BIANCHI TYPE-III COSMOLOGICAL MODEL WITH VARIABLE G AND Λ-TERM IN GENERAL RELATIVITY

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1 BIANCHI TYPE-III COSMOLOGICAL MODEL WITH VARIABLE G AND Λ-TERM IN GENERAL RELATIVITY HASSAN AMIRHASHCHI 1, H. ZAINUDDIN 2,a, ANIRUDH PRADHAN 2,3 1 Young Researchers Club, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran h.amirhashchi@mahshahriau.ac.ir 2 Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, University Putra Malaysia, Serdang, Selangor D.E., Malaysia a : hisham@putra.upm.edu.my 3 Department of Mathematics, Hindu Post-graduate College, Zamania , Ghazipur, India pradhan@iucaa.ernet.in; pradhan.anirudh@gmail.com Received August 1, 2011 Exact solution of Einstein s field equations with variable gravitational and cosmological constant is obtained in presence of perfect fluid for Bianchi III space-time. To get the deterministic solution of the field equations the expansion θ, in the model, is considered as proportional to the eigen value σ 2 2 of the shear tensor σ j i and also the fluid obeys the barotropic equation of state. The value of cosmological constant Λ for the model is found to be small and positive which is supported by the results from recent supernovae Ia observations. Moreover, it is observed that due to the combined effect of time variable Λ and G the universe evolved with deceleration as well as acceleration. The model shows that G varies with time as suggested earlier by Large Number Hypothesis proposed by Dirac. It has been found that all physical and geometric parameters of the model are in fair agreement of observational results. Some physical and geometric properties of the model are also discussed. Key words: Cosmology, Exact solution, Variable G and Λ, Perfect fluid. PACS: Es, k. 1. INTRODUCTION The Einstein field equation has two parameters, the gravitational constant G and the cosmological constant Λ. The Newtonian constant of gravitation G plays the role of a coupling constant between geometry and matter in the Einstein field equation. In an evolving universe, it appears to look at this constant as a function of time. There are significant observational evidence that the expansion of the Universe is undergoing a late time acceleration 1 15]. This, in other words, amounts to saying that in the context of Einstein s general theory of relativity some sort of dark energy, constant or that varies only slowly with time and space dominates the current composition of cosmos. The origin and nature of such an accelerating field poses a RJP Rom. 57(Nos. Journ. Phys., 3-4), Vol , Nos. 3-4, (2012) P , (c) Bucharest, 2012

2 2 Bianchi type-iii cosmological model with variable G and Λ-term in general relativity 749 completely open question. Recently, Riess et al. 16] have presented an analysis of 156 SNe including a few at z > 1.3 from the Hubble Space Telescope (HST) GOOD ACS Treasury survey. They conclude to the evidence for present acceleration q 0 < 0 (q 0 0.7). Observations 16, 17] of Type Ia Supernovae (SNe) allow us to probe the expansion history of the universe leading to the conclusion that the expansion of the universe is accelerating. Observations strongly favor a small and positive value of the effective cosmological constant at the present epoch. Among many possible alternatives, the simplest and most theoretically appealing possibility for dark energy is the energy density stored on the vacuum state of Λ 8πG all existing fields in the universe, i.e., ρ v =, where Λ is the cosmological constant. However, a constant Λ cannot explain the huge difference between the cosmological constant inferred from observation and the vacuum energy density resulting from quantum field theories. In an attempt to solve this problem, variable Λ was introduced such that Λ was large in the early universe and then decayed with evolution 18]. Cosmological scenarios with a time-varying Λ were proposed by several researchers. A number of models with different decay laws for the variation of cosmological term were investigated during last two decades 19 27]. In recent past, a number of authors have considered cosmological models with time-dependent cosmological constant (see 28 36] and references therein). On the other hand, numerous modifications of general relativity to allow for a variable G based on different arguments have been proposed 37]. First time Dirac 38 41] and Dicke 42] suggested a possible time varying gravitational constant. The Large Number Hypothesis (LNH) proposed by Dirac leads to a cosmology when G varies with time. Variation of G has many interesting consequences in astrophysics. Canuto and Narlikar 43] have shown that G-varying cosmology is consistent with whatsoever cosmological observations available at present. A modification linking the variation of G with that of variable Λ-term has been considered within the framework of general relativity by a number of workers 44 47]. This modification is appealing as it leaves the form of Einstein s equations formally unchanged by allowing a variation of G to be accompanied by a change in Λ. Cosmological models with time-dependent G and Λ in the solutions Λ R 2, Λ t 2, were first obtained by Bertolami 48,49]. Variability of G is also supported by observational results coming from Lunar Lase Ranging 50], spinning rate of pulsars 51 53], distant Type Ia Supernovae observations 54], Helioseismological data 55], and white dwarf G 117-B 15A 56, 57]. Also the discovery of the scenario of accelerating universe 2 5], the investigations within the framework of variable G is also not uncommon in the literature. The cosmological models with variable G and Λ have been recently studied by several authors 58 69]. Recently, Ray et al. 72] obtained dark energy models with time-dependent G. Mukhopadhyay et al. 73] studied higher dimensional dark energy with time variable Λ and G. Arbab 74] investigated bulk viscous dark energy

