Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Deceleration Parameter in Saez-Ballester Theory

IOSR Journal of Applied Physics (IOSR-JAP)
e-ISSN: 2278-4861.Volume 4, Issue 5 (Sep. - Oct. 2013), PP 79-84
www.iosrjournals.org
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Anisotropic Bianchi Type-III Dark Energy Model with Time-
Dependent Deceleration Parameter in Saez-Ballester Theory
Md. Azizur Rahman1
, M. Ansari2
Department of Mathematics, North-Eastern Hill University, Permanent Campus, Shillong-793022, Meghalaya,
India.
Abstract: An anisotropic Bianchi type-III cosmological model is investigated in a Saez-Ballester scalar-tensor
theory of gravitation. Three different time-dependent skewness parameters along spatial directions are
introduced to represent the deviation of pressure from isotropy. To get deterministic solutions of the field
equations, we choose variation law of scale factor 𝑆 = (𝑡 𝑟
𝑒 𝑡
)
1
𝑙 which yields a time-dependent deceleration
parameter (DP) representing a model that generates a transition of the universe from the early decelerating
phase to the present accelerating phase. Some physical and geometrical properties of the model are also
discussed.
Keywords: Bianchi type-III model, Dark energy, Scalar-tensor theory, Variable deceleration parameter
I. Introduction
In recent years, several modifications of Einstein’s theory of gravitation have been developed. Among
them scalar-tensor theories proposed by Brans and Dicke [1], Saez and Ballester [2], Nordtvedt [3], Wagoner
[4], Ross [5], Dunn [6], Barber [7] are important. Saez and Ballester [2] developed a scalar-tensor theory in
which the metric is coupled with a dimensionless scalar field in a simple manner. This coupling gives a
satisfactory description of the weak fields. Inspite of the dimensionless character of the scalar field, an anti-
gravity regime appears. This theory also suggests a possible way to solve the missing-matter problem in non-flat
FRW cosmologies.
Cosmological models in the Saez-Ballester scalar-tensor theory of gravitation have been studied by
Singh and Agrawal [8], Ram and Tiwari [9], Singh and Ram [10], Mohanty and Sahu [11], Tripathi et al. [12],
Reddy et al. [13], Rao et al. [14], Ram et al. [15] and many. Recently, Naidu et al. [16] obtained Bianchi type-III
cosmological model in Saez-Ballester theory considering a variation law for Hubble’s parameter which yields a
constant value of deceleration parameter. In literature it is common to use a constant deceleration parameter as it
duly gives a power law for metric function or corresponding quantity. But for a universe which was decelerating
in the past and accelerating at present, the DP must show signature flipping [17-19]. So, in general, the DP is not
a constant but time variable. Recently, Amirhashchi et al. [20], Pradhan et al. [21-23] and Yadav [24]
investigated Bianchi type cosmological models with time-dependent deceleration parameter.
Motivated by the above works, in the present paper, we have examined an anisotropic Bianchi type-III
dark energy model with variable EoS parameter in the framework of Saez-Ballester scalar-tensor theory of
gravitation with the help of a variation law of scale factor [20-24] which yields a time-dependent deceleration
parameter.
The field equations given by Saez and Ballester [2] for the combined scalar and tensor field
𝑅𝑖𝑗 −
1
2
𝑔𝑖𝑗 𝑅 − 𝑤∅ 𝑛
∅,𝑖∅,𝑗 −
1
2
𝑔𝑖𝑗 ∅,𝑘∅,𝑘
= −𝑇𝑖𝑗
𝑖
(1)
where 𝑤 is a dimensionless coupling constant and 𝑛 is an arbitrary constant.
The scalar field satisfies the equation
2∅ 𝑛
∅;𝑖
𝑖
+ 𝑛∅ 𝑛−1
∅,𝑘 ∅′𝑘
= 0 (2)
It can be easily proved that
𝑇;𝑗
𝑖𝑗
= 0 (3)
which are the consequences of the field equations. 𝑇;𝑗
𝑖𝑗
= 0 is the energy momentum tensor of matter. Here
comma and semi-colon designate partial and co-variant differentiation respectively with respect to cosmic time
𝑡.
II. The Metric And Field Equations
We consider the line element for spatially homogeneous and anisotropic Bianchi type-III space-time as
𝑑𝑠2
= 𝑑𝑡2
− 𝐴2
𝑑𝑥2
− 𝑒−2𝛼𝑥
𝐵2
𝑑𝑦2
− 𝐶2
𝑑𝑧2
(4)
where 𝐴, 𝐵 and 𝐶 are functions of 𝑡 alone and 𝛼 is a positive constant.

Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Deceleration Parameter in
The energy momentum tensor of fluid is defined as
𝑇𝑖
𝑗
= 𝑑𝑖𝑎𝑔 𝑇0
0
, 𝑇1
1
, 𝑇2
2
, 𝑇3
3
(5)
We can parametrize the energy momentum tensor given in (5) as follows :
𝑇𝑖
𝑗
= 𝑑𝑖𝑎𝑔 𝜌, −𝑝𝑥 , −𝑝𝑦 , −𝑝𝑧
= 𝑑𝑖𝑎𝑔 1, −𝜔𝑥 , −𝜔𝑦 , −𝜔𝑧 𝜌
= 𝑑𝑖𝑎𝑔 1, −𝜔, − 𝜔 + 𝛾 , − 𝜔 + 𝛿 𝜌 (6)
where 𝜌 is the energy density of the fluid, 𝑝𝑥 , 𝑝𝑦 , 𝑝𝑧 are the pressures and 𝜔𝑥 , 𝜔𝑦 , 𝜔𝑧 are the directional EoS
parameters along 𝑥, 𝑦 and 𝑧 axes respectively ; 𝜔 𝑡 =
𝑝
𝜌
is the deviation-free EoS parameter of the fluid. The
deviation from isotropy is parametrized by setting 𝜔𝑥 = 𝜔 and then introducing skewness parameters 𝛾 and 𝛿
which are the deviations from 𝜔, respectively along the 𝑦 and 𝑧 axes.
The Saez-Ballester field equations (1)-(3) for the metric (4) with the help of (6) lead to the following system
of equations
𝐴 𝐵
𝐴𝐵
+
𝐴 𝐶
𝐴𝐶
+
𝐵 𝐶
𝐵𝐶
−
𝛼2
𝐴2
+
𝑤
2
∅ 𝑛
∅2
= 𝜌 (7)
𝐵
𝐵
+
𝐶
𝐶
+
𝐵 𝐶
𝐵𝐶
−
𝑤
2
∅ 𝑛
∅2
= −𝜔𝜌 (8)
𝐴
𝐴
+
𝐶
𝐶
+
𝐴 𝐶
𝐴𝐶
−
𝑤
2
∅ 𝑛
∅2
= − 𝜔 + 𝛾 𝜌 (9)
𝐴
𝐴
+
𝐵
𝐵
+
𝐴 𝐵
𝐴𝐵
−
𝛼2
𝐴2
− 𝑤∅ 𝑛
∅2
= − 𝜔 + 𝛿 𝜌 (10)
𝐴
𝐴
−
𝐵
𝐵
= 0 (11)
∅ + ∅
𝐴
𝐴
+
𝐵
𝐵
+
𝐶
𝐶
+
𝑛
2
∅2
∅
= 0 (12)
Here, and also in what follows, an overhead dot designates ordinary differentiation with respect to cosmic time
𝑡. Integrating equation (11), we obtain
𝐵 = 𝑐1 𝐴 (13)
where 𝑐1 is a constant of integration which can be taken as unity without loss of generality so that
𝐵 = 𝐴 (14)
The average scale factor 𝑆 and the spatial volume 𝑉 are given by
𝑆 = 𝐴2
𝐶
1
3 (15)
𝑉 = 𝑆3
= 𝐴2
𝐶 (16)
The generalized mean Hubble’s parameter 𝐻 is defined as
𝐻 =
𝑆
𝑆
=
1
3
𝐻1 + 𝐻2 + 𝐻3 (17)
where 𝐻1 = 𝐻2 =
𝐴
𝐴
, 𝐻3 =
𝐶
𝐶
are the directional Hubble’s parameters in the directions of 𝑥, 𝑦 and 𝑧 axes
respectively.
