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Bianchi Type-V Cosmology with Magnetized Anisotropic

Dark Energy

Journal: Canadian Journal of Physics

Manuscript ID cjp-2016-0777.R1

Manuscript Type: Article

Date Submitted by the Author: 29-Nov-2016

Complete List of Authors: Shamir, M. Farasat; National University of Computer & Emerging Sciences, Sciences & Humanities Ali, Asad; National University of Computer and Emerging Sciences

Keyword: Anisotropy, Dark Energy, Exact Solutions, Bianchi V, Cosmology

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Magnetized Anisotropic Dark Energy

M. Farasat Shamir∗and Asad Ali †

Department of Sciences and Humanities,National University of Computer and Emerging Sciences,

Lahore Campus, Pakistan.

AbstractWe study anisotropic universe in the presence of magnetized dark

energy. Bianchi type-V cosmological model is considered for this pur-pose. The energy-momentum tensor consists of anisotropic fluid withuniform magnetic field of energy density ρB. Exact solutions to thefield equations are obtained without using conventional assumptionslike constant deceleration parameter. In particular, a general solutionis obtained which further provides different classes of solutions. Onlythree cases have been discussed for the present analysis, i.e. linear,quadratic and exponential. Graphical analysis of the solutions is donefor all the three classes. The behavior of the model using some im-portant physical parameters is discussed in the presence of magneticfield.

Keywords: Dark Energy; Anisotropy; Exact Solutions.PACS: : 04.20.Jb; 98.80.-k; 98.80.Jk.

1 Introduction

In last decade researchers (Supernova Cosmology Projects), provided someevidence that expansion of the universe was accelerating [1]-[4]. The universe

∗farasat.shamir@nu.edu.pk†asadali.nu@gmail.com

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consists of 70 percent of the dark energy which is a mysterious substance, butthere is no clear explanation of it. Dark energy has been traditionally charac-terized by equation of state (EoS) parameter ω = p

ρ, which is not necessarily

a constant. The simplest dark energy case is the vacuum energy, which ismathematically equivalent to cosmological constant. The other possibilitiesare quintessence, phantom energy and quintom which have time dependentEoS parameter. Also the results coming from SN Ia data collaborated withCMBR anisotropy and galaxy clustering statistics suggest −1.67<ω<− 0.62[5, 6]. There are various models available in literature that have been pro-posed to explain dark energy, for details see [7]-[13]. However, it is not at allmandatory to use a constant value of ω.

The special law of variation of Hubble parameter in Friedman Robert-son Walker (FRW) space-time was proposed by Berman [14], which gave aconstant value of deceleration parameter. Such a law of variation for Hubbleparameter is consistent with the observations and is also approximately validfor slowly time varying models. Laws governing the universe’s scale providean explicit form of FRW universe model and facilitate to describe acceler-ating as well as decelerating models of evolution of the universe. Recently,anisotropic Bianchi type-I cosmological models have been investigated in thecontext of f(R) and f(G) modified theories of gravity [15]-[17].

Pradhan and Saha [18] constructed locally rotationally symmetric (LRS)Bianchi type-I cosmological models with dynamically anisotropic dark energyand perfect fluid. Akarsu and Kilinc [19] also investigated LRS Bianchi type-I cosmology with anisotropic dark energy and they concluded that the darkenergy is slightly interacting, had dynamical energy density and anisotropicEoS parameter. The same authors [20] found the exact solutions of Bianchitype-III models with anisotropic dark energy. Yadav [21] examined Bianchitype-III anisotropic dark energy models with constant deceleration param-eter. Bianchi type-III magnetized anisotropic dark energy models with con-stant deceleration parameter and dark energy models with anisotropic fluidin Bianchi type-V I0 with time dependent deceleration parameter have beeninvestigated by Tade et al. [22] and Pradhan et al. [23] respectively. Saha[24] has studied the evolution of the universe filled with dark energy withinthe scope of a Bianchi type-V model.

