# Vector dark energy models with quadratic terms in the Maxwell tensor derivatives

###### Abstract

We consider a vector-tensor gravitational model with terms quadratic in the Maxwell tensor derivatives, called the Bopp-Podolsky term. The gravitational field equations of the model and the equations describing the evolution of the vector field are obtained and their Newtonian limit is investigated. The cosmological implications of a Bopp-Podolsky type dark energy term are investigated for a Bianchi type I homogeneous and anisotropic geometry for two models, corresponding to the absence and presence of the self-interacting potential of the field, respectively. The time evolutions of the Hubble function, of the matter energy density, of the shear scalar, of the mean anisotropy parameter, and of the deceleration parameter, respectively, as well as the field potentials are obtained for both cases by numerically integrating the cosmological evolution equations. In the presence of the vector type dark energy with quadratic terms in the Maxwell tensor derivatives, depending on the numerical values of the model parameters, the Bianchi type I Universe experiences a complex dynamical evolution, with the dust Universes ending in an isotropic phase. The presence of the self-interacting potential of the vector field significantly shortens the time interval necessary for the full isotropization of the Universe.

###### pacs:

98.80.-k, 98.80.Jk, 98.80.Es, 95.36.+x## I Introduction

Recent cosmological observations, based initially on the study of the distant Type Ia Supernovae, have shown that the cosmological paradigm according to which the Universe must decelerate due to its own gravitational attraction is not correct and that the Universe has experienced a transition to a late time, de Sitter type accelerated phase 1n (); 2n (); 3n (); 4n (). These observations have triggered a deep revision of our understanding of the cosmological dynamics and of its theoretical basis, general relativity. To explain the current observations in cosmology many theoretical ideas and suggestions have been put forward to address the intriguing facts revealed by the complex observational study of the Universe. From both theoretical and observational points of view there is a general consensus, which may be referred to as “the standard explanation of the late time acceleration,” according to which observations can be easily explained once we assume the existence of a mysterious and dominant component in the Universe, called dark energy, and is fully responsible for the observed dynamics in the late phases of the evolution of the Universe.

Another important cosmological result, based on the combination of data from the observations of high redshift supernovae, the WMAP satellite, and the recently released Planck data, convincingly show that the location of the first acoustic peak in the power spectrum of the CMBR (Cosmic Microwave Background Radiation) is entirely consistent with the important prediction of the inflationary model for the total density parameter of the Universe, according to which at the end of inflation . The important cosmological parameter , where is the total pressure and is the total density of the Universe is also strongly constrained by cosmological observations which provide detailed evidence for the behavior of the equation of state of the cosmological fluid, constraining the parameter as lying in the range acc ().

These large number of cosmological observations have led to the formulation of the CDM paradigm according to which, in order to explain the cosmological evolution, one assumes that the Universe is filled with two main components representing around 95% of its content; cold (pressureless) dark matter (CDM) and dark energy (DE), having a negative pressure. The contribution of the CDM component to the total density parameter of the Universe is of the order of P2 (). From a theoretical point of view the necessity to consider dark matter is mainly required by the necessity of explaining the unusual behavior of the galactic rotation curves as well as the formation of the large scale structure. On the other hand, dark energy is considered as representing the major component of the “chemical” composition of the Universe, giving a contribution to the total density parameter of the order of . Dark energy is the major cause determining the recent, de Sitter type, acceleration of the Universe, as confirmed by the study of high redshift type Ia supernovae acc (). The search for an explanation of the physical (or geometrical) nature and properties of dark energy has opened a very active field of research in cosmology and theoretical physics which in turn has led to a myriad of different DE models, for reviews of DE models see, for example, PeRa03 (); new1 (); Pa03 (); DEreviews (); Od (); LiM (); Mort (); Amend ().

One interesting possibility for explaining DE which has been intensively investigated is based on a number of cosmological models in which the “chemical” composition of the Universe consists of a mixture of two major components; cold dark matter and a slowly-varying, spatially inhomogeneous component, called the quintessence 8n (). In this scenario the baryonic matter plays a negligible role with a minimal influence on late time cosmological dynamics. From a formal theoretical point of view and based on some particle physics results, quintessence type cosmological models can be implemented by assuming that dark energy is the energy associated with a scalar field with a self-interacting potential Fa04 (). During the cosmological evolution, when the potential energy density of the quintessence field becomes greater than the kinetic energy density, the thermodynamic pressure , associated with the quintessence -field, becomes negative. The cosmological and astrophysical properties of the quintessential cosmological models have been intensively investigated in the literature, for a recent review see Tsu (). Quintessence models differ from cosmological models of the standard general relativity including the cosmological constant since they imply that the equation of state of the quintessence field varies dynamically with cosmic time 11n (). A number of alternative cosmological models, called essence, where the late-time acceleration of the Universe is driven by the kinetic energy of the scalar field have also been proposed kessence0 ().

