Bianchi Type-II Cosmological Model in Presence of Bulk Stress with Varying- in General Relativity

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1 ISSN (Onlne): Index Coperncus Value (2013): 6.14 Impact Factor (2013): Banch Type-II Cosmologcal Model n Presence of Bulk Stress wth Varyng- n General Relatvty V. G. Mete 1, V.D.Elkar 2 1 Department of Mathematcs, R.D.I.K. & K.D. College, Badnera- Amravat, Inda. 2 Department of Mathematcs, J.D.Patl Sangludkar Mahavdyalaya, Daryapur,Dst.Amravat, Inda. Abstract: Banch Type-II space tme s nvestgated n presence of bulk stress gven by Landau and Lfshtz. To get a soluton, a supplementary condton between metrc potentals s used. The vscosty coeffcent of bulk vscous flud s assumed to be a smple power functon of mass densty where as the coeffcent of shear vscosty s consder as proportonal to the scale of expanson n the model. The cosmologcal term s found to be a decreasng functon of tme, whch s supported by results found from recent type Ia supernovae observatons. Also some physcal and geometrcal propertes of the model are dscussed. Keywords: Banch Type-II space tme, vscous flud, varable cosmologcal constant. 1. Introducton The Ensten s feld equatons has two parameters, the Newtonan gravtatonal constant G and the cosmologcal constant The Newtonan constant of gravtaton G plays the role of a couplng constant between geometry of space and matter n Ensten s feld equaton. In an evolvng unverse, t s natural to take ths constant as a functon of tme. In the modern cosmologcal theores, the dynamc cosmologcal term remans a focal pont of nterest as t solves the cosmologcal constant problem n a natural way. There s sgnfcant observatonal evdence towards dentfyng Ensten s cosmologcal constant or a component of materal content of the unverse that vares slowly wth tme and space and so acts lke. Recent cosmologcal observatons by the Hgh-z Supernova term and the Supernova cosmologcal proect suggest the exstence of a postve cosmologcal constant wth magntude These observatons on magntude and red-shft of type Ia Supernova suggest that our unverse may be acceleratng wth a large functon of the cosmologcal densty n the form of the cosmologcal - term. Earler researchers on ths topc, are contaned n Zeldovch, Wenberg, Dolgov, Bertolam, Ratra & Peebles, Carrol, Press and Turner. Some of the recent dscussons on the cosmologcal constant and consequence on cosmology wth a tme varyng cosmologcal constant have been dscuss by Dolgov, Tsagas and Maartens,Vshwakarma, and Pradhan et. al.. Ths motvates us to study the cosmologcal models n whch vares wth tme Cosmologcal scenaros wth a tme varyng have been proposed by several researchers. A number of models wth dfferent decay laws for the varatons of cosmologcal term were nvestgated durng the last two decades.chen and Wu, Pavon Carvalho et al. Lma and Maa Lma and Trodden Arbab and Abdel-Rahman Cunha and Santos Carnero and Lma Wenberger, Heller and Klmek, Msner Collns and Stewart have studed the effect of vscosty on the evoluton of cosmologcal models. Xng-Xang Wang dscussed Kantowsk-Sachs strng cosmologcal model wth bulk vscosty n general relatvty. Also several aspects of vscous flud cosmologcal model n early unverse have been extensvely nvestgated by many authors. Bal R. et. al..have studed Banch Type-III strng cosmologcal models wth tme dependent bulk vscosty. Adhav et al. have studed Banch Type-V strng cosmologcal model wth bulk vscous flud and Kantowsk-Sachs cosmologcal model n general relatvty. Recently Verma et.al nvestgated spatally homogeneous bulk vscous flud models wth tme dependent gravtatonal constant and cosmologcal term. Acceleratng Banch Type-I unverse wth tme varyng G and been nvestgated by Pradhan et al.. -term n general relatvty have Recently Mete et.al.