Bianchi Types Cosmological Models with Cold Dark Matter ... · weak field as well as the antigravity regime appear in this theory, provide a suitable way to solve ‘missing matter

1

Bianchi Types Cosmological Models with Cold Dark Matter and

Anisotropic Dark Energy in Saez-Ballester Theory of Gravitation

V. G. Mete*, A. S. Bansod1 & P. B. Murade2

*Department of Mathematics, R.D.I.K&.K.D. College, Bandera- Amravati (Rly), (444701), INDIA. 1Department of Applied Science, V.Y.W.S. Polytechnic, Bandera (Rly), (444701), INDIA.

2Department of Applied Science, PRMIT&R, Badnera (Rly), (444701), INDIA. *Corresponding Author, e-mail: [email protected]

Abstract: This work is devoted to the investigation of Bianchi types-III, V, VI0 universe filled with Cold dark matter and anisotropic dark energy in scalar-tensor theory of gravitation formulated by Saez and Ballester [20]. In order to obtain the solutions of the field equations, we have considered a special form of deceleration parameter. The physical and geometrical properties of the desired models are also discussed.

Keywords: Bianchi types-III, V, VI0 metrics, anisotropic dark energy, Special form of deceleration parameter. PACS Numbers: 98.80.-k ; 98.80.Es ; 98.80.Cq ; 95.35.+d

1. Introduction

The recent researches made in cosmology have subsequently brought a new outlook in the study of the universe. The cosmological observations like Type Ia Supernovae [1, 2], Cosmic microwave background radiation [3, 4], Galaxy redshift surveys [5], Large scale structure [6] have shown that we are living in expanding and accelerating universe. The cosmological observations also show that the universe is dominated by two dark components called dark energy (DE) and dark matter (DM). Exotic energy having negative pressure known as dark energy, is responsible for explaining cosmic accelerating expansion of the universe [7]. The matter without pressure known as dark matter (DM), is responsible for universe structure formation as well as explaining galactic curves. The nature of DM and DE is still unknown. For this reason, several models are put forward such as cosmological constant, quintessence [8-10], the non-linear F(R) models [11-13], DGP branes [14-15], among many more. Recently, the alternative theories of gravitation are attracting more attention to describe late-time acceleration as well as existence of DM and dark energy components such as chaplygin gas, quintessence, tachyon and phantom [13, 16-18]. These modified theories of gravitation give the most natural generalizations of the general relativity (GR). The most accepted and best motivated theory put forward by Brans and Dicke [19], introduced a scalar field additionally to the metric tensor field

ijg . The scalar field has the dimension of inverse of the gravitational constant. The role of scalar field

in this theory confines to its effect on the gravitational field equations. Afterwards, Saez and Ballester [20] introduced a new scalar tensor theory of gravitation in which the metric is coupled with dimensionless scalar field in simple way. This coupling provides a satisfactory explanation of the weak field as well as the antigravity regime appear in this theory, provide a suitable way to solve ‘missing matter problem’ in non-flat FRW universe. The initial singularities, as well as inflationary universe also discussed by Saez [21]. Many authors have discussed the cosmological model in this theory [22-30]. The study of an anisotropic, as well as spatially homogeneous models, play a vital role in understanding the early stage evolution of the universe. The Bianchi type models are anisotropic as well as spatially homogeneous. The anisotropy of the dark energy within the framework of the Bianchi type models has found to be beneficial in generating arbitrary ellipsoidality to the universe as well as fine tune the observed CMBR anisotropies. Akarsu and Kilinc [31, 32] have studied

ISSN NO: 0776-3808

http://aegaeum.com/ Page No: 992

AEGAEUM JOURNAL

Volume 8, Issue 3, 2020

2

anisotropic dark energy model using the Bianchi models. Many authors have studied anisotropic dark energy cosmological model in general relativity and alternative theories of gravitation [33-42]. Dark energy models such as holographic dark energy cosmological model were studied using alternative theories of gravitation and general relativity by many authors with the help of anisotropic Bianchi type metrics [43-47].Adhav et al. [48] investigated Kaluza-Klein interacting cosmic fluid cosmological model. Asgar and Ansari [49] studied accelerating Bianchi type-VI0 bulk viscous cosmological model in Lyra geometry. Rao et al. [50] studied string cosmological models with bulk viscosity in Nordtvedt’s general scalar tensor theory of gravitation. Ghost et al. [51] studied behavior of interacting Ricci dark energy in logarithmic f(T) gravity. Mishra and Sahoo [52] investigated Kinc space-time in scalar invariant theory with wet dark fluid. The equation of state (EoS) parameter of dark energy

