###### pacs:

95.36.+x, 45.30.+s, 98.80.-kastro-ph/0607064

ABSTRACT

Weyl’s idea on scale invariance was resurrected by Cheng in 1988. The requirement of local scale invariance leads to a completely new vector field, which we call the “Cheng-Weyl vector field”. The Cheng-Weyl vector field couples only to a scalar field and the gravitational field naturally. It does not interact with other known matters in the standard model of particle physics. In the present work, the (generalized) Cheng-Weyl vector field coupled with the scalar field and its cosmological application are investigated. A mixture of the scalar field and a so-called “cosmic triad” of three mutually orthogonal Cheng-Weyl vector fields is regarded as the dark energy in the universe. The cosmological evolution of this “mixed” dark energy model is studied. We find that the effective equation-of-state parameter of the dark energy can cross the phantom divide in some cases; the first and second cosmological coincidence problems can be alleviated at the same time in this model.

## I Introduction

Since the discovery of present accelerated expansion of our universe r1 ; r2 ; r3 ; r4 ; r5 ; r6 ; r59 , dark energy r7 has been an active field in modern cosmology. One of the puzzles of the dark energy problem is the (first) cosmological coincidence problem, namely, why does our universe begin the accelerated expansion recently? why are we living in an epoch in which the energy densities of dark energy and dust matter are comparable? In order to give a reasonable interpretation to the (first) cosmological coincidence problem, many dynamical dark energy models have been proposed, such as quintessence r8 ; r9 , phantom r10 ; r11 ; r12 , k-essence r13 ; r62 etc.

The equation-of-state parameter (EoS) of dark energy plays a central role in observational cosmology, where and are its pressure and energy density respectively. Recently, by fitting the observational data, marginal evidence for at redshift has been found r14 ; r15 ; r16 . In addition, many best-fits of the present value of are less than in various data fittings with different parameterizations (see r17 for a recent review). The present observational data seem to slightly favor an evolving dark energy with crossing from above to below in the near past r15 . Obviously, the EoS of dark energy cannot cross the so-called phantom divide for quintessence or phantom alone. Although at first glance, it seems possible for some variants of k-essence to give a promising solution to cross the phantom divide, a no-go theorem, shown in r18 , shatters this kind of hope. In fact, it is not a trivial task to build dark energy model whose EoS can cross the phantom divide. To this end, a lot of efforts r19 ; r20 ; r21 ; r22 ; r23 ; r24 ; r25 ; r26 ; r27 ; r28 ; r29 ; r30 ; r31 ; r32 ; r33 ; r34 ; r35 ; r36 ; r67 ; r68 have been made. To name a few, quintom model, string theory inspired models, vector field models, crossing the phantom divide in the braneworld models, scalar-tensor models r68 , etc. However, to our knowledge, many of those models only provide the possibility that can cross . They do not answer another question, i.e., why does crossing the phantom divide occur recently? why are we living in an epoch ? This can be regarded as the second cosmological coincidence problem r33 ; r34 .

Although cosmological observations hint that , however, there is a subtle tension between observations and theory. For the canonical scalar phantom model r10 , the universe has an inevitable fate of big rip r11 , and the instability is inherent r12 . For the k-essence model with EoS less than , the spatial instabilities inevitably arise too r37 (see also r18 ). In a more general case, it is argued that there is a direct connection between instability and the violation of null energy condition (NEC) r38 ; r39 . Some models can evade this result, at the price of the lack of isotropy of the background and the presence of superluminal modes r40 ; r41 . Recently, two seemingly viable models that violate NEC without instability or other pathological features have been proposed in different ways r42 ; r43 . In particular, in r43 , a scalar field coupled with a vector field is used; and the effective Lagrangian explicitly depends on the vector filed , which avoids one of the assumptions of r38 ; r39 that the effective Lagrangian only depends on and has no dependence on the vector filed itself.

