Journal of Modern Physics, 2012, 3, 14791486 http://dx.doi.org/10.4236/jmp.2012.310182 Published Online October 2012 (http://www.SciRP.org/journal/jmp) String Cloud and Domain Walls with Quark Matter in Lyra Geometry Kamal Lochan Mahanta, Ashwini Kumar Biswal Department of Mathematics, C. V. Raman College of Engineering, Bhubaneswar, India Email: kamal2_m@yahoo.com, biswalashwini@yahoo.com Received August 2, 2012; revised September 5, 2012; accepted September 15, 2012 ABSTRACT We have constructed cosmological models for string cloud and domain walls coupled with quark matter in Lyra geome try. For this purpose we have solved the field equations using anisotropy feature of the universe, special law of variation for Hubble’s parameter proposed by Berman [78] which yields constant deceleration parameter; and time varying dis placement field . Further some properties of the obtained solutions are discussed. Keywords: Cosmic Strings; Domain Walls; Quark Matter; Lyra Geometry 1. Introduction In order to geometrize the whole of gravitation and elec tromagnetism Weyl [1] proposed a modification of Rie mannian manifold. As this theory is physically unsatis factory, later on Lyra [2] proposed a further modification of Riemannian geometry which bears a close resem blance to Weyl’s geometry. Sen [3] pointed out that the static model with finite density in Lyra manifold is simi lar to the static model in Einstein theory. Halford [4] showed that the vector field i in Lyra’s geometry plays a similar role of cosmological constant Λ in general theory of relativity. In addition he pointed out that the energy conservation law does not hold in the cosmologi cal theory based on Lyra’s geometry. The scalartensor theory of gravitation in Lyra manifold predicts the same effects, within observational limits, as in Einstein theory [5]. Many authors [619] constructed different cosmo logical models and studied various aspects of Lyra’s geometry. In field theories, topological defects are stable field configurations with spontaneously broken discrete or continuous symmetries [20,21]. Spontaneous symmetry breaking is described within the particle physics context in terms of the Higgs field. The symmetry is said to be spontaneously broken if the ground state is not invariant under the full symmetry of the Lagragian density. The broken symmetries are restored at very high temperatures in quantum field theories. The topology of the vacuum manifoldwith 2 Z 1 is called domain walls [21,22], with S2 called strings [23] and one dimensional textures, with S called monopoles and two di mensional textures, and with 3 S is called three dimensional textures. The topological defects are called local or global depending on the symmetry is whether local (gauged) or global (rigid). In the early universe these defects are expected to be remnants of phase transi tions [24]. String theory attracted the attention of many authors as it possesses the necessary degrees of freedom to describe other interactions, even a mode to describe the graviton. The presence of cosmic strings in the early universe is considered using grand unified theories [2530]. The study of various aspects of cosmic strings in different theories of relativity is available in the literature [3152]. The topological defects such as strings, domain walls and monopoles have an important role in the formation of our universe, Hill et al. [53] pointed out that the for mation of galaxies are due to domain walls produced during a phase transition after the time of recombination of matter and radiation. Vilenkin [54] and Sikivie and Ipser [55] discussed thin domain wall in Einstein’s the ory. Further Schmidt and Wang [56] discussed the same in the context of BransDicke theory. Widraw [57] ob tained that a thick domain wall with nonzero stress along a direction perpendicular to the plane of the wall does not allow static metric to be regular throughout en tire space. Goetz [58] constructed thick domain wall cosmological model where the scalar field responsible for the symmetry breaking is static while the metric de pends on time. Wang [59] obtained a class of exact solu tion to the Einstein’s field equations representing the gravitational collapse of a thick domain wall. Rahaman et al. [60] found an exact solution of the filed equations for a thick domain wall in a five dimensional KaluzaKlein C opyright © 2012 SciRes. JMP
