Bianchi Type-II, VIII & IX Universe Filled with Wet Dark Fluid in f ( R,

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1 Inernaional Journal o heoreical Mahemaical Physics 0, (5): 9-6 DOI: 059/jmp00050 ianchi ype-ii, VIII & IX Universe Filled wih We Dark Fluid in ( R, K S dhav *, M V Dawe, R G Deshmukh Deparmen o Mahemaics, San Gadge aba mravai Universiy, mravai, 60, India bsrac he purpose o his paper is o sudy he role o (DE) dark energy in he orm o we dark luid in ianchi ype-ii, ype-viii ype-ix cosmological models wihin he rame work o ( R, heory o graviy proposed by arko e al []{arxiv:0669v[gr-qc]} We have used a new orm o he dark energy componen o universe known as we dark luid which is given by he equaion o sae p * [here he parameers are aken o * be posiive we resric ourselves o 0 ] which can describe a liquid or example waer Keywords ianchi ype-ii, ype-viii ype-ix universe, We Dark Flu id, ( R, Inroducion In view o he lae ime acceleraion o he universe he exisence o he dark maer dark energy, very recenly, he (R) heory o graviy ormulaed by Nojiri & Odinsov [] (R, heory o graviy proposed by arko e al[] are he noeworhy heories o graviaion Carroll e al[] explained he presence o a lae ime cosmic acceleraion o he universe in (R) graviy proposed dark energy model or speciic ( /R ) modiied graviy owever, his model is non-realisic because i does no pass Newon law he irs (R) dark energy model which passes he Newon law was proposed by Nojiri & Odinsov[] hey also (Nojiri & Odinsov[]) demonsraed ha phanom scalar in many respecs looks like srange eecive quanum ield heory by inroducing a non-minimal coupling o phanom ield wih graviy arko e al[] developed (R, modiied heory o graviy, where he graviaional Lagrangian is given by an arbirary uncion o he Ricci scalar R o he race o he sress-energy ensor hey have obained he graviaional ield equaions in he meric ormalism, as well as, he equaions o moion or es paricles, which ollow rom he covarian divergence o he sress-energy ensor Generally, he graviaional ield equaions depend on he naure o he maer source hey have presened he ield equaions o several paricular models, corresponding o some explici orms o he uncion (R, * Corresponding auhor: ai_ksadhav@yahoocoin (K S dhav) Published online a hp://journalsapuborg/mp Copyrigh 0 Scieniic & cademic Publishing ll Righs Reserved In ( R, ) graviy heory models, he ield equaions o his heory are obained rom he ilber-einsein ype variaional principle he acion or his modiied heory o graviy is given by S g 6G ( R, L d x m () ere ) is an arbirary uncion o he Ricci scalar R o he race o he sress-energy ensor o he maer L m is he maer Lagrangian he sress- energy ensor o maer is ( g Lm ) () g g In he presen paper, we use he naural sysem o unis wih G c so ha he Einsein graviaional consan is deined as k 8 he corresponding ield equaions o he (R, graviy are ound by varying he acion wih respec o he meric : g R R ( R, ) g ( g ) ( R, 8 ( R, ( R, where R R, R, R, is he covarian derivaive R, () is he sard maer energy-momenum ensor derived rom he Lagrangian L m

