Effects of Electromagnetic Field on the Dynamics of Bianchi type V I 0 Universe with Anisotropic Dark Energy
aa r X i v : . [ g r- q c ] A ug Effects of Electromagnetic Field onthe Dynamics of Bianchi type
V I Universe with Anisotropic DarkEnergy
M. Sharif ∗ and M. Zubair † Department of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan.
Abstract
Spatially homogeneous and anisotropic Bianchi type
V I cosmo-logical models with cosmological constant are investigated in the pres-ence of anisotropic dark energy. We examine the effects of electromag-netic field on the dynamics of the universe and anisotropic behaviorof dark energy. The law of variation of the mean Hubble parameteris used to find exact solutions of the Einstein field equations. Wefind that electromagnetic field promotes anisotropic behavior of darkenergy which becomes isotropic for future evolution. It is concludedthat the isotropic behavior of the universe model is seen even in thepresence of electromagnetic field and anisotropic fluid. Keywords:
Electromagnetic Field; Dark Energy; Anisotropy.
PACS: ∗ [email protected] † [email protected] Introduction
The most remarkable advancement in cosmology is its observational evidencewhich says that our universe is in an accelerating expansion phase. Supernova I a data [1, 2] gave the first indication of the accelerated expansion of the uni-verse. This was confirmed by the observations of anisotropies in the cosmicmicrowave background (CMB) radiation as seen in the data from satellitesuch as WMAP [3] and large scale structure [4]. Today’s one of the majorconcerns of cosmology is the dark energy. Recent cosmological observations[1, 2, 3, 4] suggest that our universe is (approximately) spatially flat andits cosmic inflation is due to the matter field (dark energy) having negativepressure (violating energy conditions). The composition of universe densityis the following: 74% dark energy (DE), 22% dark matter and 4% ordinarymatter [5]. Though there is a compelling evidence that expansion of theuniverse is accelerating, yet the nature of dark energy has been under con-sideration since the last decade [6]-[10]. Several models have been proposedfor this purpose, e.g., Chaplygin gas, phantoms, quintessence, cosmologicalconstant and dark energy in brane worlds. However, none of these modelscan be regarded as being entirely convincing so far.The cosmological constant, Λ is the most obvious theoretical candidate ofDE which has the equation of state (EoS) ω = −
1. Astronomical observationsindicate that the cosmological constant is many orders of magnitude smallerthan estimated in modern theories of elementary particles [11]. Stabell andRefsdal [12] discussed the evolution of Friedmann-Lemaˆ i tre Robertson andWalker (FLRW) dust models in the presence of positive cosmological con-stant. These results are given in a more generalized form [13, 14] using thegeneral EoS. Wald [15] examined the late time behavior of expanding homo-geneous cosmological models satisfying the Einstein field equations (EFEs)with a positive cosmological constant. He found that all the Bianchi typemodels except IX showed isotropic behavior. These models exponentiallyevolve towards the de Sitter universe with a scale factor (3 / Λ) / . Goliathand Ellis [16] used the dynamical system methods to observe the spatiallyhomogeneous cosmological models with a cosmological constant. The in-clusion of cosmological constant provides an effective mean of isotropizinghomogeneous universes [15, 16].The presence of magnetic fields in galatic and intergalatic spaces is ev-ident from recent observations [17]. The large scale magnetic fields can bedetected by observing their effects on the CMB radiation. These fields would2nhance anisotropies in the CMB, since the expansion rate