The deceleration parameter 𝑞 is defined by
𝑞 = −
𝑆𝑆
𝑆2
(18)
The expansion scalar 𝜃 , shear scalar 𝜎2
and the mean anisotropy parameter 𝐴 𝑚 are given by
𝜃 = 𝑢;𝑖
𝑖
= 2
𝐴
𝐴
+
𝐶
𝐶
(19)
𝜎2
=
1
2
𝜎𝑖𝑗 𝜎 𝑖𝑗
=
1
3
𝐴
𝐴
−
𝐶
𝐶
2
(20)
𝐴 𝑚 =
1
3
∆𝐻𝑖
𝐻
2
(21)
where ∆𝐻𝑖 = 𝐻𝑖 − 𝐻 𝑖 = 1, 2, 3

III. Solutions And The Model
The field equations (7)-(10) and (12) are a system of five independent equations with seven unknown
parameters 𝐴 = 𝐵, 𝐶, 𝜌, 𝜔, 𝛾, 𝛿 and ∅. Hence two additional constraints are required to obtain explicit
solutions of these field equations.
Since the metric (4) is completely characterized by average scale factor therefore we consider
following Amirhashchi et al. [20], Pradhan et al. [21-23] and Yadav [24] that average scale factor is an
increasing function of time as following:
𝑆 = 𝑡 𝑟
𝑒 𝑡
1
𝑙 (22)
where 𝑟 and 𝑙 are positive constants.
We also assume that the shear scalar 𝜎 is proportional to the expansion scalar 𝜃 which gives
𝐴 = 𝐶 𝑚
(23)
where 𝑚 is a constant. The motive behind assuming this condition is explained with reference to Thorne (1967)
[25], the observations of the velocity red-shift relation for extragalactic sources suggest that Hubble expansion
of the universe is isotropy today within ≈ 30 percent (Kantowski and Sachs [26]; Kristian and Sachs [27]). To
put more precisely, red-shift studies place the limit
𝜎
𝐻
≤ 0.3
on the ratio of shear 𝜎 to Hubble constant 𝐻 in the neighbourhood of our galaxy today. Collin et al. (1980) [28]
have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous expansion
satisfies that the condition
𝜎
𝜃
is constant.
Solving the field equations (7)-(10) with the help of (15), (22) and (23), we obtain
𝐴 = 𝐵 = 𝑡 𝑟
𝑒 𝑡
3𝑚
𝑙(2𝑚+1) (24)
𝐶 = 𝑡 𝑟
𝑒 𝑡
3
𝑙(2𝑚+1) (25)
Hence the metric (4) of the universe takes the form
𝑑𝑠2
= 𝑑𝑡2
− 𝑡 𝑟
𝑒 𝑡
6𝑚
𝑙(2𝑚+1) 𝑑𝑥2
− 𝑒−2𝛼𝑥
𝑡 𝑟
𝑒 𝑡
6𝑚
𝑙(2𝑚+1) 𝑑𝑦2
− 𝑡 𝑟
𝑒 𝑡
6
𝑙(2𝑚+1) 𝑑𝑧2
(26)
IV. Physical And Geometrical Properties Of The Model
Equation (26) represents Bianchi type-III dark energy model in Saez-Ballester theory of gravitation. At
𝑡 = 0, all the spatial scale factors are zero which implies that the derived model has a point-type singularity at
𝑡 = 0 [29].
Equation (22) with the help of equation (18) yields a time-dependent deceleration parameter as
𝑞 =
𝑙𝑟
𝑡 + 𝑟 2
− 1 (27)
We observe that 𝑞 > 0 for 𝑡 < 𝑙𝑟 − 𝑟 and 𝑞 < 0 for 𝑡 > 𝑙𝑟 − 𝑟. For 𝑙 ≥ 3 and 𝑟 = 1, our model evolves
from decelerating phase to accelerating phase. Recent observations of type-Ia supernovae [30, 31] suggests that
the present universe is accelerating and the value of DP lies in the range −1 < 𝑞 < 0. So, our derived model is
consistent with the recent observations.
Fig. 1 : The plot of DP (q) versus time (t)
The spatial volume
𝑉 = 𝑡 𝑟
𝑒 𝑡
3
𝑙 (28)
Hubble’s parameter
0 5 10 15
0.5
0.0
0.5
1.0
1.5
2.0
2.5
t
q

𝐻 =
1
𝑙
1 +
𝑟
𝑡
(29)
The expansion scalar
𝜃 =
3
𝑙
1 +
𝑟
𝑡
(30)
Shear scalar
𝜎2
=
3 𝑚 − 1 2
𝑙2 2𝑚 + 1 2
1 +
𝑟
𝑡
2
(31)
The mean anisotropy parameter
𝐴 𝑚 =
1
3
9 2𝑚2
+ 1
2𝑚 + 1 2
+ 1 (32)
Also, we have
𝜎2
𝜃2
=
𝑚 − 1 2
3 2𝑚 + 1 2
≠ 0, 𝑚 > 1 (33)
The scalar field
∅ = ∅0
𝑛 + 2
2
𝑑𝑡
𝑡 𝑟 𝑒 𝑡
3
𝑙
2
𝑛+2
(34)
In the derived model (26), the present value of DP is estimated as
𝑞0 = −1 +
𝑟
𝑙𝐻0
2
𝑡0
2 (35)
where 𝐻0 is the present value of Hubble’s parameter and 𝑡0 is the age of the universe at present epoch. If we set
𝑙 = 3 and 𝑟 = 1 in Eq. (35), we obtain 𝑞0 = −0.67 which matches the observed value of the DP at the present
epoch [32].