Another possibility to explain the recent acceleration of the Universe and the nature of dark energy is provided by scalar fields that are minimally coupled to gravity via a negative kinetic energy. An interesting property of these fields is that they allow for values of the equation of state parameter, , of dark energy to vary in such a way as to have . These types of scalar fields are known as phantom fields, proposed as an explanation for the late time acceleration of the Universe in phan1 (). The energy density and the pressure of phantom scalar fields are given by and , respectively. Phantom cosmological models for dark energy have been investigated in detail in phan2 (); phan3 (); phan4 (); new2 (). Some recent cosmological observations seem to support the interesting result that at some instant during the evolution of the Universe the value of representing dark energy equation of state may have crossed the standard value , hence entering a de Sitter type expansion with a cosmological constant . This intriguing cosmological phenomenon is called the phantom divide line crossing phan4 (). The crossing of the phantom divide line was investigated in the case of scalar field models with cusped potentials in phan3 (). The phantom divide line crossing can also be explained in cosmological models where dark energy is represented by a scalar field, which is non-minimally coupled to gravity phan3 ().

A different line of research on dark energy is based on the assumption that instead of interpreting dark energy as a specific physical field, the cosmological dynamics of the Universe can be understood as a modification of the gravitational force itself. By following this line of thought one can assume that at very large cosmological scales general relativity cannot describe the dynamical evolution of the Universe, and therefore the acceleration of the Universe is related to an intrinsic change of the gravitational interaction. A plethora of modified gravity models, based on different extensions of general relativity, like, for example, gravity (in which the gravitational action is an arbitrary function of the Ricci scalar ) Bu70 () and mimetic- gravity models mime (), the model (where is the matter Lagrangian) Har1 (), modified gravity models (where denotes the trace of the energy-momentum tensor) Har2 (), the Weyl-Cartan-Weitzenböck (WCW) model WCW (), hybrid metric-Palatini gravity models (where is the Ricci scalar formed from a connection independent of the metric) Har3 (), type models, where is the Ricci tensor and is the matter energy-momentum tensor, respectively Har4 (), the Eddington-inspired Born-Infeld theory EIBI (), gravity HT (), implying coupling between torsion scalar and trace of the matter energy-momentum tensor, or vector Gauss-Bonnet theory vgb (), have been recently proposed in the literature. The cosmological and astrophysical properties of these models have been extensively investigated. For a recent review of the generalized gravitational models with non-minimal curvature-matter coupling and type see Revn (). For a review of hybrid metric-Palatini gravity see Revn1 (). Modified gravity models can provide convincing theoretical explanations for the late time acceleration of the Universe without advocating the existence of dark energy and can also offer some alternative explanations for the nature of dark matter.

From a field theoretical point of view however, despite the great success of the scalar field dark energy models, the possibility that dark energy has a more complex structure than allowed by the simple scalar field model cannot be ignored a priori. One promising direction in the analysis of dark energy is represented by models in which dark energy is described by a vector or Yang-Mills type field which may also couple, minimally or non-minimally, to gravity. The simplest action for a Yang-Mills type dark energy model is v1 (); new5 ()

(1) |

where , are the potentials of the Yang-Mills field, , is the covariant derivative with respect to the metric, is defined as and represents a self-interacting potential, explicitly violating gauge invariance. In the action given by Eq. (1) there are three vector fields describing dark energy. Hence Eq. (1) generalizes the Einstein-Maxwell type single vector field dark energy model. The astrophysical and cosmological applications of the single or Yang-Mills type vector dark energy models have been comprehensively investigated in v2 ().

Extended vector field dark energy models where the vector field is non-minimally coupled to the gravitational field can also be constructed v3 (). The action for such a non-minimally coupled vector dark energy model is given by

(2) |

where , is the four-potential of the vector type dark energy, which couples non-minimally to gravity, and is the mass of the massive cosmological vector field, respectively. The constants and are dimensionless coupling parameters, while the vector dark energy field tensor is defined as .

Inspired by the possible analogy between dark energy and some condensed matter concepts, a so called superconducting type dark energy model was proposed in SupracondDE (). This model describes the spontaneous breaking of U(1) symmetry of the “electromagnetic” type dark energy, and is described by the action

(3) |

where and are constants, is the Lagrangian of the total (ordinary baryonic plus dark) matter, and is the total mass current, where is the total matter density (including dark matter), and is the matter four-velocity. This model can also be interpreted and understood as unifying, in a single formalism, the scalar and vector dark energy models. The predictions of the superconducting dark energy model have been compared with observations in supobs ().