3 750 Hassan Amirhashchi, H. Zainuddin, Anirudh Pradhan 3 models with variable G and Λ. Pradhan et al. 75] also obtained FRW universe with variable G and Λ-term. Recently, Khadekar and Kamdi 76] have obtained exact solution of the Einstein s field equations in higher dimension with variable G and Λ. Recently, Singh et al. 77] and Bali et al. 78] have investigated Bianchi type- III cosmological models with variable G and Λ in presence of perfect fluid and their both solutions are particular and similar. The aforesaid survey of literature clearly indicates that there has been interest in studying Bianchi types models with variable G and Λ. Motivated by the above observations, we consider anisotropic space-time of Bianchi type-iii model in a general form with variable gravitational and cosmological constant and obtained a general and exact solution of Einstein s field equations which is new and different from other author s solutions. The out line of the paper is as follows: in Section 2, the metric and the field equations are described. Section 3 deals with the solutions of the field equations and their geometric and physical properties. Finally, conclusions are summarized in the last Section THE METRIC AND FIELD EQUATIONS We consider the space-time of general Bianchi III type with the metric ds 2 = dt 2 + A 2 (t)dx 2 + B 2 (t)e 2ax dy 2 + C 2 (t)dz 2, (1) where a is constant. The energy momentum tensor for perfect fluid distribution has the form T j i = (ρ + p)v iv j + pg j i, (2) where v i satisfies condition v i v i = 1. (3) Here p is isotropic pressure, ρ is the proper energy density and v i is the four-velocity. In a co-moving coordinate system, we have v i = (0,0,0,1) (4) The Einstein s field equations with time varying G and Λ read as R j i 1 2 Rgj i = 8πG(t)T j i + Λ(t)gj i, (5) where R j i is the Ricci tensor; R = gij R ij is the Ricci scalar, G is the gravitational constant and Λ is cosmological constant. The field equations (5) with (2) for the metric (1) subsequently lead to the following system of equations: Ä A + B B + AḂ AB a2 = 8πG(t)p + Λ(t), (6) A2

4 4 Bianchi type-iii cosmological model with variable G and Λ-term in general relativity 751 B B + C C + ḂĊ = 8πG(t)p + Λ(t), BC (7) Ä A + C C + AĊ = 8πG(t)p + Λ(t), AC (8) AĊ AC + AḂ AB + ḂĊ BC a2 = 8πG(t)ρ + Λ(t). (9) A2 ( ) A A Ḃ = 0. (10) B Here and in what follows an over dot denotes ordinary differentiation with respect to time, t. Vanishing divergence of the Einstein tensor (R j i 1 2 Rgj i ) ;j = 0, (11) leads to 8πĠ(t)ρ + Λ(t) + 8πG(t) ] A ρ + (ρ + p)( A + Ḃ B + Ċ C ) = 0. (12) We now assume that the law of conservation of energy (T ij ;j = 0) gives ( ) A ρ + (ρ + p) A + Ḃ B + Ċ = 0. (13) C Using (12) the above relation yields Ġ(t) = Λ(t) 8πρ, (14) which implies that G(t) increases or decrease as Λ(t) decreases or increases. The spatial volume for the model (1) is given by V 3 = ABCe ax. (15) We define V = (ABCe ax ) 1 3 as the average scale factor so that the Hubble s parameter is anisotropic models may be defined as H = V ( ) V = 1 A 3 A + Ḃ B + Ċ. (16) C We define the generalized mean Hubble s parameter H as H = 1 3 (H x + H y + H z ), (17)

