RESEARCH Open Access

String cosmological models with bulk viscosity inNordtvedt's general scalar-tensor theory ofgravitationUma Maheswara Rao Velagapudi* and Neelima Davuluri

Abstract

We have obtained and presented spatially homogeneous Bianchi types II, VIII, and IX string cosmological modelswith bulk viscosity in Nordtvedt's (Astrophys. J. 161:1059, 1970) general scalar-tensor theory of gravitation with thehelp of a special case proposed by Schwinger. It is observed that only the bulk viscous cosmological model existsin the case of Bianchi type IX universe. Some important features of the models, thus obtained, have been discussed.

Keywords: Bulk viscosity; Cosmic strings; Bianchi type II, VIII, and IX metrics; Nordtvedt's general scalar-tensor theory

IntroductionNordtvedt [1] proposed a general class of scalar-tensorgravitational theories in which the parameter ω of theBrans-Dicke theory is allowed to be an arbitrary (positivedefinite) function of the scalar field (ω→ω(ϕ)). The studyof scalar field cosmological models in the framework ofNordtvedt's theory with the help of a special case pro-posed by Schwinger [2] has attracted many researchworkers. This general class of scalar-tensor gravitationaltheories includes the Jordan [3] and Brans and Dicke [4]theories as special cases. These general cases of scalar-tensor theories are seen to lead to a super richness (or ar-bitrariness) of possible theories.The field equations of the general scalar-tensor theory

proposed by Nordtvedt (using geometrized units with c = 1,G = 1) are

Rij−12gijR ¼ −8πϕ−1Tij −ωϕ−2 ϕ;iϕ;j−

12gijϕ;kϕ

;k

� �

−ϕ−1 ϕi;j−gijϕ;k

;k

� �ð1Þ

ϕ;k;k ¼

8πT3þ 2ω

−1

3þ 2ωð Þdωdϕ

ϕ;iϕ;i ð2Þ

where Rij is the Ricci tensor, R is the curvature invariant,Tijis the stress energy of the matter, and the comma and

semicolon denote partial and covariant differentiation,respectively.Also, we have the energy conservation equation

Tij;j ¼ 0 ð3Þ

Several investigations have been made in higher-dimensional cosmology in the framework of differentscalar-tensor theories of gravitation. In particular, Barker[5], Ruban and Finkelstein [6], Banerjee and Santos [7,8],and Shanti and Rao [9,10] are some of the authors whohave investigated several aspects of Nordtvedt's generalscalar-tensor theory in four dimensions. Rao and SreeDevi Kumari [11] have discussed a cosmological modelwith a negative constant deceleration parameter in ageneral scalar-tensor theory of gravitation. Rao et al. [12]have obtained Kaluza-Klein radiating model in a generalscalar-tensor theory of gravitation.The study of string theory has received considerable at-

tention in cosmology. Cosmic strings are important inthe early stages of evolution of the universe before theparticle creation. Cosmic strings are one-dimensionaltopological defects associated with spontaneous sym-metry breaking whose plausible production site is cosmo-logical phase transitions in the early universe. Reddy andSubba Rao [13] have studied axially symmetric cosmicstrings and domain walls in a theory proposed by Sen[14] based on Lyra [15] geometry. Mohanty and Mahanta[16] have studied a five-dimensional axially symmetricstring cosmological model in Lyra [15] manifold. Rao and

* Correspondence: umrao57@hotmail.comDepartment of Applied Mathematics, Andhra University, Visakhapatnam,Andhra Pradesh 530003, India

© 2013 Velagapudi and Davuluri; licensee Springer. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Velagapudi and Davuluri Journal of Theoretical and AppliedPhysics 2013, 7:50http://www.jtaphys.com/content/7/1/50

