Bianchi Type-I Anisotropic Universe with Metric Potential in Saez-Ballester Theory of Gravitation

Bianchi Type-I cosmological model in the presence of Saez-Ballester theory gravitation is studied. An exact solution of the field equation is given by considering the cosmological model yield a metric potential included with a real number. The relation between the deceleration parameter and Hubble parameter and average scale factor is used in that cosmological model. The effect of the viscosity on the entropy of the universe is utilized by energy momentum tensor with bulk viscous terms in a conservative manner. We obtained a formula for calculating the entropy of the universe in terms of viscosity and used it to compare to the study. Also, various physical and kinematical properties have been discussed.

• Bianchi Type-I Space Time
• Saez-Ballester Theory
• Energy Momentum Tensor
• Bulk Viscosity
• Hubble and Deceleration Parameter

1. Introduction

Friedman Robertson-Walker (FRW) universe is one of the most generalizations of the flat universe. Likewise, the Bianchi Type-1 universe is one of the simple and elegant space times in an anisotropic flat universe. The directional scale factor makes the difference between FRW universe and Bianchi Type-1 universe. That is, FRW universe has the same scale factors. Bianchi Type-1 universe behaves like Kansor universe near the singularity. Since a universe is filled with matter, the initial anisotropy in Bianchi Type-1 universe rapidly dies away and evolves into an FRW universe. Several authors have investigated Bianchi Type-1 universe from different aspects due to its importance.

The present universe is moving through a phase of accelerated expansion that has been supported by a lot of work in astrophysics and cosmology based on observational evidence. The idea, the present universe is expanding with accelerated forward by the recent cosmological observations from a supernova [1] [2]: cosmic microwave background (Emb)) anisotropies [3] [4] [5] and large-scale structure [6] [7]. In the second self-creation theory relativity, Kantowski, R., Sachs, R.K. [8] studied spatially homogeneous and anisotropic FRW space time in the presence of viscous fluid. A five-dimensional Kaluza-Klein cosmological model in the presence of Bulk viscous fluid was investigated by Kumar and Reddy [9] and also by Mohammad Moksud Alam, Mohammad Amjad Hossain, Mohammad Ashraful Islam [10]. Anirudh Pradhan, Anil Kumar Vishwakarma, A. Dolgov [11] and Matjask, J. [12] studied Bianchi Type-1 cosmological model with bulk viscous barotrophic fluid with varying Λ and functional relation on Hubble parameter H with deceleration parameter where the metric potential is taken as a function of x and ∈ and the coefficient of bulk viscosity is assumed to be a power function of mass density was given by Anirudh Pradhan, Hare Rm Pandey [13]. The significant important is the scalar tensor theories of gravitation generated by [14] [15] [16]. Numerous versions of the scalar tensor theories are based on the introduction of a scalar field ∅ into the formulation of general relativity and cosmology. In Saez Ballester theory the metric is coupled with a dimensionless scalar field.

The metric [17] field equations are

G i j − w ∅ n ( ∅ , i ∅ , j − 1 2 g i j ∅ , k ∅ , k ) = − π T i j (1.1)

where ∅ satisfies the following conditions

2 ∅ n ∅ , i i + n ∅ n − 1 ∅ , k ∅ , k = 0 (1.2)

where G i j = R i j − 1 2 R g i j (1.3)

Equation (3) is called the Einstein tensor T_ij is the stress energy tensor of the matter, w and n are constant. Comma (,) and semicolon (;) denotes partial and co-variant differentiation respectively. A detailed explanation of Saez-Ballester cosmological model is formulated in the work of [18] [19] [20]. In this paper, we obtain Bianchi Type-1 cosmological model in scalar tensor theory of gravitation formulated by [18]. My paper is organized as follows: In Section 2, we derive field equations; in Section 3, we deal with the solution in the presence of Bulk viscous fluid. Section 4 includes the solution for the metric potential. Section 5 is mainly written with physical and kinematical properties. The last section contains the conclusion.

This research was motivated by the influence of the great scientist Albert Einstein who is the author of Relativity and another legendary scientist Stephen Hawking who is regarded as a brilliant theoretical physicist. His works on Black Holes and the Big Bang are the resources of research.

2. The Metric and Field Equations

We consider anisotropic Bianchi Type-1 space time metric is given by [21],

d s 2 = − d t 2 + A 2 ( t ) d x 2 + B 2 ( t ) d y 2 + C 2 ( t ) d z 2 (2.1)

where A, B, C are the directional scale factors and are the functions of cosmic time t. The Bianchi Type-1 space time becomes isotropic if the entire directional scale factor becomes equal and we get the usual FRW space time. The energy momentum tensor is as follows:

T j i = ( ρ + p ) u i u j + p g j i − ε θ ( g j i + u i u j ) (2.2)

ε is the coefficient of Bulk viscosity, θ is the expansion scalar of the cosmological model, ρ is the energy density and p is the isotropic pressure.

