Abstract

It has been shown earlier that Noether symmetry does not admit a form of corresponding to an action in which is coupled to scalar-tensor theory of gravity or even for pure theory of gravity taking anisotropic model into account. Here, we prove that theory of gravity does not admit Noether symmetry even if it is coupled to tachyonic field and considering a gauge in addition. To handle such a theory, a general conserved current has been constructed under a condition which decouples higher-order curvature part from the field part. This condition, in principle, solves for the scale-factor independently. Thus, cosmological evolution remains independent of the form of the chosen field, whether it is a scalar or a tachyon.

1. Introduction

Interest in theory of gravity has increased predominantly in recent years, since it appears to explain most of the presently available cosmological data unifying early inflation with late time cosmic acceleration (see [1, 2] for recent reviews and also references therein). However, most of these interesting results are the outcome of scalar-tensor equivalence under some arbitrary choice of the form of . It is, therefore, important to test if the same results are obtainable from theory of gravity without invoking scalar-tensor equivalence. But then how to choose a form of out of indefinitely large number of curvature invariant terms and how to find exact solutions are big questions. From physical ground, namely, to obtain a renormalizable theory of gravity, a form of had been found in the context of early universe, which contains ghosts [3]. A ghost-free action has also been presented in recent years [4, 5]. Likewise, the only physically meaningful technique to obtain a form of to explain late time cosmological evolution is to invoke Noether symmetry as a selection rule. This requires canonical formulation, and for a general theory of gravity, it is only possible treating as an auxiliary variable, provided (here, prime represents derivative with respect to ). In the process, it is possible to construct a point Lagrangian, and one can demand Noether symmetry to find a suitable form of . Following this technique, several authors [6–12] have found in the Robertson-Walker metric both in vacuum and pressureless dust . Although such a form of shows accelerating expansion in the matter dominated era (), nevertheless, early decelerating phase tracks as in the matter dominated era instead of usual and in the radiation dominated era () instead of usual , creating problem in explaining structure formation [13]. Thus, alone, in the absence of a linear term in the action, is not worth explaining presently available cosmological data [13]. It has been also noticed [13] that instead of the scale-factor , if one would have started with the basic variable , then becomes cyclic both in vacuum and matter dominated era, and therefore, such symmetry is independent of the choice of configuration space variables. Thus, the Noether symmetry obtained in the process is inbuilt and trivial. The situation could have been improved if the Noether symmetry would allow linear term in the action, but it does not. To obtain a better form of , scalar field has been incorporated both minimally and nonminimally, but even then Noether symmetry remains absent [14].

Despite the fact that Noether symmetry yields nothing other than in vacuum or in matter dominated era and that too only in isotropic space-time, some authors have recently claimed [15] to have obtained arbitrary form of taking into account a gauge term in the Noether theory. Note that the theory of gravity is very special in the context that it admits reparametrization invariance leading to the equation of Einstein or the so-called Hamiltonian constraint equation . This reparametrization invariance is not reflected in the Noether equations. Thus, solution obtained from the Noether equations, namely, the conserved current, must satisfy the Hamilton constraint equation. It has been shown that the result [15] is not true, since neither all the Noether equations nor the Hamilton constraint equation is satisfied for such symmetry [16]. More recently, there is yet another claim [17] that the gauge Noether symmetry yields taking tachyon field into account. It is important to review this fact since in our earlier works [13, 14] tachyon field had not been accounted for.

Rolling tachyon condensates originated from string theories and have interesting cosmological consequences [18–25] particularly because its equation of state parameter (, , ) smoothly interpolates between and [26]. As a result, it behaves like pressureless dust and cosmological constant in the limits. Such a nice feature, initiated to construct viable cosmological models treating tachyon as the inflaton field by coupling it minimally to the gravitational field taking different self-interacting potential densities, in the form of power law, exponential, and hyperbolic functions of the tachyon field [27–40]. Nevertheless, such models have been found to suffer from serious disease associated with density perturbations and reheating [41]. Tachyon fields with such kinds of potential densities were also used in order to describe the present accelerating period of the universe, where it behaves as dark energy [42–52]. Further, it has been observed that a tachyon field with an exponential potential plays the role of inflaton in the early universe and dark energy at the late [53]. Thus, it appears to be a good candidate that might explain late time cosmological evolution. Nonetheless, present observations suggest that the state parameter might even cross the phantom divide line , which is not realizable under minimal coupling. Such crossing may be possible if the tachyon field is nonminimally coupled to the gravitational field, which already exists in the literature [54, 55]. Another possibility appears if the tachyon field is coupled to theory of gravity, which is our present concern. We therefore, in following sections, proceed to check if theory of gravity being coupled to tachyon field admits Noether symmetry. The result we find is null, since the conserved current, thus obtained, does not satisfy the Hamilton constraint equation. This completes our “Tour de Noether symmetry” of theory of gravity, which now states that theory of gravity only admits the trivial symmetry in vacuum or matter dominated era and that too in isotropic space-time only.

