A New Class of LRS Bianchi Type VI 0 Universes with Free Gravitational Field and Decaying Vacuum Energy Density
Anirudh Pradhan, Shyam Sundar Kumhar, Padmini Yadav, Kanti Jotania
aa r X i v : . [ g r- q c ] F e b A New Class of LRS Bianchi Type VI Universes with Free Gravitational Fieldand Decaying Vacuum Energy Density
Anirudh Pradhan , Shyam Sundar Kumhar , Padmini Yadav and Kanti Jotania , Department of Mathematics, Hindu P. G. College, Zamania-232 331, Ghazipur, U. P., India E-mail: [email protected], [email protected] E-mail: [email protected] Department of Mathematics, P. G. College, Ghazipur-233 001, IndiaE-mail: p [email protected] Department of Physics, Faculty of Science, The M. S. University of Baroda, Vadodara-390 002, IndiaE-mail: [email protected]
Abstract
A new class of LRS Bianchi type VI cosmological models with free gravitational fields and a variablecosmological term is investigated in presence of perfect fluid as well as bulk viscous fluid. To get thedeterministic solution we have imposed the two different conditions over the free gravitational fields. In firstcase we consider the free gravitational field as magnetic type whereas in second case ‘gravitational wrench’of unit ‘pitch” is supposed to be present in free gravitational field. The viscosity coefficient of bulk viscousfluid is assumed to be a power function of mass density. The effect of bulk viscous fluid distribution inthe universe is compared with perfect fluid model. The cosmological constant Λ is found to be a positivedecreasing function of time which is corroborated by results from recent observations. The physical andgeometric aspects of the models are discussed. PACS: 98.80.Cq, 98.80.-kKeywords: Cosmology, Bianchi models, Gravitational Fields, Bulk Viscosity
The cosmological constant(Λ) was introduced by Einstein in 1917 as the universal repulsion to make the Universestatic in accordance with generally accepted picture of that time. In absence of matter described by the stressenergy tensor T ij , Λ must be constant, since the Bianchi identities guarantee vanishing covariant divergenceof the Einstein tensor, G ij ; j = 0, while g ij ; j = 0 by definition. If Hubble parameter and age of the universe asmeasured from high red-shift would be found to satisfy the bound H t > H t > g ij is shiftedonto the right-hand side of the Einstein field equation and treated as part of the matter content. In generalrelativity, Λ can be regarded as a measure of the energy density of the vacuum and can in principle lead to theavoidance of the big bang singularity that is characterized of other FRW models. However, the rather simplisticproperties of the vacuum that follows from the usual form of Einstein equations can be made more realistic ifthat theory is extended, which in general leads to a variable Λ. Recently Overduin [6, 7] has given an accountof variable Λ-models that have a non-singular origin. Liu and Wesson [8] have studied universe models withvariable cosmological constant. Podariu and Ratra [9] have examined the consequences of also incorporatingconstraints from recent measurements of the Hubble parameter and the age of the universe in the constant and Corresponding author t ) remains a focal point of interest in modern cosmological theories as itsolves the cosmological constant problem in a natural way. There are significant observational evidence for thedetection of Einstein’s cosmological constant, Λ or a component of material content of the universe that variesslowly with time to act like Λ. In the context of quantum field theory, a cosmological term corresponds tothe energy density of vacuum. The birth of the universe has been attributed to an excited vacuum fluctuationtriggering off an inflationary expansion followed by the super-cooling. The release of locked up vacuum energyresults in subsequent reheating. The cosmological term, which is measure of the energy of empty space, providesa repulsive force opposing the gravitational pull between the galaxies. If the cosmological term exists, the energyit represents counts as mass because mass and energy are equivalent. If the cosmological term is large enough,its energy plus the matter in the universe could lead to inflation. Unlike standard inflation, a universe witha cosmological term would expand faster with time because of the push from the cosmological term (Croswell[10]). In the absence of any interaction with matter or radiation, the cosmological constant remains a “con-stant”. However, in the presence of interactions with matter or radiation, a solution of Einstein equations andthe assumed equation of covariant conservation of stress-energy with a time-varying Λ can be found. This entailsthat energy has to be conserved by a decrease in the energy density of the vacuum component followed by acorresponding increase in the energy density of matter or radiation (see also Weinberg [11], Carroll et al. [12],Peebles [13], Sahni and Starobinsky [14], Padmanabhan [15, 16], Singh et al. [17], Pradhan and Pandey [18, 19],Pradhan and Singh [20], Pradhan et al. [21, 22]). Several authors (Pradhan [23, 24], Pradhan et al. [25] − [28],Abdussattar and Viswakarma [29], Kalita et al. [30], Pradhan & Jotania [31] have studied the cosmologicalmodels with decaying vacuum energy density Λ.The discovery in 1998 that the Universe is actually speeding up its expansion was a total shock to astronomers.The observations for distant Type Ia supernovae (Perlmutter et al. [32] − [34] and Riess et al. [35, 36], Garnavichet al. [37, 38], Schmidt et al. [39]) in order to measure the expansion rate of the universe strongly favour asignificant and positive value of Λ. These measurements, combined with red-shift data for the supernovae, ledto the prediction of an accelerating universe. They obtained Ω M ≈ .