will be differentdepending on the directions of the field lines [18, 19]. Matravers and Tsagas[20] found that interaction of the cosmological magnetic field with the space-time geometry could affect the expansion of the universe. If the curvatureis strong, then even the weak magnetic field will effect the evolution of theuniverse. The magneto-curvature coupling tends to accelerate the positivelycurved regions while it decelerates the negatively curved regions [20, 21].Jacobs [22] studied the spatially homogeneous and anisotropic Bianchitype I cosmological model with expansion and shear but without rotation.He discussed anisotropy in the temperature of CMB and expansion both withand without magnetic field. It was concluded that the primordial magneticfield produced large expansion anisotropies during the radiation-dominatedphase but it had negligible effect during the dust-dominated phase. Dunnand Tupper [23] discussed properties of Bianchi type V I models with perfectfluid and magnetic field. Roy et al. [24] explored the effects of cosmologicalconstant in Bianchi type I and V I models with perfect fluid and homoge-neous magnetic field in the axial direction. They found that model expandedfor negative values of the cosmological constant while it contracted for pos-itive Λ. In a recent paper [25], Sharif and Shamir explored the vacuumsolution of Bianchi I and V models in f ( R ) gravity.Rodrigues [26] proposed Bianchi I with a non-dynamical DE componentwhich yields anisotropic vacuum pressure in two ways: (i) by considering theanisotropic vacuum consistent with energy-momentum conservation; (ii) byimplementing a Poisson structure deformations between canonical momentasuch that re-scaling of scale factors is not violated. Koivisto and Mota [27]have investigated a cosmological model containing the DE fluid with non-dynamical anisotropic EoS and interacts with perfect fluid. They suggestedthat if the DE EoS is anisotropic, the expansion rate of the universe becomesdirection dependent at late times and cosmological models with anisotropicEoS can explain some of the observed anomalies in CMB.Recently, Akarsu and Kilinc [28] investigated anisotropic Bianchi type I models in the presence of perfect fluid and minimally interacting DE withanisotropic EoS parameter. They found that anisotropy of the DE did notalways promote anisotropy of the expansion. The anisotropic fluid may sup-port isotropization of the expansion for relatively earlier times in the universe.The same authors [29] have worked on the Bianchi type III model in thepresence of single imperfect fluid with dynamical anisotropic EoS parameterand dynamical energy density. They observed that anisotropy of the expan-3ion vanished and hence the universe approached isotropy for late times ofthe universe in accelerating models.It would be worthwhile to see what happens if we consider the electro-magnetic field with anisotropic DE. The main purpose of this work is tolook at the effects of electromagnetic field on the dynamics of the universein the presence of anisotropic DE for Bianchi type
V I . The layout of thepaper is as follows. In section we describe spatially homogeneous andanisotropic Bianchi type V I spacetime and formulate the EFEs in the pres-ence of anisotropic fluid and magnetic field. Section presents a speciallaw of variation for the mean Hubble parameter which yields constant de-celeration parameter. This law generates two types of solutions: power lawand exponential expansion. In section , a hypothetical form of fluid is ob-tained by making an assumption on anisotropy of the fluid. We obtain exactsolutions of the EFEs and discuss physical behavior of anisotropic DE anduniverse model. Finally, section concludes the results. The spatially homogeneous and anisotropic Bianchi type