K. L. MAHANTA, A. K. BISWAL 1480 space time within the frame work of Lyra geometry. Pradhan et al. [61] studiedplane symmetric domain wall in Lyra geometry. Rahaman and Mukherji [62] con structed two models of domain walls in Lyra geometry. In one of their model the space time is nonsingular both in its spatial and temporal behaviour and the gravitational field experienced by a test particle is attractive. In the other model they presented a spherical domain wall with non vanishing stress components in the direction perpen dicular to the plane of the wall. Rahaman et al. [63] stu died two types of thin domain wall models in Lyra ge ometry. In their first model the pressure of the domain wall is negligible along perpendicular and transverse direction to the wall. In their second model pressure of the domain wall in the perpendicular direction is negligible but transverse pressures are existed. Further they showed that the thin domain walls have no particle horizon and the gravitational force due to them is attractive. Pradhan et al. [64] obtained general solutions of the field equations for bulk viscous domain walls in Lyra geometry. It is believed that one of the transitions during the phase transitions of the universe could be Quark Gluon Plasma (QGP) hadron gas called quarkhadron phase transition when cosmic temperature was T ~200 MeV. The quark hadron phase transition in the early universe and conver sion of neutron stars into strange ones at ultrahigh densi ties are two ways of formation of strange quark matter [6567]. In quark bag models based on strong interaction theories it is considered that breaking of physical vacuum takes place inside hadrons. As a result vacuum energy densities inside and outside a hadron become essentially different and the vacuum pressure on the bag wall equi librates the pressure of quarks and stabilizes the system. It is pointed out that if the hypothesis of the quark matter is true, then some of neutron stars could actually be strange stars built entirely of strange matter [68,69]. The quark matter is modeled with an equation of state (EOS) based on phenomenological bag model of quark matter in which quark confinement is described by an energy term pro portional to the volume. In the frame work of this model the quark matter is composed of mass less u, d quarks, massive s quarks and electrons. In the simplified version of the bag model, assuming the quarks are mass less and noninteracting we have 3 q q p , where q is the quark energy density. The total energy density is given by mqc B mqc PPB (1) and the total pressure by (2) Therefore the equation of state for strange quark matter [70,71] is given by 14 3 mmC PB (3) Cosmic string is free to vibrate and different vibration modes of the string represent the different particle types since different modes are seen as different masses or spins. Therefore it is plausible to attach quark matter to the string cloud and domain walls. Yilmaz [72] studied rotating cosmological models for domain walls with strange quark matter and normal matter in the nonstatic and stationary Gödel universes with cosmological constant. Further Yilmaz [73] obtained KaluzaKlein cosmological solu tions of the Einstein’s field equations for quark matter coupled with the string cloud and domain wall. Adhav et al. [74] constructed ndimensional KaluzaKlein cosmo logical models for quark matter coupled with string cloud and domain walls in general relativity. Khadekar et al. [75] studied KaluzaKlein type Robertson Walker cosmo logical model by considering variable cosmological term Λ in the presence of strange quark matter with domain wall. Recently Mahanta et al. [76] constructed Bianchi typeIII cosmological model with strange quark matter attached to the string cloud in Barber’s second selfcrea tion theory of gravitation. Motivated by the aforesaid discussion in this paper we consider quark matter coupled to the string cloud and domain walls in the context of Lyra geometry. 