Ruban s Cosmological Modelwith Bulk Stress In General Theory of Relativity

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1 IOS Journl of Mthemtcs (IOS-JM e-issn: , p-issn: X Volume, Issue Ver IV (Jul - Aug 5, PP 5-33 wwwosrjournlsorg ubn s Cosmologcl Modelwth Bul Stress In Generl heory of eltvty VGMete, VDElr, 3 ASNmr Deprtment of Mthemtcs, DIK & KD College, Bdner-Amrvt,Ind Deprtment of Mthemtcs,JDPtl Sngludr Mhvdyly,Drypur,DstAmrvt 3 Deprtment of Mthemtcs,Shr DrGthod Arts nd ScenceCollege, Murtzpur, DstAol Abstrct: hs pper dels wth the study of ubn s cosmologcl model n presence of bul stress source n the frmewor of Generl theory of reltvty Exct soluton of Ensten s feld equtons re obtned by usng supplementry condton between metrc potentls nd curvture prmeter he vscosty coeffcent of bul vscous flud s ssumed to be smple power functon of mss densty s the coeffcent of sher vscosty s consder s proportonl to the scle of expnson n the model he tme dependent -term s found to be postve nd s decresng functon of tme Also some physcl nd geometrcl propertes of the model re dscussed Keywrds: ubn s spce tme, vscous flud models: vrbles -term I Introducton o study the evoluton of unverse mny reserchers hve constructed cosmologcl models contnng vscous fluds he effect of vscosty plys very mportnt role n the erly evoluton of the unverse here re mny crcumstnces durng the evoluton of the unverse n whch bul vscosty could rse he presence of vscosty n the flud ntroduces mny nterestng fetures n the dynmcs of homogeneous cosmologcl model he roles plyed by the vscosty nd the consequent dssptve mechnsm n cosmology hve been dscussed by severl uthors he het represented by the lrge entropy per bryon the mcrowve bcground provdes useful clue to the erly unverse nd possble explnton for ths huge entropy per bryon s tht t ws generted by physcl dssptve processes ctng t the begnnng to the evoluton of the unverse hese dssptve process my ndeed be responsble for the smoothng out of ntl nsotropes [] Msner [] suggested tht the neutrno vscosty ctng n the erly er mght hve consderbly reduced the present nsotropy of the blc body rdton durng the process of evoluton Mny reserchers [3-] hve shown nterest to study bul vscous cosmologcl model n generl reltvty Wenberger [], Heller nd Klme [], Msner [3] hve studed the effect of vscosty on the evoluton of cosmologcl models Collns nd Stewrt [] hve studed the effect of vscosty on the formton of glxes In Ensten s theory of grvty newtonn grvttonl constnt G nd cosmologcl constnt Λ re consdered s fundmentl constnts he grvttonl constnt G plys the role of couplng constnt between geometry of spce nd mtter n Ensten s feld equtons ecent cosmologcl observtons shows tht n ccelertng unverse wth vrbles G &, generlzed Ensten s theory of grvtton hve been proposed by Lu[5] he possbly of vrbles G & n Ensten s theory hs lso been studed by Dersrssn [6]In n evolvng unverse, t ppers nturl to loo t ths constnt s functon tme Drc [7] nd Dce [8] hve suggested tme-vryng grvttonl constnt he Lrge Number Hypothess (LNH proposed by Drc [9] leds to cosmology G vres wth cosmc tme here hve been mny extensons of Ensten s theory of grvtton, wth tme dependent G, n order to cheve possble unfcton of grvtton nd elementry prtcle physcs he cosmologcl model wth vrble G nd Λ hve been recently studed by severl uthors Some of the recent dscussons on the cosmologcl constnt nd on cosmology wth tme vryng cosmologcl constnt by tr nd Peebles[], Shn nd Strobnsy [], Peebles [], JPSngh et l [3-], MKVerm et l [5] nd Prdhn et l [6] ecently MKVerm nd Shr m [7-8] studed sptlly homogeneous bul vscous flud models wth tme dependent grvttonl constnt nd cosmologcl term Anrudh Prdhn et l [9] hve constructed ccelertng Bnch type-i unverse wth tme vryng G nd Λ-term n generl reltvty Also Lm nd Nobre [3-3] studed the sptlly nhomogeneous solutons of the Ensten s-mxwell equtons n the frme wor of ubn s metrc ecently Sngh et l[33] hve nvestgted nsotropc Bnch type II vscous flud model wth tme dependent G nd Λ Bul vscous nsotropc cosmologcl models wth dynmcl cosmologcl prmeters G nd Λ hve been presented by Kotmbr et l[3] Hrpreet etl[35] hve nvestgted bul DOI: 979/ wwwosrjournlsorg 5 Pge

