including Quintessence, K-essence [4, 5], Ghost Condensates [6], DGP gravity [7], and gravity, to name but a few. See Refs. [8, 9, 10, 11, 12, 13, 14] for detailed reviews of these and other models.

A common feature of the majority of dark energy and modified gravity models is that in the low energy limit, they are equivalent to general relativity coupled to one or more scalar fields, often called quintessence fields. Therefore it is useful to try to construct very general low energy effective quantum field theories of general relativity coupled to light scalar fields, in order to encompass broad classes of dark energy models. Considering dark energy models as quantum field theories is useful, even though the dynamics of dark energy is likely in a classical regime, because it facilitates discriminating against theories which are theoretically inconsistent or require fine tuning.

A similar situation occurs in the study of models of inflation, where it is useful to construct generic theories using effective field theory. Cheung et al. [15] constructed a general effective field theory for gravity and a single inflaton field, for perturbations about a background Friedman-Robertson-Walker cosmology in unitary gauge. This work was later generalized in multiple directions [16, 17] and has been very useful. An alternative approach to single field inflationary models was taken by Weinberg [18], who constructed an effective field theory to describe both the background cosmology and the perturbations. This theory consisted at leading order of a standard single field inflationary model with a potential, together with higher order terms in a covariant derivative expansion up to four derivatives. More detailed discussions of this type of effective field theory were given by Burgess, Lee and Trott [19].

When one turns from inflationary effective field theories to quintessence effective field theories, the essential physics is very similar, but there are three important differences that arise:

  • First, the hierarchy of scales is vastly more extreme in quintessence models. The Hubble parameter is typically several orders of magnitude below the Planck scale eV in inflationary models, whereas for quintessence models eV is orders of magnitude below the Planck scale. Quintessence fields must have a mass that is smaller than or on the order of . It is a well-known, generic challenge for quintessence models to ensure that loop effects do not give rise to a mass much larger than . Because of the disparity of scales, this issue is more extreme for quintessence models than inflationary models.

  • In most inflationary models, it is assumed that the dynamics of the Universe are dominated by gravity and the scalar field (at least until reheating). By contrast, for quintessence models in the regime of low redshifts relevant to observations, we know that cold dark matter gives an contribution to the energy density. Therefore there are additional possible couplings and terms that must be included in an effective field theory.

  • For any effective field theory, it is possible to pass outside the domain of validity of the theory even at energies low compared to the theory’s cutoff , if the mode occupation numbers are sufficiently large (see Sec. 5.2 below for more details). This corresponds to a breakdown of the classical derivative expansion. For quintessence theories, mode occupation numbers today can be as large as and it is possible to pass outside the domain of validity of the theory. By contrast in inflationary models, this is less likely to occur since mode occupation numbers for the perturbations are not large before modes exit the horizon. Thus, the effective field theory framework is less all-encompassing for quintessence models than for inflation models. This issue seems not to have been appreciated in the literature and we discuss it in Sec. 5.2 below.

Several studies have been made of generic effective field theories of dark energy. Creminelli, D’Amico, Noreña and Vernizzi [20] constructed a the general effective theory of single-field quintessence for perturbations about an arbitrary FRW background, paralleling the similar construction for inflation [15]. Park, Watson and Zurek constructed an effective theory for describing both the background cosmology and the perturbations, following the approach of Weinberg [18] but generalizing it to include couplings to matter [21].

The two approaches to effective field theories of quintessence – specialization to perturbations about a specific background, and maintaining covariance and the ability to describe the dynamics of a variety of backgrounds – are complementary to one another. The dynamics of the cosmological background FRW solution can be addressed in the covariant approach of Weinberg, but not in the background specific approach of Creminelli et al., which restricts attention to the dynamics of perturbations about a given, fixed background. On the other hand, a background specific approach can describe a larger set of dynamical theories for the perturbations than can a covariant derivative expansion111To see this, consider for example a term in the Lagrangian of the form , where is the quintessence field. Such a term would be omitted in the covariant derivative expansion for sufficiently large . However, upon expanding this term using , where is the background solution, one finds terms which are included in the Creminelli approach of applying standard effective field theory methods to the perturbations..

