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CHAPTER – 7*

Bianchi type-III Cosmological Model in f(R,T)

Theory of Gravity

*The work presented in this chapter is published in Astrophysics and Space Science. Volume 342, Issue-I, 249-252 (2012)

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7.1 INTRODUCTION

The most striking discovery of the modern cosmology is that the

current universe is not only expanding but also accelerating. This late

time accelerated expansion of the universe which has been confirmed by

the high red-shift supernove experiments [Riess et al. 1998, perlmutter et

al.1999;Bennet et al.(2003)]. Also, observations such as cosmic

microwave background radiation (Sprgel et al. 2003, 2007) and large

scale structure (Tegmark et al. 2004) provide an indirect evidence for the

late time accelerated expansion of the universe. In view of this it is now

believed that energy composition of universe has 4% ordinary matter,

20% dark matter and 76% dark energy. Modifications of general

relativity are attracting much attention, in recent years, to explain late

time acceleration and dark energy.

During last decade, there had been several modifications of

general relativity to provide natural gravitational alternative for dark

energy. Among the various modifications, f(R) theory of gravity is treated

most suitable due to cosmologically important f(R) models. It has been

suggested that cosmic acceleration can be achieved by replacing the

Einstein-Hilbert action of general relativity with a general function Ricci

scalar, f(R). Viable f(R) gravity models have been proposed by Nojiri and

Odintsov(2007) Multamaki and Vilja (2006,2007) and Shamir (2010)

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which show the unification of early time inflation and late time

acceleration. Very recently, Harko et al. (2011) proposed another

extension of standard general relativity, f(R,T) theory of gravity where in

the gravitational Lagrangian is given by an arbitrary function of the Ricci

scalar R and of the trace of the stress energy tensor T. It is to be noted

that the dependence from T may be induced by exotic imperfect fluids or

quantum effects. They have derived the field equations from a Hilbert-

Einstein type variational principal and also obtained the covariant

divergence of the stress-energy tensor. The f(R,T) gravity model depends

on a source term, representing the variation of the matter stress energy

tensor with respect to the metric. A general expression for this source

term is obtained as a function of the matter Lagrangian Lm so that each

choice of Lm would generate a specific set of field equations. Some

particular models corresponding to specific choices of the function f(R,T)

are also presented, they have also demonstrated the possibility of

reconstruction of arbitrary FRW cosmologies by an appropriate choice of

a function f(T). In the present model, the covariant divergence of the

stress energy tensor is non-zero. Hence, the motion of test particles is

non-geodesic and an extra acceleration due to the coupling between

matter and geometry is always present. A detailed discussion of f(R,T)

gravity has been presented in chapter 1.

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In this chapter, we investigated the perfect fluid solutions of f(R,T)

gravity in spatially homogeneous and anisotropic Bianchi type-III space-

time. The investigation of Bianchi type models in alternative or modified

theories of gravity is an interesting discussion. The anomalies found in

the cosmic microwave background (CMB) and large scale structure

observations stimulated a growing interest in anisotropic cosmological

models of the universe. Kumar and Sing (2008) investigated perfect fluid

solutions using Bianchi type-I space- time in scalar-tensor theory. Singh

and Agarwal (1991) studied some Bianchi type-III cosmological models

in scalar-tensor theory. Adhov et al. (2009) obtained an exact solution of

the vacuum Brans-Dicke (1961) field equations for a spatially

homogeneous and anisotropic metric. Paul et al. (2009) obtained FRW

models in f(R) gravity while Sharif and Shamir (2009, 2010) have studied

the solutions of Bianchi type- I and V space-times in the framework of

f(R) gravity. Recently Shamir (2010) studied the exact vacuum solutions

of Bianchi type I, III and Kantowski-Sachs space-times in the metric

version of f(R) gravity. Here we confine ourselves to Bianchi type-III

perfect fluid solutions in f(R,T) gravity. Bianchi type –III space-time has

a fundamental role in describing early stages of evolution of the universe.

The chapter is organized as follows: Explicit field equations in

f(R,T) gravity are derived using the particular form of f(T) used by Harko

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et al. (2011) with the aid of Bianchi type-III metric in the presence of

perfect fluid in section 7.2. In section 7.3, solutions of the field equations

using the special law of variation for Hubble‟s parameter proposed by

Berman (1983) are obtained. Section 7.4, deals with some physical

properties of the model and the last section contains some conclusions.