3, Ω Λ ≈ .
7, and strongly ruled out thetraditional (Ω M , Ω Λ ) = (1 ,
0) universe. This value of the density parameter Ω Λ corresponds to a cosmologicalconstant that is small, nevertheless, nonzero and positive, that is, Λ ≈ − m − ≈ − s − . An intense searchis going on, in both theory and observations, to unveil the true nature of this acceleration. It is commonly be-lieved by the cosmological community that a kind of repulsive force which acts as anti-gravity is responsible forgearing up the Universe some 7 billion years ago. This hitherto unknown exotic physical entity is termed as darkenergy . The simplest Dark Energy (DE) candidate is the cosmological constant Λ, but it needs to be extremelyfine-tuned to satisfy the current value of the DE.In classical electromagnetic theory, the electromagnetic field has two independent invariants F ij F ij and ∗ F ij F ij . The classification of the field is characterized by the property of the scalar k = ( F ij F ij ) + ( ∗ F ij F ij ) .When k = 0, the field is said to be null and for any observer | E | = | H | and E . H = 0 where E and H are theelectric and magnetic vectors respectively. When k = 0, the field is non-null and there exists an observer forwhich mE = nH , m and n being scalars. It is easy to see that if ∗ F ij F ij = 0, F ij F ij = 0, then either E = 0 or H = 0. These we call the magnetic and electric fields, respectively. If F ij F ij = 0, ∗ F ij F ij = 0 then E = ± H .This we call the ‘electromagnetic wrench’ with unit ‘pitch’. In this case Maxwell’s equations lead to an emptyelectromagnetic field with constant electric and magnetic intensities. In the case of gravitational field, the num-ber of independent scalar invariants of the second order is fourteen. The independent scalar invariants formedfrom the conformal curvature tensor are four in number. In the case of Petrov type D space-times, the number ofindependent scalar invariants are only two, viz. C hijk C hijk and ∗ C hijk C hijk . Analogous to the electromagneticcase, the electric and magnetic parts of free gravitational field for an observer with velocity v i are mentioned byEllis [40] as E αβ = C αjβi v i v j and H αβ = ∗ C αjβi v i v j . It is clear from the canonical form of the conformal curva-ture for a general Petrov type D space-time that there exists an observer for which E αβ = ( nm ) H αβ , where m, n being integers and m = 0. The field is said to be purely magnetic type for n = 0, m = 0. In this case we have E αβ = 0 and H αβ = 0. The physical significance for the gravitational field of being magnetic type is that thematter particles do not experience the tidal force. When m = 0 and also n = 0, we call that there is a ‘gravita-tional wrench’ of unit ‘pitch’ | nm | in the free gravitational field [41]. If ‘pitch’ is unity then we have E αβ = ± H αβ .The space-time having a symmetry property is invariant under a continuous group of transformations. The2ransformation equations for such a group of order r is given by X i = f i ( x , ..., x r , a , ..., a r ) (1)which satisfy the differential equations ∂X i ∂x α = ξ i ( β ) ( X ) A βα ( a ) , ( α, β = 1 , ..., r ) (2)where a , ..., a r are r essential parameters. The vectors ξ iα are the Killing vectors for the group G r of isometrysatisfying the Killing’s equation ξ ( α ) i ; j + ξ ( α ) j ; i = 0 (3)A subspace of space-time is said to be the surface of transitivity of the group if any point of this space can betransformed to another point of it by the action of this group. A space-time is said to be spatially homogeneousif it admit a group G r of isometry which is transitive on three dimensional space-like hyper-surfaces. The group G of isometry was first considered by Bianchi [42] who obtained nine different types of isometry group known asthe Bianchi types. The space-time which admits G group of isometry is known as locally rotationally symmetric(LRS) which always has a G as its subgroup belonging to one of the Bianchi type provided this G is simplytransitive on the three dimensional hyper-surface t = constant.Considerable work has been done in obtaining various Bianchi type cosmological models and their inhomoge-neous generalization. Barrow [43] pointed out that Bianchi VI models of the universe give a better explanationof some of the cosmological problems like primordial