We study the string cloud cosmological model in the context of Kantowski-Sachs space-time. For this purpose, the Einstein\u27s field equations are solved for Kantowski-Sachs space-time with strange quark matter coupled to the string cloud by using the relation between metric potentials. The physical and kinematical parameters of the models are studied.Proučavamo kozmološki model s oblakom struna u Kantowski-Sachsovom prostoruvremenu. S tim su ciljem riješene Einsteinove jednadžbe polja za Kantowski-Sachsov prostor-vrijeme, sa stranom kvarkovskom tvari vezanom s oblakom struna, primjenom relacija među metričkim potencijalima. Razmatraju se fizički i kinematički parametri modela

Adhav, K. S.

Katore, S. D.

Kurkure, V. B.

Rane, R. S.

English

We study the string cloud cosmological model in the context of Kantowski-Sachs space-time. For this purpose, the Einstein's field equations are solved for Kantowski-Sachs space-time with strange quark matter coupled to the string cloud by using the relation between metric potentials. The physical and kinematical parameters of the models are studied.Proučavamo kozmološki model s oblakom struna u Kantowski-Sachsovom prostoruvremenu. S tim su ciljem riješene Einsteinove jednadžbe polja za Kantowski-Sachsov prostor-vrijeme, sa stranom kvarkovskom tvari vezanom s oblakom struna, primjenom relacija među metričkim potencijalima. Razmatraju se fizički i kinematički parametri modela

