(1985), Reddy and venkateswarlu (1988) are some of the authors who have investigated various aspects of the four di-

Transcription

1 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March Wet dark flud Cosmologcal Model n Lyra s Manfold.S.Nmkar, M.R.Ugale.M.Pund bstract : In ths aer, we have obtaned feld equatons and ther soluton n the resence of wet dark flud n Lyra s manfold wth the ad of Banch tye-i sace tme. For solvng the Ensten feld equatons the relaton between ρ and s used. lso, some hyscal and knematcal roertes of the model are dscussed. Introducton: Lyra (1951) roosed a modfcaton of Remannan geometry by ntroducng Key words: Banch tye-i, Wet dark flud, Lyra s Manfold. gauge functon nto structureless manfold, as a result of whch the cosmologra s manfold. Sngh and Sngh mensonal cosmologcal models n Ly- cal constant arses naturally from the (1991,199,1993) have resented Banch geometry. Ths bears a remarkable resemblance to Weyl s (1918) geometry. In logcal models wth a tme-deendent Tye-I, III and Kantowsk-Sachs cosmo- subsequent nvestgaton Sen (1957) and dslacement feld and have made a Sen and Dunn (1971) formulated a new scalar-tensor theory og gravtaton and constructed an analogue of the Ensten feld equatons based on Lyra s geometry. Halford (197,197) has shown that the scalar-tensor treatment based on Lyra s geometry redcts the same effects as n general relatvty. Bhamra (1974), Kalyanshett and Waghmode (198), Reddy and Innah (1985), Reddy and venkateswarlu (1988) are some of the authors who have nvestgated varous asects of the four d- comaratve study of the Robertson- Walkar models wth a constant deceleraton theory based on Lyra s geometry. Rahaman et al (), Pradhan and Pandy (3), studed some toologcal defects wthn the framework of Lyra s geometry, Bhowmk and Raut (4) obtaned ansotroc Banch tye cosmologcal models on the bass of Lyra s geometry. Reddy DRK (5) examned

2 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March-14 3 eoch. lane symmetrc cosmc strngs n Lyra s manfold. R. Holman and Sddartha Nadu (5) studed wet dark flud () as a model for dark energy. Ths model was n the srt of the generalzed chalygn gas (GCG), where a hyscally motvated equaton of state was offered wth roertes relevant for the dark energy roblem. Here the motvaton stems In real flud negatve ressures eventually lead to a breakdown of equaton (1) as a Phemenologcal equaton. We wll show that ths model can be made consstent wth the most recent SNIa data, the WMP results as well as the constrants comng from measurements of the matter ower sectrum. The aram- * eters γ and ρ are taken to be ostve from an emrcal equaton of state roosed by Tat (1988) and we restrct ourselves to γ 1. Note and Hayword that t C s denotes the adabatc sound (1967) to treat water and aqueous soluton. The equaton of state for s seed n, then γ C s. To fnd the energy densty, we use very smle. the energy conservaton equaton * γ ( ρ ρ ) (1) ρ + 3 H ( + ρ ) () and s motvated by the fact that t s a γ * D ρ ρ +, (1+ γ ) good aroxmaton for many fluds, 1+ γ V ncludng water, n whch the nternal where D s the constant of ntegraton attracton of the molecules makes negatve ressures ossble. One of the vr- naturally ncludes two comonents, a and V s the volume exanson. tues of ths model s that the square of ece that behaves as a cosmologcal the sound seed, C S whch deends on constant as well as a ece that red shfts as a standard flud wth an equaton of, can be ostve, even whle gvng ρ state γρ. We can show that t we take rse to cosmc acceleraton n the current D >, ths flud wll never volate the

3 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March strong energy condton The relatvstc feld equatons n normal gauge n Lyra s manfold are as + ρ. Thus, we get * ( 1+ γ ) ρ γρ + ρ ( 1+ ) ( ) 1 D D γ. V +γ R k g R + φφ gφkφ 8πT (4) 4 Where φ s a dslacement feld and the other symbols have ther usual meanng The wet dark flud has been used as as n Remannan geometry. We now assume the vector dslacement feld φ to dark energy n the homogeneous, sotroc FRW case by Holman and Nadu (5). T. Sngh and R. Chaubey (8) The energy momentum tensor for wet studed n Banch tye I unverse wth dark flud s gven by wet dark flud. Recently, dhav et al T ( ρ + ) uu δ (6) (11) have been studed n detaled for where u s the flow vector satsfyng Ensten-Rosen unverse wth wet dark g u u 1 flud. Now, wth the hel of equaton (6), the In ths aer, we study the Banch tyefeld equaton (4) for the metrc (3) can I model n Lyra s manfold n resence be wrtten as of wet dark flud. B C B C β 8 (7) Metrc and solutons of feld equa- B C BC 4 tons: We consder the Banch tye-i metrc ds dt dx B dy C dz (3) Where, B,C are functons of tme t only. Ths ensures that the model s satally homogeneous. be the tme lke constant vector. φ (,,, β cons tan t) (5) B B C C β 8 (8) C C 4 44 B B β 8 (9) B B 4 44 C B C β 8πρ (1) C BC Here the suffx 4 after,b and C denotes ordnary dfferentaton wth resect to t. Equaton (7) to (1) are four