[51-53] have studed varous aspects of cosmologcal models n general theory of gravtaton. Katore et.al.[54] have nvestgated Banch I nflatonary unverse n presence of massless scalar feld wth flat potental n general relatvty. In ths paper a new ansotropc L.R.S. Banch type-ii stff flud cosmologcal model wth varable nvestgated by assumng supplementary condtons -term has been where and are metrc potental. The outlne of ths paper s as follows: Basc equaton of model are gven n Secton.2, the solutons of the feld equatons s gven n Paper ID: SUB

2 ISSN (Onlne): Index Coperncus Value (2013): 6.14 Impact Factor (2013): Secton.3.In secton 4 and 5, we have dscuss the model.we conclude our result n Secton The Metrc and Feld Equatons In an orthonormal frame, the metrc for Banch Type-II space-tme n the LRS case s gven by where the Cartan bases,, (1) are gven by,,, = (2) where and are the tme dependent metrc functons. Assumng x,y,z) as local coordnates, the dfferental one forms are gven by (3) The Energy momentum tensor for vscous flud dstrbuton n the presence of bulk stress gven by Landau and Lfshtz T as = ( ε + ρ) v v η( v ; + v; + pg + v v v; l + v v l v; l 3. Soluton of the feld Equatons Equatons (8) (10) are three ndependent equatons n seven unknowns. For the complete determnacy of the system, we assume that (12) and the coeffcent of shear velocty s proportonal to the scale expanson,.e. where n s real number. Wth ths extra condtons, equatons (8) and (9) Condton (13) where s proportonalty constant. Equaton (14) together wth (12) and (15) ( 2 3 ξ η) θ ( g + v v ), (4) where s the densty, are two coeffcents vscosty, and satsfyng the relatons s the flow vector We choose the coordnates to be commovng, so that The Ensten s feld equatons wth tme-dependent (n gravtatonal unts c=1, G=1) read as 7) For the metrc (1) and energy-momentum tensor (4) n commovng system of co-ordnates, the feld equaton (7) yelds as (5) where whch can be rewrtten as whch on ntegraton gves (19) where s a constant of ntegraton. After a sutable transformaton of coordnates, the metrc (1) reduces to the for The pressure and densty for model (20) are gven by and where suffx 4 at the symbols A and B denotes ordnary dfferentaton wth respect to t and s the shear expanson gve by where, and Paper ID: SUB

3 ISSN (Onlne): Index Coperncus Value (2013): 6.14 Impact Factor (2013): For the specfcaton of,now we assume that the flud obeys an equaton of state of where s constant. the form Thus, gven we can solve the cosmologcal parameters. In most of the nvestgaton nvolvng bulk vscosty s assume to be a smple power functon of the energy densty (Pavon, ; Maartens, ; Zmdahl, ) where and are constant. If Equaton (24) may correspond to a relatve flud (Wenberg realstc models (Santos, regme. Usng (24) n (21), we obtan + ). However, more ) are based on m lyng n the (29) quaton (27) and (29), we observe that If postve cosmologcal constant s a decreasng functon of tme and approaches a small value n the present epoch. 4. Some Physcal Aspects of the Model The straght forward calculaton the followng expresson for the scalar of expanson for the shear of the flud for the metrc (20) (30) We consder the two model correspondng to m=0 and m= Model- I : When Equaton (24) reduces to = constant. Hence n ths case Equaton (25), wth the use of (22) and (23), The expanson factor decreases as a functon of T and asymptotcally approaches zero wth and p also approaches zero as Partcular Model If we set the geometrc space tme (20) reduces to the form Elmnatng between (26) and (22), we have The pressure and densty for model (32) are gven by 3.2. Model-II When Equaton (24) reduces to. Hence n ths case Equaton (25), wth the use of (22) and (23), 4.1. Model- I : When Equaton (24) reduces to = constant. Hence n ths case Equaton (33), wth the use of (23) and (34), Elmnatng between Equaton (28) and (22), we have Elmnatng between Equaton (34) and (35), we have Paper ID: SUB