1.2 Approach and Assumptions

The purpose of this paper is to revisit, generalize and correct slightly the covariant effective field theory analysis of Park, Watson and Zurek [21]. Following Weinberg and Park et al., we restrict attention to theories where the only dynamical degrees of freedom are a graviton and a single scalar. We allow couplings to an arbitrary matter sector, but we assume the validity of the weak equivalence principle, motivated by the strong experimental evidence for this principle. We assume that the theory consists of a standard quintessence theory coupled to matter at leading order in a derivative expansion, with an action of the form

(1.1)

Here denotes a set of matter fields, and is the Planck mass. The factor in the matter action provides a leading-order non-minimal coupling of the quintessence field to matter, in a manner similar to Brans-Dicke models in the Einstein frame.

Our analysis then consists of a series of steps:

  1. We add to the action all possible terms involving the scalar field and metric, in a covariant derivative expansion up to four derivatives. We truncate the expansion at four derivatives, as this is sufficient to yield the leading corrections to the action (1.1). As described by Weinberg [18] there are ten possible terms, with coefficients that can be arbitrary functions of [see Eq. (2.3) below]. Section 5.1 below describes one possible justification of this covariant derivative expansion from an effective field theory viewpoint, starting from a set of ultralight pseudo Nambu-Goldstone bosons (PNGBs). It is likely that the same expansion can be obtained from other, more general starting points.

  2. We allow for corrections to the coupling to matter by adding to the metric that appears in the matter action all possible terms involving the metric and allowed by the derivative expansion, that is, up to two derivatives. There are six such terms [see Eq. (2.4) below.] We also add to the action terms involving the stress energy tensor of the matter fields, up to the order allowed by the derivative expansion using [see Eq. (2.3) below]. Including such terms in the action seems poorly motivated, since a priori there is no reason to expect that the resulting theory would respect the weak equivalence principle (see Appendix B). However we show in Appendix B that the weak equivalence principle is actually satisfied, to the order we are working to in the derivative expansion. In addition, all the terms in the action involving can be shown to have equivalent representations not involving the stress energy tensor, using field redefinitions (see Appendix B).

  3. The various correction terms are not all independent because of the freedom to perform field redefinitions involving , and the matter fields, again in a derivative expansion. In Sec. 3 we explore the space of such field redefinitions, finding eleven independent transformations and tabulating their effects on the coefficients in the action (see Table 1 below).

  4. Several of the correction terms that are obtained from the derivative expansion are “higher derivative” terms, by which we mean that they give contributions to the equations of motion which involve third-order or higher-order time derivatives of the fields222The precise definition of higher derivative that we use, which is covariant, is that an equation will be said not to contain any higher derivative terms if there exists a choice of foliation of spacetime for which any third-order or higher-order derivatives contain at most two time derivatives. Theories which are higher derivative in this sense are generically associated with instabilities (Ostragradski’s theorem) [22], although the instabilities can be evaded in special cases, for example gravity. For most of this paper (except for the Chern-Simons term), a simpler definition of higher derivative would be sufficient: a term in the action is “higher derivative” if it gives rise to terms in the equation of motion that involve any third-order or higher order derivatives.. Normally, such higher derivative terms give rise to additional degrees of freedom. However, if they are treated perturbatively (consistent with our derivative expansion) additional degrees of freedom do not arise. Specifically, one can perform a reduction of order procedure on the equations of motion [23, 24, 25], substituting the zeroth-order equations of motion into the higher derivative terms in the equations of motion to eliminate the higher derivatives333This is more general than requiring the solutions of the equation of motion to be analytic in the expansion parameter, as advocated by Simon [26]; see Ref. [25].. We actually use a slightly different but equivalent procedure of eliminating the higher derivative terms directly in the action using field redefinitions444This procedure is counterintuitive since normally field redefinitions do not change the physical content of a theory; here however they do because the field redefinitions themselves involve higher derivatives. (see Appendix C).