Vinutha [17] have studied axially symmetric cosmologicalmodels in a theory of gravitation based on Lyra [15]geometry.In order to study the evolution of the universe, many

authors constructed cosmological models containing aviscous fluid. The presence of viscosity in the fluid intro-duces many interesting features in the dynamics ofhomogeneous cosmological models. The possibility ofbulk viscosity leading to inflationary-like solutions ingeneral relativistic Freidmann-Robertson-Walker (FRW)models has been discussed by several authors [18-22].Bali and Dave [23], Bali and Pradhan [24], Tripathy et al.[25,26], and Rao et al. [27] have studied various Bianchi-type string cosmological models in the presence of bulkviscosity.Bianchi space-times play a vital role in understanding

and describing the early stages of the evolution of theuniverse. In particular, the study of Bianchi types II, VIII,and IX universes are important because familiar solu-tions like the FRW universe with positive curvature, thede Sitter universe, and the Taub-Nut solutions corres-pond to Bianchi types II, VIII, and IX space-times. Reddyet al. [28] studied Bianchi types II, VIII, and IX modelsin the scale-covariant theory of gravitation. Shanthi andRao [29] studied Bianchi types VIII and IX models in theLyttleton-Bondi universe. Also, Rao and Sanyasi Raju[30,31] have studied Bianchi types VIII and IX models inzero mass scalar fields and self-creation theory of gravi-tation. Rao et al. [32-34] have obtained Bianchi types II,VIII, and IX string cosmological models; perfect fluidcosmological models in the Saez-Ballester theory ofgravitation; and string cosmological models in generalrelativity as well as self-creation theory of gravitation, re-spectively. Rao and Vijaya Santhi [35,36] have obtainedBianchi types II, VIII, and IX perfect fluid magnetizedcosmological models and string cosmological models, re-spectively, in the Brans and Dicke [4] theory of gravita-tion. Recently, Rao and Sireesha [37,38] have obtainedBianchi types II, VIII, and IX string cosmological modelswith bulk viscosity in Lyra [15] and Brans and Dicke [4]theory of gravitation, respectively. Sadeghi et al. [39]have discussed the cosmic string in the BTZ black holebackground with time-dependant tension. Katore et al.[40], Motevalli et al. [41], Saadat [42], Pourhassan [43],Amani and Pourhassan [44], Saadat and Pourhassan[45,46], Motevalli et al. [47], Saadat [48], and Saadat andFarahani [49] are some more authors who havediscussed various aspects of bulk viscosity in differenttheories of gravitation.In this paper, we will discuss Bianchi types II, VIII, and

IX string cosmological models with bulk viscosity inNordtvedt's [1] general scalar-tensor theory with thehelp of a special case proposed by Schwinger [2], i.e.,3þ 2ω ϕð Þ ¼ 1

λϕ, where λ is a constant.

Metric and energy momentum tensorWe consider a spatially homogeneous Bianchi types II,VIII, and IX metrics of the form

ds2 ¼ dt2−R2 dθ2 þ f 2 θð Þdφ2� �− S2 dψ þ h θð Þdφ½ � 2

ð4Þ

where (θ, φ,ψ) are the Eulerian angles, and R and S arefunctions of t only.It represents Bianchi type II if f(θ) = 1 and h(θ) = θ,

Bianchi type VIII if f(θ) = cosh θ and h(θ) = sinh θ, andBianchi type IX if f(θ) = sin θ and h(θ) = cos θ. The energymomentum tensor for a bulk viscous fluid containing one-dimensional string is

Tij ¼ ρþ �pð Þuiuj − �pgij − λsxixj ð5Þand

�p ¼ p− 3ξH ; ð6Þ

where p = ∈ 0ρ(0 ≤ ∈ 0 ≤ 1). Here, �p is the total pressurewhich includes the proper pressure p, ρ is the rest energydensity of the system, λs is the tension in the string, ξ(t) isthe coefficient of bulk viscosity, 3ξH is usually known asthe bulk viscous pressure, H is the Hubble parameter, ui isthe four velocity vector, and xi is a space-like vector whichrepresents the anisotropic directions of the string.Here, ui and xi satisfy the equations

gijuiu j ¼ 1;

gijxix j ¼ −1;

and

uixi ¼ 0: ð7Þ

We assume the string to be lying along the z-axis. Theone-dimensional strings are assumed to be loaded withparticles, and the particle energy density is ρp = ρ − λs.In a commoving coordinate system, we get

T11 ¼ T 2

2 ¼ −�p;T33 ¼ λs − �p;T 4

4 ¼ ρ; ð8Þ

where ρ; λs; �p are functions of time t only.

