Bulg. J. Phys. 38 (2011) 145–154
Bianchi Type-VI Bulk Viscous Fluid String Cosmological Model in General Relativity S.P. Kandalkar1 , P.P. Khade2 , S.P. Gawande1 1
2
Department of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati – 444604, India Department of Mathematics, Vidyabharati Mahavidyalaya, Amravati – 444603, India
Received 19 February 2011 Abstract. We investigate the integrability of cosmic string of Bianchi type-VI space-time in presence of bulk viscous fluid by applying a new technique. The behavior of the model is reduced to the solution of single second order nonlinear differential equation. We show that this equation admits an infinite family of solutions. The physical implications of these results are also discussed. PACS codes: 04.20.-q
1
Introduction
In recent years there has been considerable interest in string cosmology. Cosmic strings are topologically stable objects which might be found during a phase transition in the early universe (Kibble [1]). Cosmic string plays an important role in the study of the early universe. This arises during the phase transition after the big bang explosion as the temperature goes down below some critical temperature as predicted by grand unified theories (Zel’dovich et al. [2]; Kibble [1,3]; Everett [4]; Vilenkin [5]). It is believed that cosmic strings give rise to density perturbations which lead to the formation of galaxies (Zel’dovich [6]). These cosmic strings have stress energy and coupled to the gravitational field. There it is interesting to study the gravitational effects that arise from strings. The general relativistic treatment of strings was initiated by Letelier [7,8] and Stachel [9]. Letelier [7] has obtained the solution of Einstein’s field equations for a cloud of strings with spherical, plane and cylindrical symmetry. Then in 1983, he solved Einstein’s field equations for a cloud of massive strings and obtained cosmological models in Bianchi type-I and Kantowski-Sachs space-times. Banerjee et al. [10] have investigated an axially symmetric Bianchi type-I string dust cosmological models with a magnetic field discussed also by Chakraborty c 2011 Heron Press Ltd. 1310–0157
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Bianchi Type-VI Bulk Viscous Fluid String Cosmological Model [11], Tikekar and Patel [12]. Patel and Maharaj [13] investigated stationary rotating world model with magnetic field. Ram and Singh [14] obtained some new exact solutions of string cosmology with and without a source free magnetic field for a Bianchi type-I space-time in different basic form considered by Carmaniti and McIntosh [15]. Exact solutions of string cosmology for Bianchi type-II, VI0 , VIII and IX space-time have been studied by Krori et al. [16] and Wang [17]. On the other hand, the matter distribution is satisfactorily described by perfect fluids due to the large scale distribution of galaxies in our universe. However, a realistic treatment of the problem requires the consideration of material distribution other than the perfect fluid. It is well-known that when neutrino decoupling occurs, the matter behaves as a viscous fluid in an early stage of universe. Viscous fluid cosmological models of early universe have been widely discussed in the literature. Recently Yadav et al. [18] have studied some Bianchi type string cosmological model with bulk viscosity. Motivated by the situation discussed above, in this paper we focus on the problem establishing formalism for studying the new integrability of cosmic strings in Bianchi type-VI space-time in the presence of bulk viscous fluid by applying new technique. 2
Metric and Field Equations
We consider the space-time of general Bianchi type-VI with the metric ds2 = −dt2 + A2 dx2 + B 2 exp(−2qx)dy 2 + C 2 exp(2qx)dz 2 ,
(1)
where q is a constant. A, B, C are functions of t. The energy momentum tensor for a cloud of string dust with a bulk viscous fluid of string is given by Letelier and Landau–Lifchitz Tij = ρui uj − λxi xj − ξui; i (gij + ui uj ),
(2)
where ui and xi satisfy condition ui ui = −xi xi = −1,
ui xi = 0.
(3)
In (2) ρ is the proper energy density for a cloud of string with particles attached to them, λ is the string tension density, ui is the four velocities of the particles and xi is a unit space-like vector representing the direction of string. If the particle density of the configuration is denoted by ρp , then we have ρ = ρp + λ.