2. Field Equations and Their Solutions for the String Cloud with Quark Matter In this section we consider the metric of the form 22222 22 ddddd tAxy Bz ijijsij Tuu xx (4) where A and B are functions of cosmic time t only. The energy momentum tensor for string cloud given by Le telier [32] and Stachel [31] is (5) Here is the rest energy density for the cloud of strings with particles attached to them, is the string tension density, i is the four velocity for the cloud of particles, 1 0,0,0, i B is the four vector which represents the strings direction which is the direction of anisotropy and s (6) where is the particle energy density. We know that string is free to vibrate. The different vibrational modes of the string represent different types of particles because these different modes are seen as different masses or spins. Therefore, in this section we take quarks instead of parti cles in the string cloud. Hence we consider strange quark Copyright © 2012 SciRes. JMP
K. L. MAHANTA, A. K. BISWAL C JMP 1481 qsc B matter energy density instead of particle energy density of the string cloud. Thus from (6) we get we consider opyright © 2012 SciRes. ijs ij uuxx i ui (7) From (5) and (7) we get the energy momentum tensor for strange quark matter attached to the string cloud as ijqs c TB (8) Moreover and satisfy the standard relations 1 ii i xx i 0 ii uxuu and (9) The Einstein’s field equations based on Lyra’s geome try proposed by Sen [3] and Sen and Dunn [16] in normal gauge are 13 3 22 4 ij iji jij RgRkk ij g T ,0,0,0t (10) where i is the displacement vector and other symbols have their usual meaning as in Riemannian geometry. Using (8) and (9) the field Equations (10) for the metric (4) leads to the following system of equations 2 2 AA AA 2 3 4 B B (11) 2 30 4 BAAB AB B (12) 2 2 3 4 2 AA AA (13) Here afterwards the dash over the field variable repre sents ordinary differentiation with respect to t. Now we find that Equations (11)(13) are there independent equa tions involving five unknowns A, B, , and . Therefore to obtain exact solution of the field equations we require two more relations. In view of the anisotropy of the space time, we assume that expansion is proportional to the components of shear tensor 2 which also represents anisotropy of the universe [77]. This leads to a polynomial relation between the metric coeffi cients n B (14) where n is a nonzero constant. In addition, with the help of special law of variation of Hubble’s parameter proposed by Berman [78] that yields constant deceleration parameter models of the universe, 2Constant RR qR (15) 213 RAB where is the overall scale factor. The con stant is taken as negative to obtain an accelerating model of the universe. From (15) we obtain 1 1q Ratb (16) 0a 0b and where are constants of integration. Equation (16) yields the condition of expansion i.e. 10q . Now we get 11 231q ABatb . (17) Using (14) in (17) we find 1321qn Batb (18) Thus 31 21nqn Aatb (19) Substituting (18) and (19) in (12) we get 2 22 22 41 31 akknnn kat b (20) 121.kqn where Now using (18)(20) in (11) and (13) we obtain string energy density 2222 22 2 1827933ananaankak at b (21) and string tension density 22 22 22 2 2 1899 33 sanan aankak katb (22) Using (18) and (19) the line element (4) is expressed as 121 22 22 12162 6 dd dd d qn qn n n tatb xy at bz (23) The model (23) represents a string cosmological model with quark matter in Lyra geometry with negative con stant deceleration parameter. For the model (23) we have string particle density 222222222 2 2 118319 39 13 ps kannak ka akank katb (24)