/p is a function of time t, where

p

is pressure and

is energy density, characterize the models of dark energy. In which p is a

vectorial quantity, is a scalar quantity and as a result, the EoS parameter of DE can be determined

separately on each spatial axis with the conservation of energy-momentum tensor. We assume a phenomenological parameterization of anisotropic dark energy in terms of its time dependent

deviation free EoS parameter ( ) along x-axis and two skewness parameter ( y , z ) along y and z

axes respectively. So, in this paper, we consider anisotropic and spatially homogeneous Bianchi types-III, V, VI0 metric with anisotropic dark energy and cold dark matter (CDM) in Saez-Ballester theory of gravitation. To obtain the solutions of field equations, we consider a special form of deceleration parameter. Physical and geometrical properties of the desired models are discussed in detail.

2. Metric and Field Equations

The Bianchi types- III, V, VI0 metric can be written as [53] 2222222222 21 dzCedyBedxAdtds xmxm , (1)

where, Bianchi type-III metric if 02 m

Bianchi type-V metric if 21 mm

Bianchi type-VI0 metric if 21 mm

and A , B and C are the cosmic scale factors and functions of the cosmic time t only and 1m , 2m are

arbitrary constants. The field equations in scalar tensor theory of gravitation given by Saez and Ballester [20] for the combined scalar and tensor fields are as ( 18 G and 1c )

)(2

1

2

1 ,,,, ijij

mkkijji

nijij TTgwRgR

(2)

and the scalar field satisfies the equations

02 ,,

1;

, kk

ni

in n , (3)

where R , ijR , ijg denote Ricci scalar, Ricci tensor, metric tensor respectively. w is a dimensionless

coupling constant and n is an arbitrary constant. ijmT and ijT are the energy momentum tensor for

cold dark matter and anisotropic dark energy respectively, which are given by

mijm diagT ]0,0,0,1[

and (4)

],,,[ pppdiagTij

],,,1[diag

)](),(,,1[ zydiag , (5)

here m , are energy densities of CDM and anisotropic dark energy. p be the pressures and be

the deviation free EoS parameters of the anisotropic dark energy in the direction of x, y, z axes. y and

ISSN NO: 0776-3808


AEGAEUM JOURNAL


3

z are the deviation along y and z axes respectively, which is deviation from deviation free EoS

parameter of anisotropic dark energy ( ) along x axis. , y , z are not necessarily constants and

may be the function of cosmic time t [32]. The field equations (2) for metric (1) with the help of equations (4) and (5) can be written as

2

221

2

nw

A

mm

BC

CB

C

C

B

B, (6)

)(2

2

2

22

ynw

A

m

CA

AC

A

A

C

C

, (7)

)(

22

2

21

znw

A

m

AB

BA

B

B

A

A

, (8)

mnw

A

mmmm

BC

CB

CA

AC

AB

BA 2

2

2221

21

2

, (9)

0)( 2121

C

Cm

B

Bm

A

Amm

, (10)

02

2

n

C

C

B

B

A

A. (11)

The conservation equation 0)( ; jijijm TT , is given as

0)1(

C

C

B

B

C

C

B

B

A

A

C

C

B

B

A

Azymm

, (12)

where overhead dot (.) denotes differentiation with respect to time t only. Here, we consider that two-fluid minimally interact with each other. So, the general form of conservation equation (12) can be written in the following form for CDM as

03 mm

a

a

, (13)

where 3/1)(ABCa denotes the average scale factor of the model.

3. Bianchi type-III model ( 02 m )

If 02 m , then above system of field equations (6)-(11) reduce to

2

2

nw

BC

CB

C

C

B

B , (14)

)(

22

ynw

CA

AC

A

A

C

C

, (15)

)(

22

2

21

znw

A

m

AB

BA

B

B

A

A

, (16)

mnw

A

m

BC

CB

CA

AC

AB

BA 2

2

21

2

, (17)

0B

B

A

A ,

(18)

02

2

n

C

C

B

B

A

A, (19)

On integrating equation (18) and keeping a constant of integration equal to one, we get

BA . (20)

ISSN NO: 0776-3808


AEGAEUM JOURNAL


4

From equations (14) and (15) with the help of equation (20), we get 0y . (21)