helium abundance and they also isotropize in a special sense.Looking to the importance of Bianchi type VI universes, many authors [44] − [48] have studied it in differentcontext. Recently Bali et al. [49] have obtained some LRS Bianchi type VI cosmological models imposing twotypes of conditions over the free gravitational fields.In this paper we have revisited and extended the work of Bali et al. [49] for bulk viscous fluid distribution.We have considered an LRS Bianchi type VI space-time and obtained models with free gravitational field ofpurely ‘magnetic type’ and also in the presence of ‘gravitational wrench’ of unit ‘pitch’ in the free gravitationalfield. It is found that the ‘magnetic’ part of the free gravitational field induces shear in the fluid flow, which iszero in the case of a ‘electric’ type free gravitational field representing an unrealistic distribution in this case.This paper is organized as follows. The introduction and motivation are laid down in Sec. 1. The metric and thefield equations are given in Sec. 2. In Sec. 3, solutions representing LRS Bianchi type VI cosmological modelswith perfect fluid and bulk viscous fluid are obtained imposing the condition when the free gravitational field ispurely magnetic type ( m = 0 , n = 0). In Sec. 4, we obtain the solution in presence of perfect fluid imposing thecondition when there is a ‘gravitational wrench’ of unit ‘pitch’ in the free gravitational field i.e. E αβ = ± H αβ .Discussion and concluding remarks are given in the last Sec 5. We consider an LRS Bianchi type VI universe for which ds = η ab θ a θ b , (4)where θ = A ( t ) dx , θ = B ( t ) exp ( x ) dy , θ = B ( t ) exp ( − x ) dz , θ = dt .The energy-momentum tensor for a perfect fluid distribution with comoving flow vector v i is given by T ji = ( ρ + p ) v i v j + pδ ji , (5)where v i = δ i , ρ and p being respectively, energy density and thermodynamic pressure of the fluid. Here weobtain T = T = T = p, T = − ρ. (6)The Einstein’s field equations (in gravitational units c = 1, G = 1) read as R ji − Rδ ji + Λ δ ji = − πT ji , (7)3or the line element (4) has been set up as2 ¨ BB + ˙ B B + 1 A + Λ = − πp, (8)¨ AA + ¨ BB + ˙ A ˙ BAB − A + Λ = − πp, (9)2 ˙ A ˙ BAB + ˙ B B − A + Λ = 8 πρ. (10)Here, and also in the following expressions a dot indicates ordinary differentiation with respect to t .The energy conservation equation T ij ; j = 0, leads to the following expression˙Λ + ˙ ρ + ( ρ + p ) ˙ AA + 2 ˙ BB ! = 0 . (11)The average scale factor S for LRS Bianchi type VI model is defined by S = ( AB ) . (12)A volume scale factor is given by V = S = ( AB ) . (13)The generalized mean Hubble parameter H is given by H = 13 ( H x + H y + H z ) , (14)where H x = ˙ AA , H y = H z = ˙ BB .The expansion scalar θ and shear scalar σ are obtained as θ = v i ; i = ˙ AA + 2 ˙ BB , (15)and σ = 1 √ ˙ AA − ˙ BB ! , (16)respectively. The average anisotropy parameter is given by A p = 13 X i =1 (cid:18) ∆ H i H (cid:19) , (17)where ∆ H i = H i − H ( i = 1 , , q ) is defined as q = − ¨ SS ˙ S S . (18)The non-vanishing physical components of C ijkl for the line-element (4) are given by C = − C = C = − C = 12 C = − C = 16 " AA − BB − A ˙ BAB + 2 ˙ B B + 4 A , (19) C = − C = C = − A " ˙ AA − ˙ BB . (20)4quations (8)-(10) are three relations in five unknowns A , B , p , ρ and Λ. For complete solutions of equations(8)-(10), we need two extra conditions. To simply the Einstein equations, we impose conditions on the Weyletensor. Since the distribution of matter determines the nature of expansion in the model, the latter is alsoaffected by the free gravitational field through its effect on the expansion, vorticity and shear in the fluid flow.A prescription of such a field may therefore be made on an a priori basis. The cosmological models of FriedmanRobertson Walker, as well as the universe of Einstein-de Sitter, have vanishing free gravitational fields. Inthe following two cases we impose different conditions over the free gravitational field to find the deterministicsolutions. In this section we have extended the solution obtained by Bali et al. [49] by revisiting their solution. When freegravitational field is purely magnetic type ( m = 0 , n = 0), we have H αβ = 0 and E αβ = 0. From (19), we obtain¨ AA − ¨ BB − ˙ A ˙ BAB + ˙ B B + 2 A = 0 . (21)Equation (8) together with (9) reduce to¨ AA − ¨ BB + ˙ A ˙ BAB − ˙ B B − A = 0 . (22)From Eqs. (21) and (22), we obtain two independent equations¨ AA − ¨ BB = 0 , (23)˙ A ˙ BAB − ˙ B B − A = 0 . (24)Let AB = U . Then Eqs. (23) and (24) take the form U U ξξ + U U ξ ˙ A − U ξ = 0 , (25)and U ξ − U U ξ ˙ A + 2 U = 0 , (26)respectively, where ξ is defined by ddt = 1 A ddξ . (27)Eqs. (25) and (26) lead to
