Int J Theor Phys (2012) 51:1442–1447DOI 10.1007/s10773-011-1020-7

Higher Dimensional Cosmological Model with Quarkand Strange Quark Matter

G.S. Khadekar · Rajani Shelote

Received: 7 September 2011 / Accepted: 8 November 2011 / Published online: 22 November 2011© Springer Science+Business Media, LLC 2011

Abstract We study higher dimensional homogeneous cosmological model in the presenceof quark and strange quark matter. The dynamical behavior of the model for the strangequark matter equation of state of the form p = 1

3 (ρ − 4Bc) are studied.

Keywords Cosmology · Higher dimensional space time · Strange quark matter

1 Introduction

A revolutionary development seems to have taken place in cosmology during the last fewyears. The latest development of super-string theory and super-gravitational theory have cre-ated interest among scientists to consider higher-dimensional space-time, for study of theearly universe. A number of authors [1–4] have studied physics of the universe in higher-dimensional space-time. Overduin and Wesson [5] have presented an excellent review ofhigher-dimensional unified theories, in which the cosmological and astrophysical implica-tions of extra-dimension have been discussed.

Kaluza-Klein theory has a long and venerable history. However, the original Kaluza ver-sion of this theory suffered from the assumption that the 5-dimensional metric does notdepend on the extra coordinate (the cylinder condition). Hence the proliferation in recentdecade of various versions of Kaluza-Klein theory, super gravity and super strings. Kaluza-Klein achievements is shown that 5-dimensional general relativity contains both Einstein’s4-dimensional theory of gravity and Maxwell’s theory electromagnetism. In the last decadenumber of authors [6–9] have considered multidimensional cosmological model.

Chatterjee and Banarjee [10] and Banarjee et al. [11] have studied Kaluza-Klein inho-mogeneous cosmological model with and without cosmological constants respectively. Sofar there has been many cosmological solution dealing with higher dimensional model con-taining a variety of matter field.

G.S. Khadekar (�) · R. SheloteDepartment of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Mahatma Jyotiba PhuleEducational Campus, Amravati Road, Nagpur 440033, Indiae-mail: gkhadekar@yahoo.com

Int J Theor Phys (2012) 51:1442–1447 1443

It was suggested that the quark matter composed of comparable members of u, d and squarks may be the true ground state of matter which is stable at zero pressure and temper-ature [12–14] in which case some or all neutron stars can turn out to be so-called strangestars [12–16]. If on the other hand strange matter is only metastable, the high pressure in thecentral regions of neutron stars may lead to formation of hybrid stars, having strange mattercores.

The possibility of the existence of quark matter dates back to early seventies Bodmer [17]and Witten [12] proposed two ways of formation of strange matter, the quark-hadron phasetransition in the early universe and conversion of neutron stars into strange ones at ultrahighdensities.

Typically, strange quark matter is modeled with an equation of state (EOS) based onthe phenomenological bag model of quark matter, in which quark confinement is describedby an energy term proportional to the volume [13]. In this model, quark are thought asdegenerate Fermi gases, which exist only in a region of space endowed with a vacuumenergy density Bc (called as a bag constant). Also, in the framework of this model, thequark matter is composed of massless u, d quark, massive s quarks and electrons. In thesimplified version of this model, on which our study is based, quarks are massless and non-interacting. Then we have quark pressure pq = ρq

3 (ρq is the quark energy density), the totalenergy density ρ = ρq + Bc and total pressure p = pq − Bc . One therefore gets equation ofstate for strange quark matter [18]:

p = 1

3(ρ − 4Bc). (1)

In this study we will examine quark matter in 5-dimensional Kaluza-Klein theory of gravi-tation.

2 Model and Field Equations

We consider five dimensional Kaluza-Klein cosmological model of the form

ds2 = dt2 − R2(t)

[dr2

1 − kr2+ r2(dθ2 + sin2 θdφ2) + (1 − kr2)dψ2

]. (2)

The Einstein field equations are given as

Gij = Rij − 1

2gijR = −8πTij , (3)

where Tij = [(p + ρ)μiμj − pgij ].From (2) and (3), we obtained the following field equations:

R2 + k = 4πρR2

3, (4)

R

R+ R2 + k

R2= −8πp

3, (5)

where the dot (.) denotes differentiation with respect to the proper time t , R(t) is the scalefactor.

To solve (4) and (5) for the equation of state for strange quark matter with and without k

i.e. p = 1

3(ρ − 4Bc), for k = 0 and k �= 0,

1444 Int J Theor Phys (2012) 51:1442–1447

where Bc is vaccum energy density or bag constant.Differentiating (4) with respect to t , we have

R

R= 8πρ

6+ 2πRρ

3R. (6)

Using (4) and (6), (5) can be rewritten as

R

R= − ρ

4(p + ρ). (7)

Using (7) we can rewrite (4) as[( −ρ

4(p + ρ)

)2

− 4πρ

3

]R2 + k = 0. (8)

We discuss the following two cases for k = 0 and k �= 0.