2 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty vscous Bnch type- I cosmologcl models wth tme dependent Λ- term n self creton theory of grvtton Very recently Mete etl [36-39] hve studed vrous cosmologcl models n presence of bul vscous flud In ths pper we dscussed ubn s cosmologcl model n presence of bul stress source wth tme dependent -term n the frmewor of generl theory of reltvty hs pper s orgnzed s follows: In secton- we hve derve the feld equtons, secton-3 dels wth the soluton of feld equtons n presence of vscous flud Some prtculr nd specl cses re dscussed n Secton- nd Secton-5he lst secton contns concludng remr II he Metrc And Feld Equtons Let us consder ubn s lne element[] ds dt Q ( x, t dx ( t( dy h dz, ( sn y f sn y h( y y f snh y f ( And s the curvture prmeter of the homogeneous -spces t nd x constnts he functons Q nd re free nd wll be determned by the Ensten feld equtons (EFE wth cosmologcl constnt j gj 8 j g j, (3 s the cc tensor; j g j j s the cc sclr; nd j s the energy momentum tensor of vscous flud gven by j j j p v v pg g ( v v v v v v v v, ( Here ( ; ; ; ; p ( 3, p, nd re the energy densty, pressure, coeffcent of sher nd bul vscostes respectvelyhe p semcolon (; ndctes covrnt dfferentton he sher nd bul vscosty nd re postve nd my be ether constnt or functon of tme or energy such s, b, nd b re constnt v s the flow vector stsfyng the reltons g v v j j (5 we choose the co ordntes to be commovng, so tht v v v 3, v (6 he feld equtons (3 for the metrc ( wth mtter dstrbuton ( yeld Q 8 p Q (7 Q Q Q 8 p Q (8 DOI: 979/ wwwosrjournlsorg 6 Pge

3 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty Q Q 8, (9 the over hed dot( t the symbol Q nd mens tme dervtve he sptl volume for the model ( s gven by V hq ( he sclr expnson, nd sher sclr re defne by v : Q Q IJ ( IJ ( III Soluton Of Feld Equtons he feld equtons (7-(9 re system of three equtons wth seven unnowns prmeters, Q,, p,, nd For complete determncy of the system, extr condtons re needed Frst we ssume relton n metrc potentl s n Q ( x (3 nd secondly we ssume tht the coeffcent of sher vscosty s proportonl to the scle of expnson, e n s constnt equton (7 nd (8 leds to Q Q Q Q, ( Q Q 6 (5 Condton ( leds to Q l, (6 Q l s proportonlty constnt equton (5 together wth (3 nd (6 yeld ( n (7 ( n 6 l( n (8 we ssume tht f ( (9 ff, ( DOI: 979/ wwwosrjournlsorg 7 Pge

4 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty df when f d Equtng (7, (9 nd ( we hve d ( f ( f ( d ( n From equton ( we obtn d dt ( n, ( s the constnt of ntegrton After sutble trnsformton of co ordntes, the metrc ( reduces to the form ds ( n d x n n dx ( dy h dz, (3 he pressure nd densty for the model (3 re gven by 8 p K 8 ( n ( ( ( n ( n ( nd (n 8 K, (5 ( K 6n( n l 6l( n 3 (n ( n, K For the specfcton of, we ssume tht the flud obeys n equton of stte of the form p, (6 ( s constnt hus, gven (t we cn solve for the cosmologcl prmeters In most of the nvestgtons nvolvng bul vscosty s ssumed to be smple power functon of the energy densty (Pvon, []; Mrtens,[]; Zmdhl, [] m t, (7 ( nd m re constnt If m equton (7 my correspond to rdtve flud (Wenberg, [3] However, more relstc models (Sntos, [] re bsed on m lyng n the regme m Usng (7 n (, we obtn 8 m p K 8 ( ( n (8 ( ( n ( n DOI: 979/ wwwosrjournlsorg 8 Pge