U U ξξ − U ξ + 2 U = 0 . (28)On substituting U = exp ( µ ), the above second order differential equation reduces to the form µ ξξ + 2 = 0 , (29)which gives µ = k ξ − ξ + k , (30)where k and k are arbitrary constants. Hence, we obtain U = k exp [ − T ( T − k )] , (31)where T stands for ξ and k is an arbitrary constant. Therefore, from Eqs. (26) and (31), we obtain A = k exp [ − T ( T − k )]( k − T ) , (32)5 = k k ( k − T ) , (33)where k is an arbitrary constant and T is given by dTdt = 1 k ( k − T ) exp [ T ( T − k )] , (34)Therefore the geometry of the universe (4) reduces to the form ds = k exp [ − T ( T − k )]( k − T ) " − dT + dx + exp [2 T ( T − k )] k (cid:8) exp (2 x ) dy + exp ( − x ) dz (cid:9) . (35)The expressions for pressure p and density ρ for the model (35) are given by8 πp = − k (cid:2) − ( k − T ) (cid:3) exp [2 T ( T − k )] − Λ( T ) , (36)8 πρ = 3 k (cid:2) k − T ) (cid:3) exp [2 T ( T − k )] + Λ( T ) . (37)For the specification of Λ( T ), we assume that the fluid obeys an equation of state of the form H T L -> Ρ o r L -> LΡ , Γ= Figure 1: The plot of energy density ρ and cosmological constant Λ Vs. time T for γ = 0. Here k = 3 . k = 1 . p = γρ, (38)where γ (0 ≤ γ ≤
1) is a constant. Using (38) in (36) and (37), we obtain8 π (1 + γ ) ρ = 6 k ( k − T ) exp [2 T ( T − k )] . (39)Eliminating ρ between Eqs. (37) and (39), we obtain(1 + γ )Λ = − γ ) k exp [2 T ( T − k )] + 3(1 − γ ) k ( k − T ) exp [2 T ( T − k )] . (40)6 .0 0.5 1.0 1.5 2.0012345 Time H T L -> Ρ o r L -> LΡ , Γ= (cid:144) Figure 2: The plot of energy density ρ and cosmological constant Λ Vs. time T for γ = . Here k = 3 . k = 1 . H T L -> Ρ o r L -> LΡ , Γ= Figure 3: The plot of energy density ρ and cosmological constant Λ Vs. time T for γ = 1. Here k = 3 . k = 1 .
0. 7sing above solutions, it can be easily seen that the energy conservation equation (11) in perfect fluid distribu-tion is satisfied.From Eq. (39), we observe that ρ ( t ) is a decreasing function of time and ρ > γ = 0 , ,
1. Figures1 , , ρ and Λ are in geometrical units in entire paper) show this behaviour of energy density for vacuum( γ = 0), radiating ( γ = ) and Zeldovice ( γ = 1) universes. From Eq. (40), we note that the cosmological termΛ is a decreasing function of time. From Figure 1 we observe that Λ, for empty universe, is a positive decreasingfunction of time and and it approaches to a positive small value at late time. Recent cosmological observations(Perlmutter et al. [32] − [34] and Riess et al. [35, 36], Garnavich et al. [37, 38], Schmidt et al. [39]) suggest theexistence of a positive cosmological constant Λ with the magnitude Λ( G ¯ h/c ) ≈ − . These observations onmagnitude and red-shift of type Ia supernova suggest that our universe may be an accelerating one with inducedcosmological density through the cosmological Λ-term.From Figure 2, we observe that the Λ, in radiating universe, decreases sharply with time and goes to a negativepoint and then increases with time approaching to a constant value near zero. This is to be taken as a repre-sentative case of physical viability of the model. From Figure 3, it is also observed that for Zeldovice universe( γ = 1), Λ is negative at initial stage but it increases very rapidly with time and ultimately approaches to a posi-tive constant near zero. Thus Λ makes a transition from negative to positive value near zero at the present epoch.The behaviour of the universe in the above models are to be determined by the cosmological term Λ, thisterm has the same effect as a uniform mass density ρ eff = − Λ / π which is constant time. A positive valueof Λ corresponds to a negative effective mass density (repulsion). Hence, we expect that in the universe witha positive value of Λ the expansion will tend to accelerate whereas in the universe with negative value of Λthe expansion will slow down, stop and reverse. In a universe with both matter and vacuum energy, there