    Weinberg [18] and Park et al. [21] use a slightly different method, consisting of substituting the leading order equations of motion directly into the higher derivative terms in the action. This method is not generally valid, but it is valid up to field redefinitions that do not involve higher derivatives, and so it suffices for the purpose of attempting to classify general theories of dark energy (see Appendix C).

  5. Another issue that arises with respect to the higher derivative terms is the following. Is it really necessary to include such terms in an action when trying to write down the most general theory of gravity and a scalar field, in a derivative expansion? Weinberg [18] suggested that perhaps a more general class of theories is generated by including these terms and performing a reduction of order procedure on them, rather than by omitting them. However, since it is ultimately possible to obtain a theory that is perturbatively equivalent to the higher derivative theory, and which has second order equations of motion, it should be possible just to write down the action for this reduced theory. In other words, an equivalent class of theories should be obtained simply by omitting all the higher derivative terms from the start. We show explicitly in Sec. 4 that this is the case for the class of theories considered here.

  6. We fix the remaining field redefinition freedom by choosing a “gauge” in field space, thus fixing the action uniquely (see Sec. 4.2).

1.3 Results and Implications

Our final action is [Eq. (4.5) below]

(1.2)

Here the coefficients , etc. of the next-to-leading order terms in the derivative expansion are arbitrary functions of , and the ellipsis refers to higher order terms with more than four derivatives. The corresponding equations of motion do not contain any higher derivative terms. This result generalizes that of Weinberg [18] to include couplings to matter.

Figure 1: The parameter space of fractional density perturbation for perturbations to the quintessence field, and cutoff scale for the effective field theory, illustrating the constraint (1.3) on the domain of validity. Near the boundary of the domain of validity the higher derivative terms in the action are potentially observable, this is labeled the “interesting regime”. Further away from the boundary the higher derivative terms are negligible and the theory reduces to a standard quintessence model with a matter coupling.

We can summarize our key results as follows:

  • The most general action contains nine free functions of : , as compared to the four functions that are needed when matter is not present [18].

  • There are a variety of different forms of the final theory that can be obtained using field redefinitions. In particular some of the matter-coupling terms in the action can be re-expressed as terms that involve only the quintessence field and metric. Specifically, the term term could be eliminated in favor of , the could be eliminated in favor of a term , or the term could be eliminated in favor of a term (see Sec. 4.2).

  • As mentioned above, one obtains the correct final action if one excludes throughout the calculation all higher derivative terms.

  • The final theory does contain terms involving the matter stress-energy tensor. Nevertheless, the weak equivalence principle is still satisfied (see Appendix B). It is possible to eliminate the stress-energy terms, but only if we allow higher derivative terms in the action (where it is assumed that the reduction of order procedure will be applied to these higher derivative terms). Thus, for a fully general theory, one must have either stress-energy terms or higher derivative terms; one cannot eliminate both (see Sec. 4.2).

  • We can estimate how all the coefficients etc. scale with respect to a cutoff scale for an effective field theory as follows (see Sec. 5.1). We assume that several ultralight scalar fields of mass arise as pseudo Nambu-Goldstone bosons (PNGBs) from some high energy theory [27, 28], and are described by a nonlinear sigma model at low energies. We then suppose that all but one of the these PNGB fields have masses that are somewhat larger than , and integrate them out. This will give rise to a theory of the form discussed above for the single light scalar, where the higher derivative terms are suppressed by powers of . The scalings for each of the coefficients in the action are summarized in Table 3. We find that the fractional corrections to the cosmological dynamics due to the higher derivative terms scale as , as one would expect.

  • Finally, we can use these scalings to estimate the domain of validity of the effective field theory (see Sec. 5.2). We find that cosmological perturbations with a density perturbation in the quintessence field must have a fractional density perturbation that satisfies

    (1.3)

    Thus perturbations can become nonlinear, but only modestly so, if is close to . The parameter space of fractional density perturbation and cutoff scale is illustrated in Fig. 1. In addition there is the standard constraint for derivative expansions

    (1.4)

    where is the length-scale or time-scale for some process. We show in Fig. 2 the two constraints (1.3) and (1.4) on the two dimensional parameter space of energy and mode occupation number .