4 ISSN (Onlne): Index Coperncus Value (2013): 6.14 Impact Factor (2013): Elmnatng between Equaton (44) and (45), we have 4.2. Model-II : When, Equaton (24) reduces to. Hence n ths case Equaton (33), wth the use of (23) and (34), 5.2. Model-II When, Equaton (24) reduces to. Hence n ths case Equaton (43), wth the use of (23) and (44), Elmnatng between equaton (37)and (34) From the equaton (46) and (48) we observe that the postve cosmologcal constant s a decreasng functon of tme and approaches small value. Some Physcal aspect of the model The scalar of expanson and the shear of the model (32) s gven by Some Physcal aspect of the model The scalar of expanson and the shear of the model (42) s gven by 5. Specal Model If we set and equaton (19) Usng the transformaton the metrc (1) takes the form The pressure and densty for the model (42) s gven by 5.1. Model- I : When Equaton (24) reduces to = constant. Hence n ths case Equaton (43), wth the use of (23) and (44), 6. Concluson We have presented Banch type-ii non statc cosmologcal model n presence of bulk stress gven by Landau L.D. and Lfshtz E.M. It s found that physcally relevant solutons are possble for the Banch-II space tme wth bulk stress n the presence of tme varyng term. For solvng the feld equatons we have assumed that the flud obey an equaton of state of the form and bulk vscous flud s assumed to be the smple power functon of mass densty gven by Generally the models are expandng, shearng and non-rotatng. The cosmologcal constant n all models gven n secton 3.1 and 3.2 are decreasng functon of tme and all approaches a small postve value at fnte large tmes (.e., the present epoch). These results are supported by the results from the supernova observatons recently obtan by the Hgh-z Supernova team and Supernova Cosmologcal proect [1-7].Thus wth our approach, we obtan a physcally relevant decay law for the cosmologcal constant unlke other nvestgators where adhoc laws were used to arrve at a mathematcal expressons for the decayng energy. Thus our models are more general than those studed earler. In all models, the Paper ID: SUB

5 ISSN (Onlne): Index Coperncus Value (2013): 6.14 Impact Factor (2013): physcal parameters pressure and densty are found to be decreasng functon of tme. Also we fnd that all the physcal quanttes the expanson scalar and the shear scalar. References [1] Garnavch,P.M., et al.: 1998a, Astrophyscs J., 493, L53, (H-z Supernova Team Collaboraton astroph/ ) [2] Garnavch, P.M.,et al.: 1998b, Astrophyscs J., 509, 74,(H-z Supernova Team Collaboraton astroph/ [3] Perlmutter, S.,et al.:1997, Astrophyscs J., 483, 565 (Supernova Cosmology proect Collaboraton astroph/ ) [4] Perlmutter,S., et al.:1998, Astrophyscs J., 483, 565 (Supernova Cosmology proect Collaboraton astroph/ ) [5] Perlmutter,S., et al.:1999, Astrophyscs J., 483, 565 (Supernova Cosmology proect Collaboraton) [6] Ress, A. G., et al.: 1998, Astrophyscs J., 116,1009. [7] Schmdt, B. P. et al.: 1998, Astrophyscs J., 507, 46. [8] Zeldovch,Ya.B.,:1968, Sov. Phys,- Uspekhb 11, 381. [9] Wenberg,S.,: 1972, Gravtaton and Cosmology, Wley, New York. [10] Dolgov,A.D.,,: 1983, n: The Very Early Unverse(eds), Gbbons,G.W, Hawkng, [11] S.W.,Sklos,S.T.C., Cambrdge Unversty Press, Cambrdge. [12] D. Dolgov,A.D., Sazhn, M.V.,and Zeldovch,Ya.B.,.:1990, Basc Modern Cosmology. [13] O.Bertolam,O,: 1986, Nuovo Cmento, B 93, 36. [14] B. Ratra,B. and Peebels,P.J.E.,: 1988, Phys, Rev, D 37, [15] S. M. Carroll, S.M., Press,W.N., and. Turner,E.L.,: 1992, Ann. Rev. Astron. Astrophys. 