7.2 METRIC AND FIELD EQUATIONS

We consider the spatially homogeneous and anisotropic Bianchi type-III

metric given by

𝑑𝑠2 = 𝑑𝑡2 − 𝐴2 𝑡 𝑑𝑥2 − 𝑒−2𝑚𝑥𝐵2 𝑡 𝑑𝑦2 − 𝐶2 𝑡 𝑑𝑧2 (7.1)

where A,B and C are cosmic scale factors and m is a positive constant.

1

, , , 8 , ,2

2ij ij ij i j R ij T ij T ijf R T R f R T g g f R T T f R T T f R T

In chapter 1, We have seen that the field equations of f(R,T) gravity can

be written as

𝑓𝑅 𝑅,𝑇 𝑅𝑖𝑗 −1

2𝑓 𝑅,𝑇 𝑔𝑖𝑗 + 𝑔𝑖𝑗▭− ∇𝑖∇𝑗 𝑓𝑅 𝑅,𝑇 = 8𝜋𝑇𝑖𝑗 − 𝑓𝑇 𝑅,𝑇 𝑇𝑖𝑗 − 𝑓𝑇 𝑅,𝑇 𝜃𝑖𝑗 (7.2)

Where 𝜃𝑖𝑗 = −2 𝑇𝑖𝑗 + 𝑔𝑖𝑗 𝐿𝑚 − 2𝑔𝑙𝑘 𝜕

2𝐿𝑚

𝜕𝑔 𝑖𝑗 𝜕𝑔 𝑙𝑚 (7.3)

Here 𝑓𝑅 =𝛿𝑓(𝑅,𝑇)

𝛿𝑅, 𝑓𝑇 =

𝛿𝑓(𝑅,𝑇)

𝛿𝑇

▭ = ∇𝑖∇𝑖 , ∇𝑖 is the covariant derivative and𝑇𝑖𝑗 is the standard matter

energy-momentum tensor derived from the Lagrangian Lm. It may be

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noted that when f(R,T) ≡f(R) the equations (7.2) 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 𝑢𝑖𝑢𝑗 − 𝑝𝑔𝑖𝑗 (7.4)

And the matter Lagrangian can be taken as Lm = -p and we have

𝑢 𝑖∇j𝑢𝑖 = 0, 𝑢 𝑖𝑢𝑖 = 1 (7.5)

Then with the use of Eq.(7.3) we obtain for the variation of stress-energy

of perfect fluid the expression

𝜃𝑖𝑗 = −2𝑇𝑖𝑗 − 𝑝𝑔𝑖𝑗 (7.6)

Generally, the field equations also depend through the tensor 𝜃𝑖𝑗 , on the

physicsl 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) (7.7)

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as the 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. (7.2) as

𝑅𝑖𝑗 −1

2𝑅𝑔𝑖𝑗 = 8𝜋𝑇𝑖𝑗 − 2𝑓

′ 𝑇 𝑇𝑖𝑗 − 2𝑓′ 𝑇 𝜃𝑖𝑗 + 𝑓(𝑇) 𝑔𝑖𝑗 (7.8)

where the prime denotes differentiation with respect to the argument.

If the matter source is a perfect fluid,

𝜃𝑖𝑗 = −2𝑇𝑖𝑗 − 𝑝𝑔𝑖𝑗 (7.9)

then the field equations become

𝑅𝑖𝑗 −1

2𝑔𝑖𝑗𝑅 = 8𝜋𝑇𝑖𝑗 + 2𝑓

′ 𝑇 𝑇𝑖𝑗 + [2𝑝𝑓′ 𝑇 + 𝑓(𝑇)] 𝑔𝑖𝑗 (7.10)

Using co-moving coordinates and the equations (7.4)-(7.6), the f(R,T)

gravity field equations, with the particular choice of the function (Harko

et al. 2011)

𝑓 𝑇 = 𝜆𝑇, 𝜆 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (7.11)

for the metric (7.1), take the form

𝐵

𝐵+

𝐶

𝐶+

𝐵 𝐶

𝐵𝐶= 8𝜋 + 3𝜆 𝑝 − 𝜆𝜌 (7.12)