Figure 2: The domain of validity of the effective field theory in the two dimensional parameter space of energy per quantum of a mode of the quintessence field, and mode occupation number . The cutoff scale must be larger than the Hubble parameter in order that the background cosmology lie within the domain of validity. Perturbation modes on length-scales that are small compared to but large compared to can be described, but only if the mode occupation number and fractional density perturbation are sufficiently small. See Sec. 5.2 for details.

Finally, in Appendix D we compare our analysis to that of Park, Watson and Zurek [21], who perform a similar computation but in the Jordan frame rather than the Einstein frame (see also Ref. [29]). The main difference between our analysis and theirs is that they use a different method to estimate the scalings of the coefficients, and as a result their final action differs from ours, being parameterized by three free functions rather than nine.

2 Class of Theories Involving Gravity and a Scalar Field

As discussed in the Introduction, our starting point is an action for a standard quintessence model with an arbitrary matter coupling, together with a perturbative correction which consists of a general derivative expansion up to four derivatives. The action is a functional of the Einstein-frame metric , the quintessence field , and some matter fields which we denote collectively by :

(2.1)

Here is the action for the matter fields, and the quantity is a formal expansion parameter. We will see in Sec. 5.1 below that can be identified as proportional to , where is a cutoff scale or the mass of the lightest of the fields that have been integrated out to obtain the low energy action. Equivalently, counts the number of derivatives in our derivative expansion, with corresponding to derivatives. The notation in the second term indicates that the perturbative correction to the action can depend on the matter fields, but only through their stress energy tensor (defined in Appendix A). Explicitly we have

(2.2)

and [18, 21]

(2.3)

Here is the Weyl tensor and is the antisymmetric tensor (our conventions for these are given in Appendix A). There are additional terms with four derivatives that one can write down, but all such terms can be eliminated by integration by parts. Finally, the metric which appears in the matter action in Eq. (2.1) is given by555We call this metric the Jordan frame metric, in an extension of the usual terminology which applies to the case when the relation (2.4) between the two metrics is just a conformal transformation.

(2.4)

All of the coefficients , and are arbitrary functions of .

Let us briefly discuss each of the perturbative terms. The terms with coefficients are corrections to the kinetic term of the scalar field. The and terms are couplings between the scalar field and the stress-energy tensor, or between curvature and the stress-energy tensor. The terms are kinetic couplings between the scalar field and gravity. The terms are quadratic curvature terms, which we have chosen to write as an term, an term, and the Gauss-Bonnet term. Any constant piece of the coefficient is a topological term and may be omitted. The term is the gravitational Chern-Simons term, which may be excluded if one wishes to introduce parity as a symmetry of the theory, and again, any constant component of is topological and may be omitted. Finally, the terms are quadratic in the stress-energy tensor.

Note that several of the terms in the action (2.3) are “higher derivative” terms, that is, they give rise to contributions to the equations of motion containing derivatives of order three or higher. The specific terms are those parameterized by the coefficients , , , , , and . As discussed in the Introduction and in Appendix C, we will choose to define our theory by treating these terms perturbatively, which excludes the extra degrees of freedom and instabilities that are normally associated with higher derivative terms.

We also note that the theory (2.1) satisfies the weak equivalence principle, to linear order in , as we show in Appendix B. That is, objects with negligible self-gravity with different compositions all experience the same acceleration. It is not a priori obvious that the principle should be satisfied since, as we show in Appendix B, violations of the principle generically arise whenever the matter stress energy tensor appears explicitly in the gravitational action, as in Eq. (2.1).

3 Transformation Properties of the Action

The description of the theory provided by Eqs. (2.1) – (2.4) is very redundant, in part because of the freedom to perform field redefinitions. In this section we derive how the various coefficients in the action (2.1) are modified under various transformations. In the next section we will use these transformation laws to derive a canonical representation of the theory, involving only nine free functions.

