Higher Dimensional Cosmology with Some Dark Energy Models in Emergent, Logamediate and Intermediate Scenarios of the Universe

Higher Dimensional Cosmology with Some Dark Energy Models in Emergent, Logamediate and Intermediate Scenarios of the Universe

Abstract

We have considered -dimensional Einstein field equations in which four-dimensional space-time is described by a FRW metric and that of extra dimensions by an Euclidean metric. We have chosen the exponential forms of scale factors and numbers of in such a way that there is no singularity for evolution of the higher dimensional Universe. We have supposed that the Universe is filled with K-essence, Tachyonic, Normal Scalar Field and DBI-essence. Here we have found the nature of potential of different scalar field and graphically analyzed the potentials and the fields for three scenario namely Emergent Scenario, Logamediate Scenario and Intermediate Scenario. Also graphically we have depicted the geometrical parameters named statefinder parameters and slow-roll parameters in the higher dimensional cosmology with the above mentioned scenarios.

pacs:

I Introduction

From recent observations it is strongly believed that the most interesting problems of particle physics cosmology are the origin due to accelerated expansion of the present Universe. The observation from type Ia supernovae [1,2] in associated with Large scale Structure [3] and Cosmic Microwave Background anisotropies(CMB) [4] have shown the evidences to support cosmic acceleration. The theory of Dark energy is the main responsible candidate for this scenario. From recent cosmological observations including supernova data [5] and measurements of cosmic microwave background radiation(CMBR) [4] it is evident that our present Universe is made up of about 4% ordinary matter, about 74% dark energy and about 22% dark matter. Several interesting mechanisms have been suggested to explain this feature of this Universe, such as Loop Quantum Cosmology (LQC) [6], modified gravity [7], Higher dimensional phenomena [8], Brans-Dicke theory [9], brane-world model [10] and many others.

Recently many cosmological models have been constructed by introducing dark energies such as Phantom [11], Tachyon scalar field [12], Hessence [13], Dilaton scalar field [14], K-essence scalar field [15], DBI essence scalar field [16], and many others. After realizing that many interesting of particle interactions need more than four dimensions for their formulation, the study of higher dimensional theory has been revived. The model of higher dimensions was proposed by Kaluza and Klein [17,18] who tried to introducing an extra dimension which is basically an extension of Einstein general relativity in 5D. The activities of extra dimensions also verified from the STM theory [19] proposed recently by Wesson et al [20]. As our space-time is explicitly four dimensional in nature so the ‘hidden’ dimensions must be related to the dark matter and dark energy which are also ‘invisible’ in nature.

Form the cosmological observation the present phase of acceleration of the Universe is not clearly understood. Standard Big Bang cosmology with perfect fluid assumption fails to accommodate the observational fact. Recently, Ellis and Maartens [21] have considered a cosmological model where inflationary cosmologies exist in which the horizon problem is solved before inflation begins, no big-bang singularity exist, no exotic physics is involved and quantum gravity regime can even be avoided. An emergent Universe model if developed in a consistent way is capable of solving the conceptual problems of the big-bang model. Actually the Universe starts out in the infinite past as an almost static Universe and expands slowly, eventually evolving into a hot big-bang era. An interesting example of this scenario is given by Ellis, Murugan and Tsagas [22], for a closed Universe model with a minimally coupled scalar field , which has a special form of interaction potential . There are several features for the emergent Universe [21,23] viz. (i) the Universe is almost static at the finite past, (ii) there is no time like singularity, (iii) the Universe is always large enough so that the classical description of space time is adequate, (iv) the Universe may contains exotic matter so that the energy condition may be violated, (v) the Universe is accelerating etc.

Here we also consider another two scenarios: (i) “intermediate scenario” and (ii) “logamediate scenario” [24-27] to study of the expanding anisotropic Universe in the presence of different scalar fields. In the first case the scale factors evolves separately as and where , , and . So the expansion of the Universe is slower than standard de Sitter inflation (arises when ) but faster than power law inflation with power greater than 1. The Harrison - Zeldovich spectrum of fluctuation arises when and . In the second case we analyze the inflation with scale factors separately of the form and with , , and . When this model reduces to power law inflation. The logamediate inflationary form is motivated by considering a class of possible cosmological solutions with indefinite expansion which result from imposing weak general conditions on the cosmological model.