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7STRING COSMOLOGICAL MODEL WITH STRANGE QUARK MATTER INKANTOWSKI-SACHS SPACE-TIMES. D. KATOREa, K. S. ADHAVa, R. S. RANEb and V. B. KURKUREaaPG Department of Mathematics, SGB Amravati University, Amravati. 444602 IndiabDepartment of Mathematics, Y. C. Science and Arts College, Mangrulpir, 444403 IndiaE-mail addresses: katore777@gmail.com, ati ksadhav@yahoo.co.in,rsrane53@rediffmail.com, vb kurkure@rediffmail.comReceived 1 March 2010; Accepted 2 February 2011Online 26 May 2011We study the string cloud cosmological model in the context of Kantowski-Sachsspace-time. For this purpose, the Einstein’s field equations are solved for Kantowski-Sachs space-time with strange quark matter coupled to the string cloud by usingthe relation between metric potentials. The physical and kinematical parametersof the models are studied.PACS numbers: 11.27.+d, 98.80.-k UDC 524.83Keywords: string cosmology, quark matter, Kantowski-Sachs space-time1. IntroductionIn recent years, attempts to unify gravity with other fundamental forces innature are an active field of research at the interface between particle physics andcosmology. According to particle physicists, one of the most important problemsof the early universe is the grand unification of the strong and electroweak forcesof nature. It was realized that this grand symmetry present in the universe at thevery high energy scale had to be broken in order to account for the electroweak andnuclear forces as different fields in the present day low-energy universe. One of theimmediate consequences of this symmetry breaking is the formation of topologicaldefects such as monopoles, cosmic strings or domain walls.Among these topological defects, cosmic strings are believed to be one-dimensional topologically stable objects, which might have been formed during aphase transition in the early universe (Kibble [1]), among others like monopoles andFIZIKA B (Zagreb) 19 (2010) 4, 323–328 323katore et al.: string cosmological model with strange quark matter in . . .domain walls. The present day configurations of the universe are not contradictedby a large-scale network of the strings in the early universe. They may be one ofthe sources of density perturbations that are required for the formation of galaxies(Vilenkin [2], Gott [3], Stachel [4], Krori et al. [5], Banerjee et al. [6], Tikekar andPatel [7], Bhattacharjee and Baruah [8]). All that work has been confined to therelativistic string models in the context of general relativity.One of the interesting consequences of phase transition in the early universeis the formation of strange quark matter. Itoch [9], Bodmar [10] and Witten [11]proposed two ways of formation of quark matter, namely, the quark-hadron phasetransition in the early universe and conversion of neutron stars into strange atultrahigh densities. In this respect, Alcock et al. [12], Haensel [13], Cheng et al.[14], Yavuz et al. [15], Yilmaz [16, 17] studied the quark matter attached to thetopological defects in general relativity.In the present paper, we solve the Einstein’s field equations for Kantowski-Sachs space-time with strange quark matter coupled to the string cloud by usingthe relation between metric potentials. The physical and kinematical parametersof the model are studied. Lastly some discussions are given.2. String with quark matter solutionsWe consider the Kantowski-Sachs space-time metric in the formds2 = −dt2 +A2dr2 +B2(dψ2 + sin2 ψdφ2) , (1)where A and B are functions of t alone. Here (x1, x2, x3, x4) corresponds to(r, ψ, φ, t). The energy momentum tensor for string cloud (Letelier [18]) is taken asTij = ρUiUj − λxixj , (2)where ρ is the rest energy for the cloud of strings with particles attached to themand λ is the string tension density. They are related byρ = ρp + λ, or ρp = ρ− λ , (3)where ρp is the particle energy density. The string is free to vibrate and differentvibration modes of the string represent the different particle types, since differentmodes are seen as different masses or spins. So we will take quarks instead ofparticles in the string cloud. From (3), we obtain in this caseρ = ρq + λ+Bc, or ρq +Bc = ρp = ρ− λ , (4)where ρq is the quark energy density and Bc is a bag constant. Using (4) intoequation (2), we have the following energy-momentum tensor for strange quarkmatter coupled to the string cloud (Yavuz et al. [15])Tij = (ρq + λ+Bc)UiUj − λxixj , (5)324 FIZIKA B (Zagreb) 19 (2010) 4, 323–328katore et al.: string cosmological model with strange quark matter in . . .together with UαUα = −1, xαxα = 1 and Uαxα = 0, where Uα is the four velocityUα = δα4 , and xα is the unit space-like vector in the radial direction xα = δα1 whichrepresent the strings directions in the cloud, i.e. the anisotropic direction. Usingco-moving coordinates systems, we haveUα = (0, 0, 0,−1) and xα = (1A, 0, 0, 0) .The Einstein’s field equations (with gravitational units 8πG = c = 1)) readRij −12Rgij = −Tij . (6)Using Kantowski-Sachs line element (1), the field equations (6) with (2) take theform2B̈B+Ḃ2B2+1B2= λ , (7)ÄA+B̈B+ȦḂAB= 0 (8)andḂ2B2+ 2ȦḂAB+1B2= ρ , (9)where the dot denotes differentiation with respect to t. The expansion scalar θ andshear