Solution of field equationsNow, with the help of (5) to (8), the field equations (1)for the metric (4) can be written as

€RRþ

€SSþ

_R _SRSþ S2

4R4 þω _ϕ2

2ϕ2 þ€ϕ

ϕþ

_R _ϕ

Rϕþ

_S _ϕ

Sϕ¼ −

8π�pϕ

ð9Þ

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2€RRþ

_R2 þ δ

R2 −3S2

4R4 þω _ϕ2

2ϕ2 þ€ϕ

ϕþ 2

_R _ϕ

Rϕ¼ 8π λs−�pð Þ

ϕ

ð10Þ2

_R _SRSþ

_R2 þ δ

R2 −S2

4R4 −ω _ϕ2

2ϕ2 þ 2_R _ϕ

Rϕþ

_S _ϕ

Sϕ¼ 8πρ

ϕ

ð11Þ_SS−

_RR

� �h θð Þ _ϕ

ϕ¼ 0 ð12Þ

€ϕ þ _ϕ 2_RRþ

_SS

� �¼ 8π

3þ 2ωð Þ ρþ λs − 3�pð Þ

−1

3þ 2ωdωdϕ

_ϕ2

ð13Þ_ρ þ ρþ �pð Þ 2

_RRþ

_SS

� �− λs

_SS¼ 0: ð14Þ

Here, the overhead dot denotes differentiation withrespect to t.When δ = 0, − 1 & + 1, the field equations (9) to (14) cor-

respond to the Bianchi types II, VIII, and IX universes.By taking the transformation dt = R2S dT, the above

field equations (9) to (14) can be written as

R″

Rþ S″

S−2

R′2

R2 −S′

2

S2−2

R′S′

RSþ 14S4

þωϕ′2

2ϕ2 þϕ″

ϕ−ϕ ′R′

ϕR¼ 8π�p

ϕR4S2�

ð15Þ

2R″

R−2

R′S′

RS−3

R′2

R2 þ δ R2S2�

−34S4 þ ωϕ ′2

2ϕ2

þ ϕ″

ϕ−ϕ″S′

φS¼ 8π

ϕλs−�pð Þ R4S2

� ð16Þ

2R′S′

RSþ R′2

R2 þ δ R2S2�

−14S4−

ωϕ ′2

2ϕ2

þ ϕ ′

ϕ2R′

Rþ S′

S

� �¼ 8π

φρ R4S2�

ð17Þ

S′

S−R′

R

� �h θð Þϕ ′

ϕ¼ 0 ð18Þ

3þ 2ωð Þϕ″ þ dωdφ

ϕ ′2 ¼ 8π ρþ λs−3�pð Þ R4S2� ð19Þ

ρ′ þ ρþ �pð Þ 2R′

Rþ S′

S

� �−λs

S′

S¼ 0; ð20Þ

where the dash denotes differentiation with respect to T,and R, S, and ϕ are functions of T only.Since we are considering the Bianchi types II, VIII, and

IX metrics, we have h(θ) = θ, h(θ) = sinh θ& h(θ) = cos θ,

respectively. Therefore, from Equation (18), we will getthe following possible cases with h(θ) ≠ 0:

1ð Þ S′

S−R′

R¼ 0 and ϕ ′≠0

2ð Þ S′

S−R′

R≠0 and ϕ ′ ¼ 0

3ð Þ S′

S−R′

R¼ 0 and ϕ ′ ¼ 0

From the above three possibilities, we will consideronly the first possibility since the other two cases willgive us cosmological models in general relativity.