Now, the field equations (14)-(19) give a system of five independent equations having seven

unknowns as A , C , m , , z , , . Therefore, additional relation is required to get the solutions

of the system of equations. In order to attain a desired model of universe, which is consistent with current observations, the Hubble parameter H is needed such that the model is in phase of an accelerating expansion. The accelerating expansion of the universe can be describe by Ellis and Mandson [54] and Singh’s [55] work, by considering the functional form of the Hubble parameter ( H ) which despite being simple. The single parameter function satisfies the above mention condition is given by

)1( naka

aH

, (22)

where 1n and 0k are the constants and a is the average scale factor. From equation (22) decelerating parameter is obtained as [56]

na

nq

11 . (23)

Solving equation (22), we get the average scale factor as

nkntea/1

1 . (24)

The uniform description of evolution of the universe has been presented by the choice of H . Adhav et al. [57] utilized the above form of average scale factor to study the Bianchi type-I, III, V, VI0 as well as Kantowski-sachs universe using variable EoS parameter in general relativity. Adhav et al. [43, 44] assumed above type of ansatz for the scale factor. For simplicity, here we have considered 1k , 1n (Sharif et al. [36])

33 )1( teaV , (25)

From equations (15) and (16), we get

dt

C

C

B

BB

m

a

c

C

C

B

B z

2

21

31 exp

. (26)

We consider following type of ansatz (Adhav [34]),

2

21

B

m

C

C

B

Bz

. (27)

Using equations (26) and (27), we have

tea

c

C

C

B

B31

. (28)

Using equations (20), (25) and (28), we obtain

213/1

2)1(6

exp)1()(t

t

e

cecBA , (29)

213/2

2)1(3

exp)1()(t

t

e

cecC , (30)

where 01 c and 02 c are the constants of integration.

Also, from equations (19) and (25), the scalar field is given by

302/

)1(

t

n

e

, (31)

where 00 is a constant of integration.

Which on integration yield, scalar field as

ISSN NO: 0776-3808


AEGAEUM JOURNAL


5

2

2

020 )1log()1(2

32

2

2

n

t

t

t

ete

en , (32)


From equations (13) and (25), we obtain the energy density of CDM as 3)1( t

m eK , (33)

where 0K is a constant of integration. The energy density of the anisotropic dark energy is obtained from field equation (17) using equations (29)-(31) and (33)

36

20

23/22

212

1

6

221

2

2

)1()1(2)1()(

)1(3exp

)1(3

)(

)1(

3

ttt

t

t

t

t

t

e

K

e

w

ec

e

cm

e

ec

e

e . (34)

The equation of state parameter of the anisotropic dark energy ( ) is obtained from field equation

(14), using equations (29)-(31) and (34) as

36

20

23/22

212

1

6

221

2

2

6

20

6

221

4

21

2

2

)1()1(2)1()(

)1(3exp

)1(3

)(

)1(

3

)1(2)1(3)1(3

)(

)1(

23

ttt

t

t

t

t

t

tt

t

t

tt

t

tt

e

K

e

w

ec

e

cm

e

ec

e

e

e

w

e

ec

e

eec

e

ee

. (35)

The skewness parameter ( z ) is obtained from field equation (16), using equations (29)-(31), (34) and

(35) as

36

20

23/22

212

1

6

221

2

2

23/22

212

1

4

21

)1()1(2)1()(

)1(3exp

)1(3

)(

)1(

3

)1()(

)1(3exp

)1(

)(

ttt

t

t

t

t

t

t

t

t

tt

z

e

K

e

w

ec

e

cm

e

ec

e

e

ec

e

cm

e

eec

. (36)

The density parameter of CDM and anisotropic dark energy are respectively as

2

2

3

2

)1(3

)1(

3

t

t

tm

m

e

e

eK

H

, (37)

2

2

36

20

23/22

212

1

6

221

2

2

2

)1(3

)1()1(2)1()(

)1(3exp

)1(3

)(

)1(

3

3

t

t

ttt

t

t

t

t

t

e

e

e

K

e

w

ec

e

cm

e

ec

e

e

H

. (38)

The total energy density parameter ( ) is given by

2

2

6

20

23/22

212

1

6

221

2

2

)1(3

)1(2)1()(

)1(3exp

)1(3

)(

)1(

3

t

t

tt

t

t

t

t

t

e

e

e

w

ec

e

cm

e

ec

e

e

. (39)

Therefore metric (1) for Bianchi type-III can be written as

2

2123/4

22123/2

222222

)1(3

2exp)1()(

)1(3exp)1())(( 1 dz

e

cec

e

cecdyedxdtds

t

t

t

txm

. (40)

ISSN NO: 0776-3808


AEGAEUM JOURNAL


6

The metric (40) together with (32)-(36) constitute Bianchi type-III cosmological model with cold dark matter and anisotropic dark energy in Saez-Ballester theory of gravitation.