Using the proportionality condition and variational law of Hubble param-eter, the exact solutions of field equations are explored by many researchers.Priyanka et al. [25] gave the exact solutions of Einstein’s field equations for aBianchi type-V I0 space-time filled with perfect fluid satisfying the barotropic

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EoS under the supposition that the expansion scalar was proportional toshear scalar. Saha [26] studied Bianchi type-V I dark energy model withvarying EoS parameter. Sharif and Shamir [27] did work on exact solutionsof Bianchi types I and V space-time in f(R) theory of gravity. The same au-thors [28] also studied the non-vacuum solutions of these models. Dynamicsof Bianchi type-I universe with magnetized anisotropic dark energy has beendiscussed by Sharif and Zubair [29]. They concluded that the universe modelas well as anisotropic fluid did not approach isotropy through the evolutionof the universe. The same authors [30] studied the effects of electromagneticfield on the dynamics of Bianchi type V I0 universe with anisotropic darkenergy. Thus, it seems interesting to explore further about the universe withanisotropic dark energy.

In this paper, we are focussed to explore Bianchi Type-V cosmology withmagnetized anistropic dark energy. It is mentioned hare that the exact solu-tions to the field equations are obtained without using conventional assump-tions like constant deceleration parameter. In particular, a general solutionis obtained which further provides different classes of solutions. Only threeclasses have been discussed for the present analysis, i.e. linear, quadratic andexponential. Graphical analysis of the solutions for the discussed classes isgiven. The plan of paper is as follows: In section 2, we give Bianchi type-Vfield equations and energy conservation equation. Section 3 provides qualita-tive analysis of some exact solutions along with discussion of some importantphysical parameters. Summary and concluding remarks are given in the lastsection.

2 Einstein Field Equations and Exact Solu-

tions

The line element for Bianchi type-V space-time is given as [27]

ds2 = dt2 − a2dx2 − b2e2mxdy2 − c2e2mxdz2 (1)

where the scale factors a and b and c are functions of cosmic time t only.Here we take energy-momentum tensor for the magnetized anisotropic DE

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fluid in the following form

T ab =

ρ+ ρB 0 0 0

0 ρB − ωρ 0 00 0 −(ω + δ)ρ− ρB 00 0 0 −(ω + γ)ρ− ρB

where ρ is the energy density of the fluid, ρB magnetic energy density, ωdeviation free EoS parameter of the fluid and δ, γ shows the deviation fromω along y and z axis respectively which are called skewness parameters. UsingEq.(1), with gravitational units, the Einstein field equations take the form

a′b′

ab+b′c′

bc+a′c′

ac− 3m2

a2= (ρB + ρ), (2)

[b′′

b+c′′

c+b′c′

bc− m2

a2] = (ρB − ωρ), (3)

[a′′

a+c′′

c+a′c′

ac− m2

a2] = −[(ω + δ)ρ+ ρB)], (4)

[a′′

a+b′′

b+a′b′

ab− m2

a2] = −[(ω + γ)ρ+ ρB)], (5)

[2a′

a− b′

b− c′

c] = 0. (6)

These are five non-linear differential equations with eight unknowns namelya, b, c, ρ, ρB, γ, δ, and ω. Therefore, we need some additional constraints inorder to find the solutions. From Eq.(6), we get

a2 = bc (7)

Energy conservation equation T µν;µ = 0 provides

ρ′ + ρ(b′

bδ + γ

c′

c) + (ρ+ ωρ)[

a′

a+b′

b+c′

c] + ρ′B + 2ρB(

b′

b+c′

c) = 0 (8)

This equation contains magnetic and non-magnetic part. Assuming magneticand non-magnetic part zero and solving its magnetic part, we get

ρB =c1

(bc)2(9)

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where c1 is an integration constant. The filed equations are highly nonlinearand it seems difficult to solve them with eight unknowns. So to solve the fieldequations, we use a physical condition that expansion scalar θ is proportionalto shear scalar σ which provides b = cn, where n is an arbitrary non-zeroreal number. We further consider c in a general power law form, i.e., c =(f(t) + β)k, where f(t) is an arbitrary function of t and k is a non-zero realnumber. Using these assumptions and Eqs.(2)-(6), we obtain

ρB =c1

(f(t) + β)2k(n+1), (10)

ρ = (kf(t)′

f(t) + β)2(n2 + 4n+ 1

2)−3[

m2

(f(t) + β)k(n+1)]− c1

(f(t) + β)2k(n+1), (11)