Case (i): For k = 0

With the help of (1), (8) takes the form

3ρ

16[ρ − Bc] = −S

√4πρ

3, (9)

where S = 1 or −1.Using (1) and (9) one can rewrite (7) as

R

R= +S

√4πρ

3. (10)

Integrating (9), one obtains

ρ = Bc

(e

St 163

√4πBc

3 + c

eSt 16

3

√4πBc

3 − c

)2

, (11)

where c is the constant of integration.Using (11), one can integrate (10) to get

R = d

[(e

St 163

√4πBc

3 − c)2

eSt 16

3

√4πBc

3

] 316

, (12)

where d is constant of integration.Thus the solution of (4) and (5) are given by (1), (11) and (12). If we set c = 0, S = 1,

then we have from (11), ρ = BC .

Case (ii): For k �= 0

Here we study (4) and (5) for unrestricted k together with a strange quark matter. Byusing (1), one can eliminate ρ from (4) and (5) to get

RR + 5

3R2 + 5

3k + 32πBcR

2

9= 0. (13)

Integrating (13) one obtains

√3R

53 R = S

√4πBcR

163 − 3kR

103 + c1, (14)

Int J Theor Phys (2012) 51:1442–1447 1445

where S = +1 or −1 and c1 is integrating constant.Using (14) from (4) one obtains

ρ = Bc + 2c1

8πR163

, for c1 �= 0. (15)

For the case of c1 = 0, one can integrate (14) to get

R = c22e

2St

√4πBc

3 + 3k

2c2√

4πBceSt

√4πBc

3

, for k �= 0,

R = c22e

2St

√4πBc

3 − 3

2c2√

4πBceSt

√4πBc

3

, for k = −1,

R = c22e

2St

√4πBc

3 + 3

2c2√

4πBceSt

√4πBc

3

, for k = 1,

(16)

where c2 = c√

4πBc .If we set c1 = 0 for the both cases k = 1 and k = −1, then we get

ρ = Bc. (17)

If c1 > 0 and Bc > 0, we have ρ > 0, i.e. subjected to the conditions on the integrationconstants and bag constant one obtains a realistic physical model.

Using (14), one obtains the model decelerations parameter q(t) = −RR

R2 as follows:

q(t) = −4πBcR2 + c1R

−103

4πBcR2 − 3k + c1R−10

3

, for c1 �= 0,

q(t) = −1

1 − A0kR−2, for c1 = 0.

(18)

For k = 0, one obtains q(t) < 1. For k = −1, one obtains q(t) = −11+A0R−2 . For k = 1, one

obtains q(t) = −11−A0R−2 , where A0 = 3

4πBc.

For

R = 0, (19)

one obtains from (14)

4πBcR163 − 3kR

103 + c1 = 0. (20)

Using (19) and (20) from (13) one obtains

RR = k − 8c1

9R103

. (21)

For c1 = 0 we obtain, RR = k. For k = −1, we have R < 0 which indicates that the modelexpands to a maximum and then contracts back. When k = 1, we have R > 0 which indicatesthat the model contract to a minimum and then it expands.

To discuss the cosmological evolution of the scale factor R(t), we consider a test particleof mass m and a linear coordinate qn under the action of potential (Haru Chand Dhara [19])

ν(q) = λqn − m

12q2. (22)

1446 Int J Theor Phys (2012) 51:1442–1447

Then the lagrangian of the test particle is given by

L(q, q) = 1

2mq2 − λqn + m

12q2, (23)

where q = dq

dt, etc. and the Euler-Lagrange equation is given by

mq + λnqn−1 − mq

6= 0, (24)

whereas the conserved total energy of the particle E is given by

E = 1

2mq2 + λqn − m

12q2. (25)

Using (25) and (24) can be rewritten as

qq − n

2q2 + nE

m+ (n − 2)

12q2 = 0. (26)

Equations (13) and (26) will describe the same dynamic behaviour only if = −8πBc , thenfollowing two conditions are satisfied

n = −10

3, E = −km

2. (27)

Thus n is directly related to the nature of the matter content filling the relatistic cosmologicalmodels.

From the relation (27) we can conclude that the model behaves relativistic open fork = −1, closed for k = 1 or flat for k = 0 if the total energy of the particle is positive, nega-tive or zero respectively.

3 Conclusion

In this work we have solved the Einstein field equations for 5-dimensional cosmologicalmodel. The dynamical behaviour of the solution for strange quark matter are studied. Wehave obtained a 5-dimensional cosmological solution in the presence of quark and strangequark matter in the frame work of Kaluza-Klein theory of gravitation. In case (I) for k = 0the quark density ρ is always positive for Bc > 0. For c = 0, we get ρ = Bc . Similarly inthe case (II) for c1 �= 0, ρ is given in (16). If c1 > 0 and Bc > 0, we have ρ > 0. For c1 = 0,in both the cases k = ±1, we get (18). In (19), we obtained deceleration parameter q(t) forc1 = 0 and c1 �= 0. It is observed that for k = 0, the deceleration parameter is q(t) < 1. Fork = ±1, the deceleration parameter q(t) are −1

1±A0R−2 .

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