It is the goal of the present paper to consider a vector-tensor type model of dark energy, based on the analogy with Bopp-Podolsky electrodynamics. The Bopp-Podolsky theory was first suggested by Bopp Bopp (), and was independently reobtained by Podolsky Podolsky (). The Bopp-Podolsky theory retains linearity of the field equations but introduces higher-derivative terms proportional to the parameter where , having the physical dimensions of mass, is a new hypothetical fundamental constant of Nature. For the Maxwell-Lorentz theory, and the Maxwell equations, are retained. The Bopp-Podolsky theory is formulated in terms of an action functional from which the field equations, which are of fourth order in the electromagnetic potential, are derived. However, as noted by both Bopp and Podolsky, in a certain gauge these fourth-order equations are equivalent to a pair of second-order equations Bopp (); Podolsky (). Different aspects of the Bopp-Podolsky type extension of classical electrodynamics were investigated in applBopp ().

To this and other ends, we start from the analogy with the Bopp-Podolsky electrodynamics and introduce a vector-tensor gravitational model where the action for the minimally coupled vector field also contains additional terms, quadratic in the Maxwell tensor derivatives. These terms correspond to the covariant form of the action of the Bopp-Podolsky electrodynamics. Moreover, a term describing the non-minimal coupling between the cosmological mass current and the four-potential of the vector field is also added to the action. The possible existence of a self-interaction potential of the vector field is also considered. From a cosmological point of view we propose to interpret the vector field as describing the dark energy component of the Universe, which is responsible for the late, de Sitter type acceleration of the Universe. We obtain the gravitational field equations of this vector dark energy model as well as the equations describing the evolution of the vector field. We investigate the Newtonian limit of the model and show that the Poisson equation as well as the Bopp-Podolsky electrodynamics can be recovered for weak fields.

The cosmological implications of this vector type dark energy model are investigated for a Bianchi type I homogeneous and anisotropic geometry. Two cases are investigated in detail, the evolution of the Universe with and without the self-interacting potential of the field, respectively. In both cases the evolution of the Hubble function, of the matter energy density, of the shear scalar, of the anisotropy parameter, of the deceleration parameter, and of the field potential are analyzed in detail. To escribe the matter content of the Universe we adopt the radiation fluid and the dust matter equations of state. We find that in the presence of the vector type dark energy with quadratic terms in Maxwell tensor derivatives the anisotropic Universe experiences a complex dynamical behavior, with the dust Universes ending in an isotropic stage, a result which is independent on the presence or absence of the self-interaction potential of the field.

The present paper is organized as follows. The field equations of the Bopp-Podolsky type vector-tensor gravitational model are derived in Section II and their Newtonian limit is also investigated. The cosmological implications of the model are investigated in Section III, where the cosmological dynamics of a Bianchi type I geometry is analyzed for both models with and without self-interaction potential of the vector field, and for two different equations of state of the cosmic matter. We discuss and conclude our results in Section IV.

## Ii Bopp-Podolsky type vector dark energy models

We first start by briefly introducing the basic theoretical ideas of the Bopp-Podolsky type electrodynamics in its standard formulation in Minkowski geometry. Then, by adopting, as a starting point, the view that higher order derivatives of the Maxwell tensor may play a significant role in vector type models of dark energy, we introduce the gravitational action for such a theoretical model. The gravitational field equations as well as the equations of the vector field are derived from the action together with an equation representing the covariant conservation of the energy-momentum tensor.

### ii.1 The Bopp-Podolsky model of electrodynamics

The Lagrangian density from which Maxwell’s equations can be obtained by the usual variational principle is LaLi ()

(4) |

where we use a system of units with . In the above equation is the Maxwell electromagnetic field tensor, is the four-vector potential of the field while denotes the electromagnetic current. This Lagrangian is a function of the field variables and of their first derivatives. There is no reason why we should restrict ourselves to only first derivatives in the action and it therefore seems natural to try a generalization of Eq. (4) of the form Bopp (); Podolsky ()

(5) |

The usual variational principle applied to this Lagrangian leads to the field equations

(6) |

where

(7) |

The simplest choice for , as proposed by Bopp and Podolsky Bopp (); Podolsky (), is

(8) |

where is a new fundamental constant with mass dimension . Using Eq. (8) in Eq. (7) we obtain