V I model is de-scribed by the line element ds = dt − A ( t ) dx − e mx B ( t ) dy − e − mx C ( t ) dz , (1)where scale factors A, B and C are functions of cosmic time t only, m = 0is a constant. The energy-momentum tensor for the electromagnetic field isgiven as [30] T ν ( em ) µ = ¯ µ [ | h | ( u µ u ν − / δ νµ ) − h µ h ν ] , (2)where u ν is the four-velocity vector satisfying g µν u µ u ν = 1 . (3) µ is the magnetic permeability and h µ is the four-magnetic flux given by h µ = √− g µ ε µναβ F αβ u ν , ( µ, ν, α, β = 0 , , ,
3) (4)where ε µναβ is the Levi-Civita tensor, F αβ is the electromagnetic field tensorand | h | = h ν h ν . We assume that magnetic field is due to an electric current4roduced along x -axis and thus it is in yz -plane. In co-moving coordinates u ν = (1 , , ,
0) and hence Eq.(4) gives h = 0 , h = h = h = 0. Usingthese values in Eq.(4), it follows that F = F = 0 , F = 0 . The electric and magnetic field in terms of field tensor are defined as [31] E µ = F µν u ν , B µ = 12 ε µνα F να . (5)According to Ohm’s law, we have h µν J ν = σF µν u ν , (6)where h µν = g µν + u µ u ν is the projection tensor orthogonal to u µ , σ is theconductivity and J µ is the four current density. In the magnetohydrodynamiclimit, conductivity takes infinitely large value while current remains finite sothat E µ → F = F = F = 0 . Thusthe only non-vanishing component of electromagnetic field tensor F µν is F .The Maxwell’s equations F µν ; α + F να ; µ + F αµ ; ν = 0 , F µν ; α = 0 (7)are satisfied by F = K = constant. (8)It follows from Eq.(4) that h = AKµBC , | h | = K µ B C . (9)Using this equation in Eq.(2), we obtain T em )0 = K µB C = T em )1 = − T em )2 = − T em )3 . Thus we have T ν ( em ) µ = diag [ K µB C , K µB C , − K µB C , − K µB C ] . (10)The energy-momentum tensor for anisotropic DE fluid is taken in thefollowing form [28, 29] T νµ = diag [ ρ, − p x , − p y , − p z ] (11)5his model of the DE is characterized by the EoS, p = ωρ , where ω is notnecessarily constant [33]. From Eq.(11), we have T νµ = diag [1 , − ( ω + δ ) , − ω, − ( ω + γ )] ρ, (12)where ρ is the energy density of the fluid; p x , p y and p z are pressures and ω x , ω y , ω z are directional EoS parameters on x, y and z axes respectively.The deviation from isotropy is obtained by setting ω x = ω + δ, ω y = ω, ω z = ω + γ, where ω is the deviation free EoS parameter and δ and γ are the deviationsfrom ω on x and z axes respectively. The EFEs with cosmological constantare given by G µν = R µν − Rg µν − Λ g µν = 8 π ( T µν + T ( em ) µν ) , (13)where R µν is the Ricci tensor, R is the Ricci scalar, T µν is the energy-momentum tensor for anisotropic fluid and T ( em ) µν is the energy-momentumtensor for the electromagnetic field.For Bianchi type V I spacetime, the EFEs become˙ A ˙ BAB + ˙ A ˙ CAC + ˙ B ˙ CBC − m A = 8 πρ + 4 πK µB C + Λ , (14)¨ BB + ¨ CC + ˙ B ˙ CBC + m A = − π ( ω + δ ) ρ + 4 πK µB C + Λ , (15)¨ AA + ¨ CC + ˙ A ˙ CAC − m A = − πωρ − πK µB C + Λ , (16)¨ AA + ¨ BB + ˙ A ˙ BAB − m A = − π ( ω + γ ) ρ − πK µB C + Λ , (17) m ( ˙ BB − ˙ CC ) = 0 , (18)where dot denotes derivative with respect to time t . Equation (18) yields B = c C, (19)where c is a constant of integration. Subtracting Eq.(17) from (16) andusing (19), we obtain γ = 0 which shows that directional EoS parameters6 y , ω z and the pressures p y , p z become equal. Using Eq.(19) and γ = 0, theEFEs (14)-(17) reduce to the following set of equations2 ˙ A ˙ BAB + ˙ B B − m A = 8 πρ + 4 πK µk B + Λ , (20)2 ¨ BB + ˙ B B + m A = − π ( ω + δ ) ρ + 4 πK µk B + Λ , (21)¨ AA + ¨ BB + ˙ A ˙ BAB − m A = − πωρ − πK µk B + Λ . (22) Here we discuss some physical and geometrical quantities for the Bianchi type