The energy density
𝜌 =
9𝑚(𝑚 + 2)
𝑙2 2𝑚 + 1 2
1 +
𝑟
𝑡
2
−
𝛼2
𝑡 𝑟 𝑒 𝑡
6𝑚
𝑙 2𝑚+1
+
𝑤∅0
2
2 𝑡 𝑟 𝑒 𝑡
6
𝑙
(36)
The EoS parameter
𝜔 =
1
𝜌
𝑤∅0
2
6
𝑙
−
9 𝑚2
+ 𝑚 + 1
𝑙2 2𝑚 + 1 2
1 +
𝑟
𝑡
2
+
3 𝑚 + 1
𝑙 2𝑚 + 1
𝑟
𝑡2
(37)
Skewness parameters
𝛾 = 0 (38)
and
𝛿 =
1
𝜌
𝑤∅0
2
6
𝑙
−
9 2𝑚2
− 𝑚 − 1
𝑙2 2𝑚 + 1 2
1 +
𝑟
𝑡
2
+
3 𝑚 − 1
𝑙 2𝑚 + 1
𝑟
𝑡2
+
𝛼2
𝑡 𝑟 𝑒 𝑡
6𝑚
𝑙 2𝑚+1
(39)
From the above results, we observe that at 𝑡 = 0, the spatial volume is zero and other parameters 𝜃, 𝜎, 𝐻
diverge. This implies that the universe starts evolving with zero volume at 𝑡 = 0 and expands with cosmic time
𝑡 which is a big bang scenario. At 𝑡 → ∞, we get 𝑞 = −1 and
𝑑𝐻
𝑑𝑡
= 0, which implies the greatest value of
Hubble’s parameter and fastest rate of expansion of the universe. The negative value of 𝑞 would accelerate and
increase the age of the universe. Figure 1 depicts the dynamics of DP versus cosmic time 𝑡. It can be seen that
initially the value of DP was positive and its value becomes negative later on. This clearly explains that the
universe had a decelerated expansion in the past and has an accelerating expansion at present as observed by the
recent observations of SN Ia. So, the derived model is fit to describe the dynamics of the late time evolution of
the universe.
Since
𝜎2
𝜃2 ≠ 0 except at 𝑚 = 1 , hence the model is anisotropic for all values of 𝑚 except for 𝑚 = 1.
The mean anisotropy parameter is uniform throughout the evolution of the universe.
It is important to note here that lim𝑡→0(
𝜌
𝜃2) turn out to be constant. Therefore the model approaches
homogeneity and matter is dynamically negligible near the origin. This is in good agreement with the result
already given by Collins [33].
The scalar field is found to be a decreasing function of time. At 𝑡 = 0, it becomes infinite and for
large value of time it vanishes.
The energy density 𝜌 and EoS parameter 𝜔 are both decreasing functions of time and they become
constant at large values of 𝑡. From Eqs. (36) and (37), we observe that EoS parameter 𝜔 is time-dependent, it

can be a function of redshift 𝑧 or scale factor 𝑆 as well. So, if the present work is compared with experimental
results [34-37], then one can find the limit of 𝜔 provided by Eq. (37) within acceptable range.
For different choices of 𝑙 and 𝑟, we can obtain a class of viable DE models in Bianchi type-III space-time and
such models are consistent with recent observations.
V. Conclusion
An anisotropic Bianchi type-III dark energy cosmological model is investigated in the framework of
Saez-Ballester scalar-tensor theory of gravitation. Exact solutions of the field equations are obtained by
assuming suitable physical conditions that yield a time-dependent DP representing a model which generates a
transition from early decelerating phase to the present accelerating phase of the universe. The EoS parameter
and skewness parameters appear to be time-dependent. The derived model can be a viable cosmological model
to explain the present acceleration of the universe.
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