Hindawi Publishing CorporationISRN Astronomy and AstrophysicsVolume 2013 Article ID 924834 11 pageshttpdxdoiorg1011552013924834

Research ArticleBianchi Type II VIII and IX Perfect FluidDark Energy Cosmological Models in Saez-Ballester andGeneral Theory of Gravitation

V U M Rao K V S Sireesha and D Neelima

Department of Applied Mathematics Andhra University Visakhapatnam AP 530003 India

Correspondence should be addressed to V U M Rao umrao57hotmailcom

Received 2 December 2012 Accepted 20 December 2012

Academic Editors H Bushouse I Goldman M Grewing and V Pierrard

Copyright copy 2013 V U M Rao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper deals with spatially homogeneous anisotropic Bianchi type II VIII and IX dark energy cosmological models filledwith perfect fluid in the framework of Saez-Ballester (1986) theory and Einsteinrsquos general relativity Assuming that the two sourcesinteract minimally and therefore their energy momentum tensors are conserved separately we have considered different cases andpresented anisotropic as well as isotropic cosmological models Some important physical and geometrical features of the modelsthus obtained have been discussed

1 Introduction

Saez and Ballester [1] formulated a scalar tensor theory ofgravitation in which the metric is coupled with a dimen-sionless scalar field in a simple manner This coupling givesa satisfactory description of the weak fields In spite of thedimensionless character of the scalar field an antigravityregime appears This theory also suggests a possible way tosolve missing matter problem in nonflat FRW cosmologies

The field equations given by Saez-Ballester [1] for thecombined scalar and tensor fields (using geometrized unitswith c = 1 8120587G = 1) are

119866119894119895minus 120596120601119903(120601119894120601119895minus

1

2

1198921198941198951206011198961206011015840119896) = minus119879

119894119895(1)

and the scalar field 120601 satisfies the equation

2120601119903120601119894

119894+ 119903120601119903minus11206011198961206011015840119896= 0 (2)

where119866119894119895= 119877119894119895minus(12)119877119892

119894119895is an Einstein tensor119877 is the scalar

curvature 120596 and 119903 are constants and 119879119894119895is the stress energy

tensor of the matterThe energy conservation equation is

119879119894119895

119895= 0 (3)

The study of cosmological models in the framework ofscalar tensor theories has been the active area of researchfor the last few decades In particular Rao et al [2] and Raoet al [3 4] are some of the authors who have investigatedseveral aspects of the cosmological models in Saez-Ballester[1] scalar tensor theory Naidu et al [5ndash8] have discussedvarious aspects of Bianchi space times in Saez-Ballester [1]scalar tensor theory

Recent observations of type Ia supernovae (SN Ia) [9ndash13]galaxy redshift surveys [14] cosmic microwave backgroundradiation (CMBR) data [15 16] and large scale structure [17]strongly suggest that the observable universe is undergoingan accelerated expansion Observations also suggest thatthere had been a transition of the universe from the earlierdeceleration phase to the recent acceleration phase [18]The cause of this sudden transition and the source of theaccelerated expansion are still unknown Measurements ofCMBR anisotropies most recently by the WMAP satelliteindicate that the universe is very close to flat For a flatuniverse its energy density must be equal to a certain criticaldensity which demands a huge contribution from someunknown energy stuffThus the observational effects like thecosmic acceleration sudden transition flatness of universeand many more need explanation It is generally believedthat some sort of ldquodark energyrdquo (DE) is pervading the whole

2 ISRN Astronomy and Astrophysics

universe It is a hypothetical form of energy that permeatesall of space and tends to increase the rate of expansion of theuniverse [19] The most recent WMAP observations indicatethat DE accounts for 72 of the total mass energy of theuniverse [20] However the nature of DE is still a mystery

Many cosmologists believe that the simplest candidate forthe DE is the cosmological constant (Λ) or vacuum energysince it fits the observational data well During the cosmo-logical evolution the Λ term has the constant energy densityand pressure 119901(de) = minus120588(de) where the superscript (de) standsfor DE However one has the reason to dislike the cosmo-logical constant since it always suffers from the theoreticalproblems such as the ldquofine-tuningrdquo and ldquocosmic coincidencerdquopuzzles [21] That is why the different forms of dynamicallychanging DE with an effective equation of state (EoS)120596(de) =119901(de)120588(de)

lt minus13 have been proposed in the literature Oth-er possible forms of DE include quintessence (120596(de) gt minus1)

[22] phantom (120596(de)

lt minus1) [23] and so forth While thepossibility 120596(de) ≪ minus1 is ruled out by current cosmologicaldata from SN Ia (Supernovae Legacy Survey Gold sam-ple of Hubble Space Telescope) [13 24] CMBR (WMAPBOOMERANG) [25 26] and large scale structure (SloanDigital Sky Survey) [27] data the dynamically evolving DEcrossing the phantom divide line (PDL) (120596(de) = minus1) is mildlyfavored SN Ia data combined with CMBR anisotropy andgalaxy clustering statistics suggest that minus133 lt 120596(de) lt minus079(see [28])

Most of the models with constant DP have been studiedby considering perfect fluid or ordinary matter in the uni-verse But the ordinary matter is not enough to describe thedynamics of an accelerating universe as mentioned earlierThis motivates the researchers to consider the models of theuniverse filled with some exotic type of matter such as theDE along with the usual perfect fluid Recently some darkenergy models with constant DP have been investigated byKumar and Singh [29] Akarsu and Kilinc [30ndash32] Yadav etal [33] A K Yadav and L Yadav [34] Pradhan et al [35]and Rao et al [36 37] Reddy et al [38] have discussed fivedimensional dark energy models in a Saez-Ballester [1] scalartensor theory

In this paper we will discuss minimally interactingperfect fluid and dark energy Bianchi type II VIII andIX space-times in a scalar tensor theory of gravitationproposed by Saez and Ballester [1] and general theory ofgravitation

2 Metric and Field Equations

Weconsider spatially homogeneous Bianchi type II VIII andIX metrics in the form

1198891199042= minus119889119905

2+ 1198772[1198891205792+ 1198912(120579) 119889120601

2] + 1198782[119889120593 + ℎ (120579) 119889120601]

2

(4)

where 120579 120601 and 120593 are the Eulerian angles Also 119877 and 119878 arefunctions of 119905 only

It represents

Bianchi type II if 119891(120579) = 1 and ℎ(120579) = 120579Bianchi type VIII if 119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579Bianchi type IX if 119891(120579) = sin 120579 and ℎ(120579) = cos 120579

The energy momentum tensor is given by

119879119895

119894= 119879(119898)119894

119895+ 119879(de)119894119895

(5)

where 119879(119898)119894119895

and 119879(de)119894119895

are the energy momentum tensors ofordinary matter and DE respectively and are given by

119879(119898)119894

119895= diag [minus120588(119898) 119901(119898) 119901(119898) 119901(119898)]

= diag [minus1 119908(119898) 119908(119898) 119908(119898)] 120588(119898)(6)

119879(de)119894119895

= diag [minus120588(de) 119901(de) 119901(de) 119901(de)]

= diag [minus1 119908(de) 119908(de) 119908(de)] 120588(de)(7)

where 120588(119898) and 119901(119898) are the energy density and pressureof the perfect fluid component or ordinary baryonic matterwhile 119908(119898) = 119901(119898)120588(119898) is its EoS parameter Similarly 120588(de)

and 119901(de) are the energy density and pressure of the DE

componentwhile119908(de) = 119901(de)120588(de) is the correspondingEoSparameter

Now with the help of (5) to (7) the field equation (1) forthe metric (4) can be written as

119877

+

119878

119878

+

119878

119877119878

+

1198782

41198774minus

120596

2

120601119903 1206012= minus119908(119898)120588(119898)minus 119908(de)120588(de)

(8)

2

119877

+

2+ 120575

1198772

minus

31198782

41198774minus

120596

2

120601119903 1206012= minus119908(119898)120588(119898)minus 119908(de)120588(de)

(9)

2

119878

119877119878

+

2+ 120575

1198772

minus

1198782

41198774+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (10)

120601 +

120601 (2

119877

+

119878

119878

) +

119903

2120601

1206012= 0 (11)

The conservation equation yields

120588(119898)+ 3 (1 + 119908

(119898)) 120588(119898)119867 + 120588

(de)+ 3 (1 + 119908

(de)) 120588(de)119867 = 0

(12)

where119867 is themean Hubble parameter

3 Solutions of the Field Equations

In order to solve the field equations completely we assumethat the perfect fluid and DE components interact minimallyTherefore the energy momentum tensors of the two sourcesmay be conserved separately

ISRN Astronomy and Astrophysics 3

The energy conservation equation 119879(119898)119894119895119895

= 0 of the perf-ect fluid leads to

120588(119898)+ 3 (1 + 119908

(119898)) 120588(119898)119867 = 0 (13)

whereas the energy conservation equation 119879(de)119894119895119895

= 0 of theDE component yields

120588(de)

+ 3 (1 + 119908(de)) 120588(de)119867 = 0 (14)

Following Akarsu and Kilinc [30] we assume that the EoSparameter of the perfect fluid is a constant that is

119908(119898)

=

119901(119898)

120588(119898)

= cons tan 119905 (15)

while119908(de) has been allowed to be a function of time since thecurrent cosmological data from SN Ia CMB and large scalestructures mildly favor dynamically evolving DE crossing thephantom divide line (PDL)

Now we assume that the shear scalar (120590) in the models isproportional to expansion scalar (120579)

This condition leads to

119878 = 119877119899 (16)

where 119877 and 119878 are the metric potentials and 119899 is a positiveconstant

Since we are looking for a model explaining an expand-ing universe with acceleration we also assume that theanisotropic distribution of DE ensures the present accelerat-ing universe Thus (8) to (12) may be rewritten as

119877

+

119878

119878

+

119878

119877119878

+

1198782

41198774minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(17)

2

119877

+

2+ 120575

1198772

minus

31198782

41198774minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(18)

2

119878

119877119878

+

2+ 120575

1198772

minus

1198782

41198774+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (19)

120601 +

120601 (2

119877

+

119878

119878

) +

119903

2120601

1206012= 0 (20)

120588(de)

+ 3 (1 + 119908(de)) 120588(de)119867 + 120588

(de)(2120574119867119909+ 120585119867119911) = 0 (21)

The third term of (21) arises due to the deviation from 119908(de)

while the first two terms of (21) are deviation free part of119879(de)119894119895

According to (21) the behavior of 120588(de) is controlled bythe deviation free part of EoS parameter of DE but deviationwill affect 120588(de) indirectly since as can be seen later they affectthe value of EoS parameter But we are looking for physicallyviable models of the universe consistent with observationsHence we constrained 120585(119905) and 120574(119905) by assuming the special

dynamics which are consistent with (21) The dynamics ofskewness parameter on 119909-axis or 119910-axis 120574 and 119911-axis 120585 aregiven by