ω = − 2

(f(t)′)2(n2 + 4n+ 1)(f(t) + β)2kn+2k − 6m2(f(t) + β)kn+k−2 − 2c1(f(t) + β)2×[

f(t)′2k(kn2 + 1)(f(t) + β)2kn+2k + f(t)′′k(n+ 1)(f(t) + β)2kn+2k+1 +

(kf(t)′)2(f(t) + β)2kn+2k −m2(f(t) + β)kn+k+2 − c1(f(t) + β)2], (12)

δ = − 1

(kf(t)′)2(n2 + 4n+ 1)− 6m2(f(t) + β)2−kn−k − 2c1(f(t) + β)2

[f(t)′2k(n+ 1)

−2k(n+ 1)− 2(f(t) + β)f(t)′′k(n+ 1)− 4(f(t)′)2kn(kn− 1)− 4f(t)′′2kn(f(t) + β)

+2(kf(t)′)2(n+ 1)− 4n(kf(t)′)2 + 8c1(f(t) + β)2−2kn−2k

], (13)

γ = − 1

(kf(t)′)2(n2 + 4n+ 1)(f(t) + β)2kn+2k − 6m2(f(t) + β)2−kn−k − 2c1(f(t) + β)2×[

f(t)′2k(n+ 1)2(f(t) + β)2kn+2k − 2k(n+ 1)(f(t) + β)2kn+2k + 2(f(t)′′)k(n+ 1)×

(f(t) + β)2kn+2k+1 − 4k(k − 1)(f(t) + β)2kn+2k+1 − 4f(t)′′(f(t) + β)2kn+2k+2

+2n(n+ 1)(f(t)′k)2(f(t) + β)2kn+k − 4n(f(t) + β)2kn+2k(kf(t)′)2 + 8c1(f(t) + β)2].

(14)

So far we have found the values of the unknowns ρ, ρB, γ, δ and ω in termsof unknown f(t).

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3 Different Cases of Some Exact Solutions

In this section, we will investigate different possibilities of solutions alongwith the graphical analysis. Polynomial form of f(t) are considered and onlylinear and quadratic cases are analyzed. The solutions for exponential f(t)are also discussed.

3.1 Linear case

When f(t) = ξt, where ξ is an arbitrary non-zero constant, the solutionmetric takes the form

ds2 = dt2 − (ξt+ β)kn+kdx2 − (ξt+ β)2kne2mxdy2 − (ξt+ β)2ke2mxdz2, (15)

while the expressions for energy density and magnetic energy density takesthe form

ρ =

(kξ

ξt+ β

)2(n2 + 4n+ 1

2

)− 3

[m2

(ξt+ β)k(n+1)

]− c1

(ξt+ β)2k(n+1), (16)

ρB =c1

(ξt+ β)2k(n+1). (17)

Its graph is shown in figure 1 (1a, 1b). The graph shows that energy densityof magnetic field increases with the increase of t and n, while energy densityof the fluid decreases with the increase of t and n. This indicates the magneticdominated universe as the time grows. The expressions for other parametersω, δ and γ take the form

ω =−2[ξ2k(kn2 + 1)(ξt+ β)2kn+2k + (kξ)2(ξt+ β)2kn+2k −m2(ξt+ β)kn+k+2 − c1(ξt+ β)2]

(ξk)2(n2 + 4n+ 1)(ξt+ β)2kn+2k − 6m2(ξt+ β)kn+k−2 − 2c1(ξt+ β)2,

(18)

δ = − 1

(kξ)2(n2 + 4n+ 1)− 6m2(ξt+ β)2−kn−k − 2c1(ξt+ β)2

[ξ2k(n+ 1)− 2k(n+ 1)

−4(ξ)2kn(kn− 1) + 2(kξ)2(n+ 1)− 4n(kξ)2 + 8c1(ξ + β)2−2kn−2k

], (19)

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γ = − 1

(kf(t)′)2(n2 + 4n+ 1)(f(t) + β)2kn+2k − 6m2(f(t) + β)2−kn−k − 2c1(f(t) + β)2×[

ξ2k(n+ 1)2(ξt+ β)2kn+2k − 2k(n+ 1)(ξt+ β)2kn+2k − 4k(k − 1)(ξt+ β)2kn+2k+1

+2n(n+ 1)(ξk)2(ξt+ β)2kn+k − 4n(ξt+ β)2kn+2k(kξ)2 + 8c1(ξt+ β)2]. (20)