In view of the above discussion, we organize the present work in the following manner. In Section 2, we review the gauge Noether symmetry of theory of gravity being coupled to tachyonic field to show that indeed such symmetry is not admissible. In Section 3, we take up a more general action and attempt Noether symmetry once again but in vain. At this end, we would like to mention that on the contrary, starting from , a priori, the Noether symmetry is obtainable for a scalar-tensor theory of gravity [56] which we have not been able to recover. This is important, since is special in the sense that the corresponding action is scale invariant which is also somehow related to Noether symmetry [57]. In Section 4, we briefly enunciate our view in this regard. Section 5 is devoted to construct a general conserved current and to explain its utility in extracting solutions choosing arbitrary form of . In view of such conserved current, we show that the form of the field in no way dictates the cosmological evolution, but rather it depends only on the form of chosen. Finally, in Section 6, we find the constraint to recover the Newtonian limit for such an action under weak field approximation. Section 7 concludes the present work.

2. Action, Field Equations, and Noether Equation

Among all the dynamical symmetries, transformations that map solutions of the equations of motion into solutions, one can single out the Noether symmetries as the continuous transformations that leave the action invariant—except for boundary terms. In formal language, the Noether symmetry states that for any regular system if there exists a vector field , such that

in the presence of a gauge function , where is the first prolongation of the vector field given by

with , , then there exists a conserved current

In some earlier works [16, 58] it has been shown that under infinitesimal coordinate and temporal transformations ( and ), time dependence may be introduced in a Lagrangian which does not contain time explicitly, through a time dependent gauge function. Invariance of Hamilton’s principal function finally yields

retaining only up to first-order term in the Taylor expansion. It is apparent that the conserved current (3) may be obtained from (4) only if the Lagrangian in the left-hand side and that in the right-hand side of (4) get cancelled. This is possible provided that the Lagrangian appearing in the left-hand side is time independent. Remember that time-dependence has been generated in the Lagrangian only through a time-dependent gauge function [16]. Thus for a time-independent Lagrangian (as in the case of gravity under consideration), the gauge function has to be time independent as in the case of harmonic oscillator. Further, it is known that the Noether integral is the Hamiltonian for the trivial Noether point symmetry . Other way round, the Hamiltonian is the generator of time translation, and so conservation of the Hamiltonian requires to be constant. Further, in gravity, the Hamiltonian is not only conserved, but it is constrained to vanish. As a result, does not play any significant role in the Noether symmetry as is clearly observed from (4) and may even be set equal to zero. Time independence of the Lagrangian, the gauge term, and automatically enforces time independence of . In the process, the integral of motion (3) reduces simply to

Essentially, if gauge turns out to be zero while solving the Noether equations, the integral of motion remains the same for the Noether symmetry without gauge, and no new result is expected. Although it is clear that time translation is unnecessary, still we keep it explicitly in the following, to keep track with the earlier work [17] performed in this context.