2 0 K S dhav e al: ianchi ype-ii, VIII & IX Universe Filled wih We Dark Fluid in ( R, One should noe ha when ( R, ) ( R) equaion () reduces o he ield equaions o (R) graviy y conracing equaion (), we ge hen R R, R + ( R, ( R, 8 ( R, ( R, R () Generally, he ield equaions also depend on [hrough he ensor ] he physical naure o he maer ield ence, several heoreical models corresponding o dieren maer sources in (R, graviy can be obained I we assume ha he uncion ( R, ) is given by ( R, ) R ( ) (5) where ( ) is an arbirary uncion o he race o he sress-energy ensor From equaion (), we ge he graviaional ield equaions in his case as G (6) R Rg 8 ( ( ( g where he prime denoes a derivaive wih respec o he argumen arko e al[] have invesigaed FRW cosmological models in his heory by choosing appropriae uncion ( hey have also discussed he case o scalar ields since scalar ields play an vial role in cos mology he equaions o moion o es paricles a rans-dicke ype ormulaion o he model are also presened Observaions o microwave background radiaion (CM) & experimenal daa sugges ha he presen day universe is largely homogeneous isoropic which is represened by FRW model owever, a he early sages o evoluion o universe here are reasons o believe ha he universe is, in general, spaially homogeneous anisoropic I is well known ha spaially homogeneous anisoropic cosmological models play a signiican role in describing he large srucure behavior o he universe Such models have been invesigaed in he ramework o general relaiviy in search o a realisic picure o universe in early sages lso, ianchi ype cosmological models are imporan in he sense ha hese are homogeneous anisoropic rom which he process o isoropisaion o he universe is sudied hrough he passage o ime complee discussion o ianchi ype models is given in Kramer e al[5] he purpose o presen paper is o sudy dieren viable cosmological models in he newly esablished exension o he sard general relaiviy which is known as he ( R, ) graviy heory In his paper, we sudy a spaially homogenous anisoropic ianchi ype-ii, ype-viii & ype-ix models in (R, graviy (proposed by arko e al[]) illed wih We Dark Fluid We Dark Fluid Riess e al [6], Perlmuar e al [7], Sahni [8] sudied he naure o he dark energy componen o he universe as one o he deepes myseries o cosmology We are moivaed o use he we dark luid () as a model or dark energy which sems rom an empirical equaion o sae proposed by ai [9] ayward [0] o rea waer aqueous soluions he equaion o sae or We Dark Fluid is p * () where he parameers are aken o be posiive * we resric ourselves o 0 We have energy conservaion equaion as p 0 () Using equaion o sae v / v in he above equaion, we ge c * v, () where c is he consan o inegraion v is he volume expansion has wo componens : one behaves as cosmological consan oher as sard luid wih equaion o sae p I we consider c 0, his luid will no violae he srong energy condiion p 0 : p * c v( ) 0 () olman Naidu [] used he we dark luid as dark energy in he homogeneous, isoropic FRW case Singh Chaubey [] sudied ianchi ype I universe wih we dark luid dhav e al[, ] sudied Einsein-Rosen ianchi ype-iii universe wih we dark luid in general relaiviy Recenly Jain e al [5] sudied axially symmeric cosmological model wih we dark luid in bimeric heory o graviaion Samana [6] sudied ianchi ype-v model in ( R, ) heory o graviy Meric Field Equaions wih heir Soluions

3 Inernaional Journal o heoreical Mahemaical Physics 0, (5): 9-6 We consider a spaially homogeneous ianchi ype merics in he combined orm as ds d d ( ) d d h( ) d, () where are he scale acors (meric ensors) which are uncions o cosmic ime only lso, are he Eulerian angles his equaion () represens (i) ianchi ype-ii meric i ( ) h ( ) (ii) ianchi ype-viii meric i ( ) cosh h ( ) sinh (iii) ianchi ype-ix meric i ( ) sin h ( ) cos he energy momenum ensor o he maer source (a we dark luid) is given by where ( p ) u u p g () j u is he low vecor saisying i j u u In a co-moving sysem o coordinaes, rom equaion () we ge Now varying he acion i S j i j g (), p, p, p ) () ( g 6 G o he graviaional ield wih respec o he meric ensor componens graviy model as ( arko e al[] ) R R, R R, g ( g m (5) g, we obain he ield equaions o ( R, ) L d x i j R 8 where,, R ere R R g Lm, i j ), (6) pg g g ; (7), is an arbirary uncion o he Ricci scalar R o he race o he sress-energy ensor o he maer Lm is he maer Lagrangian densiy Now assuming ha he uncion R where he, given by arko eal (0) R, R is an arbirary uncion o he race o he sress-energy ensor o maer Using equaion (7) (8), he ield equaions (6) ake he orm, (8) R R g 8 p ( ) g, (9) where prime indicaes derivaive wih respec o he argumen Now we choose he uncion o he race o he sress-energy ensor o he maer so ha is a consan (0) he corresponding ield equaions (9) or meric () wih he help o equaions () (0) can be wrien as 8 8 p p, where () ()