𝐴

𝐴+

𝐶

𝐶+

𝐴 𝐶

𝐴𝐶= 8𝜋 + 3𝜆 𝑝 − 𝜆𝜌 (7.13)

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𝐴

𝐴+

𝐵

𝐵+

𝐴 𝐵

𝐴𝐵−

𝑚2

𝐴2= 8𝜋 + 3𝜆 𝑝 − 𝜆𝜌 (7.14)

𝐴 𝐵

𝐴𝐵+

𝐴 𝐶

𝐴𝐶+

𝐵 𝐶

𝐵𝐶−

𝑚2

𝐴2= − 8𝜋 + 3𝜆 𝜌 + 𝜆𝑝 (7.15)

𝐴

𝐴−

𝐵

𝐵= 0 (7.16)

where an overhead dot denotes differentiation with respect to t.

Integrating eq. (7.16), we obtain

𝐵 = 𝑐1𝐴 (7.17)

where c1is an integration constant. Without any loss of generality, we

take c1=1, so that we have

𝐵 = 𝐴 (7.18)

Using Eq. (7.18), the field equations (7.12)-(7.15) will reduce to

𝐴

𝐴+

𝐶

𝐶+

𝐴 𝐶

𝐴𝐶= 8𝜋 + 3𝜆 𝑝 − 𝜆𝜌 (7.19)

2𝐴

𝐴+

𝐴

𝐴

2

− 𝑚

𝐴

2= 8𝜋 + 3𝜆 𝑝 − 𝜆𝜌 (7.20)

𝐴

𝐴

2

+ 2𝐴 𝐶

𝐴𝐶−

𝑚

𝐴

2= − 8𝜋 + 3𝜆 𝜌 + 𝜆𝑝 (7.21)

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7.3 SOLUTIONS AND THE MODEL

The field equations (7.19)-(7.21) are the system of three

independent equations in four unknowns A, C, p and ρ. Hence to obtain

determinate solution of the system we take the help of special law of

variation proposed by Bermann (1983) for Hubble‟s parameter that yields

constant deceleration parameter models of the universe. Here , it may be

noted, that most of the well known models of Einstein‟s theory and

Brans-Dicke (1961) theory including inflationary models are models with

constant deceleration parameter.

We consider constant deceleration parameter model defined by

𝑞 = −𝑎𝑎

𝑎 2= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (7.22)

where a is the scale factor of the universe. Here the constant is taken as

negative since f(R,T) gravity is about accelerated expansion of the

universe. The solution of (7.22), in view of (7.18), is

𝑎 = 𝐴2𝐶 1

3 = (𝑎𝑡 + 𝑑)1

1+𝑞 (7.23)

where𝑐 ≠ 0 and d are constants of integration. This equation implies that

the condition of accelerated expansion is 1+q>0. The average Hubble‟s

parameter H is given by

𝐻 =1

3 𝐻1 + 𝐻2 + 𝐻3 (7.24)

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where𝐻1 =𝐴

𝐴= 𝐻2, 𝐻3 =

𝐶

𝐶 are the directional Hubble parameters in

the directions of x, y and z axes respectively. Using Eqs.(7.23) and (7.24),

we obtain

𝐻 =𝑎

𝑎 (7.25)

The expansion scalar θ and the shear scalar 𝜍2 are defined as

𝜃 = 𝑢;𝑖𝑖 = 2

𝐴

𝐴+

𝐶

𝐶 (7.26)

𝜍2 =1

2𝜍𝑖𝑗𝜍

𝑖𝑗 =1

3 𝐴

𝐴−

𝐶

𝐶

2

(7.27)

where 𝜍𝑖𝑗 =1

2 𝑢𝑖;𝑘𝑗

𝑘 + 𝑢𝑗 ;𝑘𝑖𝑘 −

1

3𝜃𝑖𝑗

and𝑖𝑗 = 𝑔𝑖𝑗 − 𝑢𝑖𝑢𝑗 is the projection tensor 𝑢𝑖 = 𝑔00 (1,0,0,0) is the

four-velocity in comoving coordinates.