The graphical behavior of ω, δ and γ is shown in figures 2 and 3. It canbe seen from graph that for a certain value of anisotropic parameter n andt, ω = −1. This is interesting as it has been proved that the expansion ofthe universe is accelerating when ω ≈ −1 [31]. Moreover, the phantom likedark energy is found to be in the region where ω < −1 and the universe withphantom dark energy ends up with a finite time future singularity known ascosmic doomsday or big rip [32].

3.2 Quadratic case

For f(t) = ξt2 + ζt + λ where ξ, λ, ζ are arbitrary non-zero constants, thesolution metric becomes

ds2 = dt2−(ξt2+ζt+λ+β)kn+kdx2−(ξt2+ζt+λ+β)2kne2mxdy2−(ξt2+ζt+λ+β)2ke2mxdz2.(21)

In this case ρ, ρB, ω, δ and γ turn out to be

ρ = (k(2ξt+ ζ)

ξt2 + ζt+ λ+ β)2(n2 + 4n+ 1

2)− 3m2

(ξt2 + ζt+ λ+ β)k(n+1)− c1(ξt2 + ζt+ λ+ β)2k(n+1)

,

(22)

ρB =c1

(ξt2 + ζt+ λ+ β)2k(n+1), (23)

ω = −2

[(2ξt+ ζt)2k2(n2 + 4n+ 1)(ξt2 + ζt+ λ+ β)2kn+2k − 6m2(ξt2 + ζt+ λ+ β)kn+k−2

−2c1(ξt2 + ζt+ λ+ β)2

]−1[4kξ2t2(kn2 + 1)(ξt2 + ζt+ λ+ β)2kn+2k + 2kξ(n+ 1)×

(ξt2 + ζt+ λ+ β)2kn+2k −m2(ξt2 + ζt+ λ+ β)kn+k+2 − c1(ξt2 + ζt+ λ+ β)2

], (24)

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δ = − 1

k2(ξt+ ζ)2(n2 + 4n+ 1)− 6m2(ξt2 + ζt+ λ+ β)2−kn−k − 2c1(ξt2 + ζt+ λt+ β)2×[

(ξt+ ζ)2k(n+ 1)− 2k(n+ 1)− 4kξ(n+ 1)(ξt2 + ζt+ λ+ β)− 4kn(kn− 1)(2ξt+ ζt)2

−8kn(ξt2 + ζt+ λ+ β) + 2(k(ξt+ ζ))2(n+ 1)− 4n(k(ξt+ ζ))2

+8c1(ξt2 + ζt+ λ+ β)2−2kn−2k

], (25)

γ = −[k2(2tξ + ζ)2(n2 + 4n+ 1)(t2ξ + tζ + λ+ β)2kn+2k − 6m2(t2ξ + ζt+ λ+ β)2−kn−k

−2c1(t2ξ + tζ + λ+ β)2

]−1[(2tξ + ζ)2k(n+ 1)2(t2ξ + ζt+ λ+ β)2kn+2k − 2k(n+ 1)×

(t2ξ + ζt+ λ+ β)2kn+2k + 4kξ(n+ 1)(t2ξ + tζ + λ+ β)2kn+2k+1 − 4k(k − 1)×(t2ξ + t2ζ + λ+ β)2kn+2k+1 − 8ξ(t2ξ + tζ + λ+ β)2kn+2k+2 + 2n(n+ 1)((2tξ + ζ)k)2 ×

(t2ξ + tζ + λ+ β)2kn+k − 4(2tξ + ζ)k2 + 8c1(t2ξ + ζt+ λ+ β)2

].

(26)

The graphical behavior of all above quantities is similar as compared to thelinear case (see figures 4, 5 and 6). Thus it is conjectured that by increasingthe degree of polynomial, the resulting solutions will have same physicalbehavior. Now we discuss the exponential case.

