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Bianchi Type III Anisotropic Dark Energy Model withConstant Deceleration Parameter

Anil Kumar Yadav1 and Lallan Yadav2

1Department of Physics, Anand Engineering College,Keetham, Agra-282 007,India

E-mail: abanilyadav@yahoo.co.in, akyadav@imsc.res.in

2 Department of Physics, D. D. U. Gorakhpur University, Gorakhpur - 273 009E-mail: nisaly06@rediffmail.com

Abstract

The Bianchi type III dark energy model with constant deceleration pa-rameter is investigated. The equation of state parameter ω is found to betime dependent and its existing range for this model is consistent with therecent observations of SN Ia data, SN Ia data (with CMBR anisotropy)and galaxy clustering statistics. The physical aspects of the dark energymodel is discussed.

Keywords : Bianchi type III Universe, Dark Energy, EoS Parameter

1 Introduction

Arguments have been put forward that we live in a spatially flat low matter den-sity universe which is currently undergoing a period of accelerating expansion.If the observational evidence upon which these claims are based are rainforcedand strengthened by future experiments, the implications for cosmology will beincredible. It could then appear that the cosmological fluid is dominated bysome sort of fantastic energy density, which has negative pressure and has justbegun to play significant role today. Recent years have witnessed the emergenceof the idea of an accelerating universe. Therefore, due to some observationalresults [1]−[4] it is now established that universe is accelerating. This signifiesa remarkable shift in cosmological research from expanding universe to accel-erated expanding universe. Now, the problem lies in detecting an exotic typeof unknown repulsive force, termed as dark energy. The detection of dark en-ergy would be a new clue to an old puzzle: the gravitational effect of the zero- point energies of particles and fields. The total with other energies, that areclose to homogeneous and nearly independent of time, acts as dark energy. The

1corresponding author

1

paramount characteristic of the dark energy is a constant or slightly changingenergy density as the universe expands, but we do not know the nature of darkenergy very well [5]−[13]. Dark Energy (DE) has been conventionally character-ized by the equation of state (EoS) parameter ω = p/ρ which is not necessarilyconstant. The simplest DE candidate is the vacuum energy (ω = −1), which ismathematically equivalent to the cosmological constant (Λ). The other conven-tional alternatives, which can be described by minimally coupled scalar fields,are quintessence (ω > −1), phantom energy (ω < −1) and quintom (that canacross from phantom region to quintessence region) as evolved and have timedependent EoS parameter. Some other limits obtained from observational re-sults coming from SN Ia data [14] and SN Ia data collaborated with CMBRanisotropy and galaxy clustering statistics [15] are −1.67 < ω < −0.62 and−1.33 < ω < −0.79 respectively. However, it is not at all obligatory to use aconstant value of ω. Due to lack of observational evidence in making a distinc-tion between constant and variable ω, usually the equation of state parameter isconsidered as a constant [16, 17] with phase wise value −1, 0, −1/3 and +1 forvacuum fluid, dust fluid, radiation and stiff fluid dominated universe, respec-tively. But in general, ω is a function of time or redshift [18]−[20]. For instance,quintessence models involving scalar fields give rise to time dependent EoS pa-rameter ω [21]−[24]. Also some literature is available on models with varyingfields, such as cosmological model with variable equation of state parameter inKaluza-Klein metric and wormholes [25, 26]. In recent years various form oftime dependent ω have been used for variable Λ models [27, 28]. Recently Rayet al [29], Akarsu and Kilinc [30] and Yadav [31] have studied dark energy modelwith variable EoS parameter.

Spatially homogeneous and anisotropic cosmological models play a signifi-cant role in description of the large scale behaviour of universe. Bianchi typeIII cosmological model in presence of dark energy have been studied in generalrelativity by numerous authors. Lorentz [32] has presented a model with dustand cosmological constant. Chakraborty and Chakraborty have given a bulkviscous cosmological model with variable G and Λ [33]. Singh et al [34] haveinvestigated a model with variable G and Λ in presence of perfect fluid by as-suming a conservation law of energy-momentum tensor. Recently, Tiwari [35]has studied a model in presence of perfect fluid and time dependent λ with con-stant deceleration parameter. Bali and Tinkar [36] have investigated a modelin the presence of bulk viscous borotropic fluid with variable G and Λ. Un-like Robertson-Walkar metric, Bianchi type III can admit a dark energy thatyields an anisotropic EoS parameter according to their characteristics. The cos-mological data from the large scale structure [37] and type Ia supernova [3, 4]observations-do not rule out the possibility of anisotropic dark energy either[38, 39].