In this work, we have considered N-dimensional Einstein field equations in which 4-dimensional space-time is described by a FRW metric and that of the extra -dimensions by an Euclidean metric. We also consider the Universe is filled with K-essence scalar field, normal scalar field, tachyonic field and DBI essence and investigate the natures of the dark energy candidates for Emergent, Intermediate and Logamediate scenarios of the Universe. Here in extra dimensional phenomenon we have shown the change of the potential corresponding to the field for the dark energies mentioned above and also analyze the anisotropic Universe using the “slow roll’ parameters in Hamilton-Jacobi formalism and in terms of above mentioned scalar field and they are given by [24]

(1)

Sahni et al [28] proposed the trajectories in the {} plane corresponding to different cosmological models to depict qualitatively different behavior. The statefinder diagnostic along with future SNAP observations may perhaps be used to discriminate between different dark energy models. The above statefinder diagnostic pair for higher dimensional anisotropic cosmology are constructed from the scale factors and as follows:

(2)

where is the deceleration parameter defined by and is the Hubble parameter. Since this parameters are dimensionless so they allow us to characterize the properties of dark energy in a model independently. Finally we graphically analyzed geometrical parameters in the higher dimensional anisotropic Universe in emergent, logamediate and intermediate scenarios of the universe.

Ii Basic Equations

We consider homogeneous and anisotropic -dimensional space-time model described by the line element [29,30]

(3)

where is the number of extra dimensions and represents the line element of the FRW metric in four dimensions is given by

(4)

where and are the functions of alone represent the scale factors of 4-dimensional space time and extra -dimensions respectively. Here is the curvature index of the corresponding 3-space, so that the above Universe is described as flat, closed and open respectively.

The Einstein’s field equations for the above non-vacuum higher dimensional space-time symmetry are

(5)
(6)

and

(7)

where and are energy density and isotropic pressure respectively. Here we choose here and , so we have and . Also in this model we define the Hubble parameter as  .

Iii Emergent Scenario

At first we consider Emergent scenario, where the scale factors and are consider as the power of cosmic time are given by [23, 31, 32]

(8)

where , , , , , , and are positive constants. So the field equations (5), (6) and (7) become

(9)
(10)

and

(11)

We now consider K-essence field, Tachyonic field, normal scalar field and DBI essence field. For these four cases we analyze the behavior of the Emergent Universe in extra dimension and finally we analyze the behavior of the statefinder parameters and .

K-essence Field:

The energy density and pressure due to K-essence field are given by [15]

(12)

and

(13)

where and is the relevant potential for K-essence Scalar field .

Using equations (5)-(7), we can find the expressions for and as

(14)

and

(15)

From above forms of and , we see that can not be expressed explicitly in terms of .

Tachyonic field:

The energy density and pressure due to the Tachyonic field are given by [12]

(16)

and

(17)

where is the relevant potential for the Tachyonic field . Using equations (5)-(7), we can find the expressions for and as

(18)

and

(19)

From above forms of and , we see that can not be expressed explicitly in terms of .

Normal Scalar field:

The energy density and pressure due to the Normal Scalar field are given by [37]

(20)

and

(21)

where is the relevant potential for the Normal Scalar field . Using equations (5)-(7), we can find the expressions for and as

(22)

and

(23)


DBI-essence:

The energy density and pressure due to the DBI-essence field are given by [35,36]

(24)

and

(25)

where is given by

(26)

and is the relevant potential for the DBI-essence field .

The energy conservation equation is given by

(27)

where is the Hubble parameter in terms of scale factor as

(28)

From energy conservation equation we have the wave equation for as

(29)

where is the derivative with respect to . Now for simplicity of calculations, we consider two particular cases: = constant and constant.

Case I: = constant.

In this case, for simplicity, we assume and (). So we have .
In these choices we have the following solutions for , and from equation (29)as

(30)

from

(31)

and

(32)

where, ,, and is an integrating constants. From Fig. 7 and Fig. 8, we see that and are both exponentially decreasing with DBI scalar field .

Case II: constant.

In this case, we consider and . Using equations (5)-(7) and (24)-(26), we can find the expressions for , and as [34]

(33)

and

(34)

where is an integrating constant.

Statefinder parameters:

The geometrical parameters{} for higher dimensional anisotropic cosmology in emergent scenario can be constructed from the scale factors and as

(35)
(36)


Fig.12                   
Fig. 12 shows the variations of against , for in Emergent Scenario.

The relation between and has been shown in Fig.12. From Fig.12, we see that is negative when . The curve shows that the Universe starts from Einstein static era and goes to the model ().





Iv Logamediate Scenario

Now we consider Logamediate scenario, where the scale factors and are consider as the power of cosmic time are given by [24]

(37)

where , , and are positive constants. So the field equations (5), (6) and (7) become

(38)
(39)
(40)

We now consider K-essence field, Tachyonic field, normal scalar field and DBI essence field. For these four cases we analyze the behavior of the Logamediate Universe in extra dimension and finally we analyze the behavior of the state finder parameters and .

K-essence Field:

The energy density and pressure due to K-essence field are given by the equations (12) and (13). Using equations (38)-(40), we can find the expressions for and as

(41)

and

(42)

Tachyonic field:

The energy density and pressure due to the Tachyonic field are given by the equations (16) and (17). Using equations (38)-(40), we can find the expressions for and as

(43)

and

(44)

Normal Scalar field:

The energy density and pressure due to the Normal Scalar field are given by the equations (20) and (21). Using equations (38)-(40), we can find the expressions for and as