(9) |

so that the field Eq. (6) becomes

(10) |

where

(11) |

has the property By defining

(12) |

we obtain

(13) |

If we impose the condition Eq. (11) becomes

(14) |

By introducing the four-potential we obtain

(15) |

To summarize, an interesting result in the Bopp-Podolsky theory is that the electromagnetic field equations, the potentials and the fie1d strengths can be written as the difference, respectively, of the potentials and field strengths of two distinct fields

(16) |

(17) |

These two fields are described by two sets of separate field equations, with the first set corresponding to the standard Maxwell equations, while the second set represents Proca type field equations for particles with mass . This result also provides a physical interpretation of the new fundamental constant . However, the mass term appears with a wrong sing in the equation of motion, signaling that the massive vector field is a ghost. In order to make it clear, let us rewrite the Lagrangian (8) in the Lorentz gauge with the result

(18) |

One can easily check that the above Lagrangian is identical to

(19) |

Now using the transformation , one obtains

(20) |

It is now seen that the kinetic term of the massive vector field appears with a wrong sign, signaling that the massive vector field is a ghost. In order to make the theory healthy at the background level, one should make the massive ghost non-dynamical. In this paper, we will consider the cosmology of this model, so the Maximum energy scale of our theory is . By assuming that the ghost mass is larger than the energy scale of our theory, the ghost mass does not have any dynamics at length scales smaller than which is the desired range of doing cosmology. This means that the Kinetic energy of the ghost field is much less than its potential energy. Noting that the ghost mass squared is equal to , one can see that for values , the ghost becomes non-dynamical. This is what we consider in what follows.

### ii.2 Bopp-Podolsky type vector dark energy models

In the following we assume that the vector dark energy can be described by a Bopp-Podolsky type model, with the action given by

(21) |

where is the dark energy potential related to dark energy field strength by

(22) |

where , is the cosmological matter 4-vector current, is a constant with dimension of mass and is the action for ordinary matter. We have also added to the gravitational action the self-interacting potential of the vector field, and we have allowed for the possibility of a direct coupling between the matter current and dark energy vector potential , with the strength of the coupling described by the constant .

Varying the action (22) with respect to dark energy potential and the metric we have

(23) |

where a prime indicates derivative with respect to the argument and

(24) |

At this point, a note about the variation of the term is in order. The variation of the energy momentum tensor can be written as (see Appendix A)

while the variation of the four-velocity of the particle is

Putting all these results in the variation of , one can see that dependence vanishes from the metric equation of motion. In the following we will assume that the energy momentum tensor of ordinary matter is that of a perfect fluid

(25) |

In order to write the equation of the vector field in a form similar to the one in Bopp-Podolsky electrodynamics, we introduce a new auxiliary vector field , defined as

Then the vector field equation (23) reduces to two coupled differential equations for and as

(26) |

and

(27) |

respectively. The conservation of the energy momentum tensor is now obtained by taking the covariant divergence of the metric field equation. After some algebra, one finds

(28) |

where is the expansion parameter, is the acceleration and .

By taking the covariant derivative of equation (23) we obtain

(29) |

### ii.3 The Newtonian limit

In the following we consider the weak field limit of the Bopp-Podolsky type vector dark energy model for a static source, i.e., the Newtonian limit. In this case the only non-zero component of the energy-momentum tensor is and one may easily find that where is the Newtonian potential which is related to the metric component through . Note that in this paper we are considering the vector field as the dark energy sector of the universe which should be very small in the Newtonian limit of the theory. So, we consider as a first order perturbed field, the same order as .

The trace of the equation (II.2) can be reduced to

(30) |

where is the trace of the matter energy momentum tensor. We have to keep only the first order terms in and . This implies that the terms in the metric equation (II.2) which are quadratic in do not contribute to this limit. With these assumptions one obtains the generalized Poisson equation

(31) |

where means that we only keep terms which are linear in . This implies that only the affect the Poisson equation, which is exactly the cosmological constant. Note that the minus sign behind the energy density is because of our convention in defining the Newtonian potential in .

Let us now consider the vector field equation (23) in the Newtonian limit. In this limit, the covariant derivatives should be replaced by partial derivatives, since we have assumed that the vector field is a small quantity. One can then show that the vector field equation reduces to

(32) |

which is exactly the original Bopp-Podolsky equation.

