Astrophys Space Sci (2013) 348:247–252 DOI 10.1007/s10509-013-1540-0
O R I G I N A L A RT I C L E
Bianchi type-V bulk viscous string cosmological model in f (R, T ) gravity R.L. Naidu · D.R.K. Reddy · T. Ramprasad · K.V. Ramana
Received: 22 May 2013 / Accepted: 16 June 2013 / Published online: 2 July 2013 © Springer Science+Business Media Dordrecht 2013
Abstract In this paper, we investigate a spatially homogeneous and anisotropic Bianchi type-V cosmological model in a scalar-tensor theory of gravitation proposed by Harko et al. (Phys. Rev. D 84:024020, 2011) when the source for energy momentum tensor is a bulk viscous fluid containing one dimensional cosmic strings. To obtain a determinate solution, a special law of variation proposed by Berman (Nuovo Cimento B 74:182, 1983) is used. We have also used the barotropic equation of state for the pressure and density and bulk viscous pressure is assumed to be proportional to energy density. It is interesting to note that the strings in this model do not survive. Also the model does not remain anisotropic throughout the evolution of the universe. Some physical and kinematical properties of the model are also discussed. Keywords Bianchi-V model · f (R, T ) gravity · Bulk viscous model · String model
1 Introduction It is well known that the late time accelerated expansion of the universe has been confirmed by the high red shift supernovae experiments (Reiss et al. 1998; Perlmutter et al. 1999;
R.L. Naidu · T. Ramprasad GMR Institute of Technology, Rajam, India D.R.K. Reddy () Department of Mathematics, M.V.G.R. College of Engineering, Vizainagaram, Andhra Pradesh, India e-mail:
[email protected] K.V. Ramana Govt, Res. Polytechnic, Paderu, A.P., India
Bennett et al. 2003) and by the observations such as cosmic microwave background radiation (Spergel et al. 2003, 2007). In view of this it is now believed that the energy composition of universe has 4 % ordinary matter and 20 % dark matter and 76 % dark energy. Modifications of Einstein’s theory are attracting more and more attention, in recent years, to explain the late time acceleration and dark energy. Among the various modifications of Einstein’s theory, f (R) gravity (Akbar and Cai 2006) and f (R, T ) gravity (Harko et al. 2011) theories are attracting more and more attention during the last decade because these theories are supposed to provide natural gravitational alternatives to dark energy. It has been suggested that cosmic acceleration can be achieved by replacing Einstein-Hilbert action of general relativity with a general function f (R) where R is a Ricci scalar. Chiba et al. (2007), Nojiri and Odintsov (2007, 2010), Multamaki and Vilja (2006, 2007) are some of the authors who have investigated several aspects of f (R) gravity models which show early time inflation and late time acceleration. A comprehensive review on f (R) gravity is given by Copeland et al. (2006). Another modification of standard general relativity is f (R, T ) gravity proposed by Harko et al. (2011) wherein the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the trace of the stress energy tensor T . The gravitational field equations have been derived from the Hilbert-Einstein type variational principle √ √ 1 (1) f (R, T ) −gd 4 x + Lm −gd 4 x S= 16π where f (R, T ) is an arbitrary function of the Ricci scalar, R, T is the trace of stress-energy tensor of the matter, Tij and Lm is the matter Lagrangian density. We define the stressenergy tensor of matter as √ −2 δ( −gLm ) (2) Tij = √ −g δg ij
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and its trace by T = g ij Tij respectively. By assuming that Lm of matter depends only on the metric tensor components gij , and not on its derivatives, we obtain Tij = gij Lm − 2
∂Lm ∂g ij
(3)
Now by varying the action S of the gravitational field with respect to the metric tensor components g ij , we obtain the field equations of f (R, T ) gravity as
1 Rij − Rgij = 8πTij − 2f (T )Tij − 2f (T )θij + f (T )gij 2 (10) where the prime denotes differentiation with respect to the argument. If the matter source is a perfect fluid, θij = −2Tij − pgij then the field equations become
1 f (R, T )Rij − f (R, T )gij + (gij − ∇i ∇j )fR (R, T ) 2 = 8πTij − fT (R, T )Tij − fT (R, T )θij (4)
1 Rij − gij R = 8πTij + 2f (T )Tij + 2pf (T ) + f (T ) gij 2 (11)
where