Cosmological models in Nordtvedt's generalscalar-tensor theoryWe will get cosmological models in a general scalar-tensor theory only in the case of

S′

S−R′

R¼ 0 and ϕ ′≠0:

If S′

S −R′

R ¼ 0; we get S = R + c.Without loss of generality by taking the constant of in-

tegration c = 0, we get

S ¼ R: ð21ÞUsing (21), the above field equations (15) to (20) can

be written as

2R″

R−5

R′2

R2 þ14R4 þ ωϕ ′2

2ϕ2 þϕ″

ϕ−ϕ ′R′

ϕR¼ −8π �p

ϕR6�

ð22Þ

2R″

R−5

R′2

R2 þ δR4−34R4 þ ωϕ ′2

2ϕ2 þϕ″

ϕ

−ϕ ′R′

ϕR¼ 8π

ϕλs−�pð Þ R6

� ð23Þ

3R′2

R2 þ δR4−14R4−

ωϕ ′2

2ϕ2 þ 3ϕ ′R′

ϕR¼ 8π

ϕρ R6� ð24Þ

3þ 2ωð Þϕ″ þ dωdϕ

ϕ ′2 ¼ 8π ρþ λs−3�pð Þ R6� ð25Þ

ρ′ þ 3 ρþ �pð ÞR′

R−λs

R′

R¼ 0: ð26Þ

The field equations (22) to (25) are four independ-ent equations with six unknowns R;ω;φ; ρ; λs and �p .From Equations (22) to (25), we have

3þ 2ωð Þϕ″ þ dωdϕ

ϕ ′2 ¼ ϕ 6R″

R−12

R′2

R2 −12R4 þ 2δR4

" #

þωϕ′2

ϕþ 3ϕ″:

ð27Þ

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Here, we obtain the string cosmological models withbulk viscosity in Nordtvedt's general scalar-tensor theorywith the help of a special case proposed by Schwinger inthe form

3þ 2ω φð Þ ¼ 1λφ

; λ ¼ constant: ð28Þ

From (27) and (28), we get

1λ

ϕ″

ϕ−ϕ ′2

ϕ2

" #þ 32ϕ ′2

ϕ−3ϕ″

¼ ϕ 6R″

R−12

R′2

R2 −12R4 þ 2δR4

" #ð29Þ

From Equation (29), we get

ϕ ¼ e k1Tþk2ð Þ; ð30Þwhere k1 and k2 are arbitrary constants and

6R″

R−12

R′2

R2 −12R4 þ 2δR4 ¼ −

32k1

2 ð31Þ

Bianchi type II (δ = 0) cosmological modelFrom Equation (31), we get

R ¼ffiffiffi3

pk1 cosech k1Tð Þ

� �12: ð32Þ

From Equations (22) to (24) we get

8πρ ¼ 1

2ffiffiffi3

pk1

sinh3 k1Tð Þ e k1Tþk2ð Þ 1− coth k1Tð Þð Þ− 16λ

� �ð33Þ

8π�p ¼ 1

6ffiffiffi3

pk1

sinh3 k1Tð Þ

� e k1Tþk2ð Þ 1− coth k1Tð Þ−2 cosech2 k1Tð Þ� −

12λ

� �ð34Þ

8πp ¼ ε02ffiffiffi3

pk1

sinh3 k1Tð Þ e k1Tþk2ð Þ 1− coth k1Tð Þð Þ− 16λ

� �ð35Þ

ξ ¼ 1

12πffiffiffi3

pk21

sinh4 k1Tð Þcosh k1Tð Þ

��e k1Tþk2ð Þ

�3ε0−1ð Þ6

coth k1Tð Þ

−13cosech2 k1Tð Þ þ 1−3ε0ð Þ

6

�þ ε0−1ð Þ

12λ

�ð36Þ

8πλs ¼ −1ffiffiffi3

pk1

e k1Tþk2ð Þ sinh k1Tð Þ: ð37Þ

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

ds2 ¼ 3ffiffiffi3

pk13 cosech

3 k1Tð Þ� dT2−

ffiffiffi3

pk1 cosech k1Tð Þ�

� dθ2þ dφ2�

−ffiffiffi3

pk1 cosech k1Tð Þ�

dψ þ θdφð Þ2:ð38Þ

Thus, Equation (38) together with (30) and (33) to (37)constitutes a Bianchi type II string cosmological modelwith bulk viscosity in isotropic form in Nordtvedt's generalscalar-tensor theory of gravitation.