(4)
The Einstein field equations (in gravitational units c = 1, G = 1) read as 1 Rij − Rgij = −8πTij , 2 146
(5)
S.P. Kandalkar, P.P. Khade, S.P. Gawande where Rij is the Ricci tensor; R = g ij Rij is the Ricci scalar. In a co-moving coordinate system, we have ui = (0, 0, 0, 1).
(6)
The field equations (5) with (2) subsequently lead to the following system of equations: ¨ C¨ B˙ C˙ B q2 + + + 2 B C BC A A˙ C˙ A¨ C¨ q2 + + − 2 A C AC A ¨ A˙ B˙ A¨ B q2 + + − 2 A B AB A ˙ ˙ ˙ ˙ ˙ ˙ AB BC AC + + − AB BC AC C˙ B˙ − = 0, B C
= 8πξθ,
(7)
= 8πξθ,
(8)
= 8π(ξθ + λ),
(9)
q2 = 8πρ, A2
(10) (11)
where (·) over the symbols A, B, C denotes ordinary differentiation with respect to t. The particle density ρp is given by 8πρp =
¨ A˙ C˙ B C¨ B˙ C˙ q2 +2 − + − 2 BC AC B C A
(12)
in accordance with equation (4). The velocity field ui specified by (6) is irrotational, the scalar expansion θ and components of shear σij are given by θ= σ11 σ22 σ33 σ44
A˙ B˙ C˙ + + , A B C B˙ C˙ A2 A˙ 2 − − , = 3 A B C A˙ C˙ B 2 exp(−2qx) B˙ 2 − − , = 3 B A C A˙ B˙ C 2 exp(2qx) C˙ 2 − − , = 3 C A B = 0.
(13) (14) (15) (16) (17)
Therefore, σ2 =
B˙ C˙ A˙ C˙ B˙ 2 C˙ 2 A˙ B˙ 1 A˙ 2 − − + + − . 3 A2 B2 C2 AB BC AC
(18) 147
Bianchi Type-VI Bulk Viscous Fluid String Cosmological Model 3
Solutions of the Field Equations
The field equations (7-11) are a system of five equations with six unknown parameters A, B, C, ρ, λ and ξ. One additional constraint relating to these parameters is required to obtain explicit solutions of the system. We assume that the expansion θ in the model is proportional to the eigenvalue σ33 of the shear tensor σij . This condition leads to (19) C = α(AB)β , where α and β are arbitrary constants. Equation (11) leads to B = mC,
(20)
where m is an integrating constant. From (19) and (20), we obtain C = M AN ,
(21)
where 1
M = α /(1−β) m
β
/(1−β)
,
N=
β . 1−β
(22)
By the use of (20) field equations (9-10) reduce to A¨ C¨ + + A C A˙ C˙ 2 AC
A˙ C˙ q2 − 2 = 8π(λ + ξθ). AC A C˙ 2 q2 + − 2 = 8πρ. C A
(23) (24)
Equations (7) and (8), together with the use of (13) and (21), lead to (3N + 1)
A¨ A˙ 2 A˙ + (5N 2 − 2N ) 2 = 16πξ(2N + 1) . A A A
Let us consider
(25)
A˙ = f (A).