4. Bianchi type-VI0 model ( 21 mm )

For 21 mm , the set of field equations (6)-(11) reduce to

2

2

21

2

nw

A

m

BC

CB

C

C

B

B, (41)

)(2

2

2

21

ynw

A

m

CA

AC

A

A

C

C

, (42)

)(2

2

2

21

znw

A

m

AB

BA

B

B

A

A

, (43)

mnw

A

m

BC

CB

CA

AC

AB

BA 2

2

21

2

, (44)

0

C

C

B

B ,

(45)

02

2

n

C

C

B

B

A

A. (46)

On integrating equation (45) and keeping a constant of integration equal to one, we get CB . (47)

From equations (42) and (43) using equation (47), we get

zy (48)

In order to obtain solutions of field equations (41)-(46), extra condition is required. We assume special form of deceleration parameter given in equation (25).

From equations (41) and (42), we get

dt

B

B

A

AA

m

a

c

B

B

A

A y

2

21

33

2

exp. (49)

We consider following type of ansatz (Adhav [34]),

yA

m

B

B

A

A2

212

. (50)

With the help of equation (49) in equation (50), we obtain

tea

c

B

B

A

A33

. (51)

Therefore from equations (25), (47) and (51), we obtain

2332

4)1(3

exp)1(t

t

e

cecA , (52)

2331

4)1(6

exp)1(t

t

e

cecCB , (53)

where 03 c and 04 c are the constants of integration.

From equations (46) and (25), the scalar field is given by

302/

)1(

t

n

e

, (54)

ISSN NO: 0776-3808


AEGAEUM JOURNAL


7


which on integration ,yield the scalar field as

2

2

020 )1log()1(2

32

2

2

n

t

t

t

ete

en , (55)


From equations (13) and (25), we obtained the energy density of CDM as 3)1( t

m eK , (56)

where 0K is a constant of integration. The energy density of anisotropic dark energy is obtained from field equation (44) using equations (52)-(54) and (56) as

36

20

23/44

232

1

6

223

2

2

)1()1(2)1()(

)1(3

2exp

)1(3

)(

)1(

3

ttt

t

t

t

t

t

e

K

e

w

ec

e

cm

e

ec

e

e . (57)

The equation of state parameter of dark energy ( ) is obtained from field equation (41) using

equations (52)-(54) and (57) as

36

20

23/44

232

1

6

223

2

2

6

20

23/44

232

1

6

223

4

23

2

2

)1()1(2)1()(

)1(3

2exp

)1(3

)(

)1(

3

)1(2)1()(

)1(3

2exp

)1(3

)(

)1(3

)(2

)1(

23

ttt

t

t

t

t

t

tt

t

t

t

t

tt

t

tt

e

K

e

w

ec

e

cm

e

ec

e

e

e

w

ec

e

cm

e

ec

e

eec

e

ee

. (58)

The skewness parameter ( z ) is obtained from field equation (43), using equations (52)-(54), (57),

(58) and (48) as

36

20

23/44

232

1

6

223

2

2

23/44

232

1

4

23

)1()1(2)1()(

)1(3

2exp

)1(3

)(

)1(

3

)1()(

)1(3

2exp2

)1(

)(

ttt

t

t

t

t

t

t

t

t

tt

z

e

K

e

w

ec

e

cm

e

ec

e

e

ec

e

cm

e

eec

. (59)

The density parameter of CDM and the anisotropic dark energy are respectively given by

2

2

3

2

)1(3

)1(

3

t

t

tm

m

e

e

eK

H

, (60)

2

2

36

20

23/44

232

1

6

223

2

2

2

)1(3

)1()1(2)1()(

)1(3

2exp

)1(3

)(

)1(

3

3

t

t

ttt

t

t

t

t

t

e

e

e

K

e

w

ec

e

cm

e

ec

e

e

H

. (61)


ISSN NO: 0776-3808


AEGAEUM JOURNAL


8

2

2

6

20

23/44

232

1

6

223

2

2

)1(3

)1(2)1()(

)1(3

2exp

)1(3

)(

)1(

3

t

t

tt

t

t

t

t

t

e

e

e

w

ec

e

cm

e

ec

e

e

. (62)

Therefore metric (1) for Bianchi type VI0 can be written as

)()1(3

exp)1()( )1(3

2exp)1()( 2222

2

323/24

2

2

323/44

22 11 dzedyee

cecdx

e

cecdtds xmxm

t

t

t

t

. (63)

The metric (63) together with (55)-(59) constitute Bianchi type-VI0 cosmological model with cold dark matter and anisotropic dark energy in Saez-Ballester theory of gravitation.

