Locally Rotationally Symmetric Bianchi Type I Massive String Cosmological Models with Bulk Viscosity and Decaying Vacuum Energy Density

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1 Advances n Astrophyscs, Vol., No., August 06 Locally otatonally Symmetrc Banch Type I Massve Strng Cosmologcal Models wth Bulk Vscosty and Decayng Vacuum Energy Densty aj Bal * and Swat Sngh States Professor of Mathematcs and CSI Emertus Scentst Department of Mathematcs, Unversty of ajasthan, Japur , Inda Emal: balr5@yahoo.co.n Abstract. We examne locally rotatonally symmetrc Banch Type I massve strng cosmologcal models wth bulk vscosty and tme varyng cosmologcal term () (vacuum energy densty). To get the determnstc models of the unverse, we consder two cases: () /, = constant, ; () H,, H where s the shear, the expanson, the coeffcent of bulk vscosty, the energy densty, the vacuum energy densty and H the Hubble parameter. Both models satsfy energy condtons and represent ansotropc space-tme. The frst model represents acceleratng behavour of the unverse whle the second model represents acceleratng and deceleratng phases of the unverse. The frst model starts wth a bg-bang at T 0 and the expanson decreases wth tme. Whle the second model also starts wth bg-bang at 0 and the expanson vanshes for large value of. For both models and, these results match T the result obtaned by Beesham [7]. Both models have Pont Type sngulartes at T 0 and 0 respectvely. Keywords: Banch I, massve strng, cosmologcal, bulk vscous, vacuum energy densty. Introducton The presence of strngs n the early unverse can be explaned usng grand unfed feld theores as nvestgated by Kbble [,], Everett [] and Vlenkn [4]. It s beleved that the very early unverse underwent phase transton gvng some topologcal stable structure. Strngs have nterestng cosmologcal consequences among the varous topologcal defects that occurred durng the phase transton and before the creaton of partcles n the early unverse (Kbble []). Zel dovch [5] n hs nvestgaton has ponted out that cosmc strngs cause densty perturbatons leadng to the formaton of galaxes. These strngs have stress energy and couple wth gravtatonal feld. The poneerng work n the formulaton of energy-momentum tensor for classcal massve strngs was ntated by Leteler [6,7] who explaned that massve strngs are formed by geometrc strng wth partcles attached along ts extenson and used ths dea to fnd strng cosmologcal solutons usng Banch Type I and Kantowsk-Sachs space-tmes. As Ensten s theory of gravtaton s the theory for understandng the nature and evoluton of the large scale structure of the unverse, t s, therefore, nterestng to study the gravtatonal effects whch arse from strngs wthn the frame work of Ensten gravty. In lterature, strng cosmologcal models for dfferent Banch Types (homogeneous and ansotropc) space-tmes have been wdely studed n dfferent physcal and geometrcal contexts by many authors vz. Banerjee et al. [8], Tkekar and Patel [9,0], Ilham and Tarhan [], Bal et al. [-8], Wang [9], eddy et al. [0], ao et al. [,], Mahanto et al []. The ntroducton of vscosty n the cosmc flud content has been very useful n explanng many physcal aspects of dynamcs of homogeneous cosmologcal models. The dsspaton mechansm not only modfes the nature of sngularty but also successfully explans the large entropy per baryon n the Copyrght 06 Isaac Scentfc Publshng