V I model which are important in cosmological observations. The averagescale factor is given by a = ( k AB ) (23)while the volume is defined as V = a = k AB . (24)The mean Hubble parameter H and the directional Hubble parameters H i ( i =1 , ,
3) in x, y and z directions are H = 13 (ln V ˙) = ln ˙ a = 13 ( ˙ AA + 2 ˙ BB ) , H x = ˙ AA , H y = H z = ˙ BB . (25)The physical parameters such as scalar expansion Θ, shear scalar σ andanisotropy of the expansion ∆ are given as followsΘ = u a ; a , (26) σ = 12 σ ab σ ab , (27)∆ = 13 X i =1 ( H i − HH ) . (28)The anisotropy of expansion shows isotropic behavior for ∆ = 0.7t is mentioned here that any universe model becomes isotropic for thediagonal energy-momentum tensor when t → + ∞ , ∆ → , V → + ∞ and T > ρ >
0) [29, 32]. The law of variation of mean Hubble parameter isgiven as H = la − n = l ( k AB ) − n/ , (29)where l > n >
0. This law was initially proposed by Berman [34]for spatially homogeneous and isotropic RW spacetime which yields constantvalue of the deceleration parameter. In recent papers [28, 35], a similar lawis proposed for the homogeneous and anisotropic Bianchi models to generateexact solutions.The volumetric deceleration parameter q is the measure of rate at whichexpansion of the universe slows down due to self-gravitation. It is defined as q = − a ¨ a ˙ a . (30)Using Eqs.(25) and (29), we get˙ a = la − n +1 , ¨ a = − l ( n − a − n +1 (31)which gives constant values of the deceleration parameter as follows q = n − f or n = 0 ,q = − f or n = 0 . Recent observations show that expansion rate of the universe is acceleratingwhich may be due to the presence of DE and q <
0. Thus the sign of q indicates whether the cosmological model inflates or not. For q > n > − q< n< q = 0 for n = 1 corresponds toexpansion with constant velocity. Equations (25) and (29) yield two differentvolumetric expansion laws V = k e lt , n = 0 , (32) V = ( nlt + c ) /n , n = 0 (33)which are used to find exact solutions of the EFEs. In fact these representtwo different models of the universe. 8 Solution of the Field Equations
We can find the most general form of anisotropy parameter for the expansionof Bianchi type
V I in the presence of anisotropic fluid and electromagneticfield using Eq.(25). The anisotropy parameter of expansion can be writtenas ∆ = 29 H ( H x − H y ) , (34)where H x − H y is the difference between the expansion rates on x and y axeswhich can be found using the field equations.Subtracting Eq.(22) from (21) and after some manipulation, it followsthat ˙ AA − ˙ BB = dV + 1 V Z (8 πδρ + 2 m A − πK k µB ) V dt, (35)where d is another constant of integration. Now using this equation inEq.(34), the anisotropy parameter takes the form∆ = 29 H [ d + Z (8 πδρ + 2 m A − πK k µB ) V dt ] V − . (36)The anisotropy parameter in the presence of isotropic fluid can be obtainedby choosing δ = 0 which yields∆ = 29 H [ d + Z ( 2 m A − πK k µB ) V dt ] V − . (37)We take the value of δ so that the integrand in the above equation vanishes δ = − m πρA + K k µρB . (38)The corresponding energy-momentum tensor for anisotropic DE fluid turnsout to be T νµ = diag [1 , − ( ω − m πρA + K k µρB ) , − ω, − ω ] ρ. (39)The anisotropy parameter of the expansion reduces to∆ = 29 d H V − . (40)9e see that ∆ obtained for the Bianchi type V I in the presence of anisotropicfluid with electromagnetic field is equivalent to that found for the Bianchitype III in the presence of anisotropic fluid [29].The difference between the directional Hubble parameters becomes H x − H y = dV . (41)The most general form of the energy density is found by using Eqs.(20) and(28) as ρ = 18 π [3 H (1 − ∆2 ) − m A − Λ − πK µk B ] . (42)This shows that anisotropy of expansion, cosmological constant and electro-magnetic field reduce the energy density ρ of anisotropic DE. n = 0 ( q = − ) The spatial volume of the universe for this model is given by V = k e lt . (43)Using this value of V in Eq.