3.1 Expansion of the Matter Action

Consider first the perturbative terms parameterized by , in the definition (2.4) of the Jordan metric , which appears in the matter action . Using the definition (A.1) of the stress-energy tensor, we can eliminate these terms in favor of terms in the action involving . Specifically we have from Eq. (A.1) that

(3.1)

Choosing

(3.2)

then gives a transformation of the action (2.1) characterized by the following changes in the coefficients:

(3.3)

Here the parameters can be arbitrary functions of . Similarly choosing gives a transformation characterized by

(3.4)

3.2 Field Redefinitions Involving just the Scalar Field

Consider a perturbative field redefinition of the form

(3.5)

where the quantity can in general depend on any of the fields and their derivatives. To leading order in , the change in the action (2.1) is then proportional to the zeroth-order equation of motion (5.10b) for . Relabeling as , the change induced in the action is

(3.6)

There are three special cases that will be useful:

  1. First, choose

    (3.7)

    where is an arbitrary function666Because we are working to linear order in , it does not matter whether we take to be a function of or of . of , and is the trace of the stress-energy tensor. Substituting this into Eq. (3.6) and comparing with the general action (2.3), we find the following transformation law for the coefficients:

    (3.8)
  2. Second, we use the field redefinition

    (3.9)

    Here the second term in the square bracket is included in order to maintain canonical normalization of the scalar field, that is, to avoid generating terms in the action of the form . The resulting transformation law is

    (3.10)
  3. Third, consider the field redefinition

    (3.11)

    where is a function of and again the particular combination of terms is chosen to maintain canonical normalization. Substituting into Eq. (3.6), performing some integrations by parts and comparing with Eq. (2.3) gives the transformation law

    (3.12)

    Note that this transformation is not well defined in general in the limit , because of the factors of . However, it is well defined in the limit , with kept constant.

Coeff. Term
Gauss-Bonnet
Chern-Simons
 (potential)
Table 1: This table shows which of the terms in our action (2.2) are affected by each of the eleven field redefinitions (3.7) – (3.29) that are parameterized by the functions . The columns represent the redefinitions, and the rows represent terms. Daggers in first column indicate “higher derivative” terms, that is, terms that give contributions to the equations of motion containing time derivatives of higher than second order. Stars indicate that the coefficient of that row’s term is altered by that column’s field redefinition. We omit the coefficients and since those coefficients are degenerate with by Eqs. (3.3) and (3.4).

3.3 Field Redefinitions Involving the Metric

We now consider a more general class of field redefinitions, where in addition to redefining the scalar field via Eq. (3.5), we also perturbatively redefine the metric via

(3.13)

Here the quantity can depend on , , their derivatives and the stress energy tensor. The corresponding change in the action is proportional to the equation of motion (5.10a). Relabeling as and as , the total change in the action is

(3.14)

Note that this formula includes the effect of the change in the Jordan frame metric (2.4) caused by the transformation (3.13). We now consider seven different transformations of this type:

  1. The first case is a change to the metric proportional to . In order to maintain canonical normalization of both the metric and the scalar field, that is, to avoid terms of the form and , we need the following combination of terms in the field redefinition:

    (3.15a)
    (3.15b)

    for some function . Substituting into Eq. (3.14), performing some integrations by parts and comparing with Eq. (2.3) we obtain for the transformation law

    (3.16)
  2. Next consider changes to the metric proportional to . In order to maintain canonical normalizations we use the following combination of terms in the field redefinition:

    (3.17a)
    (3.17b)

    for some function . This gives the transformation law

    (3.18)
  3. The next case is a change to the metric proportional to . To maintain canonical normalization of the scalar field, we need in addition a change to the scalar field, with the combined transformation being

    (3.19a)
    (3.19b)

    for some function . The resulting transformation law for the coefficients is

    (3.20)
  4. Next consider changes to the metric proportional to . The required form of field redefinition that preserves canonical normalization of is

    (3.21a)
    (3.21b)

    for some function . The coefficients in the action then change according to

    (3.22)
  5. The fifth case is a change to the metric proportional to . The required form of field redefinition that preserves canonical normalization of is