(26)
5N 2 − 2N 1 2N + 1 df + f = 16πξ . dA 3N + 1 A 3N + 1
(27)
Using (26) in (25), we get
After integration, (27) reduces to f = 16πξ
148
2N + 1 A+ 5N 2 + N + 1
„
A
P 5N 2 −2N 3N +1
«,
(28)
S.P. Kandalkar, P.P. Khade, S.P. Gawande where P is an integrating constant. Integrating (28), we obtain A=
1 [k1 + k2 ξ exp(k3 ξt)]k4 , ξ k4
(29)
where S is an integrating constant. Therefore M [k1 + k2 ξ exp(k3 ξt)]k5 , ξ k5 mM B = k5 [k1 + k2 ξ exp(k3 ξt)]k5 , ξ C=
(30) (31)
where k1 = −
P (5N 2 + 5N + 1) , 16π(2N + 1)
k2 = S, 16π(2N + 1) , (3N + 1) (3N + 1) k4 = . (5N 2 + 5N + 1) k3 =
(32)
Hence the metric (1) reduces to the form k + k ξ exp(k ξt) 2k4 1 2 3 dx2 ξ k + k ξ exp(k ξt) 2k5 1 2 3 + exp(−2qx)m2 M 2 dy 2 ξ k + k ξ exp(k ξt) 2k5 1 2 3 + exp(2qx)M 2 dz 2 . (33) ξ
ds2 = −dt2 +
Using the suitable transformation k + k ξ exp(k ξt) L sin(ξτ ) 1 2 3 = , ξ ξ Lk4 x = X,
(34)
mM L y = Y, k5
M Lk5 z = Z, the metric (33) reduces to ds2 = −
sin(ξτ ) 2k4 2 L cos(ξτ ) dτ 2 + dX 2 k3 (k1 − L sin(ξτ )) ξ 2qX sin(ξτ ) 2k5 2qX sin(ξτ ) 2k5 + exp − k4 dY 2 + exp dZ 2 . (35) L ξ Lk4 ξ 149
Bianchi Type-VI Bulk Viscous Fluid String Cosmological Model The rest energy (ρ), the string tension density (λ), the particle density (ρp ), expansion (θ) and shear (σ) for the model (35) are given by 8πρ = k32 k5 (2k4 + k5 ) ξ −
2k4 ξ k1 ξ 2 − q2 . L sin(ξτ ) L sin(ξτ ) k1 ξ 8πλ = −8πξk3 (k4 + 2k5 ) ξ − L sin(ξτ ) k1 ξ 2 + k32 (k42 + k4 k5 + k52 ) ξ − L sin(ξτ ) 2k4 2 ξ k1 ξ 2 ξ − q2 − + k32 k1 (k4 +k5 ) , 2 L sin(ξτ ) L sin (ξτ ) L sin(ξτ ) k1 ξ 2 8πρp = k32 k4 (k5 − k4 ) ξ − L sin(ξτ ) k1 ξ ξ ξ− , + 8πξk3 (k4 +2k5 ) − k32 k1 (k4 +k5 ) L sin(ξτ ) L sin(ξτ ) 2 k1 ξ σ11 = k3 (k4 − k5 ) ξ − , 3 L sin(ξτ ) 1 k1 ξ σ22 = σ33 = k3 (k5 − k4 ) ξ − , 3 L sin(ξτ ) σ44 = 0, 1 k1 ξ 2 , σ 2 = k3 (k5 − k4 ) ξ − 3 L sin(ξτ ) k1 ξ . θ = k3 (k4 + 2k5 ) ξ − L sin(ξτ )
(36)
(37)
(38) (39) (40) (41) (42) (43)
From (36) and (38), we observe the energy conditions ρ 0 and ρp 0 are fulfilled provided k5 (2k4 + k5 ) ξ −
2k4 ξ k1 ξ 2 q 2 L sin(ξτ ) k3 L sin(ξτ )
and k1 ξ 2 L sin(ξτ ) ξ k1 ξ − 8πξk3 (k4 + 2k5 ) ξ − , k32 k1 (k4 + k5 ) L sin(ξτ ) L sin(ξτ )
k32 k4 (k5 − k4 ) ξ −
respectively. From (37), we observe that the string tension density λ > 0 provided
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S.P. Kandalkar, P.P. Khade, S.P. Gawande k1 ξ 2 L sin(ξτ ) ξ2 k1 ξ 2 − + k32 k1 (k4 + k5 ) L sin(ξτ ) L sin2 (ξτ ) 2k4 ξ k1 ξ > q2 + 8πξk3 (k4 + 2k5 ) ξ − . L sin(ξτ ) L sin(ξτ )