In this paper, we have investigated the anisotropic dark energy model withvariable ω. This paper is organized as follows: In section 2, the metric and fieldequations are described. The solution of field equations are presented in section3 and section 4 concludes the findings.

2

2 The Metric and Field Equations

We consider Bianchi type III metric in the form

ds2 = −dt2 +A2dx2 +B2e2αxdy2 + C2dz2 (1)

where A, B and C are the function of t only.The simplest generalisation of EoS parameter of perfect fluid may be to de-termine the EoS parameter separately on each spatial axis by preserving thediagonal form of the energy-momentum tensor in a consistent way with theconsidered metric. Thus, the energy momentum tensor of fluid is taken as

T ji = diag

[

T 00 , T

11 , T

22 , T

33

]

(2)

Then, one may parametrize it as follows,

T ji = diag [ρ,−px,−py,−pz] = diag [1,−ωx,−ωy,−ωz] ρ

= diag [1,−ω, (−ω + δ) , (−ω + δ)] ρ (3)

where ρ is the energy density of fluid,; px, py and pz are the pressures and ωx, ωy

and ωz are the directional EoS parameters along the x, y and z axes respectively.ω is the derivation-free EoS parameter of the fluid. We have parametrizedthe deviation from isotropy by setting ωx = ω and then introducing skewnessparameter δ and γ that are deviation from ω along y and z axis respectively.The Einstein field equations, in gravitational units (8πG = 1 and c = 1), are

Rij −1

2Rgij = −Tij (4)

where the symbols have their usual meaning.In a comoving co-ordinate system, Einstein’s field equation (4), for the anisotropicBianchi type III metric (1), in case of (3), lead to the following system of equa-tions

A4B4

AB+

A4C4

AC+

B4C4

BC− α2

A2= ρ (5)

B44

B+

C44

C+

B4C4

BC= −ωρ (6)

A44

A+

C44

C+

A4C4

AC= − (ω + δ) ρ (7)

A44

A+

B44

B+

A4B4

AB− α2

A2= − (ω + γ) ρ (8)

α

(

A4

A− B4

B

)

= 0 (9)

Here, the sub indices 4 in A, B, C and elsewhere denote differentiation withrespect to t.Integrating equation (9), we obtain

A = kB (10)

3

where k is the positive constant of integration. We substitute the value ofequation (10) in equation (7) and subtract the result from equation (6), weobtain that the skewness parameter on y-axis is null i.e.

δ = 0

Thus system of equations from (5) - (9) may be reduce to

A24

A2+ 2

A4C4

AC− α2

A2= ρ (11)

A44

A+

C44

C+

A4C4

AC= −ωρ (12)

2A44

A+

A24

A2− α2

A2= − (ω + γ) ρ (13)

Now we have three linearly independent equations (11) - (13) and five unknownparameters (A,B, ω, ρ, γ). Two additional constraint relating these parametersare required to obtain explicit solutions of the system.

(i) The law of variation of Hubble’s parameter that yields a constant value ofdeceleration parameter. Such type of relation have already been consideredby Berman [40] for solving FRW models. Later on many authors (Singh et al[41, 42] and references therein) have studied flat FRW and Bianchi type mod-els by using the special law of Hubble parameter that yields constant value ofdeceleration parameter.

(ii) We assume that the expansion (θ) in the model is proportional to the shear(σ). This condition leads to

B = Cn (14)

where n is proportionality constant. The motive behind assuming condition (ii)is explained with reference to Throne (1967) [43], the observations of the veloc-ity red-shift relation for extragalatic sources suggest that Hubble expansion ofthe universe is isotropy today within ≈ 30 percent (Kantowski and Sachs [44];Kristian and Sachs [45]). To put more precisely, red-shift studies place the limit

σ

H≤ 0.3

on the ratio of shear σ to Hubble constant H in the neighbourhood of our galaxytoday. Collin et al [46] have pointed out that for spatially homgeneous metric,the normal congruence to the homogeneous expansion satisfies that the condi-tion σ

θis contant.