Bianchi type VIII (δ = −1) cosmological modelFrom Equation (31), we get

R ¼ffiffiffi35

rk1 cosech k1Tð Þ

!12

: ð39Þ

From Equations (22) to (24), we get

8πρ ¼ 5ffiffiffi5

p

2ffiffiffi3

pk1

sinh3 k1Tð Þ e k1Tþk2ð Þ 1‐coth k1Tð Þð Þ− 16λ

� �ð40Þ

8π�p ¼ 5ffiffiffi5

p

3ffiffiffi3

pk1

sinh3 k1Tð Þ

��e k1Tþk2ð Þ

�34coth2 k1Tð Þ− 23

30cosech2 k1Tð Þ

−12coth k1Tð Þ− 1

4

�−

14λ

�ð41Þ

8πp ¼ 5ffiffiffi5

pε0

2ffiffiffi3

pk1

sinh3 k1Tð Þ e k1Tþk2ð Þ 1‐coth k1Tð Þð Þ− 16λ

� �ð42Þ

ξ ¼ 5ffiffiffi5

p

12πffiffiffi3

pk21

sinh4 k1Tð Þcosh k1Tð Þ

�e k1Tþk2ð Þ 1

4coth2 k1Tð Þ− 23

60cosech2 k1Tð Þ

þ 3ε0−1ð Þ6

coth k1Tð Þ þ 1−6ε0ð Þ12

0BBBBB@

1CCCCCA

þ ε0−1ð Þ12λ

26666666664

37777777775

ð43Þ

8πλs ¼ −2ffiffiffi5

pffiffiffi3

pk1

e k1Tþk2ð Þsinh k1Tð Þ ð44Þ

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

ds2 ¼ 3ffiffiffi3

p

5ffiffiffi5

p k13 cosech3 k1Tð Þ

� �dT2

−

ffiffiffi35

rk1 cosech k1Tð Þ

� �dθ2 þ cosh2θdφ2�

−

ffiffiffi35

rk1 cosech k1Tð Þ

� �dψ þ sinhθdφð Þ2:

ð45ÞThus, Equation (45) together with (30) and (40) to

(44) constitutes a Bianchi type VIII string cosmologicalmodel with bulk viscosity in isotropic form inNordtvedt's general scalar-tensor theory of gravitation.

Bianchi type IX (δ = 1) cosmological modelFrom Equation (31), we get

R ¼ k1 sech k1Tð Þð Þ12: ð46ÞFrom Equations (22) to (24), we get

8πρ ¼ 12k1

cosh3 k1Tð Þ 3e k1Tþk2ð Þ 1− tanh k1Tð Þð Þ− 12λ

� �ð47Þ

8π�p ¼ 12k1

cosh3 k1Tð Þ e k1Tþk2ð Þ 1− tanh k1Tð Þð Þ− 12λ

� �ð48Þ

8πp ¼ ε02k1

cosh3 k1Tð Þ 3e k1Tþk2ð Þ 1− tanh k1Tð Þð Þ− 12λ

� �ð49Þ

ξ ¼ 1

24πk21

cosh4 k1Tð Þsinh k1Tð Þ

� e k1Tþk2ð Þ 1−3ε0ð Þ þ 3ε0−1ð Þ tanh k1Tð Þð Þ þ ε0−1ð Þ2λ

� �ð50Þ

8πλs ¼ 0: ð51Þ

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

ds2 ¼ k13 sech3 k1Tð Þ�

dT2− k1 sech k1Tð Þð Þ

� dθ2 þ sin2θdφ2�

− k1 sech k1Tð Þð Þ

� dψ þ cosθdφð Þ2:

ð52Þ

Thus, metric (52) together with (30) and (47) to (51)constitutes a Bianchi type IX bulk viscous cosmologicalmodel in isotropic form in Nordtvedt's general scalar-tensor theory of gravitation.