5 752 Hassan Amirhashchi, H. Zainuddin, Anirudh Pradhan 5 where H x = Ȧ A, H y = Ḃ B and H z = Ċ C directions of x, y and z respectively. are the directional Hubble s parameters in the An important observational quantity is the deceleration parameter q, which is defined as q = V V V 2. (18) The velocity field v i as specified by (4) is irrotational. The scalar expansion θ, components of shear σ ij and the average anisotropy parameter A m are defined by Therefore σ 2 = 1 3 A θ = A + Ḃ B + Ċ C, (19) ] σ 11 = A2 2Ȧ 3 A Ḃ B Ċ, (20) C σ 22 = B2 e 2ax 2Ḃ 3 B A ] A Ċ, (21) C σ 33 = C2 2Ċ 3 C A ] A Ḃ, (22) B σ 44 = 0. (23) A 2 A 2 + Ḃ2 B 2 + Ċ2 C 2 AḂ AB ḂĊ BC Ċ A ]. (24) CA 3 ( ) 2 Hi, (25) A m = 1 3 where H i = H i H(i = 1,2,3). i=1 H 3. SOLUTIONS OF THE FIELD EQUATIONS The field equations (6)-(10) are a system of five equations with seven unknown parameters A, B, C, ρ, p, Λ(t) and G(t). Two additional constraints relating these parameters are required to obtain explicit solutions of the system. We firstly consider that the expansion (θ) in the model is proportional to the eigen value σ2 2 of the shear tensor σ j i. This condition leads to B = l 1 (AC) m 1, (26) where l 1 and m 1 are arbitrary constants. Equations (10) leads to A = mb, (27)

6 6 Bianchi type-iii cosmological model with variable G and Λ-term in general relativity 753 where m is an integrating constant. The equations (7) and (8) reduce to B B Ä A + ḂĊ BC AĊ = 0. (28) AC Using (27) in (28) we obtain ( ) B (1 m) B + ḂĊ = 0. (29) BC As m 0, (29) gives which on integration reduces to where k 1 is an integrating constant. From (26) and (31), we obtain 1 ( ) B B + ḂĊ = 0, (30) BC 1 m where l 2 = l 1 1 m l, l = m 1 1 m 1. Using (32) in (31) we get which on integration gives C = (l + 1) 1 ḂC = k 1, (31) B = l 2 C l, (32) C l Ċ = k 1, (33) ] 1 k1 t + k 2, (34) where k 2 is an integrating constant. Using (34) in (32) and (27) we obtain ] l B = l 2 (l + 1) l k1 t + k 2, (35) and ] l A = ml 2 (l + 1) l k1 t + k 2, (36) respectively. Hence the metric (1) reduces to the form ) l ] 2 ds 2 = dt 2 + ml 2 (l + 1) ( l k1 t + k 2 dx 2 + l 2 (l + 1) l e ax( k1 t + k 2 ) l ] 2 dy 2 + ) 1 ] 2 (l + 1) ( 1 k1 t + k 2 dz 2. (37)

7 754 Hassan Amirhashchi, H. Zainuddin, Anirudh Pradhan 7 Using the suitable transformation the metric (37) reduces to where ml 2 (l + 1) l x = X, l2 (l + 1) l y = Y, (l + 1) 1 z = Z, k 1 t + k 2 = T, (38) ds 2 = β 2 dt 2 + T 2L dx 2 + T 2L e 2a N X dy 2 + T 2L l dz 2, (39) β = k 1, M = (l + 1) 1, N = ml 2 M, L = l l + 1. Now we secondly assume that the fluid obeys barotropic equation of state (40) p = γρ, (41) where γ(0 < γ < 1) is a constant. Using (34)-(36) and (41) in (13), we obtain which by integrating leads to ρ ρ + γ)(1 + 2l) = (1, (42) lβt ρ =, (43) T (1+γ)(1+2l) where k 3 is an integrating constant. The equations (34)-(36), (9) and (14) lead to k 3 8πG = 2a2 L N 2 T L (1+3l)+γ(2l 2 +3) k 3 The equations (43), (44) and (14) also yield l 2 2L2 (l + 2) lβ 2 k 3 T γ(1+2l)+1 l. (44) Λ = 2a2 L 2 (1 + 3l) + γ(2l 2 N 2 + 3l + 1) ] T α l αk 2 L2 (l + 2) 1 3 lβ 2 k 3 T 2, (45) where α = (1 + γ)(2l 2 + l) L(1 + 3l) + γ(2l 2 + 3l + 1)]. For ρ > 0, we need k 3 > 0. Therefore, from (43), it is noted that the proper energy density ρ(t) is a decreasing function of time and it approaches a small positive value at present epoch. From (45), it can be seen that the cosmological constant Λ is a decreasing function of time and Λ > 0 when T > T c, where T c is a critical time given by T c = 2lβ 2 a 2 (1 + 3l) + γ(2l 2 + 3l + 1) ] N 2 α(l + 2) ] l 2 α+2l 2 (46)