5. Bianchi type-V model ( 21 mm )

For 21 mm , then the system of field equations (6)-(11) reduce to

2

2

21

2

nw

A

m

BC

CB

C

C

B

B, (64)

)(2

2

2

21

ynw

A

m

CA

AC

A

A

C

C

, (65)

)(2

2

2

21

znw

A

m

AB

BA

B

B

A

A

, (66)

mnw

A

m

BC

CB

CA

AC

AB

BA 2

2

21

2

3

, (67)

02 C

C

B

B

A

A , (68)

02

2

n

C

C

B

B

A

A. (69)

On integrating equation (68) and keeping a constant of integration equal to one, we get

BCA 2 . (70)

Here, we get set of five field equations (64)-(69) having eight unknowns as A , B , C , , , m , y ,

z . For solving the system of equations, two extra conditions are required. Firstly, we assume a

special form of the deceleration parameter defined in equation (25). Now, we consider the relation that the shear scalar )( is proportional to the expansion scalar )( . Which gives the following relation

as lBA , (71)

where A and B are metric potentials and 0l be a constant. Thorne [58] explained the intension behind considering this condition. The Hubble parameter of universe is isotropy in present day within

30 percent [59, 60]. This has been suggested by the reviews of the velocity-red-shift relation for extra lactic sources. To place accurately, red-shift studies place the limit

3.0H

.

On the proportion of the shear )( to the Hubble parameter )(H in neighborhood of galaxy at present

day. Usual congruence to the homogeneous expansion persuades that the relation

remains

invariant, this has been noted by Collins et al. [61]. Therefore, from equations (25), (70) and (71), we obtain

)1( teA , (72)

lteB

1

)1( , (73)

ISSN NO: 0776-3808


AEGAEUM JOURNAL


9

l

lteC

)12(

)1(

. (74)

Also, from equations (69) and (25), the scalar field is given by

302/

)1(

t

n

e

, (75)


On integration of equation (75), scalar field is obtained as

2

2

020 )1log()1(2

32

2

2

n

t

t

t

ete

en , (76)


From equations (13) and (25), we obtained the energy density of CDM as 3)1( t

m eK , (77)

where 0K is a constant of integration. The energy density of the anisotropic dark energy is obtained from field equation (67), using equations (72)-(75) and (77) as

36

20

2

21

22

22

)1()1(2)1(

3

)1(

)122(

tttt

t

e

K

e

w

e

m

el

ell . (78)

The equation of state parameter of dark energy ( ) is derived from field equation (64), using

equations (72)-(75) and (78) as

)1()1(2)1(

3

)1(

)122(

)1(2)1(

)122(

)1(

2

36

20

2

21

22

22

6

20

22

221

22

tttt

t

tt

t

t

t

e

K

e

w

e

m

el

ell

e

w

el

lmell

e

e

. (79)

The skewness parameters ( z and z ) are obtained from field equations (65)-(66), using equations

(72)-(75), (78) and (79) as

36

20

2

21

22

22

22

2

)1()1(2)1(

3

)1(

)122(

)1(

)1(2

)1(

)1(

tttt

t

t

t

t

t

y

e

K

e

w

e

m

el

ell

el

ell

el

el

, (80)

36

20

2

21

22

22

22

2

)1()1(2)1(

3

)1(

)122(

)1(

)1(2

)1(

)1(

tttt

t

t

t

t

t

z

e

K

e

w

e

m

el

ell

el

ell

el

el

. (81)

The density parameter of CDM and the anisotropic dark energy are respectively given by

2

2

3

2

)1(3

)1(

3

t

t

tm

m

e

e

eK

H

, (82)

2

2

36

20

2

21

22

22

2

)1(3

)1()1(2)1(

3

)1(

)122(

3

t

t

tttt

t

e

e

e

K

e

w

e

m

el

ell

H

. (83)


ISSN NO: 0776-3808


AEGAEUM JOURNAL


10

2

2

6

20

2

21

22

22

)1(3

)1(2)1(

3

)1(

)122(

t

t

ttt

t

e

e

e

w

e

m

el

ell

. (84)

Therefore metric (1) for Bianchi type-V can be written as

2

)12(22

222222 )1()1()1( 1 zedyeedxedtds l

ltltxmt . (85)

The metric (85) together with (76)-(81) constitute Bianchi type-V cosmological model with cold dark matter and anisotropic dark energy in Saez-Ballester theory of gravitation.