4 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March-14 3 ndeendent equatons n fve unknown, B, C,, ρ. To get a determnate soluton, one extra condton s needed. So kasner metrc to be ansotroc n Ensten s general relatvty, wth the hel of above the Equaton (13) can be wrtten n the followng exlct form we consder the equaton of state 1 ( at b), B B ( at + b) + ρ (11) fter solvng the set of equaton (7) to (1) wth the hel of equaton (11), we 1, and 3 and, B, C satsfy obtan we obtan , ( BC ) 44 (1) and BC 1 (15) Snce,B,C are non-zero, Integraton of equaton (1) gves Usng equaton (14), equaton (3) becomes BC ( at + b) (13) 1 ds dt ( at + b) dx B ( at + b) dy 3 Where a, b are constants of ntegraton. C ( at + b) dz Futher Kanser (191) unverse refers to a (16) vacuum cosmologcal model. The generalzatons The metrc can be transformed through of Kasner model were ro- a roer choce of co-ordnates nto the osed by Henekman and schuckng (1958).Msner (1968), Lfeshtzkhalatnkov (197), Belnsk (197,1971), Gron (1985, 199) has defned an analytc nondmensonal exresson for the ansotroy of the Kasner metrc. Barrow (1997), Caltaldo (), Brevk and Petterson (1997, ) roved that a vscous cosmologcal flud does not ermt the C ( at b) 3 (14) C + Wth the constant of ntegraton form ds 1 3 dz dt T dx T dy T (17) Whch s Banch tye-i metrc of the Kasner form. Some Physcal Proertes: The model (17) reresents an exact cosmologcal model n Lyra s manfold n resence of wet dark flud. The hyscal and knematcal arameters of the model (17) are

5 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March Proer volume V g T Exanson Scalar ( θ ) 7 16T a 3T Shear scalar ( σ ) Deceleraton arameter ( q ) > Concluson: In ths aer, we have consdered Lyra s feld equaton n the resence of wet dark flud for Banch tye-i sace tme. Gravt.14,83(198) 9) Reddy,D.R.K. and Innah, P.: strohys. Sace Sc.114,85 (1985). 1) Reddy,D.R.K. and venkateswarlu For solvng the feld equatons we have R.: strohys. Sc.149,87 used relaton between ρ and. The (1988). cosmologcal model thus obtaned reresents a radatng unverse n Lyra s J.Math.Phys.3,456(1991). 11) Sngh,T. nd Sngh, G.P.: theory of gravtaton. 1) Sngh,T. nd Sngh, References: G.P.:Int.J.Theor.Phys.31,1433(199). 1) Lyra, G. :math.z.54,5 (1951). 13) Sngh,T. nd Sngh, ) Weyl,Stzungsber G.P:Fortscher.Phys.41,737(1993). :reuss.kad.wss(1918). 14) Rahaman,F.Chakraborty,S.,Hossan,M.,Begu 3) Sen, D.K.:Z.Phys.149,311(1957). 4) Sen,D.K.and Dunn,K..:J.Math.Phys.1,578(197 m,n.and Beram,J.:Ind.J.Phys.B.76,747(). 1). 15) Pradhan,. and Pandy,H.R.:arXv:gr-qe/3738(3). 5) Halford,W.D.: J.Math.Phys.13,1399(197). 16) Bhowmk,B.B. and Raut, 6) Halford,W.D:ustr.J.Phys.3,863(1 97). 7) Bhamra,K.S.:ustr.J.Phys.7,541(1974). 8) Kalyanshett,S.B.andWaghmodeB.B.:Gen.Rel.

6 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March :Pramana J. Phys.6,6,1187 (4). 17) Reddy,D.R.K.: strohys. Sace Sc.3,381 (5). 18) Holman, R. and Nadu, S., ar Xv: stro-hy/481 (rernt) (5). 19) Tat, P. G. : The Voyage of HMS Challenger ( H. M. S. O., London,) ( 1988). ) Hayward,. T. J., Brt. J. l. Phys. 18, 965, (1967). 31) Gron,Q.: strohys. Sace 1) Holman, R. and Nadu, S., ar Xv: Sc.173,191 (199). stro-hy/481 (rernt) 3) Barrow,J.D.: Phys.Rev.D55,7451 (5). (1997). ) Sngh, T. and Chaubey, R. : Pramana Journal of Physcs, Vol. 71, qe/455 33) Caltaldo,M.del Camo.S.:arXv:gr- (). No. 3 (8). 3) dhav, K.S.,Mete,V.G.,Thakare,R.S., Pund,.M.: Int.J.Theory Phys, 5, 164 (11). 4) Kanser,E.:m.J.Math.43,17 (191). 5) Henekman,O., and schuckng,e.:in.edtons stoo Brassels, (1958). 6) Msner,C.W.: strohys. J.151,431 (1968). 7) Lfeshtz,E.M.,khalatnkov,I.M.:J.Ex.Theor. Phys.59,3 (197). 8) Belnsk,V..,khalatnkov,I.M, Lfeshtz,E.M.:dv.Phys.19,55 (197). 9)Belnsk,V..,khalatnkov,I.M,Lfesh tz,e.m.:j.ex.theor.phys.6,1 1(1971). 3) Gron,Q.:Phys.Rev.D3,5 (1985). 34) Brevk.I., Petterson,S.V. : Phys.Rev.D56,33 (1997). 35) Brevk.I., Petterson,S.V. : Phys.Rev.D61,1735 ().

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