K. L. MAHANTA, A. K. BISWAL 1482 Quark energy density n B 2222 2 2 18279 3 qc anan aan at b 2 3 c k ak BB (25) Quark pressure 2222 2 2 18279 3 33 q qan anaan Pat b 2 3 3 c B k ak (26) Scalar expansion 9 1 a qatb (27) Shear scalar 2 22 27 21 a qatb i jij uuPg c B 22 1 6 (28) 3. Field Equations and Their Solutions for Domain Walls with Quark Matter The energy momentum tensor of a domain wall [79] in the conventional form is given by D ij TP (29) This perfect fluid form of the domain wall includes quark matter [72] (described bymq mqc PPB w and ) as well as domain wall tension i.e. mw and mw PP . Further m and m are related by the bag model equation of state i.e. Equation (3) and equation of state P P 1 mm P 12 (30) where i u1 i uu is a constant. Here the four velocity vector is such that i . We use commoving coordinate system 0 ii u . Using the line element (4), the filed equations (10) yield 2 2 AA AA 2 3 4 B B (31) 2 3 4P AABB AB B (32) 2 2 3 4P2AA AA (33) where dash over the field variables denote differentiation with respect to t. Here Equations (31)(33) are three in dependent equations involving five unknowns A, B, (34) where n is a nonzero constant. Further we consider the power law relation between time coordinate and displacement field [61,64] 0t (35) is a constant. where Adding Equation (31) with (32) and (33) we get , P, and . Therefore, in order to obtain exact solution of the field equations two more relations connecting these va riables are required. Due to anisotropy of the spacetime we assume that scalar of expansion is proportional to the components of shear tensor 2 which gives 2 3 AB BA P AB BA (36) and 2 22 2 AAB P AAB (37) From (34), (36), and (37) we obtain 2 20,1 BB nn BB (38) Integrating Equation (38), we find 1 21 n Bctd (39) 0c where and d are constants of integration. From (34) and (39) we have 21 n n Actd (40) Substituting (35), (39) and (40) in (31)(33) we obtain 22 2 22 0 22 123 4 21 mw nc nct nctd (41) and 22 2 22 0 22 123 4 21 mw nc nc PP t nctd (42) Further we find the scalar expansion 3c ct d and shear scalar 2 2 2 3 2 c ct d . From (41) and (42) we observe that the solutions rep resent stiff domain walls. To determine the tension of the domain walls w and density and pressure of the quark matter, we will use the equations of state given by (3) and (30) separately. 3.1. Case 1 By using equation of state for quark matter i.e. Equation (3) we get Copyright © 2012 SciRes. JMP
K. L. MAHANTA, A. K. BISWAL 1483 22 2 22 32 21 nc nc nctd 22 3 24 moc tB (43) and 22 2 22 12 24 21 nc nc nctd 22 0 3 mc Pt B (44) Now from equations (1) and (2) we have 22 2 32 21 qnc nc nct 22 0 22 3 24 t d (45) and 22 2 12 21 qnc nc22 0 22 3 24 t d nc t (46) Again with the help of (41) and (43) we obtain 22 2 12 21 qnc nc P nct 22 0 22 3 24 t d (47) In this case domain walls behave like invisible matter due to their negative tension. Further we find 3 q q P as proposed by Bodmer [66] and Witten [67]. 3.2. Case 2 If we use (30) in (41) and (42) we get 22 2 12 4 21 mnc nc nct 22 0 22 3 2t d (48) 22 2 12 4 21 mnc nc22 0 22 3 2 t d nc t (49) 22 2 22 0 3 2c tB 22 12 4 21 q nc nc nctd (50) 22 2 22 0 12 43 2c nc nctB 22 21 q P nc td (51) and 222 222 22 22 00 224 63 4 nc ncnc nctt 22 21 wnc td 1 (52) In this case, when with ne tension and dust quark matter. When we get domain walls solutions egativ , n anwe have domain walls solutions with negative tensiod quark matter solution like radiation. When 2 , w clusions tained exact solutions of the field eq loud and domain walls with quark e have stiff quark matter solution and domain walls disap pear. 4. Con In this paper we ob uations for string c matter in Lyra geometry. In our solutions we observe the following properties: 1) In the case of string cloud with quark matter for 1n , we get dust quark matter solution i.e. 0 s . In this model the universe starts at an initial epoch tb a . 