(41) and then solving the EFEs (20)-(22), thescale factors become A = k e lt − dlk e − lt , (44) B = ( k k k ) e lt + dlk e − lt , (45) C = ( k k k ) e lt + dlk e − lt , (46)where k , k and k are constants of integration.The directional and the mean Hubble parameters will become H x = l + 23 dk e − lt , H y = H z = l − dk e − lt , H = l (47)while the anisotropy parameter of the expansion takes the form∆ = 29 d l k e − lt . (48)10he expansion and shear scalar are found asΘ = u a ; a = ˙ AA + 2 ˙ BB = 3 l = 3 H, (49) σ = 12 [( ˙ AA ) + 2( ˙ BB ) ] −
16 Θ = 13 d k e − lt . (50)The energy density of the DE is evaluated by using Eq.(20) with the scalefactors as ρ = 18 π [3 l − Λ − d k e − lt − m k e − lt + dlk e − lt − πK µ ( k k ) × e − lt − dlk e − lt ] . (51)The deviation free part of anisotropic EoS parameter ω can be obtained byusing Eqs.(44)-(46) and (51) in Eq.(21) ω = { k k l + d k e − lt − k k − m k e − lt + dlk e − lt + 12 πk K µ × e − lt − dlk e − lt } / { d k e − lt + 3Λ k k − k k l + 3 m k × e − lt + dlk e − lt + 12 πk K µ e − lt − dlk e − lt } . (52)Using the scale factors and the energy density in Eq.(38), the deviation inEoS parameter along x -axis δ is given as δ = { m k e − lt + dlk e − lt + 24 πK k µ e − lt − dlk e − lt } / { d k e − lt + 3Λ k × k − k k l + 3 m k e − lt + dlk e − lt + 12 πk K µ e − lt − dlk e − lt } . (53) We find that the directional Hubble parameters are dynamical where as themean Hubble parameter is constant. Also, the directional Hubble parametersbecome constant at t = 0 and when t → ∞ . These deviate from the meanHubble parameter by some constant factor at t = 0 but coincide when t → ∞ .Since the constant is positive (negative), it increases (decreases) expansion onthe x -axis and it decreases (increases) expansion on y and z axes. The volume11 (t) t (a) Ω (t) - - - - - - t (b) Figure 1: Plot of (a) ρ ( t ) and (b) ω ( t ) verses cosmic time t for m = 1 andvarying values of l as follows: solid, l = 4; dotted, l = 4 .
5; dashed, l = 5. V of the universe is finite at t = 0, expands exponentially with the increase intime t and takes infinitely large value as t → ∞ . Thus the universe evolveswith constant volume and expands exponentially. The expansion scalar isconstant for 0 ≤ t ≤∞ and hence the model represents uniform expansion.It is mentioned here that the scale factors A ( t ) , B ( t ) and C ( t ) are finiteat t = 0 which implies that the model has no initial singularity whereasthese diverge for later times of the universe. The anisotropy parameter ofthe expansion and shear scalar are found to be finite for earlier times of theuniverse where as these decrease with time and become zero as t → ∞ . Thisshows that anisotropy of the expansion is not supported by the anisotropicDE and electromagnetic field. The quantities ρ, ω and δ are dynamical andare finite at t = 0. The deviation free EoS parameter of the DE may beginin the phantom ( ω < −
1) or quintessence region ( ω > −
1) but ω → − ρ increases when ω is inphantom region and attains a constant value as ω → −
1. This is shown inFigure 1.We see from Eq.(53) that electromagnetic field favors the deviation from ω on x -axis, i.e., it contributes to anisotropic behavior of the fluid. Thedeviation parameter increases from negative values towards zero with theincrease in time and tends to zero as t → ∞ shown in Figure 2. Thus theanisotropic fluid becomes isotropic for the later times of the universe in thecase of exponential volumetric expansion. It follows from Eq.(51) that when t → ∞ , ρ → l − Λ = 3 H − Λ which implies that for t → ∞ , ρ > (t) - - - t Figure 2: Plot of δ verses cosmic time t for m = 1 and varying values of l asfollows: solid, l = 4; dotted, l = 4 .