In the commoving coordinates

u 4 = − 1 , u 4 = 1 , u i u j = − 1 , u 2 = u 3 = 0 (2.3)

Also energy conservation equation

T , j i j = 0 (2.4)

From Equation (2.2) we can write the component connection,

g 11 = − 1 ,   g 22 = A 2 ,   g 33 = B 2 ,   g 33 = C 2

g 11 = − 1 ,   g 22 = 1 A 2 ,   g 33 = 1 B 2 ,   g 44 = 1 C 2

For this metric, using the definition of affine connection, we compute the following components,

Γ 22 1 = A A ˙ ,   Γ 33 1 = B B ˙ ,   Γ 44 1 = C C ˙ ,   Γ 21 2 = Γ 12 2 = A ˙ A ,

Γ 13 3 = Γ 31 3 = B ˙ B ,   Γ 14 4 = Γ 41 4 = C ˙ C and other components vanish.

where suffix (.) at the symbol A, B, C denotes ordinary differentiation with respect to t.

Moreover, to compute Ricci scalar, using the definition of Rici tensor, we compute R₁₁, R₂₂, R₃₃, R₄₄ as follows by [22].

R j k = ∂ ∂ x k Γ j i i − ∂ ∂ x i Γ j k i + Γ r k i Γ j i r − Γ r i i Γ j k r ,

R 11 = A ¨ A + B ¨ B + C ¨ C ,

R 22 = − A A ¨ − A A ˙ B ˙ B − A A ˙ C ˙ C ,

R 33 = − B B ¨ − B A ˙ B ˙ A − B B ˙ C ˙ C ,

R 44 = − C C ¨ − C A ˙ C ˙ A − C B ˙ C ˙ B .

Therefore the Ricci scalar for this metric by [23].

R = g i j R i j

R = g 11 R 11 + g 22 R 22 + g 33 R 33 + g 44 R 44 = − 2 ( A ¨ A + B ¨ B + C ¨ C + A ˙ B ˙ A B + B ˙ C ˙ B C + C ˙ A ˙ A C )

Using the result derived just above, we have the following set of expressions using Riemann Curvature Tensor by [24].

G 11 = R 11 − 1 2 R g 11 = − ( A ˙ B ˙ A B + B ˙ C ˙ B C + C ˙ A ˙ A C ) ,

G 22 = R 22 − 1 2 R g 22 = − A 2 ( B ¨ B + C ¨ C + B ˙ C ˙ B C ) ,

G 33 = R 33 − 1 2 R g 33 = − B 2 ( A ¨ A + C ¨ C + A ˙ C ˙ A C ) ,

G 44 = R 44 − 1 2 R g 44 = − ( A ¨ A + B ¨ B + B ˙ A ˙ B A ) .

From Equation (2.2) we get the following expressions:

T 1 1 = ( ρ + P ) u 1 u 1 + p g 1 1 − ε θ ( g 1 1 + u 1 u 1 ) = − ρ ,

T 2 2 = T 3 3 = T 4 4 = p − ε θ .

Now, T 11 = T 1 1 g 11 = − ρ ,

T 22 = T 2 2 g 22 = A 2 ( p − ε θ ) ,

T 33 = T 3 3 g 33 = B 2 ( p − ε θ ) ,

T 44 = T 4 4 g 44 = C 2 ( p − ε θ ) .

In the commoving system, the fluid Equation (1.1) for the metric (2.1) with the help of energy momentum tensor (2.4) can be explicitly written as,

A ˙ B ˙ A B + B ˙ C ˙ B C + C ˙ A ˙ A C − w ∅ n 2 φ ˙ = 8 π ρ (2.5)

B ¨ B + C ¨ C + B ˙ C ˙ B C − w ∅ n 2 φ ˙ = − 8 π ( p − ε θ ) (2.6)

A ¨ A + C ¨ C + A ˙ C ˙ A C − w ∅ n 2 φ ˙ = − 8 π ( p − ε θ ) (2.7)

A ¨ A + B ¨ B + B ˙ A ˙ B A − w ∅ n 2 φ ˙ = − 8 π ( p − ε θ ) (2.8)

From Equation (2.4) we can write,

T , j i j + Γ k j i T k j + Γ k j j T i k = 0 (2.9)

or, − ρ ˙ + ( A ˙ A + B ˙ B + C ˙ C ) ( p − ε θ ) − ρ ( A ˙ A + B ˙ B + C ˙ C ) = 0 .