2 4 Advances n Astrophyscs, Vol., No., August 06 present unverse. There are several processes whch gve rse to vscous effects. These are the decouplng of neutrnos durng the radaton era and the decouplng of radaton and matter durng recombnaton era. Bulk vscosty s also assocated wth grand unfed theory phase transton and strng creaton and can lead to the nflatonary unverse scenaro. It s well known that n the early stage of the unverse when neutrno decouplng occurred, the matter behaved lke vscous flud (Klmek [4]). The coeffcent of vscosty s known to decrease as the unverse expands. The role of vscosty n cosmology has been studed by several authors vz. Pmentel [5], Berman [6], Beesham [7], Arbab [8], Pavon et al. [9], Zmdahl [0], Gron[], Saha [,], Sngh et al. [4], Bal et al. [5,6], Mostafapoor and Gron [7]. In modern cosmologcal theores, the cosmologcal term (t) s a focal pont of nterest as t solves the cosmologcal constant problem n a natural way. The current acceleratng expanson of the unverse suggests that our unverse s domnated by unknown dark energy. The cosmologcal constant s the most favoured canddate of dark energy representng energy densty of vacuum n the context of quantum feld theory. The cosmologcal constant occupes a prvleged place n the dark energy models because t provdes a good approxmaton to the present astronomcal data as studed by Zel dovch [8], Dretlen [9], Krauss and Turner [40]. The observatons for dstant type Ia supernovae (Perlmutter et al. [4,4], ess et al. [4,44], Garnavch et al. [45,46], Schmdt et al. [47]) strongly favour a postve value of n order to measure the expanson rate of unverse and now t s beleved that the unverse s not only expandng but also acceleratng. Many authors vz. Bertolam [48], Chen and Wu [49], Sahn and Starobnsky [50], Verma and am [5], Pradhan and Sngh [5], Bal et al. [5,54] nvestgated cosmologcal models wth decayng vacuum energy densty (). ecently Bal and Sngh [55] nvestgated LS Banch Type II massve strng cosmologcal model for stff flud dstrbuton wth decayng vacuum energy (). We have nvestgated some massve strng cosmologcal models wth bulk vscosty and vacuum energy densty n LS Banch Type I space-tme. To get the determnstc models of the unverse, we / consder two cases: (), = constant and ; () H, H where s shear, the expanson, the coeffcent of bulk vscosty, the vacuum energy densty, the scale factor, the energy densty, and H the Hubble parameter. In the frst case, we fnd that the model satsfes strong and weak energy condtons and the model represents acceleratng behavour of the unverse and s ansotropc. In the second case, the model also satsfes energy condtons. The second model represents acceleratng and ansotropc space-tme. The vacuum energy densty as obtaned by Beesham [7]. We have also calculated state fnder parameters {r,s} and these lead to {,0} n specal case. Metrc and Feld Equatons For smplfcaton and large scale behavour of actual unverse, LS Banch Type I models have great mportance. Ldsey [56] n hs nvestgaton has ponted out these models are equvalent to FW models whch are consdered standard models of our unverse. We consder locally rotatonally symmetrc Banch Type I space-tme n the form as ds dt A dx B (dy dz ) () where A and B are metrc potentals and are functons of t-alone. The energy-momentum tensor for a cloud of strng wth bulk vscosty s gven by Leteler [7] and Landau and Lfshtz [57] as j j j j j T v v x x (g v v ) () wth v v x x, and x 0, x 0 x x () 4 where s the proper energy densty wth partcles attached to them, the strng tenson densty, v the four velocty of partcles, x the unt space-lke vector representng the drecton of strng, the coeffcent of bulk vscosty and the expanson n the model. If the partcle densty of confguraton s denoted by then for massve strng, we have p Copyrght 06 Isaac Scentfc Publshng

3 Advances n Astrophyscs, Vol., No., August 06 5 (4) p In comovng coordnate system, we have v 0,0,0, and x,0,0,0 (5) A A The Ensten feld equatons j j j j g T g (6) for the metrc () lead to Soluton of Feld Equatons B B 44 4 (7) B B A B A B (8) A B AB A B B (9) B To get the determnstc models of unverse, we consder two cases: (), constant, (Chen and Wu [49]) / () H,, H (Barrow [58]) AB Now we consder case (): To get the determnstc soluton n terms of cosmc tme t, we assume that (shear) s proportonal to expanson (). Thus, we have A where A and B are metrc potentals and n s a constant. The motve for assumng the condton s explaned as: eferrng to Thorne [59], the observatons of velocty-red shft relaton for extra galactc sources suggest that the Hubble expanson of unverse s sotropc wthn 0% (Kantowsk and Sachs [60], n B (0) Krstan and Sachs [6]) and red-shft studes place the lmt 0.0. Also Collns et al. [6] have H ponted out that for spatally homogeneous metrc, the normal congruence to the homogeneous hypersurface satsfes the condton = constant. It s well known that coeffcent of bulk vscosty s known to decrease as unverse expands. Thus, we consder = constant as gven by Zmdahl [0]. We assume the condton as consdered by Chen and Wu [49] where s scale factor. Usng the condton (0) = constant = k and n equaton (8), we have (n) n B kb B 44 4 () n n where n AB B () and To get the soluton of (), we assume () (n) B Copyrght 06 Isaac Scentfc Publshng

4 6 Advances n Astrophyscs, Vol., No., August 06 whch leads to Thus equaton () leads to whch leads to B4 ', 44 B f f f(b) ' f df/db n d n (f ) f kb B db n B n n k 6 (n) f B B n n 4n To fnd the soluton n terms of cosmc tme t, we assume n =. Thus equaton (5) leads to Thus, we have db k f B B dt 7 / (4) (5) (6) 4/ B a snh (bt ) (7) / / A B a snh (bt ) (8) 7 4 where a, b k / 7 k After sutable transformaton of coordnates, the metrc () leads to / / ds dt a snh bt dx a snh bt(dy dz ) (9) where bt bt, x X, y Y, z Z. In the absence of bulk vscosty, the metrc (9) leads to / ds dt T dx T (dy dz ) (0) Case (): We have as consdered by Barrow [58] and The conservaton equaton leads to where A / A B / B. 4 4 Usng () and () n (), we have whch leads to Thus, we have Equaton (4) leads to where l s constant of ntegraton. We have () / H,, H bh as consdered by Arbab [8] (T g ) 0 () j j ;j ( ) 0 () 6HH (H (H) 6 HH 0 (6 6 )HH (9 9 )H 0 H ( ) 0 (4) H H t (5) Copyrght 06 Isaac Scentfc Publshng