k32 (k42 + k4 k5 + k52 ) ξ −
The model (35) represents an expanding universe when sin(ξτ ) > k1 /L. When sin(ξτ ) < k1 /L, then θ decreases with time. Therefore the model describes a shearing non-rotating expanding universe without big bang start. We can see from above discussion that the bulk viscosity plays a significant role in the evolution of the universe. σ = 0, the model does not approach isotropy for large Furthermore, since lim T →∞ θ value of τ . However, if sin(ξτ ) = k1 /L, the model (35) represents an isotropic model in presence of bulk viscosity. In absence of bulk viscosity, when ξ → 0, the metric (35) reduces to L 2 2qX dτ 2 + τ 2k4 dX 2 + exp − k4 τ 2k5 dY 2 ds2 = − k1 k3 L 2qX τ 2k5 dZ 2 . (44) + exp Lk4 The physical parameters ρ, λ, ρp and the kinematical parameters θ, σ 2 for this model are respectively given by k k 2 q2 1 3 k5 (2k4 + k5 ) − , Lτ (Lτ )2k4 k k 2 q2 1 3 (k42 + k4 k5 + k4 + k52 + k5 ) − , 8πλ = Lτ (Lτ )2k4 k k 2 1 3 8πρp = k4 (k5 − k4 + 1) + k5 , Lτ 2 k1 k3 (k5 − k4 ) , σ11 = 3 Lτ k1 k3 (k4 − k5 ) σ22 = σ33 = , 3Lτ .σ44 = 0 , 1 k1 k3 (k4 − k5 ) 2 , σ2 = 3 Lτ k1 k3 (k4 + 2k5 ) . θ=− Lτ 8πρ =
(45) (46) (47) (48) (49) (50) (51) (52)
From (45) and (47), we observe that the energy conditions ρ 0, and ρp 0, 151
Bianchi Type-VI Bulk Viscous Fluid String Cosmological Model are fulfilled provided k5 (2k4 + k5 )(Lτ )2(k4 −1)
1 , (k1 k3 )2
k5 (k4 − 1).
Respectively, from (46), we observe that the string tension density λ 0, provided q 2 . (k42 + k4 k5 + k4 + k52 + k5 )(Lτ )2(k4 −1) k1 k3 In absence of bulk viscosity, the model (44) starts expanding with a big bang at τ = 0 and the expansion in the model decreases as time increases. When τ → ∞ then shear is zero. Near the singularity τ = 0, the physical parameters σ = 0, the model does not ρ, λ, ρp are infinite, if k4 < 0. Also, since lim T →∞ θ approach isotropy for large value of τ . 4
Another Model
In general ξ is not constant throughout the fluid so that ξ cannot be taken always constant, especially when the universe is expanding. Since in general, ξ depends on temperature (T ) and pressure (p) it is reasonable to consider ξ as a function of t. In this case (25) after integration, leads to k 4 A = b0 + k4−1 h(t)dt ,
(53)
h(t) = c0 exp k3 ξ(t) dt.
(54)
where
And b0 , c0 are constants of integration. Therefore, we obtain k 5 C = M b0 + k4−1 h(t)dt , k 5 −1 B = mM b0 + k4 h(t)dt .