is acompetition between the tendency of Λ to cause acceleration and the tendency of matter to cause decelerationwith the ultimate fate of the universe depending on the precise amounts of each component. This continues tobe true in the presence of spatial curvature, and with a nonzero cosmological constant it is no longer true thatthe negatively curved (“open”) universes expand indefinitely while positively curved (“closed”) universes willnecessarily re-collapse - each of the four combinations of negative or positive curvature and eternal expansionor eventual re-collapse become possible for appropriate values of the parameters. There may even be a delicatebalance, in which the competition between matter and vacuum energy is needed drawn and the universe isstatic (non expanding). The search for such a solution was Einstein’s original motivation for introducing thecosmological constant. Some Physical and Geometric Features of the Model
The expressions for kinematics parameters i. e. the scalar of expansion θ , shear scalar σ , average scale factor S ,proper volume V and average anisotropy parameter A p for the model (35) are given by θ = 1 k (cid:2) k − T ) (cid:3) exp [ T ( T − k )] , (41) σ = 1 k √ k − T ) exp [ T ( T − k )] , (42) S = k k ( k − T ) exp [ T ( T − k )3 ] . (43) V = √− g = k exp [ − T ( T − k )] k ( k − T ) , (44) A p = 43 . (45)The directional Hubble’s parameters H x , H y and H z are given by H x = 1 k [( k − T ) + 2] exp [ T ( T − k ] , (46) H y = H z = 2 k exp [ T ( T − k ] , (47)8here the mean Hubble’s parameter is given by H = 13 k { k − T ) } exp [ T ( T − k )] (48)The model (35) starts expansion with a big-bang singularity from T = −∞ and it goes on expanding till - ´ - ´ ´ ´ Time ® S ® Figure 4: The plot of average scale factor S Vs. time. Here k = 18, k = 1 . k = 1 . T = k − respectively correspond to the cosmic time t = 0 and t = ∞ . The is found to be realistic everywherein this time interval for Λ > − k exp (cid:16) − k (cid:17) . The model behaves like a steady-state de-Sitter type universe atlate times where the physical and kinematic parameters ρ , p , θ tend to a finite value, however shear vanishesthere. The model has a point type singularity at time T = k . The singular behaviour may be close to cosmicorigin or outside the evolution. The average anisotropy parameter A p remains uniform and isotropic throughout the evolution of the universe. This would depend on physical properties of matter and radiation. This mayneed detailed study to make better quantifiable view. From Figure 4, it can be seen that in the early stages ofthe universe, i. e. , t = 0, the scale factor of the universe had been approximately constant and had increasedvery slowly. At specific time the universe had exploded suddenly and expanded to large scale. This is goodmatching with big bang scenario. This is indicated in first part (top) of Figure 4. Later singular behaviourdepends on ( k , T ). Astronomical observations of large-scale distribution of galaxies of our universe show that the distribution ofmatter can be satisfactorily described by a perfect fluid. But large entropy per baryon and the remarkable degreeof isotropy of the cosmic microwave background radiation, suggest that we should analyze dissipative effects incosmology. Further, there are several processes which are expected to give rise to viscous effect. These are thedecoupling of neutrinos during the radiation era and the recombination era [50], decay of massive super stringmodes into massless modes [51], gravitational string production [52, 53] and particle creation effect in grandunification era [54]. It is known that the introduction of bulk viscosity can avoid the big bang singularity. Thus,we should consider the presence of a material distribution other than a perfect fluid to have realistic cosmologicalmodels (see Grøn [55] for a review on cosmological models with bulk viscosity). A uniform cosmological modelfilled with fluid which possesses pressure and second (bulk) viscosity was developed by Murphy [56]. The so-lutions that he found exhibit an interesting feature that the big bang type singularity appears in the infinite past.In presence of bulk viscous fluid distribution, we replace isotropic pressure p by effective pressure ¯ p in Eq.