120574 =

minus119898119867119867119911

120588(de) (22)

120585 =

2119898119867119867119909

120588(de) (23)

where119898 is the dimensionless constant that parameterizes theamplitude of the deviation from 119908

(de) and can be given realvalues

From (17) and (18) we get

119877

+ 1198621

2

1198772minus

120575

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (24)

where 1198621= (3(119899

2minus 1) minus 119898(119899 + 2)

2)3(119899 minus 1)

31 Bianchi Type II (120575 = 0)Cosmological Model If 120575 = 0 (24)can be written as

119877

+ 1198621

2

1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (25)

From (25) we get

119877 = (1198961119905 + 1198962)1(2minus119899)

119899 = 2 (26)

where 1198992(1 + 1198961

2) + 119899((119862

1minus 2)1198962

1minus 4) + ((1 minus 119862

1)1198962

1+ 4) =

0 1198961= 0 and 119896

2are arbitrary constants

From (16) and (26) we get

119878 = (1198961119905 + 1198962)119899(2minus119899)

(27)

From (11) (16) and (26) we get

120601(119903+2)2

= [

119903+2

2

(

(119899minus2) 1198963

2119899

(1198961119905+1198962)2119899(119899minus2)

+1198964)]

119903 = minus 2 119899 = 0

(28)

where 1198963and 1198964are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198961119905 + 1198962)minus(1+119908

(119898))(119899+2)(2minus119899)

(29)

where 1205880is a constant of integration

From (19) and (26)ndash(29) we get the dark energy density

120588(de)

=

[119899 (81198962

1+ 4) + (4119896

2

1minus 4) minus 119899

2]

4(2 minus 119899)2(1198961119905 + 1198962)2

+

1205961198963

2

2(1198961119905 + 1198962)2(119899+2)(2minus119899)

minus

1205880

(1198961119905 + 1198962)(1+119908(119898))(119899+2)(2minus119899)

(30)

4 ISRN Astronomy and Astrophysics

From (22) and (30) we get the skewness parameter

120574 = 4119898119899 (119899 + 2) 1198962

1

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+ 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

]

minus1

(31)

From (23) and (30) we get the skewness parameter

120585 = 8119898 (119899 + 2) 1198962

1

times [121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

minus 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2) ]

minus1

(32)

From (17) and (26)ndash(31) we get the EoS parameter of DE

119908(de)

= minus [1198965(1198961119905 + 1198962)minus2minus 6120596119896

3

2(2 minus 119899)

2

times (1198961119905 + 1198962)2(119899+2)(119899minus2)

+ 121199081198981205880(2 minus 119899)

2

times (1198961119905 + 1198962)(1+119908(119898))(119899+2)(119899minus2)

]

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2

times (1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(119899minus2)

]

minus1

(33)

where 1198965= 1198992[41198962

1(6 + 119898) + 3] + 4119899(2119898119896

2

1minus 3) + 12(119896

2

1minus 1)

The density parameters of perfect fluid and DE are asfollows

Ω(119898)

=

120588(119898)

31198672

=

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

119899 = minus 2

Ω(de)

=

120588(de)

31198672

=

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

4(119899 + 2)21198962

1

+

31205961198962

3(2 minus 119899)

2

2(119899 + 2)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

minus

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

(34)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

8(119899 minus 1)21198962

1

+

31205961198962

3(2 minus 119899)

2

4(119899 minus 1)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

]119860119898

(35)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198961119905 + 1198962)2(2minus119899)

(1198891205792+ 1198891206012)

+ (1198961119905 + 1198962)2119899(2minus119899)

(119889120593 + 120579119889120601)2

(36)

Thus the metric (36) together with (28)ndash(33) constitutes aBianchi type II perfect fluid dark energy cosmological modelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (36) together with (28)ndash(33) constitutes a Bianchi type II perfect fluid dark energycosmological model in general relativity with 119896

3= 0

32 Bianchi Type VIII (120575 = minus1) Cosmological Model If 120575 =minus1 (24) can be written as

119877

+ 1198621

2

1198772+

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (37)

From (37) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (38)

where1198621= (9minus16119898)3 119862

2

2= 3(16119898minus9) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (38) we get

119877 = (

1198622

1198623

) sin (1198623119905) (39)

From (16) and (39) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(40)

ISRN Astronomy and Astrophysics 5

From (11) (16) and (39) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905) + 2) + 119862

5)]

119903 = minus 2

(41)

where 1198624and119862

5are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(42)

where 1205880is a constant of integration

From (19) and (39)ndash(42) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) minus 4119862

2

3cosec2 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(43)

From (22) and (43) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905) minus 3119862

2

2

+ 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(44)

From (23) and (43) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [121198622

3cosec2 (119862

3119905) + 3119862

2

2minus 60119862

2

21198622

3cot2 (119862

3119905)

minus 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

+1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(45)

From (17) and (39)ndash(44) we get the EoS parameter of DE

119908(de)

= minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(46)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

minus

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(47)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

minus

3

21198622

2sec2 (119862

3119905)]119860

119898 (48)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ cosh21205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + sinh 120579119889120601)2

(49)

Thus the metric (49) together with (41)ndash(46) constitutes aBianchi type VIII perfect fluid dark energy cosmologicalmodel in Saez and Ballester [1] theory of gravitation

Also we observe that the metric (49) together with (41)ndash(46) constitutes a Bianchi type VIII perfect fluid dark energycosmological model in general relativity with 119862

4= 0

6 ISRN Astronomy and Astrophysics

33 Bianchi Type IX (120575 = 1) Cosmological Model If 120575 = 1(24) can be written as

119877

+ 1198621

2

1198772minus

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (50)

From (50) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (51)

where1198621= (9minus16119898)3 1198622

2= 3(9minus16119898) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (51) we get

119877 = (

1198622

1198623

) sin (1198623119905) (52)

From (16) and (52) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(53)

From (11) (16) and (52) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905)+2)+119862

5)] 119903 = minus 2

(54)

where 1198624and119862

5are integrating constants One has

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(55)

where 1205880is a constant of integration

From (19) and (52)ndash(55) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) + 4119862

2

3cos 1198901198882 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(56)

From (22) and (56) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) + 12119862

2

3cos 1198901198882 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(57)

From (23) and (56) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus121198622

3cosec2 (119862

3119905)+3119862

2

2minus60119862

2

21198622

3cot2 (119862

3119905)

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(58)

From (17) and (52)ndash(57) we get the EoS parameter of DE

119908(de)

=minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [121198622

3cosec2 (119862

3119905)

minus 1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus 31198622

2+ 60119862

2

21198622

3cot2 (119862

3119905)

+61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(59)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

+

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(60)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

+

3

21198622

2

sec2 (1198623119905)]119860

119898 (61)

where 119860119898is the average anisotropy parameter

ISRN Astronomy and Astrophysics 7

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ sin 1205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + cos 120579 119889120601)2

(62)

Thus the metric (62) together with (54)ndash(59) constitutes aBianchi type IX perfect fluid dark energy cosmologicalmodelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (62) together with (54)ndash(59) constitutes a Bianchi type IX perfect fluid dark energycosmological model in general relativity with 119862

4= 0

34 Isotropic Cosmological Models To get isotropic cosmo-logical models let us assign the value unity to 119899 in (16) andthen we get

119877 = 119878 (63)

where 119877 and 119878 are the metric potentialsUsing (63) the field equations (17)ndash(20) can be written as

2

119878

119878

+

1198782

1198782+

1

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(64)

2

119878

119878

+

1198782

1198782+

120575

1198782minus

3

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(65)

3

1198782

1198782+

120575

1198782minus

1

41198782+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (66)

120601 + 3

120601 119878

119878

+

119903

2120601

1206012= 0 (67)

From (64) and (65) we get

3119898 1198782+ (120575 minus 1) = 0 (68)

35 Bianchi Type II and VIII Cosmological Models in IsotropicForm From (68) we get

119877 = 119878 = (1198626119905 + 1198627) (69)

where 1198626= ((1 minus 120575)3119898)

12 and1198627are constants

From (67) and (69) we get

120601(119903+2)2

=

119903 + 2

2

(1198629minus

1198628

2(1198626119905 + 1198627)2) 119903 = minus 2 (70)

where 1198628and 119862

9are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198626119905 + 1198627)minus3(1+119908

(119898)) (71)

where 1205880is a constant of integration

From (66) (69) and (71) we get the dark energy density

120588(de)

= 31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6minus 1205880(1198626119905 + 1198627)minus3(1+119908

(119898))

(72)

From (22) and (72) we get the skewness parameter

120574 = minus 1198981198626

2[31198626

2+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(73)

From (23) and (72) we get the skewness parameter

120585 = 21198981198626

2[31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(74)

From (64) and (69)ndash(73) we get the EoS parameter of DE

119908(de)

= minus [(1 minus 119898)1198622

6+

1

4

(1198626119905 + 1198627)minus2

minus

1205961198622

8

2

(1198626119905 + 1198627)minus6

+1205880119908(119898)(1198626119905 + 1198627)minus3(1+119908

(119898))]

times [31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(75)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

Ω(de)

=

120588(de)

31198672

= (1198626119905 + 1198627)2minus

5

121198626

2+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

minus

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

(76)

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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Journal of

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ThermodynamicsJournal of

2 ISRN Astronomy and Astrophysics

universe It is a hypothetical form of energy that permeatesall of space and tends to increase the rate of expansion of theuniverse [19] The most recent WMAP observations indicatethat DE accounts for 72 of the total mass energy of theuniverse [20] However the nature of DE is still a mystery

Many cosmologists believe that the simplest candidate forthe DE is the cosmological constant (Λ) or vacuum energysince it fits the observational data well During the cosmo-logical evolution the Λ term has the constant energy densityand pressure 119901(de) = minus120588(de) where the superscript (de) standsfor DE However one has the reason to dislike the cosmo-logical constant since it always suffers from the theoreticalproblems such as the ldquofine-tuningrdquo and ldquocosmic coincidencerdquopuzzles [21] That is why the different forms of dynamicallychanging DE with an effective equation of state (EoS)120596(de) =119901(de)120588(de)

lt minus13 have been proposed in the literature Oth-er possible forms of DE include quintessence (120596(de) gt minus1)

[22] phantom (120596(de)

lt minus1) [23] and so forth While thepossibility 120596(de) ≪ minus1 is ruled out by current cosmologicaldata from SN Ia (Supernovae Legacy Survey Gold sam-ple of Hubble Space Telescope) [13 24] CMBR (WMAPBOOMERANG) [25 26] and large scale structure (SloanDigital Sky Survey) [27] data the dynamically evolving DEcrossing the phantom divide line (PDL) (120596(de) = minus1) is mildlyfavored SN Ia data combined with CMBR anisotropy andgalaxy clustering statistics suggest that minus133 lt 120596(de) lt minus079(see [28])