Now, considering Robertson-Walker line element

the following Born-Infeld effective 4-dimensional action for a rolling tachyon field with Lagrangian density being minimally coupled to

leads to a point Lagrangian:

for the spatially flat () case, treating

as a constraint of the theory and effectively spanning the Lagrangian by a set of configuration space variables . In the above action is treated as the coupling constant required to make the kinetic part of the action dimensionless. For the above point Lagrangian the Noether equation (1) reads

where prime denotes derivative with respect to . Note that we have kept both time translation and a time-dependent gauge function to keep track with the work performed by Jamil et al. [17]. Now, equating coefficients as usual, we obtain a large overdeterminant set of the Noether equations for . These are

Before we proceed let us mention that if , the action (7) corresponds to pure theory of gravity, for which nothing other than is possible [13]. In what follows, we will assume in (22), and so it will never be possible to recover pure case. Now, in view of (11) through (15), it is apparent that , , , , and . Therefore, (16) dictates that may be at most linear in . Thus, (22) clearly states that has to be time independent (as is), and it should be linear in the scale-factor . Thus, taking , where is a constant, it is clear that has to be independent of and in view of (20) and (21) and the gauge term must be independent of as is apparent from (17). Further, since both and are time independent, therefore, the gauge term further becomes independent of the scale-factor in view of (18). Finally, (19) is satisfied provided . In the process, the gauge term turns out to be a constant and hence plays no role in the Noether symmetry. Thus, one can set , without loss of generality. In view of the previous analysis, we now have

where , and are constants. Having obtained explicit forms of , and , we are now left to find the explicit forms of , , and the potential from the set of overdeterminant equations (19) through (22) which in view of (23) now get reduced to, following four equations

2.1. Reviewing Jamil et al.’s Work

Equation (27) may now be solved for the following form of the potential:

where is a constant. However, the authors [17] claimed the following form of the potential:

which is possible only under the choices , , and in particular , which, as we will show shortly, create severe problem. One can easily check that under the choice , (26) upon integration yields (25), provided the constant of integration is zero. Thus, we are now left with two independent equations (24) and (25) to find the form of and . However, (25) now reads

Now eliminating between (24) and (30) one ends up with

in a straightforward manner, and gets solved as . At this end the authors [17] obtained two conserved currents and . is the invariance under time translation, as stated correctly by the authors (here we point out a typographical error in of [17]; should be instead, where used in [17] stands for here). Nevertheless, the third bracketed term is the Hamiltonian, which is constrained to vanish. Thus, if is substituted in (as already mentioned), the conserved current is simply

Conserved current is not an independent equation, but rather it is the first integral of certain combination of the field equations. Thus, it is essential to check if the previous conserved current obtained for and satisfies the field equations, which was not performed by the authors [17]. Now the corresponding field equations are

It is not difficult to check that the previous conserved current () satisfies the field equations only under the trivial condition together with the condition , that is, for vanishing potential, which leads to inconsistency, since it has been restricted at the beginning. Thus, is not a symmetry of the action under consideration, and the work performed by Jamil et al. [17] is completely wrong. One can even look at the consequence in a straightforward manner. It is well-known that for the Noether point symmetry , the Noether integral is the Hamiltonian; that is, is the Hamiltonian . But here, . Thus, unless , implying and , Hamiltonian is not obtained as the integral of motion. However, since , it cannot be set equal to zero at this stage, as it makes and constant, while the form of the potential (28) becomes undefined. Thus, the Hamiltonian is never recovered as the Noether integral. Therefore, to recover the Hamiltonian constraint equation as an outcome of Noether point symmetry , one should start with , a priori, which we consider in the following subsection.

2.2. No Need to Consider Time Translation, so

It must be clear by this time why earlier authors did not consider time translation. Likewise, if one starts with , which implies and , turns out to be a constant and may be set as , without loss of generality. Thus, (24) through (27) take the following forms:

Now, just multiplying (35) by and comparing it with (34), one can observe that . Thus, search for a form of by imposing the Noether symmetry in the action (7) went in vain.

3. In Search of Noether Symmetry for a More General Tachyonic Action

Having obtained null result in connection with the Noether symmetry for Born-Infeld action being coupled to , let us now turn our attention to a more general action containing a linear curvature invariant term being nonminimally coupled to Born-Infeld- action (7). However, as we have already mentioned that the gravitational Hamiltonian is not only conserved but is also constrained to vanish and therefore is a part and parcel of Einstein’s equation, namely, the component, so time translation is overall unnecessary. Further, in all our earlier analysis, we have shown that a gauge term does not contribute to the Noether equations, since it becomes constant and therefore may be set equal to zero. Therefore, in the following analysis neither do we consider time translation nor a gauge term. In the Robertson-Walker line element (6) the following action

leads to the point Lagrangian:

following the same technique as before. For the previous point Lagrangian the Noether equation (1) reads