(36) where ¯ p = p − ξv i ; i , (49)9here ξ is the coefficient of bulk viscosity.The expression for effective pressure ¯ p for the model Eq. (35) is given by8 π ¯ p = 8 π ( p − ξv i ; i ) = − k (cid:2) − ( k − T ) (cid:3) exp [2 T ( T − k )] − Λ( T ) . (50)Thus, for given ξ ( t ) we can solve for the cosmological parameters. In most of the investigation involving bulkviscosity is assumed to be a simple power function of the energy density (Pavon [57], Maartens [58], Zimdahl[59], Santos [60]) ξ ( t ) = ξ ρ n , (51)where ξ and n are constants. For small density, n may even be equal to unity as used in Murphy’s work [56] forsimplicity. If n = 1, Eq. (51) may correspond to a radiative fluid (Weinberg [11]). Near the big bang, 0 ≤ n ≤ is a more appropriate assumption (Belinskii and Khalatnikov [61]) to obtain realistic models.For simplicity sake and for realistic models of physical importance, we consider the following two cases( n = 0 , H T L -> Ρ -> Ρ , Γ= Ρ , Γ= (cid:144) Ρ , Γ= Figure 5: The plot of energy density ρ Vs. time T for γ = 0 , ,
1. Here k = 4 . k = 1 . ξ = 1 . n = 0. n = 0When n = 0, Eq. (51) reduces to ξ = ξ = constant. With the use of Eqs. (37), (38) and (41), Eq. (50) reducesto 8 π (1 + γ ) ρ = 6 k ( k − T ) exp [2 T ( T − k )]+8 πξ k { k − T ) } exp [2 T ( T − k )] . (52)Eliminating ρ ( t ) between Eqs. (37) and (52), we obtain(1 + γ )Λ = − γ ) k exp [2 T ( T − k )] + 3(1 − γ ) k ( k − T ) exp [2 T ( T − k )]+ 8 πξ k { k − T ) } exp [ T ( T − k )] . (53)10 .0 0.5 1.0 1.5 2.0 2.5012345 Time H T L -> L -> L , Γ= L , Γ= (cid:144) L , Γ= Figure 6: The plot of cosmological constant Λ Vs. time T for γ = 0 , ,
1. Here k = 4 . k = 1 . ξ = 1 . n = 0. n = 1When n = 1, Eq. (51) reduces to ξ = ξ ρ . With the use of Eqs. (37), (38) and (41), Eq. (50) reduces to ρ = 3( k − T ) exp [2 T ( T − k )]4 πk [ k (1 + γ ) − ξ { k − T ) } exp [ T ( T − k )]] . (54)Eliminating ρ ( t ) between Eqs. (37) and (54), we obtainΛ = − k exp [2 T ( T − k )] − k − T ) exp [2 T ( T − k )] k h k ( γ + 1) − ξ { k − T ) } exp [ T ( T − k )] i × h k ( γ − − ξ { k − T ) } exp [ T ( T − k )] i . (55)In the case of bulk viscous fluid, the energy conservation equation T ij ; j = 0, leads to the following expression˙Λ + ˙ ρ + ( ρ + ¯ p ) ˙ AA + 2 ˙ BB ! = 0 . (56)It is worth mentioned here that above solutions for bulk viscous fluid also satisfy the energy conservation equa-tion (56).From Eqs. (52), we observe that energy density ρ for the case n = 0, is a positive decreasing function oftime for γ = 0 , ,
1. Figure 5 depicts the variation of the energy density ρ versus time T for γ = 0 , , n = 0). The figure shows the positive decreasing function of energy density and which becomes zeroat present epoch as anticipated.From Eqs. (53), we observe that cosmological constant Λ for the case n = 0, is a decreasing function oftime for γ = 0 , . Figure 6 plots the variation of Λ versus T for γ = 0 , ,
1. Here we observe that cosmologicalterm Λ, for vacuum and radiating universe, is a decreasing function of time whereas for Zeldovice universe it isnegative and increasing function of time. We also observe that in all these three universes the Λ-term approachesto the same small negative value almost closer to zero at late time. Models with negative cosmological constanthave been investigated by Yadav [62], Saha and Boyadjiev [63], Pedram et al. [64], Biswas and Mazumdar11 .0 0.5 1.0 1.5 2.0 2.5 3.00.000.020.040.060.080.10 Time H T L -> Ρ o r L -> LΡ , Γ= Figure 7: The plot of energy density ρ and cosmological term Λ Vs. time T for γ = 0. Here k = 5 . k = 1 . ξ = 1 . n = 1. H T L -> Ρ o r L -> L Ρ , Γ= (cid:144) Figure 8: The plot of energy density ρ and cosmological term Λ Vs. time T for γ = . Here k = 5 . k = 1 . ξ = 1 . n = 0. 12 .0 0.2 0.4 0.6 0.8 1.0 1.20.0000.0050.0100.0150.020 Time H T L -> Ρ o r L -> LΡ , Γ= Figure 9: The plot of energy density ρ and cosmological term Λ Vs. time T for γ = 1. Here k = 5 . k = 1 . ξ = 1 . n = 0.