2 At initial epoch the physical parameters and di verge. As cosmic time t gradually increases 2 and decrease and finally they vanish when t. Tis is consistent with the results of BrookHave nationl laboratory [80,81]. Here we find h n a 0.408 . The present upper limit of is 10−15 obtained frndirect argu ments concernihe anisotropy of the primordial black body radiation [82]. The greater value of om i ng t for our model than the aforesaid limit indicates that e model represents the early stages of the evolution of thuniverse. th e At initial epoch tb a , the gauge function diverges. With the increase in cosmic time t, gauge fnction u decreases and drs as t . Hencehis theory isappea t te ed stiff domain wall solution. Since in the more re leads to Einstein general theory of relativity at infini time. 2) In the case of domain walls with quark matter we obtain alistic case in which the domain walls interact with the primordial plasma, the equation of state for domain walls is expected to be stiffer than that of a radiation, our solu tions correspond to the early stages of evaluation of the universe. Here we note that the universe starts at an initial epoch td c . At the initial epoch the physical parame ters 2 and diverge. With the increase in cosmic time t scalar expansion and shear scalar 2 decrease and fally thvanish as t. In case 1 of strange quark matter coupleto domain walls we get iney d 4 3 3 qq P as edpropos by Bodmer [66] and Witten [67]. Inse domain walls behave like invisi ble matter due negative tension. this ca to their In case 2 of aforesaid model when 2, we get do main walls with negative tension and dust quark matter. Copyright © 2012 SciRes. JMP
K. L. MAHANTA, A. K. BISWAL 1484 When 4 3 , we have domain walls with negative ten sion anrk matter like radiation. When 2, d qua do main walsappear and we have stiff quark matter. ls di ies o ical m REFERENCES [1] H. Weyl, Sitzuss. Berlin, 1918, p. 465. etrie,” Mathematische Zeitschrift, Vol. 54, No. 1, Here we observe that most of the propertf our models are similar to the KaluzaKlein cosmolog odels (in string cloud and domain walls coupled with quark matter) obtained earlier by Yilmaz [73] and Adhav et al. [74]. ngsber Preuss Akad Wi [2] G. Lyra, “Übereine Modifikation der Riemannschen Geom 1951, pp. 5264. doi:10.1007/BF01175135 [3] D. K. Sen, “A Static Cosmological Model,” Zeitschriftfür Physik A Hadrons and Nuclei, Vol. 149, No. 3, 1957, pp Australian Journal of Physics, Vol. 23, No. 5 urnal of Mathematical Physics Displacement F Review,” Fortschritte der Physik, Vol. 41, No . 311323. [4] W. D. Halford, “Cosmological Theory Based on Lyra’s Geometry,” , 1970, pp. 863870. [5] W. D. Halford, “ScalarTensor Theory of Gravitation in a Lyra Manifold,” Jo , Vol. 13, No. 11, 1972, pp. 16991704. [6] A. Beesham, “FLRW Cosmological Models in Lyra’s Manifold with Time Dependent ield,” Australian Journal of Physics, Vol. 41, No. 6, 1988, pp. 833842. [7] T. Singh and G. P. Singh, “Lyra’s Geometry and Cos mology: A. 8, 1993, pp. 737764. doi:10.1002/prop.2190410804 [8] G. P. Singh and K. Desikan, “A New Class of Cosmo Logical Models in Lyra Geometry,” Pramana Journal of Physics, Vol. 49, No. 2, 1997, pp. 205212. doi:10.1007/BF02845856 [9] A. Pradhan, L. Yadav and A. K. Yadav, “ mogeneous Universe with Isotropi a Bulk Viscous Fluid in Lyra c Ho Geometry,” Astrophysics and Space Science, Vol. 299, No 1, 2005, pp. 3142. doi:10.1007/s105090052795x [10] F. Rahaman, “HigherDimensional Global Monopole in Lyra’s Geometry,” Fizika B, Vol. 11, No. 4, 2002, pp. 223229. [11] F. Rahaman, S. Das, N. Begum and M. Hossain, “Higher Dimensional Homogeneouscosmology in Lyra Geome try,” Pramana Journal of Physics, Vol. 61, No. 1, 2003, pp. 153159. doi:10.1007/BF02704519 [12] G. P. Singh, R. V. Deshpande and T. Singh, “Higher Dimensional Cosmological Model with Variable Gravita tional Constant and Bulk Viscosity in Lyra Geometry,” Pramana Journal of Physics, Vol. 63, No. 5, 2004, pp. 937945. doi:10.1007/BF02704332 [13] G. Mohanty, K. L. Mahanta and R. R. Sahoo, “NonEx istence of Five Dimensional Perfect Fluid Cosmological Model in Lyra Manifold,” Astrophysics and Space Sci ence, Vol. 306, No. 4, 2006, pp. 269272. doi:10.1007/s105090069272z [14] G. Mohanty, K. L. Mahanta and B. K. Bishi, “Five Di mensional Cosmological