6. Some important features of the models

The Hubble parameter of the models (40), (63), (85) is same and is given by

1

t

t

e

e

a

aH

. (86)

The directional Hubble’s parameters for Bianchi type-III model are given by

31

)1(3)1(

t

t

t

t

yxe

ec

e

e

A

AHH

; 3

1

)1(3

2

)1(

t

t

t

t

ze

ec

e

e

C

CH

. (87)

The directional Hubble’s parameters for Bianchi type-VI0 model are given by

33

)1(3

2

)1(

t

t

t

t

xe

ec

e

e

A

AH

; 3

3

)1(3)1(

t

t

t

t

zye

ec

e

e

B

BHH

. (88)

The directional Hubble’s parameters for Bianchi type-V model are given by

)1(

t

t

xe

e

A

AH

;

)1(

t

t

yel

e

B

BH

;

)1(

)12(

t

t

zel

el

C

CH

. (89)

The expansion scalar ( ) for the models is same and it is defined and given by

133

t

t

e

eH . (90)

The deceleration parameter ( q ) for all three models is defined and given by

teHdt

dq

111

1

. (91)

The average anisotropic parameter of expansion ( ) for Bianchi type-III model is given by

4

21

3

1

2

)1(9

2

3

1

t

i

i

e

c

H

HH. (92)

The average anisotropic parameter of expansion ( ) for Bianchi type-VI0 model is given by

4

23

)1(9

2

te

c. (93)

The average anisotropic parameter of expansion ( ) for Bianchi type-V model is given by

2

2

3

)1(2

l

l . (94)

From equations (92)-(94) give the anisotropic parameter of expansion ( ) of Bianchi types- III, V, VI0. It is observed that for Bianchi types-III, VI0 the anisotropic parameter of expansion ( ) is the function of cosmic time t and is the independent of time for Bianchi type-V.

7. Discussion

(i) The deceleration parameter ( q ): The evolution of the deceleration parameter is as shown in

fig.1. The sign of q indicates whether the model decelerates or accelerates. From fig.1, it is observed

ISSN NO: 0776-3808


AEGAEUM JOURNAL


11

that, when 0t , 0q and t , 1q . So our universe starts with the constant velocity and after

finite time it remains in an accelerating phase.

Fig. 1: The deceleration parameter ( q ) versus time ( t ) for Bianchi type-III, V, VI0

(ii) The anisotropic parameter of expansion ( ): The evolution of the anisotropy parameter of expansion is as shown in fig. 2. In Bianchi types-III, VI0 for 0t , and then it attains the isotropy soon after inflation. And for Bianchi type-V universe, the anisotropic parameter of expansion is constant throughout the evolution.

Fig. 2: The anisotropic parameter of expansion ( ) versus time ( t ), for Bianchi type-III, 11 c , 12 c ,

for Bianchi type-VI0 , 13 c , 14 c and Bianchi type-V, 5.1l .

(iii) The EoS parameter of anisotropic dark energy ( ) :

For Bianchi types-III, VI0 : The evolution of is as shown in fig.3. It is observed that as 0t ,

. As time t increases starts from phantom region and for some finite time, it reaches to

1 (cosmological constant), which indicates the model reduces to ΛCDM. After some finite time

it goes in to quintessence region )3

11( , then it again converges to 1 for late time.

ISSN NO: 0776-3808


AEGAEUM JOURNAL


12

For Bianchi type-V : As 0t , starts from quintessence region, and then after some time it

reaches to 1 . Finally it enters into phantom region and remains in phantom region for late time.

Fig. 3: The EoS parameter of anisotropic dark energy ( ) versus time (t),

for Bianchi type-III : 11 c , 12 c 1w , 10 , 11 m , 5.0K ,

for Bianchi type-VI0 : 13 c , 14 c 1w , 10 , 11 m , 5.0K ,

for Bianchi type-V : 5.1l , 1w , 10 , 11 m , 5.0K .

(iv) The skewness parameter along z-axis: The evolution of the skewness parameter ( z ) is as

shown in fig.4 Initially as 0t , z and as t increases 0z for Bianchi types-III, VI0; which

implies that the anisotropic fluid isotropize. But in Bianchi type-V as time increases constantz ,

which implies that the anisotropy of fluid remain constant throughout the evolution of the universe.