[65], Jotania et al . [66]. Really at present the estimation of Λ is not only complicated but it is uncertain andindirect too. However, the Einstein-Maxwell theory indicates to a different approach which looks simpler andmore significant, since a possibility is illustrated for Λ ≤ ρ and cosmological constant Λ, for the case ( n = 1)are a decreasing function of time and both are small positive at late time for vacuum, radiating and Zeldoviceuniverses. Figures 7, 8 and 9 depict the energy density ( ρ ) and Λ-term versus time T for empty, radiating andZeldovice universe respectively. The nature of ρ and Λ can be seen in these figures.The effect of bulk viscosity is to produce a change in perfect fluid and therefore exhibits essential influenceon the character of the solution. A comparative inspection of Figures show apparent evolution of time due toperfect fluid and bulk viscous fluid. It is apparent that the vacuum energy density ( ρ ) decays much fast in latercase. It also shows the effect of uniform viscosity model and linear viscosity model. Even in these cases, thedecay of vacuum energy density is much faster than uniform. So, the coupling parameter ξ would be relatedwith physical structure of the matter and provides mechanism to incorporate relevant property. In order to saymore specific, detailed study would be needed which would be reported in future. Similar behaviour is observedfor the cosmological constant Λ. We also observe here that Murphy’s [56] conclusion about the absence of a bigbang type singularity in the infinite past in models with bulk viscous fluid in general, is not true. The resultsobtained by Myung and Cho [67] also show that, it is not generally valid since for some cases big bang singularityoccurs in finite past. For both models, it is observed that the effect of viscosity prevents the shear and the freegravitational field from withering away. In this case there is a ‘gravitational wrench’ of unit ‘pitch’ in the free gravitational field i.e. E αβ = ± H αβ .Therefore we have E αβ = κH αβ , κ = 1 . (57)In this case, Bali et al. [49] have investigated the solution given by ds = (2 τ + 3 κτ + 2) τ exp (cid:18) κ √ − (4 τ + 3 κ ) √ (cid:19)" − dτ (2 τ + 3 κτ + 2)6213 dx + (2 τ + 3 κτ + 2) exp (cid:18) − κ √ − (4 τ + 3 κ ) √ (cid:19) { e x dy + e − x dz } . (58)The expressions for pressure p and energy density ρ for the model (58) are obtained as8 πp = (12 κτ + 29 τ + 72 κτ − τ + 3 κτ + 2) exp (cid:18) − κ √ − (4 τ + 3 κ ) √ (cid:19) − Λ( τ ) (59)8 πρ = (12 κτ + 39 τ + 72 κτ + 48)4(2 τ + 3 κτ + 2) exp (cid:18) − κ √ − (4 τ + 3 κ ) √ (cid:19) + Λ( τ ) (60)For the specification of Λ( τ ), we assume that the fluid obeys an equation of state of the form (38). Using Eqs.(38) in (59) and (60), we obtain8 π (1 + γ ) ρ = (cid:20) κτ + 17 τ + 36 κτ (2 τ + 3 κτ + 2) (cid:21) exp (cid:26) − κ √ − (cid:18) τ + 3 κ √ (cid:19)(cid:27) . (61)Eliminating ρ between (59) and (61), we obtainΛ = − (cid:20) γ − κ ( τ + 6) τ + T (39 γ −
29) + 48( γ + 1)4( γ + 1)(2 T + 3 κT + 2) (cid:21) × exp (cid:26) − κ √ − (cid:18) T + 3 κ √ (cid:19)(cid:27) (62)From preliminary study, we find that both energy density and cosmological constant are negative. Hence it willnot be studied. It also shows singular behaviour in energy density at later stage of the evolution. Hence, themodel is unphysical for further study. In this paper, we have studied properties of the free gravitational field and their invariant characterizationsand obtained LRS Bianchi type VI cosmological models imposing different conditions on the free gravitationalfield. In first case, where fee gravitational field is purely magnetic type, we observe that the energy density ρ andcosmological constant Λ are well behaved. Also the effect of bulk viscous fluid distribution in the universe iscompared with perfect fluid model. We observe that due to presence of bulk viscous fluid, the rate of decrease inenergy density is faster compared to perfect fluid model. The linear relation of the coefficient of bulk viscositywith mass density provides further enhance decrease rate (see figures). Since we had considered extension ofthe various models, detailed physical parameter study would be reported in future. Important incorporationis time-dependence of cosmological constant Λ. The similar enhance decrease rate is also observed for Λ (seefigures). The scale factor dependence is also reported. Recent observational data [68] − [70] reveal the presenceof a non-vanishing positive cosmological term Λ as we have found in our present theoretical study. Acknowledgments
Authors (A. Pradhan & K. Jotania) would like to thank IUCAA, Pune, India for providing facility and supportunder associateship program where part of this work was carried out.