5 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty 3Model I: When (t s constnt When m, equton (7 reduces to ( t constnt Hence n ths cse equton (8 wth the use of (5 nd (6, leds to (n 8 ( K ( n ( n (n 8 ( ( n ( n ( (9 Elmntng (t between (5 nd (9, we obtn ( n (n ( n ( K ( 8 ( n ( n ( (n (3 3 Model II : When(t When m, equton (7 reduces to ( t, hence n ths cse equton (8 wth the use of (5 nd (6, leds to 8 ( ( n ( ( n (n K ( n ( n (n ( (3 Elmntng (t between (5 nd (3,we obtn ( ( n ( n ( (n K ( n ( n (n ( K (n ( (3 From equtons (3 nd (3, we observe tht when the tme dependent -term s decresng functon of tme nd pproches smll vlue n the present epoch Some physcl spects of the models Wth regrds to the nemtcl propertes of the velocty vector v n metrc (3 the sclr expnson ( nd sher sclr ( of the flud re gven by ( n (33 ( ( n ( n ( 3 ( n (3 DOI: 979/ wwwosrjournlsorg 9 Pge

6 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty IV Prtculr Models If we set n then the geometry of spce tme (3 reduces to the form ds d x dx ( dy h dz (36l 3 6 l s n ntegrtng constnt he pressure nd densty of the model (35 re gven by 6 8 p 6l ( l l (3 6l 3 ( 6l 3 6l (6l (35 ( (37 ( 6 3 6l 8 l Model- I : When (t s constnt When m, equton (7 reduces to ( t constnt Hence n ths cse equton (36 wth the use of (37 nd (6, leds to ( 6l (3 6 l l 3 6 l (8l 3 ( 6 l (6l 3 6l (38 Elmntng (t between (37 nd (38, we obtn 6 5 ( 6l ( (3 6 l l 3 6l (6 8l 5 3 (39 (6l 6(6l 3 6l Model- II: When(t When m, equton (7 reduces to ( t, hence n ths cse equton (36 wth the use of (37 nd (6, leds to 8 ( 6( 6l 3 6l 6 5 6l (3 6 l l 3 6 l Elmntng (t between (37 nd (,we obtn (8l ( (6l DOI: 979/ wwwosrjournlsorg 3 Pge

7 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty ( 6( 6l 3 6l 6 5 6l (3 6 l l 3 6 l 5 5 8(6l 3 6l (8l (6l ( From equtons (39 nd (, we observe tht the tme dependent -term s decresng functon of tme nd pproches smll vlue n the present epoch Some physcl spects of the models Wth regrds to the nemtcl propertes of the velocty vector ( nd sher sclr ( of the flud re gven by (3 6l 8(6 l v n metrc (3 the sclr expnson ( 8( 6l 3 (3 6l (3 If we set n nd d dt V Specl Models l, equton ( leds to 3 ( whch on ntegrton gves t b, (5 b ( b nd b s n ntegrtng constnt, hence we obtn Q x x ( t b (6 Usng the trnsformtons t b, x X, y Y the metrc ( tes the form ds d ( b X dx ( dy h he pressure nd densty for the model (7 re gven by 5 3 8p 3 3 dz (7 (8 nd DOI: 979/ wwwosrjournlsorg 3 Pge

8 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty (9 5Model -I :When (t s constnt When m, equton (7 reduces to ( t constnt Hence n ths cse equton (8 wth the use of (6 nd (9, leds to ( (5 3 3 Elmntng (t between (9 nd (5, we obtn ( (5 5 Model- II: When(t When m, equton (7 reduces to ( t, hence n ths cse equton (8 wth the use of (6 nd (9, leds to ( (5 Elmntng (t between (5 nd (5, we obtn ( (53 From equtons (5 nd (53, we observe tht the tme dependent -term s decresng functon of tme nd pproches smll vlue n the present epoch Some physcl spects of the models Wth regrds to the nemtcl propertes of the velocty vector ( nd sher sclr ( of the flud re gven by v n metrc (3 the sclr expnson (5 3 (55 VI Concluson In ths pper, we hve obtned ubn s cosmologcl model n the presence of bul stress source wth tme dependent -term In ll these models we observe tht they do not pproch sotropy for lrge vlue of tme For smplcty flud obey n equton of stte of the form p nd bul vscosty s ssume to be smple power functon of energy densty gven by m ( t he tme dependent -term n ll models re decresng functon of tme nd they ll pproches smll postve vlue s tme n progress(e present epochit s observe tht s decresng functon of nd pproches to zero s lso n ll models energy densty nd pressure tends to zero s nd DOI: 979/ wwwosrjournlsorg 3 Pge