Most of the models with constant DP have been studiedby considering perfect fluid or ordinary matter in the uni-verse But the ordinary matter is not enough to describe thedynamics of an accelerating universe as mentioned earlierThis motivates the researchers to consider the models of theuniverse filled with some exotic type of matter such as theDE along with the usual perfect fluid Recently some darkenergy models with constant DP have been investigated byKumar and Singh [29] Akarsu and Kilinc [30ndash32] Yadav etal [33] A K Yadav and L Yadav [34] Pradhan et al [35]and Rao et al [36 37] Reddy et al [38] have discussed fivedimensional dark energy models in a Saez-Ballester [1] scalartensor theory

In this paper we will discuss minimally interactingperfect fluid and dark energy Bianchi type II VIII andIX space-times in a scalar tensor theory of gravitationproposed by Saez and Ballester [1] and general theory ofgravitation

2 Metric and Field Equations

Weconsider spatially homogeneous Bianchi type II VIII andIX metrics in the form

1198891199042= minus119889119905

2+ 1198772[1198891205792+ 1198912(120579) 119889120601

2] + 1198782[119889120593 + ℎ (120579) 119889120601]

2

(4)

where 120579 120601 and 120593 are the Eulerian angles Also 119877 and 119878 arefunctions of 119905 only

It represents

Bianchi type II if 119891(120579) = 1 and ℎ(120579) = 120579Bianchi type VIII if 119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579Bianchi type IX if 119891(120579) = sin 120579 and ℎ(120579) = cos 120579

The energy momentum tensor is given by

119879119895

119894= 119879(119898)119894

119895+ 119879(de)119894119895

(5)

where 119879(119898)119894119895

and 119879(de)119894119895

are the energy momentum tensors ofordinary matter and DE respectively and are given by

119879(119898)119894

119895= diag [minus120588(119898) 119901(119898) 119901(119898) 119901(119898)]

= diag [minus1 119908(119898) 119908(119898) 119908(119898)] 120588(119898)(6)

119879(de)119894119895

= diag [minus120588(de) 119901(de) 119901(de) 119901(de)]

= diag [minus1 119908(de) 119908(de) 119908(de)] 120588(de)(7)

where 120588(119898) and 119901(119898) are the energy density and pressureof the perfect fluid component or ordinary baryonic matterwhile 119908(119898) = 119901(119898)120588(119898) is its EoS parameter Similarly 120588(de)

and 119901(de) are the energy density and pressure of the DE

componentwhile119908(de) = 119901(de)120588(de) is the correspondingEoSparameter

Now with the help of (5) to (7) the field equation (1) forthe metric (4) can be written as

119877

+

119878

119878

+

119878

119877119878

+

1198782

41198774minus

120596

2

120601119903 1206012= minus119908(119898)120588(119898)minus 119908(de)120588(de)

(8)

2

119877

+

2+ 120575

1198772

minus

31198782

41198774minus

120596

2

120601119903 1206012= minus119908(119898)120588(119898)minus 119908(de)120588(de)

(9)

2

119878

119877119878

+

2+ 120575

1198772

minus

1198782

41198774+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (10)

120601 +

120601 (2

119877

+

119878

119878

) +

119903

2120601

1206012= 0 (11)

The conservation equation yields

120588(119898)+ 3 (1 + 119908

(119898)) 120588(119898)119867 + 120588

(de)+ 3 (1 + 119908

(de)) 120588(de)119867 = 0

(12)

where119867 is themean Hubble parameter

3 Solutions of the Field Equations

In order to solve the field equations completely we assumethat the perfect fluid and DE components interact minimallyTherefore the energy momentum tensors of the two sourcesmay be conserved separately

ISRN Astronomy and Astrophysics 3

The energy conservation equation 119879(119898)119894119895119895

= 0 of the perf-ect fluid leads to

120588(119898)+ 3 (1 + 119908

(119898)) 120588(119898)119867 = 0 (13)

whereas the energy conservation equation 119879(de)119894119895119895

= 0 of theDE component yields

120588(de)

+ 3 (1 + 119908(de)) 120588(de)119867 = 0 (14)

Following Akarsu and Kilinc [30] we assume that the EoSparameter of the perfect fluid is a constant that is

119908(119898)

=

119901(119898)

120588(119898)

= cons tan 119905 (15)

while119908(de) has been allowed to be a function of time since thecurrent cosmological data from SN Ia CMB and large scalestructures mildly favor dynamically evolving DE crossing thephantom divide line (PDL)

Now we assume that the shear scalar (120590) in the models isproportional to expansion scalar (120579)

This condition leads to

119878 = 119877119899 (16)

where 119877 and 119878 are the metric potentials and 119899 is a positiveconstant

Since we are looking for a model explaining an expand-ing universe with acceleration we also assume that theanisotropic distribution of DE ensures the present accelerat-ing universe Thus (8) to (12) may be rewritten as

119877

+

119878

119878

+

119878

119877119878

+

1198782

41198774minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(17)

2

119877

+

2+ 120575

1198772

minus

31198782

41198774minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(18)

2

119878

119877119878

+

2+ 120575

1198772

minus

1198782

41198774+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (19)

120601 +

120601 (2

119877

+

119878

119878

) +

119903

2120601

1206012= 0 (20)

120588(de)

+ 3 (1 + 119908(de)) 120588(de)119867 + 120588

(de)(2120574119867119909+ 120585119867119911) = 0 (21)

The third term of (21) arises due to the deviation from 119908(de)

while the first two terms of (21) are deviation free part of119879(de)119894119895

According to (21) the behavior of 120588(de) is controlled bythe deviation free part of EoS parameter of DE but deviationwill affect 120588(de) indirectly since as can be seen later they affectthe value of EoS parameter But we are looking for physicallyviable models of the universe consistent with observationsHence we constrained 120585(119905) and 120574(119905) by assuming the special

dynamics which are consistent with (21) The dynamics ofskewness parameter on 119909-axis or 119910-axis 120574 and 119911-axis 120585 aregiven by

120574 =

minus119898119867119867119911

120588(de) (22)

120585 =

2119898119867119867119909

120588(de) (23)

where119898 is the dimensionless constant that parameterizes theamplitude of the deviation from 119908

(de) and can be given realvalues

From (17) and (18) we get

119877

+ 1198621

2

1198772minus

120575

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (24)

where 1198621= (3(119899

2minus 1) minus 119898(119899 + 2)

2)3(119899 minus 1)

31 Bianchi Type II (120575 = 0)Cosmological Model If 120575 = 0 (24)can be written as

119877

+ 1198621

2

1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (25)

From (25) we get

119877 = (1198961119905 + 1198962)1(2minus119899)

119899 = 2 (26)

where 1198992(1 + 1198961

2) + 119899((119862

1minus 2)1198962

1minus 4) + ((1 minus 119862

1)1198962

1+ 4) =

0 1198961= 0 and 119896

2are arbitrary constants

From (16) and (26) we get

119878 = (1198961119905 + 1198962)119899(2minus119899)

(27)

From (11) (16) and (26) we get

120601(119903+2)2

= [

119903+2

2

(

(119899minus2) 1198963

2119899

(1198961119905+1198962)2119899(119899minus2)

+1198964)]

119903 = minus 2 119899 = 0

(28)

where 1198963and 1198964are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198961119905 + 1198962)minus(1+119908

(119898))(119899+2)(2minus119899)

(29)

where 1205880is a constant of integration

From (19) and (26)ndash(29) we get the dark energy density

120588(de)

=

[119899 (81198962

1+ 4) + (4119896

2

1minus 4) minus 119899

2]

4(2 minus 119899)2(1198961119905 + 1198962)2

+

1205961198963

2

2(1198961119905 + 1198962)2(119899+2)(2minus119899)

minus

1205880

(1198961119905 + 1198962)(1+119908(119898))(119899+2)(2minus119899)

(30)

4 ISRN Astronomy and Astrophysics

From (22) and (30) we get the skewness parameter

120574 = 4119898119899 (119899 + 2) 1198962

1

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+ 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

]

minus1

(31)

From (23) and (30) we get the skewness parameter

120585 = 8119898 (119899 + 2) 1198962

1

times [121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

minus 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2) ]

minus1

(32)

From (17) and (26)ndash(31) we get the EoS parameter of DE

119908(de)

= minus [1198965(1198961119905 + 1198962)minus2minus 6120596119896

3

2(2 minus 119899)

2

times (1198961119905 + 1198962)2(119899+2)(119899minus2)

+ 121199081198981205880(2 minus 119899)

2

times (1198961119905 + 1198962)(1+119908(119898))(119899+2)(119899minus2)

]

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2

times (1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(119899minus2)

]

minus1

(33)

where 1198965= 1198992[41198962

1(6 + 119898) + 3] + 4119899(2119898119896

2

1minus 3) + 12(119896

2

1minus 1)

The density parameters of perfect fluid and DE are asfollows

Ω(119898)

=

120588(119898)

31198672

=

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

119899 = minus 2

Ω(de)

=

120588(de)

31198672

=

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

4(119899 + 2)21198962

1

+

31205961198962

3(2 minus 119899)

2

2(119899 + 2)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

minus

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

(34)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

8(119899 minus 1)21198962

1

+

31205961198962

3(2 minus 119899)

2

4(119899 minus 1)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

]119860119898

(35)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198961119905 + 1198962)2(2minus119899)

(1198891205792+ 1198891206012)

+ (1198961119905 + 1198962)2119899(2minus119899)

(119889120593 + 120579119889120601)2

(36)

Thus the metric (36) together with (28)ndash(33) constitutes aBianchi type II perfect fluid dark energy cosmological modelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (36) together with (28)ndash(33) constitutes a Bianchi type II perfect fluid dark energycosmological model in general relativity with 119896

3= 0

32 Bianchi Type VIII (120575 = minus1) Cosmological Model If 120575 =minus1 (24) can be written as

119877

+ 1198621

2

1198772+

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (37)

From (37) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (38)

where1198621= (9minus16119898)3 119862

2

2= 3(16119898minus9) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (38) we get

119877 = (

1198622

1198623

) sin (1198623119905) (39)

From (16) and (39) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(40)

ISRN Astronomy and Astrophysics 5

From (11) (16) and (39) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905) + 2) + 119862

5)]

119903 = minus 2

(41)

where 1198624and119862

5are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(42)

where 1205880is a constant of integration

From (19) and (39)ndash(42) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) minus 4119862

2

3cosec2 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(43)

From (22) and (43) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905) minus 3119862

2

2

+ 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(44)