Now, equating coefficients as usual, we obtain the following overdeterminant set of Noether equations:

3.1. Solutions

Equations (41) through (48) imply , , and , where is a constant. Equation (42) then determines as a linear function of the scale-factor, namely,

and (42) gets reduced to

Equation (45) is then satisfied only for spatially flat case and thus reduces to

forcing to be a function of only; that is, . Equation (46) then in view of (45) makes a linear function of as ; is a constant. Thus, search for the Noether symmetry for again went in vain. Nevertheless, the coupling parameter gets solved as

where is a constant. Equation (47) is then trivially satisfied while (48) yields the following form of the potential:

The previous forms of the potential and the coupling parameter along with the conserved current had been found earlier by de Souza and Kremer [55] as a consequence of the Noether symmetry for linear gravity, and further, they had explicitly studied the cosmological evolution. Hence, we leave our discussion here. Nonetheless, we observe that theory of gravity does not admit the Noether symmetry even for a general Born-Infeld action and so our earlier conclusion that “it is not possible to find a form of other than the trivial and very special one, namely, , by imposing Noether symmetry” stands.

4. On the Absence of Noether Symmetry and Exploring the Special Feature of

We have mentioned that starting from , a priori, the Noether symmetry is obtainable for a scalar-tensor theory of gravity [56]. Further, it has been proved that leads to a scale-invariant action [57]. So, it is expected that starting from arbitrary form of if the Noether symmetry is claimed, it should end up with . But we have not been able to recover this result. This contradiction puts up doubt in treating as an auxiliary variable for canonical formulation of theory of gravity, since an auxiliary variable different from has been considered for canonical formulation of theory of gravity. Let us briefly describe the issue.

In the quantum domain observable depends on the choice of momentum, while momentum is different for different choice of auxiliary variable. The canonical formulation of gravity in view of the auxiliary variable ( being the action) leads to a Schrödinger-like quantum dynamics, with a hermitian effective Hamiltonian leading to the straightforward probability interpretation [59–64], provided that the total derivative terms in the action are taken care of a priori. From metric variation principle, it is known that theory of gravity must be supplemented by a boundary term , where symbols have their usual meaning. It was shown [63, 64] that to obtain Schrödinger-like quantum equation as mentioned, it is required first to express the action in terms of the first fundamental form and then to split the above boundary term into , where and Canonical programme then follows by eliminating the available total derivative term from the action, which gets cancelled with the boundary term , and then introducing the auxiliary variable, , as suggested by Horowitz [65] thereafter. Thus, is different from in the theory of gravity. It is not possible to follow such technique for an action containing a general theory of gravity. It is also important to mention that classical field equations require derivative of momentum, and an arbitrary choice of auxiliary variable reproduces correct classical field equations. However, as for quantization one requires momentum (rather than its derivative, which is the reason for obtaining different quantum dynamics with different auxiliary variables); likewise, for Noether symmetry one again requires momentum instead of its derivative, and this may be the reason why it could not reproduce Noether symmetry for theory of gravity already available in the literature [56]. All the beauty of theory of gravity observed in the context of cosmological data fitting, are outcome of scalar-tensor equivalence, since otherwise it is almost impossible to find exact solutions. Scalar-tensor equivalence is a mathematical artifact, and it gives totally different quantum description [66] in comparison to one discussed earlier [62, 63]. The difference at the classical level has also been established to some extent [67, 68]. Thus, the prospect of theory of gravity appears to be at stake, unless one can find a better way out to handle the theory. Indeed a better technique has been expatiated earlier by finding a general (non-Noether) conserved current for theory of gravity being minimally coupled to scalar-tensor theory of gravity [69]. The technique was also found useful to extract exact solutions of the theory [14, 70].

The obvious question is how then was found in vacuum and in radiation dominated era? Let us brief the beauty of in isotropic space-time [13]. Remember that in metric variation technique, the field equations are obtained by varying the action with respect to , while the scale-factor () is taken as the basic variable to express the canonical action. Instead, if we take to be the basic variable, then and so the action

for reads

Introducing an auxiliary variable

(which is clearly different from ) in the action and removing appropriate surface terms, the canonical form of the action is