scalar σ have the following expressionsθ = uα;α =ȦA+ 2ḂB(10)andσ2 =12σijσij =23(ȦA−ḂB)2. (11)Equations (7) – (9) are three independent equations in four unknown quantitiesA, B, ρ and λ. Therefore, we need one more relation to obtain exact solutionsconnecting the unknown quantities. We assume that the scalar of expansion (θ) inthe model is proportional to the eigenvalue σ11 of the shear tensor σii . This conditionleads to the following relation between the metric coefficients,A = µBn, (12)where µ and n are non-zero constants. If we substitute Eq. (12) into Eq. (8), wehaveB̈B+n2n+ 1ḂB= 0 ,FIZIKA B (Zagreb) 19 (2010) 4, 323–328 325katore et al.: string cosmological model with strange quark matter in . . .which on integration givesB = (at+ b)γ , (13)where a and b are constants of integration. From Eqs. (12) and (13), we getA = µ(at+ b)nγ , (14)where γ =n+ 1n2 + n+ 1is a constant. Hence the geometry of the universe forKantowski-Sachs string cloud cosmological model can be written asds2 = −dt2 + µ2(at+ b)2nγdr2 + (at+ b)2γ(dψ2 + sin2 ψdφ2) . (15)3. Physical and kinematical parametersThe model (15) represents the cosmological model in time with strange quarkmatter attached to the string cosmology. This model has no initial singularity.Substituting equations (13) and (14) into equations (7), (9), (10) and (11), weobtain the expressions for the physical and kinematical quantities, the rest energydensity (ρ), the string tension density (λ), the particle energy density (ρp), thescalar of expansion (θ) and the shear scalar (σ) for the model (15):ρ =a2γ(2n2 + 3n+ 1)(n2 + n+ 1)(at+ b)2+1(at+ b)2γ,λ =a2γ(−2n2 + n+ 1)(n2 + n+ 1)(at+ b)2+1(at+ b)2γ,ρq +Bc = ρp = ρ− λ =2na2(2n2 + 3n+ 1)(n2 + n+ 1)2(at+ b)2,θ =a(n+ 1)(n+ 2)(n2 + n+ 1)(at+ b),andσ2 =2a2(n2 − 1)23(n2 + n+ 1)2(at+ b)2.4. Discussion and conclusionWe considered the solutions of the Einstein’s field equations in Kantowski-Sachsspace-time when the strange quark matter is attached to the string. For the model326 FIZIKA B (Zagreb) 19 (2010) 4, 323–328katore et al.: string cosmological model with strange quark matter in . . .(15), we observe that initially for t = −b/a, the quantities like the scalar of ex-pansion (θ) and shear scalar (σ2) diverge. As t increases, θ and σ2 decrease andfinally vanish at large t. This agrees with the results obtained by Back et al. [19],Adams et al. [20], Adcox et al. [21] and Yilmaz [17]. Further, we observe that thestring tension density (λ) and the rest energy density (ρ) have initial singularities.However as t increases, the parameters ρ and λ decrease, and finally vanish whent→ ∞. But for t→ −b/a, ρ and λ are infinite.For n = −1/2, the matter disappears, and ρp = 0 and ρ = λ = (at + b)−4/3,which represents the geometric string solution. In this case, ρp = −Bc (negativequark energy density). Hence in the geometric string (i.e. ρ = λ), we observe thatthe negative quark energy density is proportional to the bag constant Bc of stringcosmology. So the bag constant Bc may have negative quark energy density becausethe bag model is repulsive.AcknowledgementsThe authors are thankful to acknowledge the UGC, New Delhi for sanctioningresearch project and financial support. They are also thankful to IUCAA, Puneand TIFR, Mumbai for sending preprints and papers.References[1] T. W. B. Kibble, J. Phys. A 9 (1976) 1387.[2] A. Vilenkin, Phys. Rev. D 23 (1981) 853.[3] G. R. Gott, Appl. Phys. J. 288 (1985) 422.[4] J. Stachel, Phys. Rev. D 21 (1980) 2171.[5] K. D. Krori, T. Choudhary, C. R. Mahanta, Gen. Rel. Grav. 22 (1990) 123.[6] S. Banerjee, B. Bhui, Mon. Not. R. Astron. Soc. 247 (1990) 57.[7] R.Tikekar, L. K.Patel, Gen. Rel. Grav. 26 (1994) 647.[8] R. Bhattacharjee, K. K. Baruah, Pure Appl. Math. 32 (2001) 47.[9] N. Itoch, Prog. Theor. Phys. 44 (1970) 291.[10] A. R. Bodmar, Phys. Rev. D 4 (1971) 1601.[11] E. Witten, Phys. Rev. D 30 (1984) 272.[12] C. Alcock, E. Farthi, A. Olinto, Astrophys. J. 310 (1986) 261.[13] P. Haensel, J. L. Zdunik, R. Sehaeffer, Astron. Astrophys. 160 (1986) 121.[14] K. S. Cheng, Z. G. Dai, T. Lu, Int. J. Mod. Phys. D 7 (1998) 139.[15] I. Yavuz, I. Yilmaz, H. Baysal, Int. J. Mod. Phys. D 14 (2005) 1365.[16] I. Yilmaz, Phys. Rev. D 71 (2005) 103503.[17] I. Yilmaz, Gen. Rel. Grav. 38 (2006) 1397.[18] P. S. Letelier, Phys. Rev. D 28 (1983) 2414.[19] B. B. Back et al., PHOBOS Collab, Nucl. Phys. A 757 (2005) 28.[20] J. Adams et al., STAR Collab, Nucl Phys. A 757 (2005) 102.FIZIKA B (Zagreb) 19 (2010) 4, 323–328 327katore et al.: string cosmological model with strange quark matter in . . .[21] K. Adcox et al., PHENIX Collab, Nucl Phys. A 757 (2005) 184.KOZMOLOŠKI MODEL SA STRUNAMA I STRANOM KVARKOVSKOMTVARI U KANTOWSKI-SACHSOVOM PROSTORU-VREMENUProučavamo kozmološki model s oblakom struna u Kantowski-Sachsovom prostoru-vremenu. S tim su ciljem riješene Einsteinove jednadžbe polja za Kantowski-Sachsov prostor-vrijeme, sa stranom kvarkovskom tvari vezanom s oblakom struna,primjenom relacija med–u metričkim potencijalima. Razmatraju se fizički i kine-matički parametri modela.328 FIZIKA B (Zagreb) 19 (2010) 4, 323–328

Kozmološki model sa strunama i stranom kvarkovskom tvari u Kantowski-Sachsovom prostoru-vremenu

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