From (23) and (43) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [121198622

3cosec2 (119862

3119905) + 3119862

2

2minus 60119862

2

21198622

3cot2 (119862

3119905)

minus 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

+1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(45)

From (17) and (39)ndash(44) we get the EoS parameter of DE

119908(de)

= minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(46)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

minus

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(47)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

minus

3

21198622

2sec2 (119862

3119905)]119860

119898 (48)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ cosh21205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + sinh 120579119889120601)2

(49)

Thus the metric (49) together with (41)ndash(46) constitutes aBianchi type VIII perfect fluid dark energy cosmologicalmodel in Saez and Ballester [1] theory of gravitation

Also we observe that the metric (49) together with (41)ndash(46) constitutes a Bianchi type VIII perfect fluid dark energycosmological model in general relativity with 119862

4= 0

6 ISRN Astronomy and Astrophysics

33 Bianchi Type IX (120575 = 1) Cosmological Model If 120575 = 1(24) can be written as

119877

+ 1198621

2

1198772minus

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (50)

From (50) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (51)

where1198621= (9minus16119898)3 1198622

2= 3(9minus16119898) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (51) we get

119877 = (

1198622

1198623

) sin (1198623119905) (52)

From (16) and (52) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(53)

From (11) (16) and (52) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905)+2)+119862

5)] 119903 = minus 2

(54)

where 1198624and119862

5are integrating constants One has

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(55)

where 1205880is a constant of integration

From (19) and (52)ndash(55) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) + 4119862

2

3cos 1198901198882 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(56)

From (22) and (56) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) + 12119862

2

3cos 1198901198882 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(57)

From (23) and (56) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus121198622

3cosec2 (119862

3119905)+3119862

2

2minus60119862

2

21198622

3cot2 (119862

3119905)

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(58)

From (17) and (52)ndash(57) we get the EoS parameter of DE

119908(de)

=minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [121198622

3cosec2 (119862

3119905)

minus 1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus 31198622

2+ 60119862

2

21198622

3cot2 (119862

3119905)

+61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(59)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

+

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(60)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

+

3

21198622

2

sec2 (1198623119905)]119860

119898 (61)

where 119860119898is the average anisotropy parameter

ISRN Astronomy and Astrophysics 7

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ sin 1205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + cos 120579 119889120601)2

(62)

Thus the metric (62) together with (54)ndash(59) constitutes aBianchi type IX perfect fluid dark energy cosmologicalmodelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (62) together with (54)ndash(59) constitutes a Bianchi type IX perfect fluid dark energycosmological model in general relativity with 119862

4= 0

34 Isotropic Cosmological Models To get isotropic cosmo-logical models let us assign the value unity to 119899 in (16) andthen we get

119877 = 119878 (63)

where 119877 and 119878 are the metric potentialsUsing (63) the field equations (17)ndash(20) can be written as

2

119878

119878

+

1198782

1198782+

1

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(64)

2

119878

119878

+

1198782

1198782+

120575

1198782minus

3

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(65)

3

1198782

1198782+

120575

1198782minus

1

41198782+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (66)

120601 + 3

120601 119878

119878

+

119903

2120601

1206012= 0 (67)

From (64) and (65) we get

3119898 1198782+ (120575 minus 1) = 0 (68)

35 Bianchi Type II and VIII Cosmological Models in IsotropicForm From (68) we get

119877 = 119878 = (1198626119905 + 1198627) (69)

where 1198626= ((1 minus 120575)3119898)

12 and1198627are constants

From (67) and (69) we get

120601(119903+2)2

=

119903 + 2

2

(1198629minus

1198628

2(1198626119905 + 1198627)2) 119903 = minus 2 (70)

where 1198628and 119862

9are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198626119905 + 1198627)minus3(1+119908

(119898)) (71)

where 1205880is a constant of integration

From (66) (69) and (71) we get the dark energy density

120588(de)

= 31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6minus 1205880(1198626119905 + 1198627)minus3(1+119908

(119898))

(72)

From (22) and (72) we get the skewness parameter

120574 = minus 1198981198626

2[31198626

2+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(73)

From (23) and (72) we get the skewness parameter

120585 = 21198981198626

2[31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(74)

From (64) and (69)ndash(73) we get the EoS parameter of DE

119908(de)

= minus [(1 minus 119898)1198622

6+

1

4

(1198626119905 + 1198627)minus2

minus

1205961198622

8

2

(1198626119905 + 1198627)minus6

+1205880119908(119898)(1198626119905 + 1198627)minus3(1+119908

(119898))]

times [31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(75)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

Ω(de)

=

120588(de)

31198672

= (1198626119905 + 1198627)2minus

5

121198626

2+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

minus

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

(76)

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

ISRN Astronomy and Astrophysics 3

The energy conservation equation 119879(119898)119894119895119895

= 0 of the perf-ect fluid leads to

120588(119898)+ 3 (1 + 119908

(119898)) 120588(119898)119867 = 0 (13)

whereas the energy conservation equation 119879(de)119894119895119895

= 0 of theDE component yields

120588(de)

+ 3 (1 + 119908(de)) 120588(de)119867 = 0 (14)

Following Akarsu and Kilinc [30] we assume that the EoSparameter of the perfect fluid is a constant that is

119908(119898)

=

119901(119898)

120588(119898)

= cons tan 119905 (15)

while119908(de) has been allowed to be a function of time since thecurrent cosmological data from SN Ia CMB and large scalestructures mildly favor dynamically evolving DE crossing thephantom divide line (PDL)

Now we assume that the shear scalar (120590) in the models isproportional to expansion scalar (120579)

This condition leads to

119878 = 119877119899 (16)

where 119877 and 119878 are the metric potentials and 119899 is a positiveconstant

Since we are looking for a model explaining an expand-ing universe with acceleration we also assume that theanisotropic distribution of DE ensures the present accelerat-ing universe Thus (8) to (12) may be rewritten as

119877

+

119878

119878

+

119878

119877119878

+

1198782

41198774minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(17)

2

119877

+

2+ 120575

1198772

minus

31198782

41198774minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(18)

2

119878

119877119878

+

2+ 120575

1198772

minus

1198782

41198774+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (19)

120601 +

120601 (2

119877

+

119878

119878

) +

119903

2120601

1206012= 0 (20)

120588(de)

+ 3 (1 + 119908(de)) 120588(de)119867 + 120588

(de)(2120574119867119909+ 120585119867119911) = 0 (21)

The third term of (21) arises due to the deviation from 119908(de)

while the first two terms of (21) are deviation free part of119879(de)119894119895

According to (21) the behavior of 120588(de) is controlled bythe deviation free part of EoS parameter of DE but deviationwill affect 120588(de) indirectly since as can be seen later they affectthe value of EoS parameter But we are looking for physicallyviable models of the universe consistent with observationsHence we constrained 120585(119905) and 120574(119905) by assuming the special

dynamics which are consistent with (21) The dynamics ofskewness parameter on 119909-axis or 119910-axis 120574 and 119911-axis 120585 aregiven by

120574 =

minus119898119867119867119911

120588(de) (22)

120585 =

2119898119867119867119909

120588(de) (23)

where119898 is the dimensionless constant that parameterizes theamplitude of the deviation from 119908

(de) and can be given realvalues

From (17) and (18) we get

119877

+ 1198621

2

1198772minus

120575

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (24)

where 1198621= (3(119899

2minus 1) minus 119898(119899 + 2)

2)3(119899 minus 1)

31 Bianchi Type II (120575 = 0)Cosmological Model If 120575 = 0 (24)can be written as

119877

+ 1198621

2

1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (25)

From (25) we get

119877 = (1198961119905 + 1198962)1(2minus119899)

119899 = 2 (26)

where 1198992(1 + 1198961

2) + 119899((119862

1minus 2)1198962

1minus 4) + ((1 minus 119862

1)1198962

1+ 4) =

0 1198961= 0 and 119896

2are arbitrary constants

From (16) and (26) we get

119878 = (1198961119905 + 1198962)119899(2minus119899)

(27)

From (11) (16) and (26) we get

120601(119903+2)2

= [

119903+2

2

(

(119899minus2) 1198963

2119899

(1198961119905+1198962)2119899(119899minus2)

+1198964)]

119903 = minus 2 119899 = 0

(28)

where 1198963and 1198964are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198961119905 + 1198962)minus(1+119908

(119898))(119899+2)(2minus119899)

(29)

where 1205880is a constant of integration

From (19) and (26)ndash(29) we get the dark energy density

120588(de)

=

[119899 (81198962

1+ 4) + (4119896

2

1minus 4) minus 119899

2]

4(2 minus 119899)2(1198961119905 + 1198962)2

+

1205961198963

2

2(1198961119905 + 1198962)2(119899+2)(2minus119899)

minus

1205880

(1198961119905 + 1198962)(1+119908(119898))(119899+2)(2minus119899)

(30)

4 ISRN Astronomy and Astrophysics

From (22) and (30) we get the skewness parameter

120574 = 4119898119899 (119899 + 2) 1198962

1

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+ 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

]

minus1

(31)

From (23) and (30) we get the skewness parameter

120585 = 8119898 (119899 + 2) 1198962

1

times [121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

minus 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2) ]

minus1

(32)

From (17) and (26)ndash(31) we get the EoS parameter of DE

119908(de)

= minus [1198965(1198961119905 + 1198962)minus2minus 6120596119896

3

2(2 minus 119899)

2

times (1198961119905 + 1198962)2(119899+2)(119899minus2)

+ 121199081198981205880(2 minus 119899)

2

times (1198961119905 + 1198962)(1+119908(119898))(119899+2)(119899minus2)

]

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2

times (1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(119899minus2)

]

minus1

(33)

where 1198965= 1198992[41198962

1(6 + 119898) + 3] + 4119899(2119898119896

2

1minus 3) + 12(119896

2

1minus 1)

The density parameters of perfect fluid and DE are asfollows

Ω(119898)

=

120588(119898)

31198672

=

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

119899 = minus 2

Ω(de)

=

120588(de)

31198672

=

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

4(119899 + 2)21198962

1

+

31205961198962

3(2 minus 119899)

2

2(119899 + 2)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

minus

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

(34)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

8(119899 minus 1)21198962

1

+

31205961198962

3(2 minus 119899)

2

4(119899 minus 1)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

]119860119898

(35)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198961119905 + 1198962)2(2minus119899)

(1198891205792+ 1198891206012)

+ (1198961119905 + 1198962)2119899(2minus119899)

(119889120593 + 120579119889120601)2

(36)

Thus the metric (36) together with (28)ndash(33) constitutes aBianchi type II perfect fluid dark energy cosmological modelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (36) together with (28)ndash(33) constitutes a Bianchi type II perfect fluid dark energycosmological model in general relativity with 119896