8 8 Bianchi type-iii cosmological model with variable G and Λ-term in general relativity 755 Fig. 1 The plot of cosmological constant Λ vs. T Fig. 2 The plot of gravitational constant G vs. T

9 756 Hassan Amirhashchi, H. Zainuddin, Anirudh Pradhan 9 This nature of Λ is clearly shown in Figure 1 as a representative case with appropriate choice of constants of integration and other physical parameters using reasonably well known situations. Also from (44), it can be seen that the gravitational constant G > 0 if N 2 ] l 2 L(l + 2) L(1+3l)+γ(2l 2 +3)] lγ(1+2l)+1] T >. (47) lβ 2 a 2 We have observed that G is an increasing function of time which can also be seen in Figure 2. The physical quantities ρ and Λ tend to infinity at T = 0 and 0 at T = whereas the gravitational constant G 0 as T 0 and as T. The model (39) therefore starts with a big-bang at T = 0 and it goes on expanding until it comes to rest at T =. We also note that T = 0 and T = respectively correspond to the proper time t = 0 and t =. There is a point type singularity (MacCallum 79]) in the model at T = 0. The expressions for the scalar of expansion θ, magnitude of shear σ 2, the average anisotropy parameter A m, deceleration parameter q and proper volume V for the model (39) are given by ( (l 1)L θ = (2l + 1)L, σ 2 = 1 lβt 3 lβt q = lβ (2l + 1), V = N 2 M m ) 2 ( ) l 1 2, A m = 2, 2l l T L(2) l. The rate of expansion H i in the direction of x, y and z are given by (48) H x = H y = L βt, H z = L lβt. (49) Hence the average generalized Hubble s parameter is given by H = L(2l + 1) 3lβT. (50) From the above results, it can be seen that the spatial volume is zero at T = 0 and it increases with the increase of T. This shows that the universe starts evolving with zero volume at T = 0 and expands with cosmic time T. From equation (49), we observe that all the three directional Hubble parameters are zero at T. In derived model, the energy density tend to infinity at T = 0. The model has the point-type singularity at T = 0. The shear scalar diverse at T = 0. As T, the scale factors A(t), B(t) and C(t) tend to infinity. The energy density becomes zero as T. The expansion scalar and shear scalar all tend to zero as T. The mean anisotropy parameter are uniform throughout whole expansion of the universe when l 1 2