Harko et al. (2011) have discussed several aspects of this theory including FRW dust universe Adhav (2012) has obtained Bianchi type-I cosmological model in f (R, T ) gravity. Reddy et al. (2012a, 2012b) have discussed Bianchi type-III and Kaluza-Klein cosmological models in f (R, T ) gravity while Reddy and Shantikumar (2013a, 2013b) studied some anisotropic cosmological models and Bianchi type-III dark energy model, respectively, in f (R, T ) gravity. Recently, Rao and Neelima (2013) have obtained Bianchi type-VI0 perfect fluid model in this theory. Bulk viscosity is very important in cosmology since it has a greater role in getting accelerated expansion of the universe popularly known as inflationary phase. There are many circumstances in the evolution of the universe in which bulk viscosity could arise (Ellis 1971). When neutrinos decouple from the cosmic fluid (Misner 1968), at the time of formation of galaxies and during particle creation in the early universe (Hu 1983) viscosity arises. Bulk viscous cosmological models in general relativity have been discussed by several authors (Barrow 1986; Padmanabhan and Chitre 1987; Pavon et al. 1991; Maartens et al. 1995; Lima et al. 1993; Roy and Tiwari 1983; Mohanthy and Pradhan 1992; Mohanty and Pattanaik 1991; Singh and Shreeram 1996; Singh 2005). Also, Wang (2004, 2005, 2006) Bali and Dave (2002), Bali and Pradhan (2007), Tripathy et al. (2009, 2010) have studied the Bianchi type cosmological models in the presence of cosmic strings and bulk viscosity in Einstein’s theory of gravitation. Bulk viscous cosmological models have been discussed in Brans-Dicke theory of gravitation by Johri and Sudharsan (1989). Pimental (1994), Banerjee and Beesham (1996), Singh et al. (1997). Recently, Rao et al. (2011), Naidu et al. (2012), Reddy et al. (2013a, 2013b, 2013c) have investigated Bianchi typeI, LRS Bianchi type-II and Kaluza-Klein bulk viscous string cosmological models in Saez and Ballester (1986) theory and in f (R, T ) modified theory of gravitation proposed by Harko et al. (2011). Very recently Naidu et al. (2013) discussed FRW viscous fluid cosmological model in f (R, T ) gravity.
θij = −2Tij + gij Lm − 2g lk
∂ 2 Lm ∂g ij ∂g lm
(5)
) δf (R,T ) Here fR = δf (R,T = ∇ i ∇i , ∇i is the coδR , fT = δT variant derivative and Tij is the standard matter energymomentum tensor derived from the Lagrangian Lm . It may be noted that when f (R, T ) ≡ f (R) the equations (4) yield the field equations of f (R) gravity. The problem of the perfect fluids described by an energy density ρ, pressure p and four velocity ui is complicated since there is no unique definition of the matter Lagrangian. However, here, we assume that the stress energy tensor of the matter is given by
Tij = (ρ + p)ui uj − pgij
(6)
and the matter Lagrangian can be taken as Lm = −p and we have ui ∇j ui = 0,
ui ui = 1
(7)
Then with the use of Eq. (5) we obtain for the variation of stress-energy of perfect fluid the expression θij = −2Tij − pgij
(8)
Generally, the field equations also depend through the tensor θij , on the physics nature of the matter field. Hence in the case of f (R, T ) gravity depending on the nature of the matter source, we obtain several theoretical models corresponding to each choice of f (R, T ). Assuming f (R, T ) = R + 2f (T )
(9)
as a first choice where f (T ) is an arbitrary function of the trace of stress-energy tensor of matter, we get the gravitational field equations of f (R, T ) gravity from Eq. (4) as
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Immediately after the Big-Bang, it appears that spontaneous symmetry breaking in elementary particle physics has given rise to topological defects known as cosmic strings, domain walls and monopoles. Strings, which are line like structures with particles attached to them, play a vital role in cosmology since they are considered as possible seeds for galaxy formation at the early stages of evolution of the universe. Hence the study of string cosmological models, both in general relativity and in alternative theories of gravitation, are attracting more and more attention of research workers. Stachel (1980), Letelier (1983), Vilenkin et al. (1987), Krori et al. (1990), Mahanta and Mukharjee (2001) and Battacharjee and Baruah (2001) are some of the authors who have studied several aspects of string cosmological models in general relativity while Reddy (2003a, 2003b), Reddy and Naidu (2007a, 2007b), Rao et al. (2008a, 2008b) have investigated some string cosmological models in scalar-tensor theories of gravitation. In this paper, we investigate Bianchi type-V bulk viscous string cosmological model in f (R, T ) gravity. The study of spatially homogeneous and anisotropic models are important because they are generalization of the open universe in FRW cosmology and they play a key role in the discussion of large scale structure of the universe and they have been studied in general relativity to realize the picture of the universe at its early stages. It may be noted that f (R, T ) gravity is a modification of general relativity. This paper is organized as follows: explicit field equations in this theory of gravity are derived with the help of Bianchi type-V metric in the presence of bulk viscous fluid with one dimensional cosmic strings in Sect. 2. Section 3 deals with the solution of the field equations and presenting the model. In Sect. 4, some important physical and kinematical properties of the model are discussed. The last section contains some conclusions.