3= 0

32 Bianchi Type VIII (120575 = minus1) Cosmological Model If 120575 =minus1 (24) can be written as

119877

+ 1198621

2

1198772+

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (37)

From (37) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (38)

where1198621= (9minus16119898)3 119862

2

2= 3(16119898minus9) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (38) we get

119877 = (

1198622

1198623

) sin (1198623119905) (39)

From (16) and (39) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(40)

ISRN Astronomy and Astrophysics 5

From (11) (16) and (39) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905) + 2) + 119862

5)]

119903 = minus 2

(41)

where 1198624and119862

5are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(42)

where 1205880is a constant of integration

From (19) and (39)ndash(42) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) minus 4119862

2

3cosec2 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(43)

From (22) and (43) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905) minus 3119862

2

2

+ 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(44)

From (23) and (43) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [121198622

3cosec2 (119862

3119905) + 3119862

2

2minus 60119862

2

21198622

3cot2 (119862

3119905)

minus 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

+1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(45)

From (17) and (39)ndash(44) we get the EoS parameter of DE

119908(de)

= minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(46)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

minus

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(47)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

minus

3

21198622

2sec2 (119862

3119905)]119860

119898 (48)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ cosh21205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + sinh 120579119889120601)2

(49)

Thus the metric (49) together with (41)ndash(46) constitutes aBianchi type VIII perfect fluid dark energy cosmologicalmodel in Saez and Ballester [1] theory of gravitation

Also we observe that the metric (49) together with (41)ndash(46) constitutes a Bianchi type VIII perfect fluid dark energycosmological model in general relativity with 119862

4= 0

6 ISRN Astronomy and Astrophysics

33 Bianchi Type IX (120575 = 1) Cosmological Model If 120575 = 1(24) can be written as

119877

+ 1198621

2

1198772minus

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (50)

From (50) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (51)

where1198621= (9minus16119898)3 1198622

2= 3(9minus16119898) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (51) we get

119877 = (

1198622

1198623

) sin (1198623119905) (52)

From (16) and (52) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(53)

From (11) (16) and (52) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905)+2)+119862

5)] 119903 = minus 2

(54)

where 1198624and119862

5are integrating constants One has

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(55)

where 1205880is a constant of integration

From (19) and (52)ndash(55) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) + 4119862

2

3cos 1198901198882 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(56)

From (22) and (56) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) + 12119862

2

3cos 1198901198882 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(57)

From (23) and (56) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus121198622

3cosec2 (119862

3119905)+3119862

2

2minus60119862

2

21198622

3cot2 (119862

3119905)

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(58)

From (17) and (52)ndash(57) we get the EoS parameter of DE

119908(de)

=minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [121198622

3cosec2 (119862

3119905)

minus 1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus 31198622

2+ 60119862

2

21198622

3cot2 (119862

3119905)

+61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(59)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

+

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(60)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

+

3

21198622

2

sec2 (1198623119905)]119860

119898 (61)

where 119860119898is the average anisotropy parameter

ISRN Astronomy and Astrophysics 7

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ sin 1205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + cos 120579 119889120601)2

(62)

Thus the metric (62) together with (54)ndash(59) constitutes aBianchi type IX perfect fluid dark energy cosmologicalmodelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (62) together with (54)ndash(59) constitutes a Bianchi type IX perfect fluid dark energycosmological model in general relativity with 119862

4= 0

34 Isotropic Cosmological Models To get isotropic cosmo-logical models let us assign the value unity to 119899 in (16) andthen we get

119877 = 119878 (63)

where 119877 and 119878 are the metric potentialsUsing (63) the field equations (17)ndash(20) can be written as

2

119878

119878

+

1198782

1198782+

1

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(64)

2

119878

119878

+

1198782

1198782+

120575

1198782minus

3

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(65)

3

1198782

1198782+

120575

1198782minus

1

41198782+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (66)

120601 + 3

120601 119878

119878

+

119903

2120601

1206012= 0 (67)

From (64) and (65) we get

3119898 1198782+ (120575 minus 1) = 0 (68)

35 Bianchi Type II and VIII Cosmological Models in IsotropicForm From (68) we get

119877 = 119878 = (1198626119905 + 1198627) (69)

where 1198626= ((1 minus 120575)3119898)

12 and1198627are constants

From (67) and (69) we get

120601(119903+2)2

=

119903 + 2

2

(1198629minus

1198628

2(1198626119905 + 1198627)2) 119903 = minus 2 (70)

where 1198628and 119862

9are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198626119905 + 1198627)minus3(1+119908

(119898)) (71)

where 1205880is a constant of integration

From (66) (69) and (71) we get the dark energy density

120588(de)

= 31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6minus 1205880(1198626119905 + 1198627)minus3(1+119908

(119898))

(72)

From (22) and (72) we get the skewness parameter

120574 = minus 1198981198626

2[31198626

2+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(73)

From (23) and (72) we get the skewness parameter

120585 = 21198981198626

2[31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(74)

From (64) and (69)ndash(73) we get the EoS parameter of DE

119908(de)

= minus [(1 minus 119898)1198622

6+

1

4

(1198626119905 + 1198627)minus2

minus

1205961198622

8

2

(1198626119905 + 1198627)minus6

+1205880119908(119898)(1198626119905 + 1198627)minus3(1+119908

(119898))]

times [31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(75)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

Ω(de)

=

120588(de)

31198672

= (1198626119905 + 1198627)2minus

5

121198626

2+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

minus

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

(76)

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ThermodynamicsJournal of

4 ISRN Astronomy and Astrophysics

From (22) and (30) we get the skewness parameter

120574 = 4119898119899 (119899 + 2) 1198962

1

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+ 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

]

minus1

(31)

From (23) and (30) we get the skewness parameter

120585 = 8119898 (119899 + 2) 1198962

1

times [121205880(2 minus 119899)

2(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

minus 6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2) ]

minus1

(32)

From (17) and (26)ndash(31) we get the EoS parameter of DE

119908(de)

= minus [1198965(1198961119905 + 1198962)minus2minus 6120596119896

3

2(2 minus 119899)

2

times (1198961119905 + 1198962)2(119899+2)(119899minus2)

+ 121199081198981205880(2 minus 119899)

2

times (1198961119905 + 1198962)(1+119908(119898))(119899+2)(119899minus2)

]

times [ (12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

+6120596(2 minus 119899)21198962

3(1198961119905 + 1198962)4119899(119899minus2)

minus121205880(2 minus 119899)

2

times (1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(119899minus2)

]

minus1

(33)

where 1198965= 1198992[41198962

1(6 + 119898) + 3] + 4119899(2119898119896

2

1minus 3) + 12(119896

2

1minus 1)

The density parameters of perfect fluid and DE are asfollows

Ω(119898)

=

120588(119898)

31198672

=

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

119899 = minus 2

Ω(de)

=

120588(de)

31198672

=

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

4(119899 + 2)21198962

1

+

31205961198962

3(2 minus 119899)

2

2(119899 + 2)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

minus

3(2 minus 119899)21205880

(119899 + 2)21198962

1

(1198961119905 + 1198962)((2minus3119899)minus119908

(119898)(119899+2))(2minus119899)

(34)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

(12 (1198962

1minus 1) + 12119899 (2119896

2

1+ 1) minus 3119899

2)

8(119899 minus 1)21198962

1

+

31205961198962

3(2 minus 119899)

2

4(119899 minus 1)21198962

1

(1198961119905 + 1198962)4119899(119899minus2)

]119860119898

(35)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198961119905 + 1198962)2(2minus119899)

(1198891205792+ 1198891206012)

+ (1198961119905 + 1198962)2119899(2minus119899)

(119889120593 + 120579119889120601)2

(36)

Thus the metric (36) together with (28)ndash(33) constitutes aBianchi type II perfect fluid dark energy cosmological modelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (36) together with (28)ndash(33) constitutes a Bianchi type II perfect fluid dark energycosmological model in general relativity with 119896

3= 0

32 Bianchi Type VIII (120575 = minus1) Cosmological Model If 120575 =minus1 (24) can be written as

119877

+ 1198621

2

1198772+

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (37)

From (37) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (38)

where1198621= (9minus16119898)3 119862

2

2= 3(16119898minus9) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (38) we get

119877 = (

1198622

1198623

) sin (1198623119905) (39)

From (16) and (39) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(40)

ISRN Astronomy and Astrophysics 5

From (11) (16) and (39) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905) + 2) + 119862

5)]

119903 = minus 2

(41)

where 1198624and119862

5are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(42)

where 1205880is a constant of integration

From (19) and (39)ndash(42) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) minus 4119862

2

3cosec2 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(43)

From (22) and (43) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905) minus 3119862

2

2

+ 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(44)

From (23) and (43) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [121198622

3cosec2 (119862

3119905) + 3119862

2

2minus 60119862

2

21198622

3cot2 (119862

3119905)

minus 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

+1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(45)

From (17) and (39)ndash(44) we get the EoS parameter of DE

119908(de)

= minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(46)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

minus

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(47)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

minus

3

21198622

2sec2 (119862

3119905)]119860

119898 (48)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ cosh21205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + sinh 120579119889120601)2

(49)

Thus the metric (49) together with (41)ndash(46) constitutes aBianchi type VIII perfect fluid dark energy cosmologicalmodel in Saez and Ballester [1] theory of gravitation

Also we observe that the metric (49) together with (41)ndash(46) constitutes a Bianchi type VIII perfect fluid dark energycosmological model in general relativity with 119862

4= 0

6 ISRN Astronomy and Astrophysics

33 Bianchi Type IX (120575 = 1) Cosmological Model If 120575 = 1(24) can be written as

119877

+ 1198621

2

1198772minus

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (50)

From (50) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (51)

where1198621= (9minus16119898)3 1198622

2= 3(9minus16119898) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (51) we get

119877 = (

1198622

1198623

) sin (1198623119905) (52)

From (16) and (52) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(53)

From (11) (16) and (52) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905)+2)+119862

5)] 119903 = minus 2

(54)

where 1198624and119862

5are integrating constants One has

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(55)

where 1205880is a constant of integration

From (19) and (52)ndash(55) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) + 4119862

2

3cos 1198901198882 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(56)

From (22) and (56) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) + 12119862

2

3cos 1198901198882 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(57)

From (23) and (56) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus121198622

3cosec2 (119862

3119905)+3119862

2

2minus60119862

2

21198622

3cot2 (119862

3119905)

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(58)

From (17) and (52)ndash(57) we get the EoS parameter of DE

119908(de)

=minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [121198622

3cosec2 (119862

3119905)

minus 1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus 31198622

2+ 60119862

2

21198622

3cot2 (119862

3119905)

+61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(59)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

+

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(60)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