30, 499. [16] Dolgov, A. D. and Slk, J.: 1993, Phys. Rev. D47, [17] Dolgov, A. D.: 1997, PR, D55, [18] DolgovA.D.J,,: 1993, Phys. Rev. D48, [19] Tsagas,C.G., and Maartens,R.,: 2000, gr- qc/ [20] R. G. Vshwakarma,R.G., and Abdussattar: 1999, Phys. Rev. D60, [21] R. G. Vshwakarma,R.G.,: 2000, Class. Quant. Grav. 17, [22] R. G. Vshwakarma,R.G.,2001,Class. Quant. Grav. 18, [23] Vshwakarma,R.G., : 2001, Gen. Rel. Grav. 33, [24] Vshwakarma,R.G., : 2002, Mon. Not. R. Astron. Soc. 331, 776. [25] R. G. Vshwakarma : 2002, Class. Quant. Grav. 19, [26] Pradhan, A., Yadav V.K., and Chakrabarty,I : 2001, Int. J. Mod. Phys. D10, 339. [27] Pradhan, A, Yadav,V.K.,. and Saste,N.N., : 2002, Int. J. Mod. Phys. D11, 857. [28] Pradhan, A, Yadav,V.K.,: 2002, Int. J. Mod. Phys. D11, 839. [29] Pradhan,A.,and Aotemsh,I. : 2002, Int. J. Mod. Phys. D11, [30] Pradhan,A. and Pandey,H.R.,: 2003, Int. J. Mod. Phys. D12, 941. [31] Pradhan,A., Pandey,O.P. : 2003, Int. J. Mod. Phys. D12, [32] Chen,W,Y., Wu,S. : 1990, PR, D41, 695. [33] Pavon,D.: 1991, PR, D43, 375. [34] Pavon,D., Bafaluy,J. and Jou,D.: 1991, Class. Quant. Grav. 8, 357. [35] Carvalho,,J.C., Lma,,J.A.S, Waga,I., :1992, PR, D46, 2404 [36] Lma,. Maa, J. M. F.: 1994, PR, D49, [37] Lma,J.A.S., M. Trodden,M. : 1996, PR, D53, [38] Arbab,A.I., Abdel-Rahman : 1994, PR, D50, [39] Cunha, J.V., Santos,R.C., : 2004, IJMP, D13,1321. [40] Carnero,S., Lma,J.A.S., : 2005,IJMP, A20, [41] Wenberg,S.,: 1971, Astrophys, J. 168, 175. [42] Heller, M, Klmek,Z. : 1975, Astrophys, space Sc, 53, 37. [43] Msner,C.W., : 1967 Nature 214, 40. [44] Msner,C.W. : 1968, Astrophys, J, 151, 431. [45] Collns,C.B., Stewart,J.M., :1971 mon Not R Astron, soc 153, 419. [46] Wang Xng Xang. : 2004, Astrophys, Space Sc XXX, 1-8. [47] Bal,R. and Dave,S.:2001 Pramana J. Phys 56, 513. [48] Adhav,K.S., Nmkar,A.S., Ugale,N.R., Raut,V.B.,: 2009,Fzlea B :,18,2, [49] Adhav,K.S.,Mete,V.G.,Nmkar,A.S.,Pund, A.M., :2008,Int.J.Theor.Phys.,47, [50] Verma,M.K.,and Shr Ram: 2011, Appled Mathematcs 2, [51] Pradhan,A,Yadav, L.S.,Yadav,L.T., : 2013 ARDN ournal of Scence and Technology 3, 4, [52] Mete,V.G.,Umarkar,V.M.,Pund,A.M.,:2013 Int.J.Theor.Phys.,52, [53] Mete,V.G.,Elkar,V.D.,Bawane,V.S.:2014, Multlogc n Scence, Vol.III, Issue VIII, [54] Mete,V.G.,Umarkar,V.M.,Bawane,V.S.: 2014, Int. J. of Scence and research.,3 (12), [55] Katore,S.D.,Adhav,K.S.,Mete,V.G.,Shakh, A.Y.,: 2012,Pramana Journal of Physcs,78 (1), [56] Landau,L.D., and Lfshtz,E.M., : 1963, Flud Mechancs 6, 505. [57] Maartens,R., : 1995, Class Quantum Gravt. 12, [58] W. Zmdahl,W., : 1996, Phys, Rev, D 53, [59] Santos,N.O., Das,R.S., and A. Baneree : 1985, J. Math. Phys. 26, 878. Author Profle Dr. V. G. Mete, Ph.D. s workng as a Assocate Professor and Head n P.G. Deptt. of Mathematcs, R.D.I. K. & K.D. College, Badnera- Amravat. He has experence more than 24 years n the feld of Relatvty, cosmology and theores of gravtaton. He has publshed more than 40 research papers n nternatonal ournals and 8 research scholars are workng under hs gudance. He has completed two Mnor research proects, funded by U.G.C. He receved Best teacher award. V.D.Elkar, M.Phl. workng as a Assstant Professor & Head n Deptt. of Mathematcs, J.D.Patl Sangaludkar Mahavdyalaya, Daryapur-Amravat. He has more than 16 years teachng experence n U.G. level.presently he s workng as research fellow (F.I.P.) under the gudance of Dr.V.G.Mete. Paper ID: SUB