Models in Lyra Geom Time Dependent Displacement F etry with ield,” Astrophysics and Space Science, Vol. 310, No. 34, 2007, pp. 273276. doi:10.1007/s1050900795139 [15] G. Mohanty and K. L. Mahanta, “FiveDimensional Axi ally Symmetric String Cosmological Model in L Manifold,” Astrophysics and Spacyra e Science, Vol. 312, No. 34, 2007, pp. 301304. doi:10.1007/s1050900796915 [16] D. K. Sen and K. A. Dunn, “A ScalarTensor Theory of Gravitation in a Modified Riemannian Manifold,” Jour nal of Mathematical Physics, Vol. 12, 1971, pp. 578586. doi:10.1063/1.1665623 [17] T. Singh and G. P. Singh, “Bianchi TypeI Cosmological Models in Lyra’s Geometry,” Journal of Mathematical Physics, Vol. 32, No. 9, 1991, pp. 24562458. doi:10.1063/1.529495 [18] T. Singh and G. P. Singh, “Bianchi TypeIII and Kan towskiSachs Cosmological Models in Lyra’s G International Journal oeometry,” f Theoretical Physics, Vol. 31, No. 8, 1992, pp. 14331446. doi:10.1007/BF00673976 [19] J. K. Singh and Sri Ram, “Spatially Homogeneous Cos mological Models in Lyra’s Geometry,” Il Nuovo Ci mento B, Vol. 112, No. 8, 1997, pp.11571162. [20] A. Vilenkin and E. P. S. Shellard, “Cosmic Strings and other Topological Defects,” Cambridge University Press, Cambridge, 1994. [21] A. Vilenkin, “Cosmic Strings and Domain Walls,” Phys ics Reports, Vol. 121, No. 5, 1985, pp. 263315. doi:10.1016/03701573(85)90033X o Gold rs, Vol. 48, No. 26, [22] A. Vilenkin and A. E. Everett, “Cosmic Strings and Do main Walls in Models with Goldstone and Pseud stone Bosons,” Physical Review Lette 1982, pp. 18671870. doi:10.1103/PhysRevLett.48.1867 [23] R. Rajaraman, “Solutions and Instantons,” NorthHolland, Amsterdam, 1987. [24] T. H. R. Skyrme, “Particle States of a Quantized Meson Field,” Proceedings of the Royal Society A, Vol. 262, No. 1309, 1961, pp. 237245. 58 [25] T. W. B. Kibble, “Topology of Cosmic Domains and Strings,” Journal of Physics A, Vol. 9, No. 8, 1976, p. 1387. [26] A. Vilkenkin, “Cosmic Strings,” Physical Review D, Vol. 24, No. 8, 1982, pp. 20822089. [27] A. E. Everett, “Cosmic Strings in Unified Gauge Theo ries,” Physical Review D, Vol. 24, No. 4, 1981, pp. 858 868. doi:10.1103/PhysRevD.24.8 dinburgh, 1981, p. 201. [28] Ya. B. Zeldovich, I. Yu. Kobzarev and L. B. Okun, Zh. Eksp. Teor. Fiz., Vol. 67, 1974, p. 3 [29] Ya. B. Zeldovich, I. Yu. Kobzarev and L. B. Okun, Sov. Phys. JETP, Vol. 40, 1975, p. 1. [30] J. Ellis, “Gauge Theories and Experiments at High Ener gies,” Proceedings of the 21st Scottish University Sum mer School, SUSSP Publication, E [31] J. Stachel, “Thichening the String. I. The String Perfect Dust,” Physical Review D, Vol. 21, No. 8, 1980, pp. Copyright © 2012 SciRes. JMP
K. L. MAHANTA, A. K. BISWAL 1485 21712181. doi:10.1103/PhysRevD.21.2171 [32] P. S. Letelier, “String Cosmologies,” Physical Review D, Vol. 28, No. 10, 1983, pp. 24142419. doi:10.1103/PhysRevD.28.2414 [33] C. J. Hogan and M. J. Rees, “Gravitational Interactions of Cosmic Strings,” Nature, Vol. 311, 1984, pp. 109114. doi:10.1038/311109a0 [34] Y. S. Myung, B. H. Cho, Y. Kim and Y. J. Park, “Entropy Production of Superstrings in Thevery Early Universe Physical Review D, Vol ,” . 33, No. 10, 1986, pp. 29442947. doi:10.1103/PhysRevD.33.2944 [35] K. D. Krori, T. Chaudhury, C. R. Mahanta and A. Ma zumdar, “Some Exact Solutions Instring Cosmology,” General Relativity and Gravitation, Vol. 22, No. 2, 1990, pp. 123130. doi:10.1007/BF00756203 [36] C. Gundalach and M. E. Ortiz, “JordanBransDicke Cosmic Strings,” Physical Review D, Vol. 42, No. 8, 1990, pp. 25212526. doi:10.1103/PhysRevD.42.2521 . [37] A. Barros and C. Romero, “Cosmic Vacuum Strings and Domain Walls in BransDicke Theory of Gravity,” Jour nal of Mathematical Physical, Vol. 36, No. 10, 1995, pp 58005804. doi:10.1063/1.531287 [38] I. Yavuz anf I. Yilmaz, “Some Exact Solutions of String Cosmology with Heat Flux in Bianchi Type III Space time,” Astrophysics and Space Science, Vol. 245, No. 1, 1996, pp. 131138. doi:10.1007/BF00637808 [39] A. Banerjee, N. Banerjee and A. A. Sen, “Static and Non static Global String,” Physical Review D, Vol. 53, No. 10, 1996, pp. 55085512. doi:10.1103/PhysRevD.53.5508 [40] D. R. K. Reddy, “Plane Symmetric Cosmic Strings in Lyra Manifold,” Astrophysics and Space Science, Vol. 300, No. 4, 2005, pp. 381386. doi:10.1007/s1050900547164 [41] J. M. Nevin, “String Dust Solutions of the Einstein’s Field Equations with Spherica Symmetry,” General Relativity a l and Static Cylindrical nd Gravitation, Vol . 