5 Advances n Astrophyscs, Vol., No., August 06 7 Thus A B A B 4 4 4B as A B B Also H Ths leads to 4B4 B t Equaton (7) leads to We have and 4 (6) B4 B 4( t ) / B ( t ) / A B ( t ) After sutable transformaton of coordnates, the metrc () leads to where t, x X, y Y, z Z. (7) (8) (9) / / ds d dx (dy dz ) (0) 4 Some Physcal and Geometrcal Features The energy densty (), total energy densty (+), vacuum energy densty (), the strng tenson densty (), the partcle densty ( p ), the spatal volume ( ), the shear (), the expanson (), the Hubble parameter (H), the deceleraton parameter (q) for the model (9) are gven by 45b coth T 6 a () a 45b coth T () 6 (coth T ) a () b b coth T k 6 a a (4) b b coth T k (5) p 8 a snh T (6) A B 4 4 b cotht (7) A B 4 A B 4B A B B b coth T b coth T (8) 0 4 (9) H b cotht (40) Copyrght 06 Isaac Scentfc Publshng

6 8 Advances n Astrophyscs, Vol., No., August 06 H H The above mentoned quanttes for the model (0) are gven by 45 6 q tan h T (4) (4) 45 6 (4) (44) 7 6 (45) 9 8 p / AB A B A B 4 (46) (47) 4 4 (48) (49) 4 (50) H (5) H 0 f q H 0 f (5) beng constant of ntegraton. 5 State Fnder Parameters {r,s} The state fnder parameters effectvely dfferentate between forms of dark energy and provde smple dagnoss on whether a partcular model fts nto the basc observatonal data. Followng Sahn et al. [6], the state fnder dagnostc par {r,s} s gven by H H r (5) H H r / s (54) q where l s constant of ntegraton. However, f l 0 then r, s 0 whch agrees wth CDM model. 6 Concluson The weak energy condton (), 0 as well as strong energy condton () 0, < 0 gven by Hawkng and Ells [64] for the model (9) are satsfed f Copyrght 06 Isaac Scentfc Publshng

7 Advances n Astrophyscs, Vol., No., August 06 9 () 4(b k) b b coth T and, k b 6 a a () 45b b b, coth T k 6 a 6 a a Also for the model (0) these condtons lead to 9 7 () and (), 6 6 For the model (9), energy densty (), the strng tenson (), the partcle densty ( p ) are ntally large but for large value of T, these tend to fnte quantty. The spatally volume ncreases exponentally representng nflatonary scenaro. Snce 0, hence the model s ansotropc space-tme throughout. The model starts wth a bg-bang at T = 0 and the expanson decreases wth tme. whch T matches the result as obtaned by Beesham [7]. The deceleratng parameter q<0 whch shows that the model represents acceleratng unverse. The model has Pont Type sngularty at T = 0 (MacCallum [65]). In the absence of bulk vscosty, the model s well defned and energy condtons are satsfed. Also for the model (0), the energy densty (), the strng tenson (), the partcle densty ( p ) are ntally large but tend to zero when. The spatal volume ncreases wth tme. Snce 0, hence the model represents ansotropc space-tme, but at late tme, t sotropzes. Ths result matches recent astronomcal observatons. The model starts wth a bg-bang at = 0 and the expanson decreases wth tme. The model represents acceleratng and deceleratng phases of unverse f l and l respectvely. The vscosty prevents the matter densty to vansh for the model (0). Also the model has Pont Type sngularty at = 0. The vacuum energy densty whch matches the result as obtaned by Beesham [7]. We have also calculated state fnder parameters {r,s} for the model (0). These parameters agree wth CDM model n specal case. Acknowledgments. The authors are thankful to the eferee for valuable comments and suggestons. eferences. T.W.B. Kbble, "Topology of cosmc domans and strngs", J. Phys. A, vol. 9, pp , T.W.B. Kbble, "Some mplcatons of a cosmologcal phase transton", Phys. eports, vol. 67, pp. 8-99, A.E. Everett, "Cosmc strngs n unfed gauge theores", Phys. ev. D, vol. 4, pp , A. Vlenkn, "Cosmc strngs", Phys. ev. D, vol. 4, pp , Ya. B. Zel dovch, "Cosmologcal perturbatons produced near a sngularty", Mon. Not. oy. Astron. Soc., vol. 9, pp , P.S. Leteler, "Clouds of strngs n general relatvty", Phys. ev. D, vol. 0, pp. 94-0, P.S. Leteler, "Strng Cosmologes", Phys. ev. D, vol. 8, pp , A. Banerjee, A.K. Sanyal and S. Chakravorty, "Strng cosmology n Banch Type I space-tme", Pramana J.Phys., vol. 4, pp. -, Tkekar and L.K. Patel, "Some exact soluton of strng cosmology n Banch type III space tme", Gen. elatv. Gravt., vol. 4, pp , Tkekar and L.K. Patel, "Some exact soluton n Banch type VI0 strng cosmology", Pramana-J. Phys., vol. 4, pp , 994. Copyrght 06 Isaac Scentfc Publshng

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