+

3

21198622

2

sec2 (1198623119905)]119860

119898 (61)

where 119860119898is the average anisotropy parameter

ISRN Astronomy and Astrophysics 7

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ sin 1205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + cos 120579 119889120601)2

(62)

Thus the metric (62) together with (54)ndash(59) constitutes aBianchi type IX perfect fluid dark energy cosmologicalmodelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (62) together with (54)ndash(59) constitutes a Bianchi type IX perfect fluid dark energycosmological model in general relativity with 119862

4= 0

34 Isotropic Cosmological Models To get isotropic cosmo-logical models let us assign the value unity to 119899 in (16) andthen we get

119877 = 119878 (63)

where 119877 and 119878 are the metric potentialsUsing (63) the field equations (17)ndash(20) can be written as

2

119878

119878

+

1198782

1198782+

1

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(64)

2

119878

119878

+

1198782

1198782+

120575

1198782minus

3

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(65)

3

1198782

1198782+

120575

1198782minus

1

41198782+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (66)

120601 + 3

120601 119878

119878

+

119903

2120601

1206012= 0 (67)

From (64) and (65) we get

3119898 1198782+ (120575 minus 1) = 0 (68)

35 Bianchi Type II and VIII Cosmological Models in IsotropicForm From (68) we get

119877 = 119878 = (1198626119905 + 1198627) (69)

where 1198626= ((1 minus 120575)3119898)

12 and1198627are constants

From (67) and (69) we get

120601(119903+2)2

=

119903 + 2

2

(1198629minus

1198628

2(1198626119905 + 1198627)2) 119903 = minus 2 (70)

where 1198628and 119862

9are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198626119905 + 1198627)minus3(1+119908

(119898)) (71)

where 1205880is a constant of integration

From (66) (69) and (71) we get the dark energy density

120588(de)

= 31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6minus 1205880(1198626119905 + 1198627)minus3(1+119908

(119898))

(72)

From (22) and (72) we get the skewness parameter

120574 = minus 1198981198626

2[31198626

2+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(73)

From (23) and (72) we get the skewness parameter

120585 = 21198981198626

2[31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(74)

From (64) and (69)ndash(73) we get the EoS parameter of DE

119908(de)

= minus [(1 minus 119898)1198622

6+

1

4

(1198626119905 + 1198627)minus2

minus

1205961198622

8

2

(1198626119905 + 1198627)minus6

+1205880119908(119898)(1198626119905 + 1198627)minus3(1+119908

(119898))]

times [31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(75)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

Ω(de)

=

120588(de)

31198672

= (1198626119905 + 1198627)2minus

5

121198626

2+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

minus

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

(76)

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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AstronomyAdvances in

International Journal of

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Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

ISRN Astronomy and Astrophysics 5

From (11) (16) and (39) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905) + 2) + 119862

5)]

119903 = minus 2

(41)

where 1198624and119862

5are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(42)

where 1205880is a constant of integration

From (19) and (39)ndash(42) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) minus 4119862

2

3cosec2 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(43)

From (22) and (43) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905) minus 3119862

2

2

+ 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(44)

From (23) and (43) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [121198622

3cosec2 (119862

3119905) + 3119862

2

2minus 60119862

2

21198622

3cot2 (119862

3119905)

minus 61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

+1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(45)

From (17) and (39)ndash(44) we get the EoS parameter of DE

119908(de)

= minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [601198622

21198622

3cot2 (119862

3119905) minus 12119862

2

3cosec2 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(46)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

minus

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(47)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

minus

3

21198622

2sec2 (119862

3119905)]119860

119898 (48)

where 119860119898is the average anisotropy parameter

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ cosh21205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + sinh 120579119889120601)2

(49)

Thus the metric (49) together with (41)ndash(46) constitutes aBianchi type VIII perfect fluid dark energy cosmologicalmodel in Saez and Ballester [1] theory of gravitation

Also we observe that the metric (49) together with (41)ndash(46) constitutes a Bianchi type VIII perfect fluid dark energycosmological model in general relativity with 119862

4= 0

6 ISRN Astronomy and Astrophysics

33 Bianchi Type IX (120575 = 1) Cosmological Model If 120575 = 1(24) can be written as

119877

+ 1198621

2

1198772minus

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (50)

From (50) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (51)

where1198621= (9minus16119898)3 1198622

2= 3(9minus16119898) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (51) we get

119877 = (

1198622

1198623

) sin (1198623119905) (52)

From (16) and (52) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(53)

From (11) (16) and (52) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905)+2)+119862

5)] 119903 = minus 2

(54)

where 1198624and119862

5are integrating constants One has

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(55)

where 1205880is a constant of integration

From (19) and (52)ndash(55) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) + 4119862

2

3cos 1198901198882 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(56)

From (22) and (56) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) + 12119862

2

3cos 1198901198882 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(57)

From (23) and (56) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus121198622

3cosec2 (119862

3119905)+3119862

2

2minus60119862

2

21198622

3cot2 (119862

3119905)

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(58)

From (17) and (52)ndash(57) we get the EoS parameter of DE

119908(de)

=minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [121198622

3cosec2 (119862

3119905)

minus 1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus 31198622

2+ 60119862

2

21198622

3cot2 (119862

3119905)

+61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(59)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

+

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(60)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

+

3

21198622

2

sec2 (1198623119905)]119860

119898 (61)

where 119860119898is the average anisotropy parameter

ISRN Astronomy and Astrophysics 7

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ sin 1205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + cos 120579 119889120601)2

(62)

Thus the metric (62) together with (54)ndash(59) constitutes aBianchi type IX perfect fluid dark energy cosmologicalmodelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (62) together with (54)ndash(59) constitutes a Bianchi type IX perfect fluid dark energycosmological model in general relativity with 119862

4= 0

34 Isotropic Cosmological Models To get isotropic cosmo-logical models let us assign the value unity to 119899 in (16) andthen we get

119877 = 119878 (63)

where 119877 and 119878 are the metric potentialsUsing (63) the field equations (17)ndash(20) can be written as

2

119878

119878

+

1198782

1198782+

1

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(64)

2

119878

119878

+

1198782

1198782+

120575

1198782minus

3

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(65)

3

1198782

1198782+

120575

1198782minus

1

41198782+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (66)

120601 + 3

120601 119878

119878

+

119903

2120601

1206012= 0 (67)

From (64) and (65) we get

3119898 1198782+ (120575 minus 1) = 0 (68)

35 Bianchi Type II and VIII Cosmological Models in IsotropicForm From (68) we get

119877 = 119878 = (1198626119905 + 1198627) (69)

where 1198626= ((1 minus 120575)3119898)

12 and1198627are constants

From (67) and (69) we get

120601(119903+2)2

=

119903 + 2

2

(1198629minus

1198628

2(1198626119905 + 1198627)2) 119903 = minus 2 (70)

where 1198628and 119862

9are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198626119905 + 1198627)minus3(1+119908

(119898)) (71)

where 1205880is a constant of integration

From (66) (69) and (71) we get the dark energy density

120588(de)

= 31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6minus 1205880(1198626119905 + 1198627)minus3(1+119908

(119898))

(72)

From (22) and (72) we get the skewness parameter

120574 = minus 1198981198626

2[31198626

2+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(73)

From (23) and (72) we get the skewness parameter

120585 = 21198981198626

2[31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(74)

From (64) and (69)ndash(73) we get the EoS parameter of DE

119908(de)

= minus [(1 minus 119898)1198622

6+

1

4

(1198626119905 + 1198627)minus2

minus

1205961198622

8

2

(1198626119905 + 1198627)minus6

+1205880119908(119898)(1198626119905 + 1198627)minus3(1+119908

(119898))]

times [31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(75)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

Ω(de)

=

120588(de)

31198672

= (1198626119905 + 1198627)2minus

5

121198626

2+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

minus

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

(76)

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 ISRN Astronomy and Astrophysics

33 Bianchi Type IX (120575 = 1) Cosmological Model If 120575 = 1(24) can be written as

119877

+ 1198621

2

1198772minus

1

(119899 minus 1) 1198772+

1198772119899minus4

(119899 minus 1)

= 0 119899 = 1 (50)

From (50) for 119899 = 2 and with suitable substitution we get

2= 1198622

2minus 1198622

31198772 (51)

where1198621= (9minus16119898)3 1198622

2= 3(9minus16119898) and1198622

3= 3(12minus

16119898) 119898 = 34 916From (51) we get

119877 = (

1198622

1198623

) sin (1198623119905) (52)

From (16) and (52) we get

119878 = [(

1198622

1198623

) sin (1198623119905)]

2

(53)

From (11) (16) and (52) we get

120601(119903+2)2

= [

119903 + 2

2

(

minus1198624

3

(

1198623

1198622

)

4

cot (1198623119905)

times (cosec2 (1198623119905)+2)+119862

5)] 119903 = minus 2

(54)

where 1198624and119862

5are integrating constants One has

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880[(

1198622

1198623

) sin (1198623119905)]

minus4(1+119908(119898))

(55)

where 1205880is a constant of integration

From (19) and (52)ndash(55) we get the dark energy density

120588(de)

=

1

41198622

2

[201198622

21198622

3cot2 (119862

3119905) + 4119862

2

3cos 1198901198882 (119862

3119905)

minus 1198622

2+ 2120596 119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus412058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

(56)

From (22) and (56) we get the skewness parameter

120574 = 321198981198622

21198622

3cot2 (119862

3119905)

times [601198622

21198622

3cot2 (119862

3119905) + 12119862

2

3cos 1198901198882 (119862

3119905)

minus 31198622

2+ 6120596119862

2

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

minus1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

]

minus1

(57)

From (23) and (56) we get the skewness parameter

120585 = 321198981198622

21198622

3cot2 (119862

3119905)

times [1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus121198622

3cosec2 (119862

3119905)+3119862

2

2minus60119862

2

21198622

3cot2 (119862

3119905)

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(58)

From (17) and (52)ndash(57) we get the EoS parameter of DE

119908(de)

=minus [31198622

2(1minus12119862

2

3)+4119862

2

21198622

3cot2 (119862

3119905) (12+8119898)

+ 12119908(119898)12058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

times [121198622

3cosec2 (119862

3119905)

minus 1212058801198622

2((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

minus 31198622

2+ 60119862

2

21198622

3cot2 (119862

3119905)

+61205961198622

41198622

2((

1198622

1198623

) sin (1198623119905))

minus8

]

minus1

(59)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

Ω(de)

=

120588(de)

31198672=

15

16

+

3

161198622

2

sec2 (1198623119905)

minus

31205880

161198622

3

tan2 (1198623119905) ((

1198622

1198623

) sin (1198623119905))

minus4(1+119908(119898))

(60)