23, No. 3, 1991, pp. 253260. doi:10.1007/BF00762288 [42] S. Chakraborty, “A Study on BianchiIX Cosmological Model,” Astrophysics and Space Science, Vol. 180, No. 2, 1991, pp. 293303. doi:10.1007/BF00648184 [43] R. Tikekar and L. K. Patel, “Some Exact Solutions of String Cosmology in Bianchi III Spacetime,” General Relativity and Gravitation, Vol. 24, No. 4, 1992, pp. 397 404. doi:10.1007/BF00760415 [44] G. Mohanty, R. R. Sahoo and K. L. Mahanta, “Five Di mensional LRS Bianchi TypeI String Cosmological Model in Saez and Ballester Theory,” Astrophysics and Space Science, Vol. 312, No. 34, 2007, pp. 321324. doi:10.1007/s105090079697z [45] G. Mohanty and R. R. Sahoo, “Incompatibility of Five Dimensional Lrs Bianchi TypeV String and Meso String Cosmological Models in nic General Relativity,” As trophysics and Space Science, Vol. 315, No. 14, 2008, pp. 319322. doi:10.1007/s1050900898352 [46] R. Bali and R. D. Upadhaya, “An LRS Bianchi Type I Bulk Viscous Fluid String Cosmological Model in Gen eral Realtivity,” Astrophysics and Space Science 288, No. 3, 2003, pp. 287292. , Vol. doi:10.1023/B:ASTR.0000006037.79305.66 [47] X. X. Wang, “Exact Solutions for String Cosmology,” Chinese Physical Letters, Vol. 20, No. 5, 2003, pp. 615 617. [48] X. X. Wang, “KantowskiSachs String Cosmological Model with Bulk Viscosity in General Relativity,” As trophysics and Space Science, Vol. 298, No. 3, 2005, pp. 433440. doi:10.1007/s1050900558339 [49] A. Pradhan and P. Mathur, “Magnetized String Cosmo logical Model in Cylindrically Symmetric Inhomogene ous UniverseRevisited,” Astrophysics and Space Science, Vol. 318, No. 34, 2008, pp. 255261. doi:10.1007/s1050900899313 [50] A. Pradhan, “Some Magnetized Bulk Viscous String Cosmological Models in Cylindrically Symmetric Inho mogeneous Universe with Variable Λ Term cations in Theoretical Physics, V ,” Communi ol. 51, No. 2, 2009, pp. 367374. doi:10.1088/02536102/51/2/36 [51] H. Amirashchi and H. Zainuddin, “Magnetized Bianchi Type III Massive String Cosmological Models in General Relativity,” International Journal of Theoretical Physics, Vol. 49, No. 11, 2010, pp. 28152828. doi:10.1007/s1077301004743 [52] S. K. Tripathy. S. K. Sahu and T. R. Routray, “String Cloud Cosmologies for Bianchi TypeIII Models with Electromagnetic Field,” Astrophysics a Vol. 315, No. 14, 2008, pp. 105110. nd Space Science, doi:10.1007/s1050900898058 [53] C. T. Hill, D. N. Schramm and J. N. Fry, “Cosmological Structure Formation from Soft Topological Defects,” Comments on Nuclear and Particle 1989, p. 25. Physics, Vol. 19, [54] A. Villenkin, “Gravitational Field of Vacuum Domain Walls,” Physics Letters B, Vol. 133, No. 34, 1983, pp. 177179. [55] J. Ipser and P. Sikivie, “Gravitationally Repulsive Do main Wall,” Physical Review D, Vol. 30, No. 4, 1984, pp. 712719. doi:10.1103/PhysRevD.30.712 [56] H. Schmidt and A. Wang, “Plane Domain Walls When Coupled with the BransDicke Scalar Fields,” Physical Review D, Vol. 47, No. 10, 1984, pp. 44254432. [57] L. M. Widrow, “GeneralRelativistic Domain Walls,” Physical Review D, Vol. 39, No. 12, 1989, pp. 35713575. doi:10.1103/PhysRevD.39.3571 [58] G. Goetz, “The Gravitational Field of Plane Symmetric Thick Domain Walls,” Journal of Mathematical Physics, Vol. 31, No. 11, 1990, pp. 26832687. doi:10.1063/1.528969 [59] A. Wang, “Gravitational Collapse of Thick Domain Walls: An Analytic Model,” Modern Physics Letters A, Vol. 9, No. 39, 1994, pp. 36053609. doi:10.1142/S0217732394003440 , No. 34, 2003, 5131560 [60] F. Rahaman, P. Ghosh, S. Shekhar and S. Mal, “Higher Dimensional Thick Domain Wall in Lyra Geometry,” As trophysics and Space Science, Vol. 286 pp. 373379. doi:10.1023/A:102635 [61] A. Pradhan, I. Aotemshi and G. P. Singh, “Plane Sym metric Domain Wall in Lyrageometry,” Astrophysics and Copyright © 2012 SciRes. JMP