The average scale factor of Bianchi type III metric is given by

R3 = ABCeαx (15)

4

We define, the generalised mean Hubble’s parameter H as

H =1

3(H1 +H2 +H3) (16)

where H1 = A4

A, H2 = B4

Band H3 = C4

Care the directional Hubble’s parameter

in the direction of x, y and z respectively.Equation (15), may be reduces to

H =R4

R=

1

3

(

A4

A+

B4

B+

C4

C

)

(17)

Since the line-element (1) is completely characterized by Hubble’s parameter H.Therefore, let us consider that mean Hubble parameter H is related to averagescale factor R by following relation

H = k1R−s (18)

where k1 > 0 and s ≥ 0, are constant.The deceleration parameter is defined as

q = −RR44

R24

(19)

From equations (17) and (18), we get

R4 = k1R−s+1 (20)

R44 = −k21(s− 1)R−2s+1 (21)

Using equations (15) and (19), equation (17) leads to

q = s− 1 (constant) (22)

The sign of q indicates whether the model inflates or not. The positive signof q corresponds to standard decelerating model where as the negative sign ofq indicates inflation. However the current observations of SN Ia and CMBRfavour accelerating models i.e. q < 0.From equation (19), we obtain the law of average scale factor R as

R =

[

(Dt+ c1)1s when s 6= 0

c2ek1t when s = 0

(23)

where c1 and c2 are the constant of integration.From equation (22), for s 6= 0, it is clear that the condition for expansion ofuniverse is s > 0 i.e. q + 1 > 0. Therefore for expansion model of universe thedeceleration parameter (q) should not be greater than −1.

5

3 Solutions of the Field Equations and Discus-sion

3.1 Case(i): when s 6= 0

Equations (9), (17) and (23) lead to

B = l (Dt+ c1)1r (24)

Equation (14) and (24) lead to

C = l1 (Dt+ c1)1rn (25)

From equations (10) and (24), we obtain

A = L (Dt+ c1)1r (26)

where co is the constant of integration and l = (c0)3n

2n+1 , L = kl, l1 = l1n , r =

(2n+1)s3n .

Thus the Hubble’s parameter H , scalar of expansion θ and shear scalar σ aregiven by

H =k1

(Dt+ c1)(27)

θ = 3H =3k1

(Dt+ c1)(28)

σ2 =D2(n− 1)2

3n2r2(Dt+ c1)2(29)

Equations (28) and (29) lead to

σ

θ=

D(n− 1)√3nrk1

(30)

Using equations (11), (25) and (26), the energy density of the fluid is obtainedas

ρ =D2(n+ 2)

nr2 (Dt+ c1)2 − α2

L2 (Dt+ c1)2r

(31)

Using equations (12), (25), (26) and (31), the equation of state parameter ω isobtained as

ω =

D2[n2+(n+1)(1−rn)]n2r2(Dt+c1)

2

α2

L2(Dt+c1)2r

− D2(n+2)

nr2(Dt+c1)2

(32)

Using equations (13), (26), (31) and (32), the skewness parameter γ (i.e. devi-ation from ω along z-axis) is given by

γ =

D2[n2(2−r)+n(r−1)−1]n2r2(Dt+c1)

2 − α2

L2(Dt+c1)2r

α2

L2(Dt+c1)2r

− D2(n+2)

nr2(Dt+c1)2

(33)

6

Figure 1: The plot of energy density (ρ) vs. time (t) with D = 1, n = 0.25,s = 0.5 C1 = .05, α = 0.3 and L = 1.5

From equation (32), it is observed that the equation of state parameter ω is timedependent, it can be function of redshift z or scale factor R as well. The red-shift dependence of ω can be linear like ω(z) = ω0 + ω1z with ω1 =

(

dωdz

)

z= 0

[47, 48] or nonlinear as ω(z) = ω0 + ω1z1+z

[49, 50]. The SN Ia data suggeststhat −1.67 < ω < −0.62 [14] while the limit imposed on ω by a combina-tion of SN Ia data (with CMB anisotropy) and galaxy clustering statistics is−1.33 < ω < −0.79 [15]. So, if the present work is compared with experimentalresults mentioned above then, one can conclude that the limit of ω provided byequation (32) may accommodated with the acceptable range of EoS parameter.

Also we see that for s = 3(n2+n+1)(n+1)(2n+1) , the ω vanishes. Thus at this particular

value of s, our model represents dusty universe.

The variation of equation of state parameter (ω) with cosmic time (t) isclearly shown in figures 1, as a representative case with appropriate choice ofconstants of integration and other physical parameters using reasonably wellknown situations. From Fig. 1, we conclude that in early stage, the EoS pa-rameter ω was positive (i.e, the universe was matter dominated) and at latetime it is evolving with negative value (i.e. at the present time). The earlierreal matter later on converted to the dark energy dominated phase of universe.