2 Metric and field equations We consider the spatially homogeneous and anisotropic Bianchi type-V space-time described by the line element ds 2 = −dt 2 + A2 dx 2 + e2αx B 2 dy 2 + C 2 dz2
gij ui uj = −x i xj = −1 and ui xi = 0.
(12)
(13)
f (T ) = λT ,
λ, a constant
B¨ C¨ B˙ C˙ α2 + + − 2 = −p(8π + 7μ) + λ(8π + 3μ) + μρ B C BC A (17) A¨ C˙ A˙ α2 C¨ + + − 2 = −p(8π + 7μ) + λμ + μρ C A CA A
(18)
A¨ B¨ A˙ B˙ α2 + + − 2 = −p(8π + 7μ) + λμ + μρ A B AB A
(19)
A˙ B˙ B˙ C˙ C˙ A˙ 3α 2 + + − = ρ(8π + 7μ) − 5pμ + μλ (20) A B B C C A A2 A˙ B˙ C˙ − − =0 (21) A B C where an overhead dot denotes differentiation with respect to t. Spatial volume and the scale factor for the metric (12) are, defined respectively, by 2
V 3 = ABC
(22) 1
(23)
The physical quantities of observational interest in cosmology are the expansion scalar θ , the mean anisotropy parameter Ah and shear scalar σ 2 which are defined as ˙ C˙ A B˙ + + , (24) θ = 3H = 3 A B C where H is the mean Hubble Parameter. 3Ah =
(14)
(16)
for the metric (12) take the form
and p = p − 3ζ H
(15)
Here we also consider ρ, p and λ as functions of time t only. Using co moving coordinates and Eqs. (13)–(15), the f (R, T ) gravity field Eq. (11) with the particular choice of the function (Harko et al. 2011)
a = (ABC) 3
where A, B, C are functions of cosmic time t and α is a constant. We consider the energy momentum tensor for a bulk viscous fluid containing one dimensional cosmic strings as Tij = (ρ + p )ui uj + pgij − λxi xj
where ρ is the rest energy density of the system, ζ (t) is the coefficient of bulk viscosity, 3ζ H is usually known as bulk viscous pressure, H is Hubble’s parameter ui is the four velocity of the fluid, x i is the direction of the string and λ is the string tension density. Also, ui = δ4i is a four-velocity vector which satisfies
3 Hi 2 i=1
H
,
Hi = Hi − H, i = 1, 2, 3
(25)
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2σ 2 = σ ik σik
3
Hi2 − 3H 2 = 3Ah − H 2
(26)
where ε = ε0 − β
i=1
(0 ≤ ε0 ≤ 1),
p = ε0 ρ
(36)
3 Solutions and the model
and ε0 and β are constants. Now from Eqs. (23), (31), (32) and (34) we obtain
The field Eqs. (17)–(21) reduce to the following independent equations
A = (ct + d) 1+q ,
B˙ A¨ B˙ C˙ C˙ A˙ − + − = λ(8π + μ) B A BC CA A˙ B˙ B˙ C˙ C˙ A˙ 3α 2 + + − 2 AB BC CA A = ρ(8π + 3μ) − 5pμ + μλ
1
2m
B = (ct + d) (m+1)(q+1)
2
(37)
C = (ct + d) (m+1)(1+q) (27)
(28)
C¨ C˙ A˙ A˙ B˙ B˙ − − + =0 B C CA AB
(29)
A2 = kBC
(30)
Using Eq. (32) in Eq. (29), we obtain m = 1, since 1 + q > 0. Using this value of m in Eq. (37) and by a suitable choice of coordinates and constants (i.e. taking d = 0 and c = 1) the metric (12) can be written as 2 ds 2 = −dt 2 + t (1+q) dx 2 + e2αx dy 2 + dz2
(38)
4 Some physical properties of the model where k is a constants of integration. The constant k, without loss of generality, can be chosen as unity so that we have, from Eq. (30), A2 = BC
(31)
Now Eqs. (27)–(30) are a system of four independent equations in six unknowns, A, B, C, p, ρ, and λ. Also the equations are highly non-linear. Hence to find a determinate solution we use the following physically plausible conditions: (i) The shear scalar σ 2 is proportional to scalar expansion θ so that we can take Collins et al. (1980) B =C
m
Spatial volume 3
V 3 = t 1+q
a¨ q = −a 2 = constant a˙
(33)
which admits the solution 1 1+q
θ=
3 (1 + q)t
(40)
The mean Hubble parameter H=
1 (1 + q)t
(41)
The mean anisotropy parameter Ah = 0
(42)
The shear scalar (34)