## Iii Cosmological implications in the presence of Bopp-Podolsky type vector fields

### iii.1 The Isotropic Cosmology

In this section we want to consider the cosmological implications of the theory. First, let us assume that the geometry of the Universe is described by the Fiedmann-Robertson-Walker metric. With this choice the possible form of the vector field should have the form

to preserve homogeneity and isotropy. However, with this choice the vector field strength tensor and therefore the Bopp-Podolsky term vanishes in our theory. One can easily find that the Friedmann and the vector field equations in the absence of matter fields in this case can be written as

(33) | |||

(34) | |||

(35) |

The simplest possibility to satisfy the last equation is that the potential becomes constant. This is the standard de Sitter type theory, with constant Hubble parameter . One can however drive a self accelerated expanding universe by choosing other forms for the potential . In these cases the -component of the vector field should be constant in order to satisfy the equation . For example, in the case that , one should have .

### iii.2 Anisotropic Cosmology - Bianchi I model

In order to make the theory non-trivial, we should assume that the vector field has a spatial component. So, we will assume that the Universe can be described by the Bianchi-I type metric of the form

(36) |

and the vector field can then be written as

(37) |

One should note that we have assumed that the component of the vector field is zero. This is because this component does not contribute to the strength tensor .

Also, we assume that the matter content of the Universe consists of a perfect cosmological fluid, with energy momentum tensor

(38) |

where is the total matter density (dark plus baryonic), and is the matter thermodynamic pressure.

For later convenience, we will define the directional Hubble factors , the mean Hubble factor , the anisotropy parameter , the shear scalar and the deceleration parameter as def ()

(39) | ||||

(40) | ||||

(41) | ||||

(42) | ||||

(43) |

With the above definitions, one can easily see that the quantity vanishes. In this case, the time component of the conservation equation (28) leads to the usual conservation equation of the form

(44) |

while the -component of the conservation equation gives . We can then assume that , a condition which further implies that the Bopp-Podolsky term vanishes, or one should conclude that , i.e. no matter/vector field coupling. We will choose the second choice and in the following we will assume that and then the conservation equation for the ordinary matter field hold. With these assumptions, one can see that equation (29) is satisfied identically.

#### iii.2.1 The cosmological field equations

Let us introduce a new variable , defined as

(45) |

With the above assumptions, only the component of the vector field equation of motion becomes non-zero, which can be written as

(46) |

The Friedmann equations can then be simplified to

(47) |

(48) |

and

(49) |

where we have defined .

#### iii.2.2 The case of massless vector field

Let us now investigate the cosmological implications of the Bopp-Podolsky theory with a massless vector field. In this case the potential term vanishes. In order to simplify the mathematical formalism, let us introduce a set of dimensionless variables , defined as

(50) |

where is the present day value of the Hubble function. By using the above set of variables, the cosmological evolution equations for the Bopp-Podolsky type vector dark energy model can be written as

(51) |

(52) |

(53) |

and

(54) |

and the energy conservation equation becomes

(55) |

where now, “dot” represents derivative with respect to . One should note that because we want to make the ghost degree of freedom non-dynamical, one should assume .

From Eq. (52) we can obtain as

(56) |

After substituting this expression of into Eq. (53), we can solve Eqs. (53) and (III.2.2) to obtain the expressions of and , respectively. Therefore the system of equations describing the evolution of the anisotropic Bianchi type I Universe in the presence of Bopp-Podolsky type vector dark energy can be written as

(57) |

(58) |

(59) | |||||

(60) | |||||

(61) |

(62) |

After adopting an equation of state for the cosmological matter, the system of differential equations Eqs. (57)-(61) must be solved by choosing some appropriate initial conditions, which we take as , , , , and , respectively. In the dimensionless variables introduced above the deceleration parameter is given by

(63) |

In the following we will assume that the age of the Universe is of the order of , which gives for the dimensionless time the maximum value of .

#### iii.2.3 Approximate anisotropic solution with constant

As an example of a simple exact solution of the cosmological evolution equation (51)-(III.2.2) we will consider the case , a condition which gives for the evolution of the vector field potential an equation of the form . With this choice for the evolution equations (51)-(III.2.2) become

(64) |

(65) |

(66) |

(67) |

Eqs. (65)-(67) can be solved to give the matter energy density and the pressure as

(68) |

(69) |

(70) |

By assuming , Eq. (64) can be immediately integrated to give

(71) |

where we have denoted

(72) |

and we have used the initial condition and , respectively. Then the requirement of the equality of the pressures in Eqs. (69) and (70) gives for the evolution equation

(73) |

Eq. (73) is a Riccati type equation, which generally cannot be solved exactly. For the matter energy density and pressure we obtain

(74) |

(75) |

The solutions are periodic, with the period , where is the present day age of the Universe. Hence one period describes roughly the entire cosmological history. In the rescaled dimensionless time this corresponds to a time interval .