The mean anisotropic parameter is

𝐴𝛼 =1

3

∆𝐻𝑙

𝐻

2 (7.28)

where∆𝐻𝑙 = 𝐻𝑙 − 𝐻 (𝑙 = 1,2,3)

Since the field equations (7.19)-(7.21) are highly non-linear we use

Eq.(7.23) and a physical condition that expansion scalar θis proportional

to shear scalar ζ which gives

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𝐴 = 𝐶𝑛 , 𝑛 ≠ 1 (7.29)

Using this condition we solve the field equations (7.19)-(7.21) and obtain

metric coefficients as

𝐴 = 𝐵 = 𝑐𝑡 + 𝑑 3𝑛

1+𝑞 2𝑛+1 , 𝐶 = 𝑐𝑡 + 𝑑 3 1+𝑞 (2𝑛+1) (7.30)

By a suitable choice of coordinates and constants (i.e., c=1,d=0) the

metric (7.1) with the help of equation (7.30) can, now, be written as

𝑑𝑠2 = 𝑑𝑡2 − 𝑡6𝑛

1+𝑞 (2𝑛+1) 𝑑𝑥2 + 𝑒−2𝑚𝑥 𝑑𝑦2 − 𝑡6 1+𝑞 (2𝑛+1) 𝑑𝑧2 (7.31)

7.4 PHYSICAL PROPERTIES OF THE MODEL

Eq. (7.31) represents Bianchi type-III cosmological model, with

perfect fluid source, in f(R,T) gravity which is physically significant for

the discussion of the early stages of evolution of the universe.

The physical and kinematical parameters which are important to

discuss the physics of the cosmological model (7.31) are

The Spatial volume is

𝑉3 = 𝐴2𝐶 = 𝑡3

1+𝑞 (7.32)

This shows the late - time accelerated expansion of the universe since

1+q>0.

The Hubble‟s parameter H is

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𝐻 = 1

1+𝑞

1

𝑡 (7.33)

The scalar of expansion θ is

𝜃 = 3

1+𝑞

1

𝑡 (7.34)

The shear acalar 𝜍2 is

𝜍2 =3

2

1

(1+𝑞)21

𝑡2 (7.35)

The mean anisotropy parameter 𝐴𝛼 is

𝐴𝛼 =4

3 (7.36)

The physical parameters, energy density 𝜌 and pressure p in the model

are

𝜌 =6𝑛𝜆

(1+𝑞)2(2𝑛+1)2(8𝜋+2𝜆)(8𝜋+4𝜆) 3𝑛 − 1 + 𝑞 2𝑛 + 1 −

3𝑛 (4𝜋+𝜆 )

𝜆−

3(8𝜋+3𝜆)

𝜆

1

𝑡2+

𝑚2

8𝜋+4𝜆

1

𝑡6𝑛

1+𝑞 (2𝑛+1) (7.37)

𝑝 =6𝑛

(1+𝑞)2(2𝑛+1)2(8𝜋+3𝜆) 3𝑛 − 1 + 𝑞 2𝑛 + 1 +

3𝑛

2

1

𝑡2−

𝑚2

8𝜋+3𝜆

1

𝑡6𝑛

1+𝑞 2𝑛+1 +

6𝑛𝜆2

(1 + 𝑞)2(2𝑛 + 1)2(8𝜋 + 2𝜆)(8𝜋 + 4𝜆) 3𝑛 − 1 + 𝑞 2𝑛 + 1 −

3𝑛(4𝜋 + 𝜆)

𝜆−

3(8𝜋 + 3𝜆)

𝜆

1

𝑡2+

𝑚2𝜆

8𝜋+4𝜆

1

𝑡6𝑛

1+𝑞 2𝑛+1 (7.38)

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It may be observed that the model (7.38) has no initial singularity, ie at

t=0. The kinematical parameters H, 𝜃 and ζ diverge at initial epoch while

they vanish for large t. the physical parameters ρ and p diverge at the

initial epoch and they vanish for large values of t. However, the volume

scale factor increases with time showing the late time acceleration of the

universe.

7.5 CONCLUSIONS

The modified theories of gravity, especially the f(R) and f(R,T)

have attracted much attention in the last decade. Hence, we have studied

Bianchi type-III cosmological model in the presence of perfect fluid in

f(R,T) theory of gravity. It is observed that the model has no initial

singularity and shows the late time accelerated expansion of the universe

for large t. It is also observed that all the physical parameters diverge at

initial epoch while they approach zero for large t. The model obtained, it

is believed, throws some light on our understanding of f(R,T) cosmology.