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= [

15

2

+

3

21198622

2

sec2 (1198623119905)]119860

119898 (61)

where 119860119898is the average anisotropy parameter

ISRN Astronomy and Astrophysics 7

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ sin 1205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + cos 120579 119889120601)2

(62)

Thus the metric (62) together with (54)ndash(59) constitutes aBianchi type IX perfect fluid dark energy cosmologicalmodelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (62) together with (54)ndash(59) constitutes a Bianchi type IX perfect fluid dark energycosmological model in general relativity with 119862

4= 0

34 Isotropic Cosmological Models To get isotropic cosmo-logical models let us assign the value unity to 119899 in (16) andthen we get

119877 = 119878 (63)

where 119877 and 119878 are the metric potentialsUsing (63) the field equations (17)ndash(20) can be written as

2

119878

119878

+

1198782

1198782+

1

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(64)

2

119878

119878

+

1198782

1198782+

120575

1198782minus

3

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(65)

3

1198782

1198782+

120575

1198782minus

1

41198782+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (66)

120601 + 3

120601 119878

119878

+

119903

2120601

1206012= 0 (67)

From (64) and (65) we get

3119898 1198782+ (120575 minus 1) = 0 (68)

35 Bianchi Type II and VIII Cosmological Models in IsotropicForm From (68) we get

119877 = 119878 = (1198626119905 + 1198627) (69)

where 1198626= ((1 minus 120575)3119898)

12 and1198627are constants

From (67) and (69) we get

120601(119903+2)2

=

119903 + 2

2

(1198629minus

1198628

2(1198626119905 + 1198627)2) 119903 = minus 2 (70)

where 1198628and 119862

9are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198626119905 + 1198627)minus3(1+119908

(119898)) (71)

where 1205880is a constant of integration

From (66) (69) and (71) we get the dark energy density

120588(de)

= 31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6minus 1205880(1198626119905 + 1198627)minus3(1+119908

(119898))

(72)

From (22) and (72) we get the skewness parameter

120574 = minus 1198981198626

2[31198626

2+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(73)

From (23) and (72) we get the skewness parameter

120585 = 21198981198626

2[31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(74)

From (64) and (69)ndash(73) we get the EoS parameter of DE

119908(de)

= minus [(1 minus 119898)1198622

6+

1

4

(1198626119905 + 1198627)minus2

minus

1205961198622

8

2

(1198626119905 + 1198627)minus6

+1205880119908(119898)(1198626119905 + 1198627)minus3(1+119908

(119898))]

times [31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(75)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

Ω(de)

=

120588(de)

31198672

= (1198626119905 + 1198627)2minus

5

121198626

2+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

minus

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

(76)

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

ISRN Astronomy and Astrophysics 7

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ ((

1198622

1198623

) sin (1198623119905))

2

(1198891205792+ sin 1205791198891206012)

+ ((

1198622

1198623

) sin (1198623119905))

4

(119889120593 + cos 120579 119889120601)2

(62)

Thus the metric (62) together with (54)ndash(59) constitutes aBianchi type IX perfect fluid dark energy cosmologicalmodelin Saez and Ballester [1] theory of gravitation

Also we observe that the metric (62) together with (54)ndash(59) constitutes a Bianchi type IX perfect fluid dark energycosmological model in general relativity with 119862

4= 0

34 Isotropic Cosmological Models To get isotropic cosmo-logical models let us assign the value unity to 119899 in (16) andthen we get

119877 = 119878 (63)

where 119877 and 119878 are the metric potentialsUsing (63) the field equations (17)ndash(20) can be written as

2

119878

119878

+

1198782

1198782+

1

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120574) 120588(de)

(64)

2

119878

119878

+

1198782

1198782+

120575

1198782minus

3

41198782minus

120596

2

120601119903 1206012

= minus119908(119898)120588(119898)minus (119908(de)

+ 120585) 120588(de)

(65)

3

1198782

1198782+

120575

1198782minus

1

41198782+

120596

2

120601119903 1206012= 120588(119898)+ 120588(de) (66)

120601 + 3

120601 119878

119878

+

119903

2120601

1206012= 0 (67)

From (64) and (65) we get

3119898 1198782+ (120575 minus 1) = 0 (68)

35 Bianchi Type II and VIII Cosmological Models in IsotropicForm From (68) we get

119877 = 119878 = (1198626119905 + 1198627) (69)

where 1198626= ((1 minus 120575)3119898)

12 and1198627are constants

From (67) and (69) we get

120601(119903+2)2

=

119903 + 2

2

(1198629minus

1198628

2(1198626119905 + 1198627)2) 119903 = minus 2 (70)

where 1198628and 119862

9are integrating constants

From (13) we get the energy density of the perfect fluid

120588(119898)

= 1205880(1198626119905 + 1198627)minus3(1+119908

(119898)) (71)

where 1205880is a constant of integration

From (66) (69) and (71) we get the dark energy density

120588(de)

= 31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6minus 1205880(1198626119905 + 1198627)minus3(1+119908

(119898))

(72)

From (22) and (72) we get the skewness parameter

120574 = minus 1198981198626

2[31198626

2+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(73)

From (23) and (72) we get the skewness parameter

120585 = 21198981198626

2[31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(74)

From (64) and (69)ndash(73) we get the EoS parameter of DE

119908(de)

= minus [(1 minus 119898)1198622

6+

1

4

(1198626119905 + 1198627)minus2

minus

1205961198622

8

2

(1198626119905 + 1198627)minus6

+1205880119908(119898)(1198626119905 + 1198627)minus3(1+119908

(119898))]

times [31198622

6+

(4120575 minus 1)

4

(1198626119905 + 1198627)minus2

+

1205961198622

8

2

(1198626119905 + 1198627)minus6

minus1205880(1198626119905 + 1198627)minus3(1+119908

(119898))]

minus1

(75)

The density parameters of perfect fluid and DE are as follows

Ω(119898)

=

120588(119898)

31198672

=

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

Ω(de)

=

120588(de)

31198672

= (1198626119905 + 1198627)2minus

5

121198626

2+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

minus

1205880

31198622

6

(1198626119905 + 1198627)minus(1+3119908

(119898))

(76)

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 ISRN Astronomy and Astrophysics

The overall density parameterΩ is given by

Ω = Ω(119898)+ Ω(de)

= (1198626119905 + 1198627)2minus

5

121198622

6

+

1205961198622

8

61198622

6

(1198626119905 + 1198627)minus4

(77)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ (1198626119905 + 1198627)2[1198891205792+ 119891 (120579) 119889120601

2]

+ (1198626119905 + 1198627)2[119889120593 + ℎ (120579) 119889120601]

2

(78)

Themetric (78) together with (70) to (75) constitutes Bianchitype II and VIII perfect fluid dark energy cosmologicalmodels in isotropic form of Saez and Ballester [1] theory ofgravitation respectively with 120575 = 0 119891(120579) = 1 and ℎ(120579) = 120579and 120575 = 0119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

Also we can observe that the metric (78) together with(70) to (75) constitutes Bianchi type II and VIII perfect fluiddark energy cosmological models in isotropic form of generalrelativity with 119862

8= 0 respectively for 120575 = 0 119891(120579) =

1 and ℎ(120579) = 120579 and 120575 = minus1119891(120579) = cosh 120579 and ℎ(120579) = sinh 120579

36 Bianchi Type IX (120575 = 1) Cosmological Model in IsotropicForm From (68) if 120575 = 1 we get

119877 = 119878 = cons tan 119905 (119904119886119910 11986210) (79)

From (67) and (79) we get

120601(119903+2)2

= [

119903 + 2

2

(11986211119905 + 11986212)] (80)

The metric (4) in this case can be written as

1198891199042= minus 119889119905

2+ 1198622

10(1198891205792+ sin 120579 1198891206012)

+ 1198622

10(119889120593 + cos 120579119889120601)2

(81)

The metric (81) together with (80) constitutes flat cosmolog-ical model in Saez and Ballester [1] theory of gravitation

4 Some Other Important Features ofthe Models

41 Bianchi Type II Anisotropic Cosmological Model (120575 = 0)The spatial volume for the model (36) is

119881 = 1198863= (1198961119905 + 1198962)(119899+2)(2minus119899)

(82)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 = (

119899 + 2

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(83)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) = (

119899 minus 1

2 minus 119899

)

21198962

1

3(1198961119905 + 1198962)2 (84)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

4 (1 minus 119899)

(119899 + 2)

119899 = 1 2amp minus 2 (85)

The recent observations of SN Ia reveal that the present uni-verse is accelerating and the value of deceleration parameterlies somewhere in the range minus1 lt 119902 lt 0 From (85) we cansee that for 119899 gt 2 119902 is negative which may be attributed tothe current accelerated expansion of the universe

The components of Hubble parameter 119867119909 119867119910 and119867

119911

are given by

119867119909= 119867119910=

119877

= (

1

2 minus 119899

)

1198961

(1198961119905 + 1198962)

119867119911=

119878

119878

= (

119899

2 minus 119899

)

1198961

(1198961119905 + 1198962)

(86)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) = (

119899 + 2

2 minus 119899

)

1198961

3 (1198961119905 + 1198962)

(87)

The average anisotropy parameter is defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

2(119899 minus 1)2

(119899 + 2)2 (88)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673= (1 +

3 (119899 minus 2)

(119899 + 2)

)(1 +

6 (119899 minus 2)

(119899 + 2)

)

119904 =

119903 minus 1

3 (119902 minus (12))

=

1 minus (1 + 3 (119899 minus 2) (119899 + 2)) (1 + 6 (119899 minus 2) (119899 + 2))

9 ((119899 minus 2) (119899 + 2) + 12)

(89)

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

ISRN Astronomy and Astrophysics 9

Thedynamics of state finder 119903 119904depends on 119899 It follows thatin the derived model one can choose the pair of state finderwhich can successfully differentiate between a wide varietyof DEmodels including cosmological constant quintessencephantom quintom the Chaplygin gas braneworld modelsand interacting DE models For example if we put 119899 = 2the state finder pair will be 1 0 which yields the Λ119862119863119872(cosmological constant cold dark matter) model

42 Bianchi Type VIII (120575 = minus1) and IX (120575 = 1) AnisotropicCosmological Models Thespatial volume for both the models(49) and (62) is

119881 = 1198863= ((

1198622

1198623

) sin (1198623119905))

4

119891 (120579) (90)

where119891(120579) = cosh 120579 and sin 120579 for Bianchi type VIII and IXrespectively

The expression for expansion scalar 120579 and the shear 120590 forthe models (49) and (62) are given by

120579 = 3119867 = 41198623cot (119862

3119905)

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

1198622

3cot2 (119862

3119905)

3

(91)

The deceleration parameter 119902 for the models (49) and (62) isgiven by

119902 =

119889

119889119905

(

1

119867

) minus 1 =

3

4

sec2 (1198623119905) minus 1 (92)