K. L. MAHANTA, A. K. BISWAL Copyright © 2012 SciRes. JMP 1486 Space Science, Vol. 288, No. 3, 2003, pp. 315325. doi:10.1023/B:ASTR.0000006061.77421.c9 [62] F. Rahaman and R. Mukherji, “Domain Walls in Lyra Geometry,” Astrophysics and Space Science, Vol. 288, No. 4, 2003, pp. 389397. doi:10.1023/B:ASTR.0000005095.28348.69 [63] F. Rahaman, M. Kalam and R. Mandal, “Thin Domain Walls in Lyra Geometry,” Astrophysics and Space Sci ence, Vol. 305, No. 4, 2006, pp. 337340. doi:10.1007/s1050900690636 [64] A. Pradhan, K. K. Rai and A. K. Yadav, “Plane Symmet ric Bulk Viscous Domain Wall in Lyra Geometry,” Bra zilian Journal of Physics, Vol. 37, No. 10841093. 3b, 2007, pp. [65] N. Itoh, “Hydrostatic Equilibrium of Hypothetical Quark Stars,” Progress of Theoretical Physics, Vol. 44, No. 1, 1970, pp. 291292. doi:10.1143/PTP.44.291 [66] A. R. Bodmer, “Collapsed Nuclei,” Physical Review D, Vol. 4, No. 6, 1971, pp. 16011606. doi:10.1103/PhysRevD.4.1601 [67] E. Witten, “Cosmic Separation of Phases,” Physical Re view D, Vol. 30, No. 2, 1984, pp. 272285. doi:10.1103/PhysRevD.30.272 [68] C. Alcock, E. Farhi and A. Olinto, “Strange Stars trange Quark s, Vol. 160, No. 1, 1986 m ls by Gravitational Wave Observation,” ,” The Astrophysical Journal, Vol. 310, 1986, pp. 261272. [69] P. Haensel, J. L. Zdunik and R. Schaefer, “S stars,” Astronomy & Astrophysic , pp. 121128. [70] J. Kapusta, “Finite Temperature Field Theory,” Ca bridge University Press, Cambridge, 1994. [71] H. Sotani, K. Kohri and T. Harada, “Restricting Quark Matter Mode Physical Review D, Vol. 69, No. 8, 2004, Article ID: 084008. doi:10.1103/PhysRevD.69.084008 [72] I. Yilmaz, “Domain Walls Solutions in the NonStatic and Stationary Godel Universes with a Cosmological Constant,” Physical Review D, Vol. 71, No. 10, 2005, Ar ticle ID: 103503. doi:10.1103/PhysRevD.71.103503 [73] I. Yilmaz, “String Cloud and Domain Walls with Quark Matter in 5D KaluzaKlein Cosmological Model,” Gen eral Relativity and Gravitation, Vol. 38, No. 9, 2006, pp. 13971406. doi:10.1007/s1071400603221 [74] K. S. Adhav, A. S. Nimkar and M. V. Dawande, “String Cloud and Domain walls with Quark Matter in nDimen sional KaluzaKlein Cosmological Model,” International Journal of Theoretical Physics, Vol. 47, No. 7, 2008, pp. 20022010. doi:10.1007/s1077300796443 [75] G. S. Khadekar, R. Wanjari and C. Ozel, “Domain Wall with Strange Quark Matter in KaluzaKlein Type Cos mological Model,” International Journal of Theoretical Physics, Vol. 48, No. 9, 2009, pp. 25502557. doi:10.1007/s107730090040z [76] K. L. Mahanta, S. K. Biswal, P. K. Sahoo and hikary, “String Cloud with Quark M. C. Ad Matter in Self Creation Cosmology,” International Journal of Theoretical Phys ics, Vol. 51, No. 5, 2012, pp. 15381544. doi:10.1007/s1077301110314 [77] R. Bali, “Magnetized Cosmological M tional Journal of Theoretical Podel,” Interna hysics, Vol. 25, No. 7, 1986, pp. 755761. doi:10.1007/BF00668721 [78] M. S. Berman, “A Special Law of Variation for Hubble’s Parameter,” Il Nuovo Cimento B, Vol. 74, No. 2, 1983, pp. rld and Nonsingular Cosmological Models,” 182186. [79] N. Okuyama and K. Maeda, “Domain Wall Dynamics in Brane Wo Physical Review D, Vol. 70, No. 6, 2004, Article ID: 064030. doi:10.1103/PhysRevD.70.064030 [80] B. B. Back, et al., “The PHOBOS Perspective on Dis coveries at RHIC,” Nuclear Physics A, Vol. 757, No. 12, ch for the QuarkGluon Plasma: The gies,” General Relativ 2005, pp. 28101. [81] J. Adam, et al., “Experimental and Theoretical Chal lenges in the Sear STAR Collaboration’s Critical Assessment of the Evi dence from RHIC Collisions,” Nuclear Physics A, Vol. 757, No. 12, 2005, pp. 102183. [82] C. B. Collins, E. N. Glass and D. A. Wilkinson, “Exact Spatially Homogeneous Cosmolo ity and Gravitation, Vol. 12, No. 10, 1980, pp. 805823. doi:10.1007/BF00763057