Motivated by the work of r43 , it is interesting to study the case of a scalar field coupled with a vector field. In fact, the vector field has been used in modern cosmology in many cases, see e.g. r44 ; r34 and references therein. It is worth noting that comparing to the ones investigated in e.g. r69 ; r70 , the vector fields used in r44 ; r34 have fairly different motivations and forms. In r28 , a single dynamical scalar field is coupled with an a priori non-dynamical background vector field with a constant zeroth-component. In that model, the effective EoS can cross the phantom divide . However, the appearance of the a priori non-dynamical vector field has no clear physical motivation.

In r45 ; r46 , Weyl’s old idea of scale invariance r47 ; r48 was resurrected by Cheng in 1988, almost 60 years later (See r49 for an independent rediscovery). The requirement of local scale invariance leads to the existence of a completely new vector field, which we call the “Cheng-Weyl vector field” throughout this paper, in honor of the proposer Cheng and Weyl (a great mathematician and physicist r50 ). The Cheng-Weyl vector field only couples to the scalar field and the gravitational field. It does not interact with other known matters in the standard model of particle physics, such as quarks, leptons, gauge mesons, and so on. In particular, it has no interaction with photons and electrons. So, it is “dark” in this sense. As mentioned above, the fact that Cheng-Weyl vector field naturally couples to the scalar field makes it very interesting, especially when the scalar field is considered as dark energy candidate. Required by the local scale invariance, the potential term of the scalar field has to be of form, while the coupling form between the Cheng-Weyl vector field and the scalar field is fixed also, and the form is different from the ones of r28 ; r43 . Interestingly, the effective Lagrangian also explicitly depends on the Cheng-Weyl vector field itself naturally. In Sec. II, we will give a brief review of the work of Cheng r45 ; r46 , in which the Cheng-Weyl vector field was proposed.

In the present work, the (generalized) Cheng-Weyl vector field coupled with a scalar field and its cosmological application are investigated. We regard a mixture of the scalar field and a so-called “cosmic triad” of three mutually orthogonal Cheng-Weyl vector fields as the dark energy in the universe. We derive the effective energy density and pressure of the “mixed” dark energy, and the equations of motion for the scalar field and the Cheng-Weyl vector field respectively. The cosmological evolution of this “mixed” dark energy is studied. We find that the effective EoS of dark energy can cross the phantom divide in some cases; the first and second cosmological coincidence problems can be alleviated at the same time in this model.

This paper is organized as follows. In Sec. II, we will briefly present the main points of the Cheng-Weyl vector field proposed in r45 ; r46 . In Sec. III, the effective energy density, pressure, and the equations of motion are obtained. In Sec. IV, The cosmological evolution of the “mixed” dark energy is investigated by means of dynamical system r51 ; the first and second cosmological coincidence problems are discussed. Finally, a brief conclusion is given in Sec. V.

Throughout this paper, we use the units and the notation , and adopt the metric convention as .

## Ii The Cheng-Weyl vector field

Following r45 ; r46 , here we give a brief review of the so-called Cheng-Weyl vector field. The arguments are based on the local scale invariance. It is important to distinguish the scale invariance from the gauge invariance. The scale invariance is the invariance of the action under the change of the magnitude rather than the phase of the fields. To be definite, let us consider the distance between two neighboring spacetime points, . We change the scale of the distance, for instance, changing the unit of length from meter to inch. With this change, the distance remains the same, but is measured in a different unit r46 . That is,

(1) |

where is a constant for the global scale invariance, and is a function of space and time for the local scale invariance. Then we have , and . And, , where is the determinant of the metric . So, the action is invariant under the scale transformations, provided that the Lagrangian density satisfies

(2) |

In this case, the forms of all equations in the theory remain the same.