Fig. 4: The skewness parameter along z axis ( z ) versus time (t) ,

for Bianchi type-III : 5.01 c , 12 c 1w , 10 , 11 m , 5.0K ,

for Bianchi type-VI0 : 13 c , 14 c 1w , 10 , 11 m , 5.0K ,

for Bianchi type-V: 5.1l , 1w , 10 , 11 m , 5.0K .

ISSN NO: 0776-3808


AEGAEUM JOURNAL


13

(v) The scalar field ( ): For Bianchi types-III, V, VI0 the scalar field starts with zero and then it

increases after some finite time and then again reaches to zero for late time. Which indicates that the Saez-Ballester model reduces to general relativity after late time.

Fig. 5: The scalar field ( ) versus time (t) for 1.00 , 10 .

Conclusion

In this paper, we have studied Bianchi type-III, V, VI0 universe filled with cold dark matter and the anisotropic dark energy in scalar-tensor theory of gravitation proposed by Saez and Ballester. We have used a special form of the deceleration parameter, to obtain the solutions of field equations. In these models, the universe is expanding rapidly and remains in accelerating phase as the deceleration parameter attains the value 1 . For the Bianchi types-III, VI0 the space approaches to isotropy and the anisotropy parameter of expansion remains constant throughout the evolution for Bianchi type-V. It is found that, the EoS parameter of the anisotropic dark energy for the Bianchi types-III, VI0 models starts from phantom region, after some finite time it goes to quintessence region and finally converges

to 1 for late time whereas the EoS parameter of the anisotropic dark energy starts from

quintessence region, finally enters and remains in phantom region for the Bianchi type-V model. The anisotropic fluid isotropies for the Bianchi types-III, VI0 models whereas the anisotropy of fluid remains constant throughout the evolution of the universe for the Bianchi type-V model. References [1] A. G. Riess et al. , Astron. J. 116, (1998),pp.1009-1038. [2] S. Perlmutter et al. ,Astrophys. J. 517, (1999),pp.565-585. [3] R. R. Caldwell, Phys. Rev. D. 69, 103517 (2004) [4] Z. Y. Huang et al., Cosmol. Astropart. Phys. 05, 013 (2006) [5] C. Fedeli et al. , Astron. Astrophys. 500, (2009),pp.667-679. [6] S. F. Daniel et al. ,Phys. Rev. D 77, 103513 (2008) [7] P. J. E. Peebles, B Ratra ,Rev. Mod. Phys. 75 ,559 (2003) [8] A. R. Liddle, R. J. Scherrer, Phys. Rev. D 59, 023509 (1999) [9] R. R. Caldwell, R. Dave, P. J.Steinhardt, Phys. Rev. D 80, 1582 (1998) [10] P. J. Steinhardt, L .Wang, I .Zlatev, Phys. Rev. D 59. 123504 (1999) [11] S. Capozziello, S. Carloni, A .Troisi, http://arxiv.Org/abs/astro-ph/0303041 (2003) [12] S. M. Carroll et al. , Phys. Rev. D 70. 043528 (2004) [13] S. Nojiri, S. D. Odintsov, Phys. Rev. D 68. 123512 (2003) [14] G. Dvali, G. Gabadadze , M.Porrati, Phys. Lett. B 484, (2000) ,pp.112-118. [15] C. Deffayet , Phys. Lett. B 502, (2001),pp.199208. [16] B. Feng, X. L. Wang, X. M Zhang, Phys. Lett. B 607, 35 (2005),pp.35-41. [17] R. R .Caldwell, Phys. Lett. B 545, (2002),pp.23-29. [18] M. C. Bento, O. Bertolami, A. A. Sen, Phys. Rev. D 66 ,043507 (2002)