Some important features of the modelsFor Bianchi types II and VIII (δ = 0, −1) cosmologicalmodelsThe spatial volume for model (38) is given by

V ¼ −gð Þ12 ¼ffiffiffi3

pk1 cosech k1Tð Þ

� �32f θð Þ: ð53Þ

The spatial volume for model (45) is given by

V ¼ −gð Þ12 ¼ffiffiffi35

rk1 cosech k1Tð Þ

!32

f θð Þ: ð54Þ

The expression of the expansion scalar θ calculated forthe flow vector ui for both models is given by

θ ¼ ui;i ¼ −3k12

coth k1Tð Þ; ð55Þ

and the shear σ is given by

σ2 ¼ 12σ ijσ ij ¼ 5k12

8tanh2 k1Tð Þ: ð56Þ

The deceleration parameter q is given by

q ¼ −3θ−2�

θ;iui þ 1

3θ2

� �¼ − −2 cosech2 k1Tð Þ þ 1

� �: ð57Þ

Hubble's parameter H is given by

H ¼ −k12

tanh k1Tð Þ: ð58Þ

The average anisotropy parameter is

Am ¼ 13

X3i¼1

ΔHi

H

� �2

¼ 0;whereΔHi

¼ Hi−H i ¼ 1; 2; 3ð Þ: ð59Þ

For Bianchi type IX (δ = 1) cosmological modelThe spatial volume for model (52) is given by

V ¼ −gð Þ12 ¼ k1sech k1Tð Þð Þ32f θð Þ: ð60Þ

The expression of the expansion scalar θ calculated forthe flow vector ui is given by

θ ¼ ui;i ¼ −3k12

tanh k1Tð Þ; ð61Þ

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and the shear σ is given by

σ2 ¼ 12σ ijσ ij ¼ 5k12

8tanh2 k1Tð Þ: ð62Þ

The deceleration parameter q is given by

q ¼ −3θ−2�

θ;iui þ 1

3θ2

� �¼ − −2cosech2 k1Tð Þ þ 1

� �: ð63Þ

Hubble's parameter H is given by

H ¼ −k12

tanh k1Tð Þ: ð64Þ

The average anisotropy Am parameter is

Am ¼ 13

X3i¼1

ΔHi

H

� �2

¼ 0;

where ΔHi ¼ Hi−H i ¼ 1; 2; 3ð Þ:ð65Þ

ConclusionsIn this paper, we have presented Bianchi types II, VIII, andIX string cosmological models with bulk viscosity inNordtvedt's [1] general scalar-tensor theory with the helpof a special case proposed by Schwinger [2]. It is observedthat all the models are isotropic, and in the case of Bianchitype IX, we will get only the bulk viscous cosmologicalmodel. The models presented here are free from singular-ities, and the spatial volume decreases as time T increases,i.e., all the three models are contracting. We observe thatas T approaches to infinity, the expansion scalar θ leads toa constant value. This shows that the universe expandshomogeneously. We also observe that the shear scalar σand the Hubble parameter H are constants as T tends toinfinity. The energy density, total pressure, string tensiondensity, and coefficient of bulk viscosity diverge with theincrease of time.For these models, the deceleration parameter q tends

to −1 for large values of T. Also, in recent observationsof type Ia supernovae, Perlmutter et al. [50] and Riesset al. [51-53] proved that the decelerating parameter ofthe universe is in the range − 1 ≤ q ≤ 0 and the present-dayuniverse is undergoing an accelerated expansion. There-fore, the present models not only represent acceleratinguniverses but also conform to the well-known fact that thescalar field and the bulk viscosity will play a vital role ingetting an accelerated universe. Since Am = 0, the modelsalways represent isotropic universes. These exact modelsare new and more general and represent the present stageof the universe.

Competing interestsBoth authors declare that they have no competing interests.

Authors’ contributionsAll calculations and derivations of the various results and their verificationswere carried out by VUMR and DN. Both authors read and approved thefinal manuscript.

Authors’ informationVUMR is working as a professor, and DN is working as a research scholar inthe Department of Applied Mathematics, Andhra University, Visakhapatnam,India.

AcknowledgementsDN is grateful to the Department of Science and Technology (DST), NewDelhi, India, for providing INSPIRE fellowship.

Received: 14 March 2013 Accepted: 2 September 2013Published: 16 September 2013

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doi:10.1186/2251-7235-7-50Cite this article as: Velagapudi and Davuluri: String cosmological modelswith bulk viscosity in Nordtvedt's general scalar-tensor theory ofgravitation. Journal of Theoretical and Applied Physics 2013 7:50.

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