    (3.23a)
    (3.23b)

    for some function . The coefficients in the action then change according to

    (3.24)
  6. Next consider a change in the metric proportional to . To preserve canonical normalization of we use the redefinitions

    (3.25a)
    (3.25b)

    for some function . The coefficients in the action then change according to

    (3.26)
  7. A simple case is when the change in the metric is proportional to , for which no change to the scalar field is required. The redefinition is

    (3.27)

    for some function . The transformation law for the coefficients is

    (3.28)
  8. Similarly, no transformation to the scalar is required for the case of a change in the metric proportional to . The redefinition is

    (3.29)

    for some function , and the corresponding transformation law is

    (3.30)

The eleven777We could also consider a twelfth redefinition given by , . However this redefinition is not independent of the first eleven; the same effect can be achieved by choosing , , , . field redefinitions (3.7) – (3.29) are summarized in Table 1, which shows which coefficients are modified by which transformations.

4 Canonical Form of Action

In this section, we derive our final, reduced action (1.2) from the starting action (2.1), using the transformation laws derived in Sec. 3. There is some freedom in which terms we choose to eliminate and which terms we choose to retain. We choose to eliminate all terms that give higher derivatives in the equations of motion, so that the final theory is not a “higher derivative” theory. However, even after this has been accomplished, there is still some freedom in how the final theory is represented. We discuss this further in Sec. 4.2 below. The order of operations in the derivation is important, since we need to take care that terms which we have already set to zero are not reintroduced by subsequent transformations. Table 2 summarizes our calculations and their effects on the coefficients in the action at each stage in the computation.

4.1 Derivation

The steps in the derivation are as follows:

Step 1 2 3 3 4 5 6 6 7 Final
Transformations
Coeff. Term in Action
Gauss-Bonnet
Chern-Simons
 (potential)
Coeff. Term in
(conf. factor)
Table 2: This table shows the progression of manipulations we make in this section. The second column on the left lists the various terms in the action (2.3), or in the Jordan-frame metric (2.4). The first column lists the corresponding coefficients; daggers indicate higher derivative terms. The numbers in the first row along the top refer to the numbered steps in the derivation in Sec. 4.1. The second row shows which transformation functions are used in each step. In the table, a star indicates that the corresponding row’s term receives a contribution from the corresponding column’s reduction process, while indicates that the term has been eliminated. The check marks in the last column indicate the remaining terms that are non-zero in the final action (4.5). Finally, circles in the last column indicate terms that are nonzero in alternative forms of the final action obtained using the transformations (3.11) or (3.23), as discussed in Sec. 4.2.
  1. Elimination of Derivative Terms in the Jordan Frame Metric: The transformation (3.3) can be used to eliminate all of the terms involving derivatives in the Jordan frame metric (2.4), which are parameterized by the coefficients . This changes the coefficients of the terms in the action that depend linearly on the stress energy tensor, namely . As discussed in Appendix B, these terms involving the stress-energy tensor look like they might violate the weak equivalence principle, but in fact they do not.

  2. Elimination of Higher Derivative, Quadratic in Curvature Terms: We next consider the terms in the action that are quadratic functions of curvature, whose coefficients are , , and . The Chern-Simons term () and the Gauss-Bonnet term () give rise to well behaved equations of motion (in the sense that they not increase the number of degrees of freedom), so we do not attempt to eliminate these terms. By contrast, the terms proportional to the squares of the Ricci scalar and Ricci tensor, parameterized by and , do increase the number of degrees of freedom. We can eliminate these terms by using the transformations (3.15) and (3.17), with parameters chosen to be

    (4.1)

    These transformations will then modify the coefficients , , , and , as well as the potential (see Table 1).

  3. Elimination of some of the Linear Stress-Energy Terms: We next turn to terms which depend linearly on the stress-energy tensor, parameterized by . First, we can eliminate the term by using the transformation (3.25) with . This gives rise to changes in the coefficients , , , as well as to the potential . Second, we can eliminate the terms parameterized by and by using the transformations (3.27) and (3.29) with the parameters , . This changes the coefficients , ,