Let us first see the case of the global scale invariance, i.e. is a constant. The Lagrangian density of a scalar field is given by

(3) |

where is a dimensionless constant. The Lagrangian density of a gauge meson is

(4) |

where , or (here is a coupling constant) for the Yang-Mills theory. The Lagrangian density for a fermion coupled with the electromagnetic field and the gravitational field is

(5) |

where , , and is the tetrad satisfying . It is easy to verify that the above Lagrangian densities satisfy Eq. (2) under the scale transformation Eq. (1) and

(6) |

Next we consider the case with the theory being scale invariant locally, i.e. is a function of space and time. Similar to the well-known arguments used to deduce the existence of gauge fields, one can find that a completely new vector field , namely the so-called Cheng-Weyl vector field, is required by the local scale invariance, while the replacements

(7) |

are also required in these Lagrangian densities, where is a dimensionless constant. One can verify that these Lagrangian densities with the replacements Eq. (7) satisfy Eq. (2) under the scale transformation Eqs. (1), (6) and

(8) |

With the replacements Eq. (7), the Lagrangian density of the scalar field Eq. (3) becomes

(9) |

Thus, the scalar field is coupled with naturally by the scale invariance. However, with the replacements Eq. (7), the Lagrangian densities of the gauge meson and the fermion, i.e. Eqs. (4) and (5) respectively, need not to be altered, since the terms involving completely cancel one another r45 ; r46 . Therefore, we conclude that the gauge meson and the fermion do not couple with . With identical arguments, the quarks and leptons etc. do not couple with as well. After all, we would like to mention that the Lagrangian density of itself r45 ,

(10) |

also satisfies Eq. (2) under the transformations Eqs. (1), (6) and (8), where

(11) |

We close this section with some remarks. First, required by the local scale invariance, the potential term of the scalar field has to be of form, as in Eqs. (3) and (9). Second, it is easy to see from Eq. (9) that the coupling form between the Cheng-Weyl vector field and the scalar field is fixed naturally. Note that the form is quite different from the ones considered in r28 ; r43 . For more details on the Cheng-Weyl vector field, please see r45 ; r46 . In addition, one may also refer to r60 ; r61 for relevant papers on the scale invariance.

## Iii Applying the Cheng-Weyl vector field to cosmology

In modern cosmology, the scalar field is used extensively. Actually, the scalar field is one of the leading dark energy candidates. If the nature respects the local scale invariance, the so-called Cheng-Weyl vector field must exist, and couples to the scalar field inherently. If the scalar field is indeed the cause driving the accelerated expansion of the universe, we argue that the dark energy should be a mixture of the scalar field and the Cheng-Weyl vector field, which can be considered as the partner of the scalar field. This seems quite plausible when the fact that the Cheng-Weyl vector field does not interact with other known matters (so, it is “dark” to them) is taken into account. Therefore it is quite interesting to study the cosmological consequence of the Cheng-Weyl vector.

We begin with the action

(12) |

where and are the actions for gravitational field and matters respectively, and

(13) |

Naively, one may write the Lagrangian density as

directly from Eqs. (9) and (10). In order to be compatible with homogeneity and isotropy, the can be chosen as , where only depends on the cosmic time . However, in this case . From this Lagrangian density , for the case of homogeneous , one finds that which is not dynamical, and the Lagrangian density is zero actually. Thus, unfortunately, the naive approach is not viable.

Enlightened by the work of r44 (see also r34 ), we can describe the dark energy as a mixture of a scalar field and a so-called “cosmic triad” (in the terminology of r44 ) of three mutually orthogonal Cheng-Weyl vector fields. In this case, the Lagrangian density is given by

(14) | |||||

where

(15) |

, , and being dimensionless constants. Latin indices label the different Cheng-Weyl vector fields () and Greek indices label different spacetime components (). Actually, the number of Cheng-Weyl vector fields is dictated by the number of spatial dimensions and the requirement of isotropy r44 ; r34 . The Latin indices are raised and lowered with the flat “metric” . It is worth noting that the Lagrangian density in Eq. (14) satisfies the requirement of the local scale invariance. Note that in Eq. (14), we have generalized the original scalar field to include the cases of quintessence () and phantom (), while the Lagrangian density for the Cheng-Weyl vector fields has also been generalized by introducing the constant .