ISSN NO: 0776-3808


AEGAEUM JOURNAL


14

[19] C. H. Brans, R. H. Dicke, Phys. Rev. A 124, 925 (1961) [20] D. Saez, V. J. Ballester, Phys. Lett. A 113, (1985),pp.467-470. [21] D. Saez , Phys. Rev. D 35, 2027 (1987) [22] T. Singh, A. K. Agarwal, Astrophys. Space Sci. 182 , (1991),pp.289-312. [23] T. Singh, A. K. Agarwal, Astrophys. Space Sci. 191, (1992),pp.61-88. [24] S. Ram, J. K. Singh, Astrophys. Space Sci. 234, (1995),pp.325-334. [25] D. R. K. Reddy, V. Rao, Astrophys. Space Sci. 277, (2001),pp.461-472. [26] S. Kumar, C. P. Singh, Astrophys. Space Sci. 312, (2007),pp.57-62. [27] D. R. K. Reddy, B. Satyanarayana, R. L. Naidu, Astrophys. Space Sci. 339 , (2012),pp.401-404. [28] R. L. Naidu, B. Satyanarayana, D. R. K. Reddy, Int. J. Theor. Phys. 51, (2012),pp.2857-2862. [29] R. L. Naidu, B. Satyanarayana, D. R. K. Reddy, Astrophys. Space Sci. 388, (2012),pp.333-336. [30] V. U. M. Rao, D. Neelima, Int. Sch. Res. Not. 174741, 6 (2013) [31] O. Akarsu, C. B. Kilinc, Gen. Relativ.Gravit. 42, 119 (2010) [32] O. Akarsu, C. B. Kilinc, Astrophys. Space Sci. 326, (2010),pp.315-322. [33] K. S. Adhav et al. ,Astrophys. Space Sci. 332, (2011),pp.497-502. [34] K. S. Adhav, Int. J. Astro. Astro. 1, (2011),pp.204-209. [35] S. Ram, S. Chandel, M. K. Verma, Chin. J. Phys. 54, (2016),pp.953-059. [36] M. Sharif, M. Zubir, Int. J. Mod. Phys. D 19, (2010),pp.1957-1972. [37] H. Amirhashchi, A. Pradhan, B. Saha, Astrophys. Space Sci. 333,(2011),pp.295-303. [38] S. Kumar, A. K. Yadav, Mod. Phys. Lett. A 26, (2011),pp.647-659. [39] A. K. Yadav, Astrophys.Space Sci. 335, (2011),pp.565-575. [40] A. Pradhan, H. Amirhashchi, Astrophys. Space Sci. 332, 441 (2011),pp.441-448. [41] S. D. Katore, K. S. Adhav, M. M. Sancheti, Astrophys. Space Sci. 337, (2012),pp.393-400. [42] V. U. M. Rao, D. C. P. Rao, Astrophys. Space Sci. 357 ,65 (2015) [43] K. S. Adhav et al. , Astrophys. Space Sci. 353,(2014),pp.249-257. [44] K. S. Adhav et al., Astrophys. Space Sci. 359, 24 (2015) [45] S. Sarkar, Astrophys. Space Sci. 352, (2014),pp.589-866. [46] M. Kiran et al., Astrophys. Space Sci. 356, (2014),pp.407-411. [47] M. Kiran et al. ,Astrophys. Space Sci. 360, 54 (2015) [48] K. S. Adhav et al., J. Theor. Appl. Phys. 6 ,33 (2012) [49] A. Asgar, M. Ansari ,J. Theor. Appl. Phys. 8 , (2014),pp.219-224. [50] U. M. Rao, N. Davuluri, J. Theor. Appl. Phys. 7, 50 (2013) [51] R. Ghost et al., J. Theor. Appl. Phys. 7, 48 (2013) [52] B. Mishra, P. K. Sahoo, J. Theor. Appl.Phys. 7, 36 (2013) [53] M. V. Santhi et al. ,Astrophys. Space Sci. 361, 142 (2016) [54] G. F. R. Ellis, M. Madson, Class Quantum Grav. 8, 667 (1991) [55] J. P. Singh, Astrophys. Space Sci. 318, (2008),pp.103-107. [56] N. Banerjee, S. Das, Gen. Relativ.Gravit. 37, 1695 (2005),pp.1695-1703. [57] K. S. Adhav et al. , Astrophys. Space Sci. 345, (2013),pp.405-413. [58] K. S. Thorne, Astrophys. J. 148, (1967),pp.51-68. [59] R. Kantowski, R. K. Sachs, J. Math. Phys. 7, (1966),pp.443-446. [60] J. Kristian, R. K. Sachs, Astrophys. J. 143, 379 (1966),pp.379-399. [61] C. B. Collins, E. N. Glass, D. A. Wilkinson, Gen. Rel. Grav. 12, 805(1980),pp.805-823.

ISSN NO: 0776-3808


AEGAEUM JOURNAL


Documents

Bianchi Types Cosmological Models with Cold Dark Matter ... · weak field as well as the antigravity regime appear in this theory, provide a suitable way to solve ‘missing matter