3.3 Exponential case

Exponential form of f(t), i.e., f(t) = ψet, where ψ is an arbitrary non-zeroreal number, yields the solution metric

ds2 = dt2 − (ψet + β)k(n+1)dx2 − (ψet + β)2kne2mxdy2 − (ψet + β)2ke2mxdz2.(27)

Here the expressions for the energy density and magnetic energy density turnout to be

ρ = (k(ψet)

ψet + β)2(n2 + 4n+ 1

2)− 3[

m2

(ψt2 + ψet + β)k(n+1)]− c1

(ψet + β)2k(n+1),(28)

ρB =c1

(ψet + β)

2k(n+1)

. (29)

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Figure (7) shows that ρB decreases gradually with the increase of t and n andby setting t and n sufficiently large ρB → 0. Energy density of the universeρ increases gradually as the time grows. The other parameters for this caseare as follows:

ω = −2

[k2(ψet)2(n2 + 4n+ 1)(ψet + β)2kn+2k − 6m2(ψet + β)kn+k−2 − 2c1(ψe

t + β)2]−1[(ψet)2k(kn2 + 1)(ψet + β)2kn+2k + (ψet)2k(n+ 1)(ψet + β)2kn+2k +

(ψetk)2(ψet + β)2kn+2k −m2(ψet + β)kn+k+2 − c1(ψet + β)2

], (30)

δ = − 1

(k(ψet))2(n2 + 4n+ 1)− 6m2(ψet + β)2−kn−k − 2c1(ψet + β)2

[(ψet)2k(n+ 1)

−2k(n+ 1)− 2k(ψet)(n+ 1)(ψet + β)− 4kn(kn− 1)(ψet)2 −

4kn(ψet + β) + (k(ψet))2(n+ 1)− 4n(k(ψet))2 + 8c1(ψet + β)2−2kn−2k

], (31)

γ = − 1

(k(ψet))2(n2 + 4n+ 1)(ψet + β)2kn+2k − 6m2(ψet + β)2−kn−k − 2c1(ψet + β)2×[

(ψet)2k(n+ 1)2(ψet + β)2kn+2k − 2k(n+ 1)(ψet + β)2kn+2k + 2kψet(n+ 1)×

(ψet + β)2kn+2k+1 − 4k(n+ 1)(ψet + β)2kn+2k+1 − 4ψet(ψet + β)2kn+2k+1 +

2n(n+ 1)(ψetk)2(ψet + β)2kn+2k + 2n(n+ 1)((2ψt+ ζ)k)2(ψt2 + ζt+ λ+ β)2kn+k

−4n(ψet)k2(ψet + β)2kn+2k + 8c1(ψe

t + β)2]. (32)

The graphical behavior of ω, δ and γ is given in figures (8) and (9). It canbe seen from graph that ω possesses negative values justifying expansion ofuniverse and existance of the phantom like dark energy. It is interestingthat skewness parameter δ attains both positive and negative values depend-ing upon n. However, γ approaches to 0 with the increase in time. Nowwe discuss some important cosmological parameters in the context of exactsolutions.

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3.4 Some Important Cosmological Parameters

The volume scale factor V for Bianchi type V spacetime is defined as:

V = abc e2mx. (33)

For linear, quadratic and exponential case, the Volume scale factor turn outto be

V = (ξt+ β)32k(n+1)e2mx, (34)

V = (ξt2 + ζt+ λ+ β)32k(n+1)e2mx, (35)

V = (ψet + β)32k(n+1)e2mx. (36)

The average scale factor A is defined as:

A = (abce2mx)13 . (37)

The expressions for average scale factor for linear, quadratic and exponentialcase turn out to be

A = (ξt+ β)k(n+1)

2 e2mx3 , (38)

A = (ξt2 + ζt+ λ+ β)k(n+1)

2 e2mx3 , (39)

A = (ψet + β)k(n+1)

2 e2mx3 . (40)

The Average Hubble Parameter H is defined as:

H =H1 +H2 +H3

3. (41)

where H1 = a′

a, H2 = b′

band H3 = c′

care directional Hubble parameters.