Varying the action (13) with Eq. (14), one can get the energy-momentum tensor of the “mixed” dark energy as

(16) |

where

(17) |

From the action (13) with Eq. (14), one can also obtain the equations of motion for and , namely

(18) |

and

(19) |

respectively. We consider a spatially flat Friedmann-Robertson-Walker (FRW) universe with metric

(20) |

where is the scale factor. In this work, we assume the scalar field is homogeneous, namely . Similar to r44 , an ansatz for the Cheng-Weyl vectors is

(21) |

Thus, the three Cheng-Weyl vectors point in mutually orthogonal spatial directions, and share the same time-dependent length, i.e. . Hence, the equations of motion (18) and (19) become, respectively,

(22) |

and

(23) |

where is the Hubble parameter, and a dot denotes the derivative with respect to the cosmic time . From Eqs. (16), (14) and (17), we find that

(24) |

where is the energy density of dark energy, while

(25) |

and

(26) |

where

(27) |

It is worth noting here that in fact, even adopting the “cosmic triad” of three mutually orthogonal Cheng-Weyl vector fields, Lagrangian (14) is not invariant under rotation in the internal space. Thus, the energy-momentum tensor is not strictly diagonal. One can see this point by noting that , due to the second term in the right hand side of Eq. (17). To overcome this inconsistence with the isotropy, the spatial volume-average procedure has to be employed here as done in r63 ; r64 . In those two papers the authors considered the nonlinear electromagnetic field as the source driving the accelerated expansion of the universe. There, in order to obtain an energy-momentum tensor consistent with the FRW metric, the spatial volume-average procedure in the large scale r65 ; r66 has been used. By using this procedure, the spatial volume-averaged non-diagonal components of the energy-momentum tensor become zero, namely , while the diagonal components are kept unchanged. Therefore, in our model, the energy-momentum tensor is a spatial volume-averaged one on the cosmological scale. In this way, the energy-momentum tensor is compatible with isotropy.

The corresponding pressure of dark energy is given by

(28) |

The Friedmann equation and Raychaudhuri equation read, respectively,

(29) |

and

(30) |

where and are the pressure and energy density of the matters, respectively.

From Eqs. (24) and (28), we obtain

(31) |

Obviously, the EoS of dark energy is always larger than for the case of and , while for the case of and . Crossing the phantom divide is impossible for both cases. However, for the case of and having opposite signs, can be larger than or smaller than . Of course, for this case, crossing the phantom divide is possible.

## Iv Dynamical system and cosmological evolution

In this section, we investigate the cosmological evolution of the “mixed” dark energy by means of dynamical system r51 . Our main aim is to see whether this model can alleviate the coincidence problems. Like many considerations in the literature, we allow the existence of interaction between the dark energy and the background matter (usually the cold dark matter). The cases of the scalar field and vector field interacting with background matter are studied extensively, see, for examples, r52 ; r53 ; r54 ; r55 ; r56 ; r57 ; r58 ; r34 . Although the Cheng-Weyl vector field does not interact with the known matters in the particle physics standard model, nothing precludes the possibility of the Cheng-Weyl vector field interacting with the cold dark matter, since the nature of cold dark matter is also unknown so far.

### iv.1 Dynamical system

We assume that the dark energy and background matter interact through interaction terms and , namely

(33) | |||

(34) |

which keep the total energy conservation equation . The background matter is described by a perfect fluid with barotropic equation of state

(35) |

where the barotropic index is a dimensionless constant and satisfies . In particular, and correspond to dust matter and radiation, respectively. Due to the interaction terms and , the equations of motion for and , namely Eqs. (22) and (23), are altered. Seeing from Eqs. (32) and (33), they are given by

(36) |

respectively. Following r51 ; r52 ; r53 ; r54 ; r55 ; r56 , we introduce following dimensionless variables

(37) |

With the help of Eqs. (29), (30), (24) and (28), the evolution equations (36) and (34) can be rewritten as a dynamical system r51 , i.e.

(38) | |||

(39) | |||

(40) | |||

(41) | |||

(42) |

where

(43) |

and

(44) |

(45) |

a prime denotes derivative with respect to the so-called -folding time .

The fractional energy densities of the background matter and dark energy are given by

(46) |

and

(47) |

respectively. On the other hand, from Eq. (29), one has

(48) |

Hence, from Eqs. (47) and (48), one can find out

(49) |

where is given byEq. (45). The effective EoS of the whole system is

(50) |

where and are the EoS of dark energy and background matter, respectively.