(55) (56)
Hence, in this case, the metric (1) reduces to 2k4 ds2 = −dt2 + b0 + k4−1 h(t)dt dx2
2k5 −1 + exp(−2qx)(mM ) b0 + k4 dy 2 h(t)dt 2k5 + M 2 b0 + k4−1 h(t)dt exp(2qx)dz 2 . (57) 2
152
S.P. Kandalkar, P.P. Khade, S.P. Gawande The physical parameters ρ, λ, ρp and the kinematical parameters θ, σ 2 for this model are respectively given by 2 2k4 k5 h(t) k5 −1 2 2+ b 8πρ = − q + k , (58) h(t)dt 0 4 k4 k4 b0 + k4−1 h(t)dt −2k4 h(t) 2k5 8πλ = −q 2 b0 +k4−1 h(t)dt − 8πξ(t) 1+ k4 b0 +k4−1 h(t)dt h(t) k5 k 2 + 1+ + 52 −1 k4 k4 b0 + k4 h(t)dt −1 b h(t)dt k3 ξ(t)h(t) − k4−1 h2 (t) + k 0 4 k5 + 1+ , (59) 2 k4 b0 + k4−1 h(t)dt k2 h(t) 2k5 k5 − 1 − 52 − 8πρp = 8πξ(t) 1 + k4 k4 k4 b0 + k4−1 h(t)dt 2 k5 h(t) k5 2+ + −1 k4 k4 b0 + k4 h(t)dt −1 h(t)dt k3 ξ(t)h(t) − k4−1 h2 (t) + k b 0 4 k5 , (60) − 1+ 2 k4 −1 b0 + k4 h(t)dt 2k5 h(t) θ = 1+ , (61) −1 k4 h(t)dt b0 + k4 h(t) k5 2 σ11 = , (62) 1− 3 k4 b0 + k4−1 h(t)dt 1 k5 h(t) σ22 = σ33 = , (63) −1 3 k4 b0 + k4−1 h(t)dt σ44 = 0, k2 1 2k5 σ 2 = 1 + 52 − 3 k4 k4
5
2
h (t) 2 . b0 + k4−1 h(t)dt
(64) (65)
Conclusion
We have presented a new class of Bianchi type-VI string cosmological models in the presence and absence of bulk viscosity. In our solution, we have obtained a relation between metric coefficients from our field equation in a natural way. In Section 4, we have obtained a general solution that has a rich structure and admits many number of solutions by suitable choice of function ξ(t). Here the choice of ξ(t) is quit arbitrary but since we look for physically viable models of the universe, one can choose ξ(t), such that (54) is integrable. 153
Bianchi Type-VI Bulk Viscous Fluid String Cosmological Model It is observed that the bulk viscosity plays significant role in the evolution of the universe. In presence of bulk viscosity the model represents an expanding, shearing and non-rotating universe without the big bang start. But, in absence of bulk viscosity, the model starts expanding with a big bang at τ = 0. References [1] T.W.B. Kibble (1976) J. Phys. A Math. Gen. 9 1387. [2] Ya.B. Zel’dovich, I.Yu. Kobzarev, L.B. Okun (1975) Zh. Eksp. Teor. Fiz. 67 3.; Ya.B. Zel’dovich, I.Yu. Kobzarev, L.B. Okun (1975) Sov. Phys. – JETP 40 1. [3] T.W.B. Kibble (1980) Phys. Rep. 67 183. [4] A.E. Everett (1981) Phys. Rev. 24 858. [5] A. Vilenkin (1981) Phys. Rev. D 24 2082. [6] Ya.B. Zel’dovich (1980) Mon. Not. R. Astron. Soc. 192 663. [7] P.S. Letelier (1979) Phys. Rev. D 20 1249. [8] P.S. Letelier (1983) Phys. Rev. D 28 2414. [9] J. Stachel (1980) Phys. Rev. D 21 2171. [10] A. Banerjee, A.K. Sanyal, S. Chakraborty (1990) Prarnana – J. Phys. 34 1. [11] S. Chakraborty (1991) Ind. J. Pure Appl. Phys. 29 31. [12] R. Tikekar, L.K. Patel (1994) Gen. Rel. Grav. 24 397. [13] R. Tikekar, L.K. Patel (1992) Pramana – J. Phys. 42 483. [14] L.K. Patel, S.D. Maharaj (1996) Pramana – J. Phys. 47 1. [15] S. Ram, T.K. Singh (1995) Gen. Rel. Grav. 27 1207. [16] J. Carminati, C.B.G. Mclntosh (1980) J. Phys. A: Math. Gen. 13 953. [17] K.D. Krori, T. Chaudhury, C.R. Mahanta, A. Mazumdar (1990) Gen. Rel. Grav. 22 123. [18] X.X. Wang (2003) Chin. Phys. Lett. 20 615. [19] M.K. Yadav, A. Rai, A. Pradhan (2007) IJTP 7 9381.
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