where c = 0 and d are constants of integration. This equation implies that the condition for accelerated expansion of the universe is 1 + q > 0. (iii) For a barotropic fluid, the combined effect of the proper pressure and the bulk viscous pressure can be expressed as p = p − 3ςH = ερ
(39)
Scalar of expansion
(32)
where m = 0 is a constant. (ii) Variation of Hubble’s parameter proposed by Berman (1983) that yields constant deceleration parameter models of the universe defined by
a = (ct + d)
Equation (38) represents the Bianchi type-V bulk viscous string cosmological model in f (R, T ) modified theory of gravitation with the following physical kinematical parameters which play a vital role in the discussion of cosmology:
(35)
σ2 = 0
(43)
The string tension density λ=0
(44)
The energy density ρ=
−2 1 3 2 (1+q) − 3α t 8π + μ(3 − 5ε) [(1 + q)t]2
(45)
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The pressure
References
p=
−2 3 ε0 − 3α 2 t (1+q) 2 8π + μ(3 − 5ε) [(1 + q)t]
(46)
The coefficient of bulk viscosity ζ=
−(1−q) ε − ε0 3 2 (1+q) − 3α (1 + q)t 3[8π + μ(3 − 5ε) [(1 + q)t]2 (47)
The above results are useful to discuss the behavior of f (R, T ) gravity cosmological model given by Eq. (38). The result (39) shows that the model is expanding with time since 1 + q > 0. It can be observed the model has no initial singularity, i.e. at t = 0. Equation (44) gives that strings in this model do not survive. We observe from Eqs. (42) and (43) the model in this theory becomes isotropic and shear free during the evolution of the universe. That is, there is a transition from decelerated phase to accelerated phase in accordance with the observations (Caldwell et al. 2006). It can also be observed that θ, H, ρ, p, and ζ decrease with time and approach zero as t → ∞ while they all become infinitely large as t approaches zero (i.e. at the initial epoch accordance with the well known fact that bulk viscosity decreases with time and leads to inflationary model (Padmanabhan and Chitre 1987).
5 Conclusions Here we have discussed spatially homogeneous and anisotropic Bianchi type-V space-time in the presence of bulk viscous fluid with one dimensional cosmic strings in f (R, T ) gravity formulated by Harko et al. (2011) by modifying general relativity to explain the challenging problem of late time acceleration of the universe. To obtain a determinate solution of the highly non-linear field equations of this theory, we have taken the help of special law of variation for Hubble’s parameter proposed by Berman (1983). We have also used a barotropic equation of state for pressure and energy density and the bulk viscous pressure is assumed to be proportional to the energy density. It is interesting to observe that, in this case, strings in this universe do not survive. It is also interesting to note that the average anisotropy parameter vanishes so that the model does not remain anisotropic through out the evolution of the universe and the model becomes shear free. The model is expanding and does not have initial singularity. The bulk viscosity decreases with time so that we get, ultimately, inflationary model. Also, very recently, Kiran and Reddy (2013) established the non existence of spatially homogeneous anisotropic Bianchi type-III bulk viscous string cosmological model in f (R, T ) gravity. Hence one can prefer homogeneous isotropic models for discussion in this theory of gravity.
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