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 (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
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