Hence, − ρ ˙ + ( p − ε θ − ρ ) ( A ˙ A + B ˙ B + C ˙ C ) = 0 (2.10)

3. Solution of the Field Equations

To obtain the expression of Saez-Ballester scalar field ∅ , we obtain the equation from (1.2).

∅ n ∅ , i i + n 2 ∅ n − 1 ∅ , , k ∅ , k = 0 (3.1)

or, φ ¨ φ + ( A ˙ A + B ˙ B + C ˙ C ) + n 2 φ φ ˙ = 0

∴ φ ˙ ( A B C ) n 2 = k (3.2)

where, k is the integrating constant. We denote the average scale factor of the Bianchi-I universe by a(t) which is given by:

a ( t ) = ( A B C ) 1 3 = V

or, a 3 = A B C

∴ a ˙ a = 1 3 ( A ˙ A + B ˙ B + C ˙ C )

So, H = a ˙ a = 1 3 ( A ˙ A + B ˙ B + C ˙ C ) = 1 3 ( H 1 + H 2 + H 3 )

where, ( H 1 = A ˙ A , H 2 = B ˙ B , H 3 = C ˙ C ) are the Hubble Parameters in different and usual direction by [25].

So, Equation (3.2) can be written as,

φ ˙ ( a 3 ) n 2 = k (3.3)

∴ φ = k 1 exp [ k ∫ ​ d t ( a 3 ) n 2 ] (3.4)

where, k₁ is the integrating constant.

Equation (3.3) can be writing,

φ ˙ = k ( a 3 ) n 2

or, log φ = ∫ k ( a 3 ) − n 2 d t + log k 1

∴ φ = k 1 exp [ k ∫ ( a 3 ) − n 2 d t ] (3.5)

where, k₁ is the integrating constant.

Equation (2.10) can be written

− ρ ˙ + 3 ( p − ε θ − ρ ) H = 0

or, − ρ ˙ + ( p − ε θ − ρ ) a a = 0

or, − ρ ˙ ρ + ( p − ϵ θ ρ − 1 ) a a = 0

or, log ρ + log k 2 = ( p − ϵ θ ρ − 1 ) log a

∴ a ( p − ϵ θ ρ − 1 ) = k 2 β (3.6)

where, β = 1 ρ ,

In terms of the Hubble parameter in the axial direction the Equation (2.5-2.8) can be expressed by [26].

H 1 H 2 + H 2 H 3 + H 3 H 1 − w ∅ n 2 φ ˙ = 8 π ρ (3.7)

H ˙ 2 + H ˙ 3 + H 2 2 + H 3 2 + H 2 H 3 − w ∅ n 2 φ ˙ = − 8 π ( P − ε θ ) (3.8)

H ˙ 1 + H ˙ 3 + H 3 2 + H 1 2 + H 1 H 3 − w ∅ n 2 φ ˙ = − 8 π ( P − ε θ ) (3.9)

H ˙ 1 + H ˙ 3 + H 2 2 + H 1 2 + H 2 H 1 − w ∅ n 2 φ ˙ = − 8 π ( P − ε θ ) (3.10)

Subtract Equation (2.6) from (2.7),

A ¨ A + A ˙ C ˙ A C − B ¨ B − B ˙ C ˙ B C = 0 (3.11)

Again subtract Equation (2.7) from (2.8),

B ¨ B + A ˙ B ˙ A B − C ¨ C − A ˙ C ˙ A C = 0 (3.12)

Now by adding Equation (3.11) and (3.12),

A ¨ A + A ˙ B ˙ A B − C ¨ C − B ˙ C ˙ B C = 0

or, B ˙ B = − C A ¨ − A C ¨ A ˙ C − C ˙ A

By integrating,

C A ˙ − A C ˙ = k 3 B , where k₃ is the integrating constant.

∴ A ˙ A − C ˙ C = k 3 V , where, V = ABC.

Again integrating both sides

A C = k 3 ∫ ​ d t V + k 4 (3.13).

∴ A C = k 4 exp [ k 3 ∫ ​ d t V ] (3.14)

Similarly we can write,

B C = k 6 exp [ k 5 ∫ ​ d t V ] (3.15)

And A B = k 8 exp [ k 7 ∫ ​ d t V ] (3.16)

where, where k₄, k₅, k₆, k₇,k₈ are the integrating constant.

4. Solution of Cosmological Model with Metric Potential

We assume a relation in metric potential by [27].

A = B m (4.1)

A ˙ A = m B ˙ B and A ¨ A = m [ B ¨ B + m − 1 B 2 B ˙ 2 ] .