10 10 Bianchi type-iii cosmological model with variable G and Λ-term in general relativity 757 but for l = 1 2 it tends to infinity. This shows that the universe is expanding with the increase of cosmic time but the rate of expansion and shear scalar decrease to zero and tend to isotropic. At the initial stage of expansion, when ρ is large, the Hubble parameter is also large and with the expansion of the universe H, θ decrease as does ρ. Since σ2 = constant provided l 1 θ 2 2, the model does not approach isotropy at any time. The cosmological evolution of Bianchi type-iii space-time is expansionary, with all the three scale factors monotonically increasing function of time. The dynamics of the mean anisotropy parameter depends on the value of l. From (48) we observe that (i) for l < 1 2, q > 0 i.e., the model is decelerating and (ii) for l > 1 2, q < 0 i.e., the model is accelerating. Moreover, it is observed that due to the combined effect of time variable Λ and G the universe evolved with deceleration as well as acceleration. Recent observations of type Ia supernovae 1 5,16] and references therein) reveal that the present universe is in accelerating phase and deceleration parameter lies somewhere in the range 1 < q 0. It follows that our model of the universe is consistent with the recent observations. 4. CONCLUDING REMARKS In this paper we have presented a new exact solution of Einstein s field equations for anisotropic Bianchi type-iii space-time in presence of perfect fluid with time varying gravitational constant G and cosmological constant Λ which is different from the other author s solutions. In general the model is expanding, shearing and nonrotating. The model starts with a big-bang at T = 0 and it goes on expanding until it comes out to rest at T =. It is worth mentioned here that T = 0 and T = correspond to the proper time t = 0 and t = respectively. The initial singularity in the model is the Point Type 79]. Our universe starts evolving with zero volume at T = 0 and expand with cosmic time T. We observe that σ2 is constant provided l 1 θ 2 2, the model does not approach isotropy at any time. Our model is in accelerating phase which is consistent to the recent observations. In some cases, it is observed that G is an increasing function of time. When the universe is required to have expanded from a finite minimum volume, the critical density assumption and conservation of energymomentum tensor dictate that G increases in a perpetually expanding universe. The possibility of an increasing G has been suggested by several authors. The behavior of the universe in our models will be determined by the cosmological term Λ ; this term has the same effect as a uniform mass density ρ eff = Λ/4πG, which is constant in time. A positive value of Λ corresponds to a negative effective mass density (repulsion). Hence, we expect that in the universe with a positive value of Λ, the expansion will tend to accelerate; whereas in the universe with negative value of Λ, the expansion will slow down, stop and reverse. In a universe with both matter and vacuum energy, there is a competition between the tendency of

11 758 Hassan Amirhashchi, H. Zainuddin, Anirudh Pradhan 11 Λ to cause acceleration and the tendency of matter to cause deceleration with the ultimate fate of the universe depending on the precise amounts of each component. This continues to be true in the presence of spatial curvature, and with a non-zero cosmological constant it is no longer true that the negatively curved ( open ) universes expand indefinitely while positively curved ( closed ) universes will necessarily recollapse-each of the four combinations of negative or positive curvature and eternal expansion or eventual re-collapse become possible for appropriate values of the parameters. There may even be a delicate balance, in which the competition between matter and vacuum energy is needed drawn and the universe is static (not expanding). The search for such a solution was Einstein s original motivation for introducing the cosmological constant. Recent cosmological observations 1 8, 16] suggest the existence of a positive cosmological constant Λ with the magnitude Λ(G /c 3 ) These observations on magnitude and red-shift of type Ia supernova suggest that our universe may be an accelerating one with induced cosmological density through the cosmological Λ-term. In our derived model, we have observed that the Λ-term decreases as time increases and it approaches to a small and positive value at the present epoch. Thus, our models are consistent with the results of recent observations. Also (14) implies that G(t) increases or decrease as Λ(t) decreases or increase. In our case, G increases as Λ decreases with time. It is also worth mention here that for l = 1, H 0 T 0 = 1 2β. For 5 13 < β < 5 8, we obtain the current limits for the universe age 0.8 < T 0H 0 < 1.3 which is in good agreement with the best estimation T 0 H 0 1 (see 61]). Thus it has been found that all physical and geometric parameter of the model (39) are in fair agreement of observational results. Acknowledgments. This work has been supported by the FRGS Grant by the Ministry of Higher Education, Malaysia under the Project Number FR. H. Amirhashchi and A. Pradhan also thank the Laboratory of Computational Sciences and Mathematical Physics, Universiti Putra Malaysia for providing facility where this work was done. REFERENCES 1. S. Perlmutter et al., Astrophys. J. 483, 565 (1997). 2. S. Perlmutter et al., Nature 391, 51 (1998). 3. S. Perlmutter et al., Astrophys. J. 517, 5 (1999). 4. A. G. Riess et al., Astron. J. 116, 1009 (1998). 5. A. G. Riess et al., Publ. Astron. Soc. Pacific (PASP) 112, 1284 (2000). 6. P. M. Garnavich et al., Astrophys. J. 493, L53 (1998). 7. P. M. Garnavich et al., Astrophys. J. 509, 74 (1998). 8. B. P. Schmidt et al., Astrophys. J. 507, 46 (1998). 9. G. Efstathiou et al., Mon. Not. R. Astron. Soc. 330, L29 (2002). 10. D. N. Spergel et al., Astrophys. J. Suppl. Ser. 148, 175 (2003).

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