5; dashed, l = 5.if H > p Λ / → V → ∞ . Thus the model approaches toisotropy for its future evolution. Here q = − , dH/dt = 0 which gives thelargest value of the Hubble parameter and accelerating rate of expansion. n = 0 ( q = n − ) The initial time of the universe is found by using Eq.(33) t ∗ = − c/nl f or n = 0 . (54)We re-define the cosmic time as t ′ = nlt + c (55)such that the initial time turns out to be zero, i.e., t ′ = 0. For this value ofcosmic time, we can re-define the Bianchi model in the form ds = ( nl ) − dt ′ − A ( t ′ ) dx − e mx B ( t ′ ) dy − e − mx C ( t ′ ) dz . (56)The corresponding volume will become V = t ′ n . (57)13sing this value of V in Eq.(41) and then solving the EFEs (20)-(22), thescale factors turn out to be A ( t ′ ) = k t ′ n e ndn − t ′ − n , (58) B ( t ′ ) = ( k k k ) t ′ n e − ndn − t ′ − n , (59) C ( t ′ ) = ( k k k ) t ′ n e − ndn − t ′ − n . (60)The directional and mean Hubble parameters take the form H = ( nt ′ ) − , H x = 1 nt ′ + 2 d t ′− n , H y = H z = 1 nt ′ − d t ′− n . (61)The corresponding anisotropy parameter of the expansion turns out to be∆ = 29 n d t ′ − n (62)while the expansion and shear scalar areΘ = 3 nt ′ = 3 H, σ = 13 d t ′− n . (63)The energy density can be found from Eq.(20) by using the scale factorsEq.(58)-(60) as ρ ( t ′ ) = 18 π [3( nt ′ ) − − Λ − d t ′− n − m k t ′− n e − ndn − t ′ − n − πK µ × ( k k ) t ′− n e ndn − t ′ − n ] . (64)Using this equation and the scale factors in Eq.(21), we obtain the deviationfree EoS parameter ωω ( t ′ ) = { (9( nt ′ ) − + d t ′− n − nt ) − − k k + 3 t ′− n e ndn − t ′ − n × ( 4 πk K µ − m k t ′ n e − ndn − t ′ − n ) } / { ( d t ′− n + 3Λ − nt ′ ) − ) × k k + 3 t ′− n e ndn − t ′ − n ( 4 πk K µ + m k t ′ n e − ndn − t ′ − n ) } . (65)14 (t') t' ® (a) B(t'), C(t') t' ® (b) Figure 3: Evolution of (a) A ( t ′ ) and (b) B ( t ′ ) (dashed line), C ( t ′ ) (dottedline) for n > , d > t ′ → ∞ .Finally, the deviation parameter δ can be obtained using the value of ρ ( t ′ )along with Eqs.(58)-(60) in (38) δ ( t ′ ) = { t ′− n e ndn − t ′ − n ( 4 πk K µ + m k t ′ n e − ndn − t ′ − n ) } / { ( d t ′− n + 3Λ − nt ′ ) − ) k k + 3 t ′− n e ndn − t ′ − n ( 4 πk K µ + m k t ′ n × e − ndn − t ′ − n ) } . (66) The universe model accelerates for 0 < n <
1, decelerates for n > n = 1. The mean Hubble parameter,shear scalar and directional Hubble parameters are infinite at the initialepoch and tend to zero for later times of the universe. If n > , d > A ( t ′ ) takes infinitely large value while both B ( t ′ ) and C ( t ′ ) vanish as t ′ → ∞ .This indicates that spacetime exhibits ”pancake” type singularity which isshown in Figure 3. If n > , d <
0, then A ( t ′ ) decreases to zero where asboth B ( t ′ ) and C ( t ′ ) continue to increase as t ′ → ∞ , leading to a ”cigar”singularity shown in Figure 4. For n <
3, the scale factors A ( t ′ ) , B ( t ′ ) and C ( t ′ ) become infinite as t ′ → ∞ given in Figure 5.We note that spatial volume V is zero at t ′ = 0 and it takes infinitelylarge value as t ′ → ∞ . The expansion scalar is infinite at t ′ = 0 and de-creases with the increase in cosmic time. Thus the universe starts evolvingwith zero volume at the initial epoch with an infinite rate of expansion andexpansion rate slows down for the later times of the universe. The dynamics15 (t') t' ® (a) B(t'), C(t') t' ® (b) Figure 4: Evolution of (a) A ( t ′ ) and (b) B ( t ′ ) (dashed line), C ( t ′ ) (dottedline) for n > d < t ′ → ∞ . A(t'), B(t'), C(t') t' ® Figure 5: Evolution of A ( t ′ ), solid line; B ( t ′ ) dashed; C ( t ′ ) dotted for n < t ′ → ∞ . 16 (t') t' (a) Æ (t') t' (b) Figure 6: Plot of ∆ verses cosmic time t ′ ; solid line for n < n > t ′ → ∞ (b) t ′ → n . ∆tends to zero as t ′ → ∞ and diverges as t ′ → n < n > n = 3 shown in Figure 6.Now we determine such values of n which satisfy ρ > n < n >
3. When n <
3, we have ρ < t ′ → t ′ → ∞ , it follows that ρ > n >
3, we obtain ρ < t ′ → ∞ and it becomes positive as t ′ →