From (92) we can observe that the deceleration parameter119902 is always negative and hence they represent acceleratinguniverses

The components of the Hubble parameter 119867119909 119867119910 and

119867119911for the models (49) and (62) are given by

119867119909= 119867119910=

119877

= 1198623cot (119862

3119905)

119867119911=

119878

119878

= 21198623cot (119862

3119905)

(93)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

4

3

1198623cot (119862

3119905) (94)

The average anisotropy parameters for the models (49) and(62) are defined by

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

=

1

8

(95)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

A cosmological diagnostic pair 119903 119904 called state finder asproposed by Sahni et al [39] is given by

119903 =

119886

1198861198673=

minus1

8

(1 + 9tan21198623119905)

119904 =

119903 minus 1

3 (119902 minus (12))

=

minus1

2

sec21198623119905

(tan21198623119905 minus 1)

(96)

43 Bianchi Type II (120575 = 0) and VIII (120575 = minus1) CosmologicalModels in Isotropic Form Thespatial volume for the abovemodels is given by

119881(1198626119905 + 1198627)3 (97)

The expression for expansion scalar 120579 calculated for the flowvector 119906119894 is given by

120579 = 3119867 =

31198626

(1198626119905 + 1198627)

(98)

and the shear 120590 is given by

1205902=

1

2

(

3

sum

119894=1

1198672

119894minus

1

3

1205792) =

minus31198626

2

(1198626119905 + 1198627)2 (99)

The deceleration parameter 119902 is given by

119902 =

119889

119889119905

(

1

119867

) minus 1 = 0 (100)

The components of Hubble parameter 119867119909 119867119910 and119867

119911are

given by

119867119909= 119867119910= 119867119911=

1198626

(1198626119905 + 1198627)

(101)

Therefore the generalized mean Hubble parameter (119867) is

119867 =

1

3

(119867119909+ 119867119910+ 119867119911) =

1198626

(1198626119905 + 1198627)

(102)

The average anisotropy parameter is

119860119898=

1

3

3

sum

119894=1

(

Δ119867119894

119867

)

2

= 0 (103)

where Δ119867119894= 119867119894minus 119867 (119894 = 1 2 3)

5 Conclusions

In this paper we have presented spatially homogeneousanisotropic Bianchi type II VIII and IX as well as isotropicspace times filled with perfect fluid and DE possessingdynamical energy density in Saez-Ballester [1] scalar-tensortheory of gravitation and general relativity Studying theinteraction between the ordinary matter and DE will openup the possibility of detecting DE For Bianchi type IIcosmological model we observe that at 119905 = minus119896

21198961 the

spatial volume vanishes while all other parameters diverge for1 lt 119899 lt 2 Thus the derived model starts expanding with bigbang singularity at 119905 = minus119896

21198961which can be shifted to 119905 = 0 by

choosing 1198962= 0 The model has point type singularity at 119905 =

minus11989621198961when 0 lt 119899 lt 2 (119899 = 1) Also the model has cigar type

singularity at 119905 = minus11989621198961when 119899 lt 0 and also for 119899 gt 2 For

Bianchi type VIII and IX cosmological models we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = 0 The expansionscalar 120579 and the shear scalar 120590 decrease as time increases In

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 ISRN Astronomy and Astrophysics

the derivedmodels the EoS parameter ofDE119908(de) is obtainedas time varies and it is evolving with negative sign which maybe attributed to the current accelerated expansion of universe

For the isotropic cosmological models (78) we observethat the spatial volume increases as time increases and alsothe models have no initial singularity at 119905 = minus119862

71198626 Thus

the derived models start expanding with big bang singularityat 119905 = minus119862

71198626which can be shifted to 119905 = 0 by choosing

1198627= 0The expansion scalar 120579 shear scalar120590 and theHubble

parameter 119867 decrease as time increases Since 119902 = 0 theexpansion of the isotropic universes proceeds at a constantrate It may be noted that Bianchi type II VIII and IX darkenergy cosmologicalmodels filledwith perfect fluid representthe cosmos in its early stage of evolution and isotropicmodelsrepresent the present universe

Acknowledgment

The second author (K V S Sireesha) is grateful to theDepartment of Science and Technology (DST) New DelhiIndia for providing INSPIRE fellowship

References

[1] D Saez and V J Ballester ldquoA simple coupling with cosmologicalimplicationsrdquo Physics Letters A vol 113 no 9 pp 467ndash470 1986

[2] V U M Rao T Vinutha and M Vijaya Santhi ldquoAn exactBianchi type-V cosmological model in Saez-Ballester theory ofgravitationrdquoAstrophysics and Space Science vol 312 no 3-4 pp189ndash191 2007

[3] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype II VIII and IX string cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 314 no 1ndash3 pp 73ndash77 2008

[4] V U M Rao M Vijaya Santhi and T Vinutha ldquoExact Bianchitype-II VIII and IX perfect fluid cosmological models in Saez-Ballester theory of gravitationrdquo Astrophysics and Space Sciencevol 317 no 1-2 pp 27ndash30 2008

[5] R L Naidu B Satyanarayana andD R K Reddy ldquoLRS Bianchitype-II Universe with cosmic strings and bulk viscosity ina scalar tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 338 no 2 pp 351ndash354 2012

[6] R L Naidu B Satyanarayana and D R K Reddy ldquoLRSBianchi type-II dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 338 no 2 pp333ndash336 2012

[7] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-III dark energy model in a Saez-Ballester scalar-tensortheoryrdquo International Journal of Theoretical Physics vol 51 no9 pp 2857ndash2862 2012

[8] R L Naidu B Satyanarayana and D R K Reddy ldquoBianchitype-V dark energy model in a scalar-tensor theory of gravita-tionrdquo International Journal of Theoretical Physics vol 51 no 7pp 1997ndash2002 2012

[9] S Perlmutter S Gabi G Goldhaber et al ldquoMeasurements ofthe cosmological parameters Ω and and from the first sevensupernovae at 119911 ge 035rdquoThe Astrophysical Journal vol 483 no2 p 565 1997

[10] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 pp 51ndash54 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andand from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[13] A G Riess L-G Strolger J Tonry et al ldquoType Ia SupernovaDiscoveries at 119911 gt 1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astronomical Journal vol 607 no 2 p 665 2004

[14] C Fedeli L Moscardini and M Bartelmann ldquoObserving theclustering properties of galaxy clusters in dynamical dark-energy cosmologiesrdquo Astronomy amp Astrophysics vol 500 no 2pp 667ndash679 2009

[15] R R Caldwell and M Doran ldquoCosmic microwave back-ground and supernova constraints on quintessence concor-dance regions and target modelsrdquo Physical Review D vol 69Article ID 103517 6 pages 2004

[16] Z-Y Huang B Wang E Abdalla and R-K Sul ldquoHolographicexplanation of wide-angle power correlation suppression in thecosmicmicrowave background radiationrdquo Journal of Cosmologyand Astroparticle Physics vol 2006 article 013 2006

[17] S F Daniel R R Caldwell A Cooray andAMelchiorri ldquoLargescale structure as a probe of gravitational sliprdquo Physical ReviewD vol 77 no 10 Article ID 103513 12 pages 2008

[18] R R Caldwell W Komp L Parker and D A T VanzellaldquoSudden gravitational transitionrdquo Physical ReviewD vol 73 no2 Article ID 023513 8 pages 2006

[19] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[20] G Hinshaw J L Weiland N Odegard et al ldquoFive-yearWilkin-son Microwave Anisotropy Probe observations data processingsky maps and basic resultsrdquoThe Astrophysical Journal vol 180no 2 p 225 2009

[21] E J Copeland M Sami and S Tsujikava ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 p 1753 2006

[22] P J Steinhardt L M Wang and I Zlatev ldquoCosmologicaltracking solutionsrdquo Physical Review D vol 59 Article ID123504 13 pages 1999

[23] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[24] P Astier J Guy N Regnault et al ldquoThe supernova legacysurvey measurement of ΩM Ωand and w from the first year datasetrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006

[25] D J Eisentein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

[26] C J MacTavish P A R Ade J J Bock et al ldquoCosmologicalparameters from the 2003 flight of BOOMERANGrdquoThe Astro-physical Journal vol 647 no 2 p 799 2006

[27] E Komatsu J Dunkley M R Nolta et al ldquoFive-yearWilkinsonMicrowave Anisotropy Probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

ISRN Astronomy and Astrophysics 11

[28] M Tegmark M R Blanton M A Strauss et al ldquoThe three-dimensional power spectrum of galaxies from the sloan digitalsky surveyrdquo The Astrophysical Journal vol 606 no 2 p 7022004

[29] S Kumar and C P Singh ldquoAnisotropic Bianchi type-I modelswith constant deceleration parameter in general relativityrdquoAstrophysics and Space Science vol 312 no 1-2 pp 57ndash62 2007

[30] O Akarsu and C B Kilinc ldquoLRS Bianchi type I models withanisotropic dark energy and constant deceleration parameterrdquoGeneral Relativity and Gravitation vol 42 no 1 pp 119ndash1402010

[31] O Akarsu and C B Kilinc ldquoBianchi type III models withanisotropic dark energyrdquoGeneral Relativity andGravitation vol42 no 4 pp 763ndash775 2010

[32] O Akarsu and C B Kilinc ldquode Sitter expansion with anisotropicfluid in Bianchi type-I space-timerdquo Astrophysics and SpaceScience vol 326 no 2 pp 315ndash322 2010

[33] A K Yadav F Rahaman and S Ray ldquoDark energy models withvariable equation of state parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 871ndash881 2010

[34] A K Yadav and L Yadav ldquoBianchi type III anisotropic darkenergy models with constant deceleration parameterrdquo Interna-tional Journal of Theoretical Physics vol 50 no 1 pp 218ndash2272010

[35] A Pradhan H Amirhashchi and B Saha ldquoBianchi type-I anisotropic dark energy model with constant decelerationparameterrdquo International Journal of Theoretical Physics vol 50no 9 pp 2923ndash2938 2011

[36] V U M Rao G Sreedevi Kumari and D Neelima ldquoA darkenergy model in a scalar tensor theory of gravitationrdquo Astro-physics and Space Science vol 337 no 1 pp 499ndash501 2012

[37] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[38] D R K Reddy B Satyanarayana and R L Naidu ldquoFivedimensional dark energy model in a scalar-tensor theory ofgravitationrdquo Astrophysics and Space Science vol 339 no 2 pp401ndash404 2012

[39] V Sahni T D Saini A A Starobinsky and U AlamldquoStatefindermdasha new geometrical diagnostic of dark energyrdquoJournal of Experimental and Theoretical Physics Letters vol 77no 5 pp 201ndash206 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of