In absence of any curvature, matter energy density (Ωm) and dark energy

7

density (ΩΛ) are related by the equation

Ωm +ΩΛ = 1 (34)

where Ωm = ρ3H2 and ΩΛ = Λ

3H2

Thus equation (34), reduce to

ρ

3H2+

Λ

3H2= 1 (35)

Using equations (27) and (31), in equation (35), the cosmological constant isobtained as

Λ =3k21

(Dt+ c1)2− D2(n+ 2)

nr2(Dt+ c1)2+

α2

L2(Dt+ c1)2r

(36)

Measurements of redshift of Ia supernova SN 1997 ff indicate the present ac-celeration of universe. Recently, Carmeli and Kuzmenko [51] have shown thatthe cosmological relativistic theory predicts the value for cosmological constantΛ = 1.934 × 10−35s−2. This value of Λ is excellent in agreement with themeasurements recently obtained by the High-z Supernovae Cosmology Project.Recent supernovae Ia [3, 4] observations suggest that the cosmological constantis decreasing function of time and they approach to small positive values astime increases (i.e. the present epoch). From equation (36), we observe thatcosmological constant (Λ) decreases as time increases.

3.2 Case(ii): when s = 0

Equations (9), (14), (17) and (23) lead to

B = l0ek2t (37)

Equations (14) and (37) lead to

C = l2ek2t (38)

From equations (10) and (37), we obtain

A = L0ek2n

t (39)

where l0 is constant of integration and L0 = kl0, l2 = l1n

0 ., k2 = 3nk1

2n+1 .Thus the Hubble’s parameter H , scalar of expansion θ and shear scalar σ aregiven by

H =(2n+ 1)k2

3n(40)

θ =(2n+ 1)k2

n(41)

σ2 =k22(n− 1)2

3n2(42)

8

Figure 2: The plot of EoS parameter (ω) vs. time (t) with α = 0.3, n =0.25, k2 = 0.5 and L0 = 0.10

Equations (41) and (42) lead to

σ

θ=

(n− 1)√3(2n+ 1)

(43)

Using equations (11), (38) and (39), the energy density of the fluid is obtainedas

ρ =(n+ 2)k22L

20 − nα2e−2k2t

nL20

(44)

Using equations (12), (38), (39) and (44), the equation of state parameter ω isobtained as

ω =

(

n2 + n+ 1)

L20k

22

n [nα2e−2k2t − (n+ 2)k22L20]

(45)

Using equations (13), (39), (44) and (45), the skewness parameter γ (i.e. devi-ation from ω along z-axis) is given by

γ =(2n2 − n− 1)L2

0k22 − n2α2e−2k2t

n [nα2e−2k2t − (n+ 2)k22L02 ](46)

The Eos parameter ω is found to be negative and its time varying nature isclearly shown in figure 2.

9

Using equations (40) and (44), in equation (35), the cosmological constant isobtained as

Λ =3(2n+ 1)2K2

9n2− (n+ 2)k2L

20 − nα2e−2k2t

nL20

(47)

From equation (47), it is clear that cosmological constant Λ is decreasing func-tion of time and approaches to small positive value at late time which is sup-ported by results from supernova observations recently obtained by High-z Su-pernova Team and Supernova cosmological project [3, 4].

4 Concluding Remarks

In this paper, we have studied anisotropic dark energy model with variableEoS parameter ω, considering two cases, (3.1) and (3.2) for n 6= 0 and n = 0respectively. In both cases, ω is found to be time varying which is consistentwith recent observations [14, 15]. In case (i), we have shown that, in early stage,the equation of state parameter ω was positive i.e. the universe was matterdominated and at late time, it is evolving with negative values i.e. present epoch(fig.1). Where as in case (ii), for n = 0, from equation (41), we have obtained

that, at cosmic time t = 12k2

ln n2α2

(n−1)L20k20

, ω = −1 (i.e. cosmological constant

dominated universe), when t < 12k2

ln n2α2

(n−1)L20k20

, ω > −1. (i.e. quintessence),

and when t > 12k2

ln n2α2

(n−1)L20k20

, ω < −1. (i.e. super quintessence or phantom fluid

dominated universe) [52], representing the different phase of universe throughout the evolving process. Since in both cases, σ

θ= constant, the model does not

approach isotropy at any time. Therefore, we can not rule out the passibility ofanisotropic nature of DE at least in Bianchi type III framework.

Acknowledgements

Authors would like to thank The Institute of Mathematical Science (IMSc),Chennai, India for providing facility and support where part of this work wascarried out. Author (AKY) is also thankful to Prof. D. R. Somashekar, DirectorAEC. Agra, for kind supports.

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