For linear, quadratic and exponential case, the average Hubble parameter Htakes the form

H =ξtk(n+ 1)

2(ξt+ β), (42)

H =k(n+ 1)(2ξt+ ζ)

2(ξt2 + ζt+ λ+ β), (43)

H =ψetk(n+ 1)

2(ψet + β). (44)

The expansion scalar θ and shear scalar σ are given by

θ = (a′

a+b′

b+c′

c), (45)

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1

2[(a′

a)2 + (

b′

b)2 + (

c′

c)2]− (

θ2

6), (46)

respectively. For linear, quadratic and exponential case, the expansion andshear scalar become

θ =3k(n+ 1)ξ

2(ξt+ β), σ2 = (

ξk

ξt+ β)2(n2 − 2n+ 1

4), (47)

θ =3k(n+ 1)(2ξt+ ζ)

2(ξt2 + ζt+ λ+ β), σ2 = (

2(ξt+ ζ)k

ξt2 + ζt+ λ+ β)2(n2 − 2n+ 1

4), (48)

θ =3ψetk(n+ 1)

2(ψet + β), σ2 = (

ψetk

ψet + β)2(n2 − 2n+ 1

4). (49)

The deceleration, jerk and snap parameters are defined as:

q = − 1

H2

A′′

A, j = − 1

H3

A′′′

A, s = − 1

H4

A′′′′

A. (50)

The value of these parameters for our solutions turn out to be

q = −(k(n+ 1)− 1

k(n+ 1)), j = −(

(kn+ k − 2)(kn+ k − 4)

k2(n+ 1)2),

s = −((kn+ k − 2)(kn+ k − 4)(kn+ k − 6)

k3(n+ 1)3). (51)

4 Summary and Conclusion

This paper is devoted to exploring the exact solutions for spatially homoge-neous and anisotropic Bianchi type-V cosmological model with magnetizedanisotropic DE fluid having anisotropic EoS. The parameters ρ, ρB, γ, δ andω are obtained by assuming that conservation equation of DE consists of twoseparate conserved parts i.e. magnetic part and anisotropic part.

We have discussed some physical aspects of the model in the presence ofmagnetic field. Assuming magnetic part to be zero, we found ρB. We haveconsidered only three cases, linear, quadratic, and exponential for the presentstudy. It is conjectured that the solutions for cubic case and higher degreepolynomials in general will have same physical behavior as that of linear andquadratic cases. We have calculated the unknowns ρ, ρB, γ, δ and ω and theirgraphical analysis is given. It is worth mentioning here that we have not used

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1a 1b

Figure 1: Behavior of ρ and ρB for Linear Case

the conventional assumption of constant deceleration parameter to explorethe exact solutions of field equations. However, for the sake of simplicity, wehave used the condition that expansion scaler proportional to shear scalarwhich gives b = cn. We further considered the metric coefficient c(t) in ageneral form, i.e. c(t) = (f(t) + β)k, where f(t) is an arbitrary analyticfunction of t. This assumption allows us to consider different forms of f(t)to analyze different solution of field equations. The cosmological parametersare calculated for the given model. Moreover, the isotropy condition σ2

θ→ 0

as t → ∞ is satisfied for linear and quadratic cases. For exponential case,σ2

θapproaches to a constant value as t→ ∞ and the isotropy condition may

be verified for some suitable values of n. It is interesting to mention thatEoS parameter ω possesses negative values for all three cases. It can beseen from graphs for a certain value of anisotropic parameter n, ω = −1.This is interesting as it has been proved that the expansion of the universeis accelerating when ω ≈ −1 [31]. Moreover, the phantom like dark energyis found to be in the region where ω < −1 and the universe with phantomdark energy ends up with a finite time future singularity known as cosmicdoomsday or big rip [32].

We have discussed the exact solutions with only three cases. Many othersolutions can be explored by assuming some other forms of f(t). This studywill throw some light on the structure formation of the universe, which hasastrophysical significance. Similar analysis for other Bianchi type models is

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Figure 2: Behavior of ω and δ for Linear Case

1e

Figure 3: Behavior of γ for Linear Case

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Figure 4: Behavior of ρ and ρB for Quadratic Case

2c 2d

Figure 5: Behavior of ω and δ for Quadratic Case

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Figure 6: Behavior of γ for Quadtratic Case

3a 3b

Figure 7: Behavior of ρ and ρB for Exponential Case

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Figure 8: Behavior of ω and δ for Exponential Case

3e

Figure 9: Behavior of γ for Exponential Case

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