References [1] L. M. Krauss and M. S. Turner, Gen. Rel. Gravit. , 1137 (1995).[2] S. L. Adler, Rev. Mod. Phys. , 729 (1982).[3] S. Weinberg, Phys. Rev. Lett. , 1264 (1967).[4] A. D. Linde, Sov. Phys. Lett. , 183 (1974). 145] J. M. Overduin and F. I. Cooperstock, Phys. Rev. D , 043506 (1998).[6] J. M. Overduin, Ap. J. , L1 (1999).[7] J. M. Overduin,
Phys. Rev. D , 102001 (2000).[8] H. Liu and P. S. Wesson, Astrophys. J. , 1 (2001).[9] S. Podariu and B. Ratra,
Astrophys. J. , 109 (2000).[10] K. Croswell,
New Scientist
April
18 (1994).[11] S. Weinberge,
Gravitation and Cosmology , Wiley, New York (1972).[12] S. M. Caroll, W. H. Press and E. L. Turner,
Ann. Rev. Astron. Astrophys. , 499 (1992).[13] P. J. E. Peebles, Rev. Mod. Phys. , 559 (2003).[14] V. Sahani and A. Starobinsky, Int. J. Mod. Phys. D , 373 (2000).[15] T. Padmanabhan, Phys. Rep. , 235 (2003).[16] T. Padmanabhan,
Gen. Rel. Grav. , 529 (2008).[17] C. P. Singh, S. Kumar and A. Pradhan, Class. Quantum Grav. , 455 (2007).[18] A. Pradhan and O. P. Pandey, Int. J. Mod. Phys. D , 1299 (2003).[19] A. Pradhan and P. Pandey, Astrophys. Space Sci. , 221 (2006).[20] A. Pradhan and S. Singh,
Int. J. Mod. Phys. D , 503 (2004).[21] A. Pradhan, A. K. Singh and S. Otarod, Roman. J. Phys. , 415 (2007).[22] A. Pradhan, K. Jotania and A. Singh, Braz. J. Phys. , 167 (2008).[23] A. Pradhan, Fizika B (Zagreb) , 205 (2007).[24] A. Pradhan, Commun. Theor. Phys. , 367 (2009).[25] A. Pradhan and A. Kumar, Int. J. Mod. Phys. D , 291 (2001).[26] A. Pradhan and V. K. Yadav, Int. J. Mod Phys. D , 983 (2002).[27] A. Pradhan and H. R. Pandey, Int. J. Mod. Phys. D , 941 (2003).[28] A. Pradhan and P. Pandey, Czech. J. Phys. , 749 (2005).[29] Abdussattar and R. G. Vishwakarma, Pramana - J. Phys. , 41 (1996).[30] S. Kalita, H. L. Duorah and K. Duorah, Ind. J. Phys. , 629 (2010).[31] A. Pradhan and K. Jotania, Ind. J. Phys. , 497 (2011).[32] S. Perlmutter et al., Astrophys. J. , 565 (1997).[33] S. Perlmutter et al.,
Nature , 51 (1998).[34] S. Perlmutter et al.,
Astrophys. J. , 565 (1999).[35] A. G. Riess et al.,
Astron. J. , 1009 (1998).[36] A. G. Riess et al.,
Astrophys. J. , 665 (2004).[37] P. M. Garnavich et al.,
Astrophys. J. , L53 (1998).[38] P. M. Garnavich et al.,
Astrophys. J. , 74 (1998).[39] B. P. Schmidt et al.,
Astrophys. J. , 46 (1998).1540] G. F. R. Ellis,
General Relativity and Cosmology , ed. R. K. Sachs, New York: Academic Press (1971).[41] S. R. Roy and S. K. Banerjee,
Astrophys. Space Sci. , 81 (1990).[42] L. Bianchi,
Lezioni Sull a Teoria Deigruppi Continui di Transformazioni Pisa , Editiozi Speern (1918).[43] J. D. Barrow,
Mon. Not. R. Astron. Soc. , 221 (1984).[44] S. R. Roy and J. P. Singh,
Acta Physica Austriaca , 57 (1983).[45] R. Tikekar and L. K. Patel, Pramana - J. Phys. , 483 (1994).[46] R. Bali, R. Banerjee and S. K. Banerjee, Astrophys. Space Sci. , 21 (2008).[47] R. Bali, A. Pradhan and A. Hassan,
Int. J. Theor. Phys. , 2594 (2008).[48] A. Pradhan and R. Bali, EJTP , 91 (2008).[49] R. Bali, R. Banerjee and S. K. Banerjee, EJTP , 165 (2009).[50] E. W. Kolb and M. S. Turner, The Early Universe , Addison - Wesley, U S A (1990).[51] S. Myung and B. M. Cho,
Mod. Phys. Lett. A , 37 (1986).[52] N. Turok, Phys. Rev. Lett. , 549 (1988).[53] J. D. Barrow, Nucl. Phys. B , 243 (1988).[54] C. Wolf, S.-Afr. Tydskr. , 68 (1991).[55] Ø. Grøn, Astrophys. Space Sci. , 191 (1990).[56] G. L. Murphy,
Phys. Rev. D , 4231 (1973).[57] D. Pavon, J. Bafaluy and D. Jou, Class. Quant. Grav. , 357 (1996).[58] R. Maartens, Class Quantum Gravit. , 1455 (1995).[59] W. Zimdahl, Phys. Rev. D , 5483 (1996).[60] N. O. Santos, R. S. Dias and A. Banerjee, J. Math. Phys. , 878 (1985).[61] U. A. Belinskii and I. M. Khalatnikov, Sov. Phys. JETP , 205 (1976).[62] A. K. Yadav, arXiv:0911.0177.[63] B. Saha, T. Boyadjiev, Phys. Rev. D , 124010 (2004).[64] P. Pedram, M. Mirzaei, S. Jalalzadeh, S. S. Gousheh, Gen. Rel. Grav. , 1663 (2008).[65] T. Biswas, A. Mazumdar, Phys. Rev. D , 023519 (2009).[66] K. Jotania, P. Yadav, S. A. Faruqi, Int. J. Theor. Phys. , 1424 (2011).[67] S. Myung, B. M. Cho, Mod. Phys. Lett. A , 37 (1986).[68] BOOMERanG (P. de Bernardis et al.), Nature , 955 (2000).[69] MAXIMA (S. Hanany et al.),
Astrophys. J. Lett. , 5 (2000).[70] D. N. Sperger,
Astrophys. J. Suppl.148