Equation (3.16) can be written as,

B = [ k 8 exp [ k 7 ∫ ​ d t V ] ] 1 1 − m (4.2)

We have,

H = a ˙ a = 1 3 ( A ˙ A + B ˙ B + C ˙ C )

or, a 3 = A B C = V

or, a ˙ a = 1 3 [ ( m + 1 ) B ˙ B + C ˙ C ]

or, a 3 = k 9 B m + 1 c

or, A = k 9 B m

∴ A = k 10 [ exp [ k 7 ∫ ​ d t V ] ] m 1 − m (4.3)

where, k 10 = k 9 ( k 8 ) m m + 1 is a constant.

Again,

a 3 = A B C

or, C = a 3 A B C = a 3 B m + 1

or, C = a 3 [ k 8 exp [ k 7 ∫ ​ d t V ] ] m + 1 1 − m

∴ C = V k 11 [ exp [ k 7 ∫ ​ d t V ] ] m + 1 1 − m (4.4)

where, k 11 = k 9 ( k 8 ) m m + 1 is a constant.

Now, a 3 = A B C = B m + 1

a = V 1 3 k 12 [ exp [ k 7 ∫ ​ d t V ] ] 2 ( 1 + m ) 3 ( 1 − m ) (4.5)

5. Physical and Kinematical Properties

Shear Scalar,

σ 2 = 1 2 σ i j σ i j = 1 2 ∑ i = 1 3 ( H i − H ) 2 = 1 2 ( H 1 2 + H 2 2 + H 3 2 − 3 H 2 )

σ = 1 2 ( H 1 2 + H 2 2 + H 3 2 − 3 H 2 ) (5.1)

Deceleration Parameter,

q = − a ¨ a H 2 = − 1 − H ˙ H 2 (5.2)

Anisotropic Pressure,

A = 1 3 ( ∇ H i H 2 ) 2 = 1 3 [ H 1 2 + H 2 2 + H 3 2 − 3 H 2 H 2 ] (5.3)

Time to Time Component Field Equation.

From Equation (2.6), (5.1) we get,

H 1 2 + H 1 2 + H 1 2 − ( H 1 + H 2 + H 3 ) 2 − w ∅ n 2 φ ˙ = 8 π ρ

or, 3 H 2 = σ 2 − w ∅ n 2 φ ˙ − 4 π ρ

∴ 3 H 2 M p 2 = σ 2 M p 2 − w ∅ n 4 φ ˙ M p 2 − ρ 2 G (5.4)

We know Plank Mass by [28].

M p = 1 8 π G (5.5)

By adding Equation (3.8)-(3.10) we get plank mass equation with deceleration and Hubble parameter, Bulk Viscosity and scalar expansion by [29].

2 ( H ˙ 1 + H ˙ 2 + H 2 2 ) + 2 ( H 1 2 + H 1 2 + H 1 2 ) + ( H 1 H 2 + H 2 H 3 + H 3 H 1 − 3 w ∅ n 2 φ ˙ ) = − 24 π ( p − ϵ θ ) ,

Hence, H 2 ( 2 q − 1 ) 8 π G = σ 2 M p 2 + 3 ( p − ε θ ) G − ρ G − w φ n φ ˙ M p 2 (5.6)

6. Conclusions

In this paper, we summarize our findings throughout Sections 1 to 5.

We applied anisotropic Bianchi Type-1 space time metric in the presence of Saez-Ballester’s theory of gravitation. By using energy momentum tensor with Bulk Viscosity, energy conservation equation and commoving vector, we get a new cosmological field equation. Moreover, we have considered Hubble parameter, so we get a special solution of the field equation. Furthermore, we have used new metric potential and we determined the time-to-time component equation and Plank Mass.

The obtained result of this paper clearly defines entropy and isotropy of the universe utilizing the new Plank Mass equation with deceleration and Hubble parameter, Bulk Viscosity and scalar expansion. The significant result is that an anisotropic universe with higher anisotropy transits to a late accelerating phase before a universe with lower anisotropy. Numerically we identified that new exponent result plays an important role in identifying the nature of the universe.

In the future, this research work will help to investigate more realistic cosmology. For example, we can take the bilinear deceleration parameter in a suitable form and some other assumptions that may explain the phase transition of the universe [30] more effectively.

Acknowledgements

I am very much thankful to Professor Dr. Mohammed Ashraful Islam, Department of Mathematics, University of Chittagong, Bangladesh. I am giving more thanks to Professor Dr. Mohammed Aman Ullah, Professor and Head, Department of Mathematics, University of Chittagong, Bangladesh, for his great contribution.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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