### iv.2 Interaction terms and critical points

In this subsection, we obtain all critical points of the dynamical system (38)–(42). A critical point satisfies the conditions . Before giving the particular interaction terms and , let us first find the general features of the critical points of dynamical system (38)–(42). From Eq. (42) and , it is easy to see that

(51) |

If this dynamical system has some critical points, their corresponding , , , , and should be constants. Therefore, from Eqs. (49) and (45), the corresponding Hubble parameter . From Eq. (44),

(52) |

follows. Substituting into Eq. (41), requires

(53) |

since . Hence, from Eqs. (52), (44), (51) and (53), we have

(54) |

So, is required for non-vanishing real and . By using Eqs. (38)–(40) and Eqs. (51)–(54), become, respectively,

(55) | |||||

(56) | |||||

(57) |

where

(58) |

which comes from Eqs. (45), (51), (53), and (54). Then, one can find out the remaining , and from Eqs. (54)–(57). Obviously, only three of them are independent of each other.

So far, the above results are independent of particular interaction terms and . To find out , and , we have to choose proper and here. The interaction forms extensively considered in the literature (see r51 ; r52 ; r53 ; r54 ; r55 ; r56 ; r57 ; r58 ; r34 ; r32 for instance) are

Noting that Eqs. (53), (37) and the definition of in Eq. (43), we have to choose

(59) |

to avoid the divergence of in Eq. (55), where is a dimensionless constant. In this case, from Eq. (43), one has

(60) |

From Eqs. (53) and (60), we find that in Eq. (57). Noting that Eqs. (51), (53), (46) and the definition of in Eq. (43), we cannot choose or , to avoid the solution from Eq. (57), since our main aim is to alleviate the coincidence problems. Therefore, we choose Case (I) or Case (II) , where and are dimensionless constants.

#### Case (I)

In this case, from Eq. (43), one has

(61) |

Noting that in Eq. (57), we find out

(62) |

One can check that Eq. (56) is equivalent to Eq. (57) for this case. Then, one can find out and from Eqs. (54) and (55), by using Eqs. (51), (53) and (62). We do not present them here, since the final results are involved and tedious. One can work them out with the help of Mathematica. Instead we would like to give several particular examples to support our statement. Example (I.1), for parameters , , , , , and , we have , . Example (I.2), for parameters , , , , , and , we find that , .

From Eqs. (46), (48), and (62), the fractional energy densities of background matter and dark energy are given by

(63) |

respectively. For reasonable and , it is easy to see that is required. As mentioned above, at the critical point , the Hubble parameter . From Eq. (30), this means

(64) |

From Eq. (50), we find that the EoS of the dark energy is given by

(65) |

Obviously, .

#### Case (II)

In this case, the corresponding and read

(66) |

respectively. Solving Eq. (57) with , we get

(67) |

Again, one can check that Eq. (56) is equivalent to Eq. (57) for this case. Then, one can find out and from Eqs. (54) and (55), by using Eqs. (51), (53) and (67). Once again, we do not present the long and involved expressions here. We only give some particular examples. Example (II.1), for parameters , , , , , and , we find that , . Example (II.2), for parameters , , , , , and , we have , .

From Eqs. (46), (48), and (67), the fractional energy densities of background matter and dark energy are given by, respectively,

(68) |

which requires . Following a similar argument, we have also. And then, from Eq. (50), we find that the EoS of the dark energy is given by

(69) |

which is also smaller than .

### iv.3 Stability analysis

In this subsection, we discuss the stability of these critical points. An attractor is one of the stable critical points of the autonomous system. To study the stability of these critical points, we substitute linear perturbations , , , and about the critical point into the dynamical system Eqs. (38)–(42) and linearize them. We get the evolution equations for the fluctuations as

(70) | |||||

(71) | |||||

(72) | |||||

(73) | |||||

(74) |

where , , , , and are the linear perturbations coming from , , , , and , respectively. The five eigenvalues of the coefficient matrix of the above equations determine the stability of the corresponding critical point.