0. Thus this model can representthe universe only for the earlier times of the universe by assigning suitablevalues to the constants.If n < , ρ > ω begins inquintessence region and then passes into the phantom region. It remains inthe phantom region with the increase in time and becomes − t ′ → ∞ . If n > ρ > ω begins in quintessenceregion, passes into the phantom region for small interval of time then it passesback into the quintessence region and becomes − t ′ → ∞ .Thus we may examine the behavior of anisotropy of the fluid and universe17odel for n < ρ > t ′ → ∞ . One can observe that mag-netic field may increase anisotropic behavior of the DE since it contributes to δ which decreases with the increase in time and it tends to zero as t ′ → ∞ .Thus the anisotropy of the DE vanishes in the presence of magnetic field forlater times of the universe with negative cosmological constant. We note thatanisotropy parameter of the expansion is not supported by anisotropy of theDE and magnetic field for later times of the universe since for n < , ∆ → t ′ → ∞ . It is observed that V →∞ and ρ > t ′ → ∞ for n < We have obtained two exact solutions of the dynamical equations for the spa-tially homogeneous and anisotropic Bianchi type
V I model with magneticfield, anisotropic DE and cosmological constant. The dark energy componentis dynamical which yield anisotropic pressure. Assuming the law of variationof the mean Hubble parameter, the cosmological models are given for n = 0and n = 0. The physical and geometrical properties of the models are dis-cussed. We have found the explicit form of scale factors and have explainedthe nature of singularities.The model represents uniform expansion for the exponential expansionwhile in the case of power law expansion, universe expands with an infiniterate of expansion which slows down for the later times of the universe. It isfound that electromagnetic field affects anisotropies in the CMB, in partic-ular, it increases anisotropic behavior. Our results show that even the fluidis anisotropic which yields anisotropic EoS parameter with the electromag-netic field, its anisotropy vanishes for future evolution of the universe in bothcases. The expansion of the universe becomes isotropic due to the isotropicbehavior of the fluid when t ′ → ∞ .The model with zero deceleration parameter can approach to isotropy as t ′ → ∞ with the condition that H > p Λ /
3, Λ >
0. It is shown that ω is inthe phantom region which tends to constant value − n < , Λ ω begins in quintessence region then passes into the phantom region,but it remains in the phantom region and tends to − t ′ → ∞ whichcan result in accelerating the expansion. However, due to the presence ofnegative cosmological constant, it starts contracting. One can observe thatthis behavior may re-collapse the universe and hence the model representsdecelerating expansion of the universe.Finally, we would like to mention here that the model (the exponentialexpansion law) represents expanding universe for both present and futureevolution which fits with the current observations and hence the standardΛCDM model. Also, the observational data favors the power law ΛCDMmodel [1]. We have found that the model represents an accelerating universefor the power law expansion. Both the expansion models approach to theEoS of cosmological constant for future evolution. The anisotropy of theuniverse and DE vanish for the period of the accelerated expansion. Thusthe isotropy is observed for the future evolution of the universe. However,there is still a possibility of DE component with anisotropic EoS in presentepoch of the universe. References [1] Perlmutter, S. et al.: Astrophys. J. (1997)565; Perlmutter, S. et al.:Nature (1998)51; Perlmutter, S. et al.: Astrophys. J. (1999)565.[2] Riess, A.G. et al.: Astron. J. (1998)1009.[3] Bennett, C.L. et al.: Astrophys. J. Suppl. (2003)1; Spergel, D.N. etal.: Astrophys. J. Suppl. (2003)175.[4] Verde, L. et al.: Mon. Not. R. Astron. Soc. (2002)432; Hawkins, E.et al.: Mon. Not. Roy. Astr. Soc. (2003)78; Abazajian, et al.: Phys.Rev.