9 ubn s Cosmologcl Modelwth Bul Stress In Generl heory Of eltvty he model s not sotropy for lrge vlue of he model s expndng, sherng nd non rottng nd hs no ntl sngulrty for =,, + Acnowledgement One of the uthor ( VDElr s grteful to the Unversty Grnts Commsson, New Delh, Ind for provdng fellowshp under FIP eferences [] S Wenberg: Grvtton nd cosmology, Wley Newyor Chp,5,97 [] CWMsner, he sotropy of the unverse,:astrophys J 5, 3-57, 968 [3] Soy nd APwr,: Ind J Pure nd Appl Mths, 33-3, 983 [] JDBrrow, : Phys Letter B, 8,, , 986 [5] DPvon, J Bfluy nd DJou, Clsscl nd Quntum grvty, 8,,37,99 [6] GMohnty nd Pttn,: IntJherot Phys3,,39-, 99 [7] GMohnty nd BD Prdhn,: IntJherot Phys, 3,, 5 6, 99 [8] JASLm, ASM Germno, nd LWAbrmo,: Physcl evew D, 53, 87-97, 993 [9] JKSngh, : II Nuovo Cmento B,,, 5, 5 [] Mrttens,: Clsscl nd Quntum Grvty,, 6,55-65,995 [] S Wenberg, : Astrophys, J 68, 75, 97 [] M Heller,Z Klme, :, Astrophys, spce Sc, 53, 37, 975 [3] CWMsner, Nture,, 967 [] CB Collns, JM Stewrt, : mon Not Astron, soc 53, 9,97 [5] YK Lu, : Astroln, Journl Physcs, 38,, [6] M Dersrssn,: Nue, Cm, B 88, 9, 985 [7] PAMDrc, : Nture 39, 33 (937 [8] HDce,,:Nture 9, (96 [9] PAMDrc, : Proc, Soc Lon 65, 99 (938 [] B tr, nd PJE Peebels : Phys, ev, D 37, 36, 998 [] V Shn, nd A Strobnsy,: Int,J Mod, Phys D-9, 373 ; gr-qc/99398, [] PJEPeebles, nd B tr, : ev Mod Phys, Vol 75,559, 3 [3] JP Sng,,A Prdhn, nd AK Sngh :Astrophys, Spce Sc, 3, 83-88, 8 [] GPSng,nd AY Kle, : Int, J heor, Phys, ,9 [5] MK Verm, nd Shr m : Int, J heor, Phys 9, 693-7, [6] A Prdhn, nd S Lt, : Elect J heor Phys,vol8, no, 5, pp 53-68, [7] M K Verm, nd Shr m : Adv Studes heor, Phys, 5, 8, , [8] M K Verm, nd Shr m : Appled Mthemtcs, 38-35, [9] A Prdhn,, LS Ydv,, L Ydv, : ADN journl of Scence nd echnology 3,, -9, 3 [3] JAS Lm, MAS Nobre,u Dr Xver Sgud 5-Urc, 9 o de Jnero, J, Brzl [3] JAS Lm,: Phys Lett, 6A, 986 [3] JAS Lm,nd J omno,: Gen el Gev, 57, 988 [33] MKSngh, MKVerm, Shr m,: Int J heor Phys,, 77-83,3 [3] SKotmr, GPSngh, Kelr,: Nturl Sc,7, [35] Hrpreet Kur, een Behl, DPShul, : Int J Phys And MthsSc, 5,, 9, 5 [36] VGMete, ASNmr, VD Elr: IntJ heor PhysVol5, N6 (5 DOI 7/s [37] VGMete, VDElr: IntJ of Sc nd eserch, Vol Issue, 8-8, 5 [38] VGMete, VDElr : IntJ of Sc nd eserch, Vol Issue3,88-9, 5 [39] VGMete, VDElr,VS Bwne: Multlogc n Scence, VolIII, IssueVIII,3-39, [] DPvon,J,Bfluy, nd DJou, :, Clss Qunt Grv 8, 357, 99 [] Mrtens,: Clss Quntum Grvt, 55, 995 [] MZndhl,: Phys ev D 53, 583, 996 [3] SWenberg,: Grvtton nd Cosmology, Wley, New Yor, 97 [] NOSntos, SDs nd ABnergee,: J Mh Phys, 6, 878, 985 DOI: 979/ wwwosrjournlsorg 33 Pge