D69 (2004)103501.[5] Hinshaw, G. et al.: Astrophys. J. Suppl. (2009)225.[6] Sahni, V. and Starobinsky, A.A.: Int. J. Mod. Phys. D9 (2000)373.[7] Carroll, S.M.: Living Rev. Rel. (2001)1.198] Peebles, P.J.E. and Ratra, B.: Rev. Mod. Phys. (2002)559.[9] Padmanabhan, T.: Phys. Rep. (2003)235.[10] Sahni, V.: Lecture Notes Physics (2004)141.[11] Weinberg, S.: Rev. Mod. Phys. (1989)1.[12] Stabell, R. and Refsdal, S.: Mon. Not. R. Astron. Soc. (1966)279.[13] Madsen, M.S. and Ellis, G.F.R.: Mon. Not. R. Astron. Soc. (1988)67.[14] Madsen, M.S. et al.: Phys. Rev. D46 (1992)1399.[15] Wald, R.M.: Phys. Rev.
D28 (1983)2118.[16] Goliath, M. and Ellis, G.F.R.: Phys. Rev.
D60 (1999)023502.[17] Grasso, D. and Rubinstein, H.R.: Phys. Rep. (2001)163; Maartens,R.: Pramana (2000)575.[18] Madsen, M.S.: Mon. Not. R. Astron. Soc. (1989)109.[19] King, E.J. and Coles, P.: Class. Quantum Grav. (2007)2061.[20] Matravers, D.R. and Tsagas, C.G.: Phys. Rev. D62 (2000)103519.[21] Tsagas, C.G. and Barrow, J.D.: Class. Quantum Grav. (1997)2539; (1998)3523; Matravers, D.R. and Tsagas, C.G.: Phys. Rev. D61 (2000)083519.[22] Jacobs, K.C.: Astrophys. J. (1968)661; Jacobs, K.C.: Astrophys. J. (1969)379.[23] Dunn, K.A. and Tupper, B.O.J.: Astrophys. J. (1976)322.[24] Roy, S.R., Singh, J.P. and Narin, S.: Astrophys. Space Sci. (1985)389; ibid. Aust. J. Phys. (1985)239.[25] Sharif, M. and Shamir, M.F.: Class. Quantum Grav. (2009)235020.[26] Rodriguse, D.C.: Phys. Rev. D77 (2008)023534-7.2027] Koivisto, T. and Mota, D.F.: Astrophysical J. (2008)1.[28] Akarsu, O. and Kilinc, C.B.: Gen. Relativ. Gravit. (2010)1.[29] Akarsu, O. and Kilinc, C.B.: Gen. Relativ. Gravit. (2010)763.[30] Lichnerowicz, A.: Relativistic Hydrodynamics and Magnetohydrodynam-ics (Benjamin, New York, 1967)p.13[31] Maartens, R.: Pramana. J. Phys. (2000)576.[32] Collins, C.B. and Hawking, S.W.: Astrophys. J. (1973)317.[33] Carroll, S.M. Hoffman, M. and Trodden, M.: Phys. Rev. D68 (1992)023509.[34] Berman, M.S.: Nuovo Cimento B (1983)182; Berman, M.S. and Go-mide, F.M.: Gen. Relativ. Gravit. (1988)191.[35] Singh, C.P. and Kumar, S.: Int. J. Mod. Phys. D15 (2006)419; Singh,C.P., Zeyauddin, M. and Ram, S.: Int. J. Theor. Phys. (2008)3162;ibid. Astrophys. Space Sci. (2008)181.[36] Spergel, D.N. et al.: Astrophys. J. Suppl.170