4 K S dhav e al: ianchi ype-ii, VIII & IX Universe Filled wih We Dark Fluid in ( R, 8 p where he overhead do ( ) denoes derivaive wih respec o he cosmic ime lso, or 0,,, he above ield equaions correspond o ianchi ype-ii, ianchi ype-viii ianchi ype-ix respecively he ield equaions ()-() are only hree independen equaions wih our unknowns,, p So in order o ge a deerminisic soluion we consider he physical condiion ha he Shear scalar is proporional o Scalar expansion, which leads o he linear relaionship beween he meric poenials ie From equaions (), () (), we obain n n 0, n n ( n) () n, where n is an arbirary consan () (5) ianchi ype-ii Cosmological Model: 0 For 0, he above equaion (5) reduces o From equaions () (), we obain scale acor as n n 0, n ( n) c d, () n n () n n c d, n ere c & n saisy c n( n) ( n) 0, where c 0 & d are arbirary consans y a suiable choice o coordinaes consans [ consider c =, d = 0 ], he meric () wih he help o equaions () () can be wrien as he spaial volume is given by ds d n n d d ( ) d d n ( ) () () V n n (5) he Generalized ubble parameer is ound o be V ( n) (6) V ( n) he expansion scalar is ound as ( n) (7) ( n) he deceleraion parameer q is given by he anisoropic parameer o he expansion q d ( n), n, & d ( n ) (8) is ound o be i ( n ) i ( n ) (9)

5 Inernaional Journal o heoreical Mahemaical Physics 0, (5): 9-6 where i,,, i represen he direcional ubble parameers in he direcion o x, y z respecively he Shear scalar is given by n (0) n Using () () in ()-(), we ge pressure energy densiy as 6( n ) n n (9 ) p, () ( ) ( ) ( n) ( ) ( )[n 0n 6] n (8 ) (9 ) 60 6 n( ) 8 ( n ) () ( ) ( ) ( n) he ensor o roaion { he voriciy ensor is a measure o he roaion o he local res- rame relaive o he compass o ineria} 0 is idenically zero ence, his universe is non-roaing i, j j, i 5 ianchi ype-viii Cosmological Model: For, he equaion (5) reduces o ( n) n ( n) n 0, n We can solve he above equaion ge he deerminisic soluion only or n= So, rom equaion (5) wih suiable subsiuion, we ge (5) where w From equaions (5), we ge From equaions () (5), we ge w w sin, (5) (5) w sin he meric () wih he help o equaions (5) (5) can be wrien as ds d w sin he spaial volume is given by where cosh he componen o he ubble parameer x (5) w d cosh d sin d sinh d y (55) w V sin ( ), (56) x, y z co( ) he Generalized mean ubble parameer is ound o b are given by (57), co( ) z

6 K S dhav e al: ianchi ype-ii, VIII & IX Universe Filled wih We Dark Fluid in ( R, co( ) (58) he expansion scalar is ound o be co( ) (59) he deceleraion parameer q is ound o be he anisoropic parameer o he expansion q d sec ( ) d (50) is ound o be i i 8 (5) i i,,, represen he direcional ubble parameers in he direcion x, y z respecively where o he Shear scalar is ound o be co ( ) (5) Using (5) (5) in ()-(), we ge pressure energy densiy as p w cos ec ( ) 7 ( w ) cosec ( ) w w ( ) (5) w ( )( ) cos ec ( ) 7 ( w )cosec ( ) w (8 ) w w ( ) w ( )( ) (5) 7 cosec ( ) he ensor o roaion { he voriciy ensor is a measure o he roaion o he local res- rame relaive o he compass o ineria} 0 is idenically zero ence, his universe is non-roaing i, j j, i 6 ianchi ype-ix Cosmological Model: For, he equaion (5) reduces o ( n) n ( n) n 0, n We can solve he above equaion ge he deerminisic soluion only or n= So, rom equaion (6) wih suiable subsiuion, we ge (6) where w From equaions (6), we ge w w sin, (6) (6)

7 Inernaional Journal o heoreical Mahemaical Physics 0, (5): From equaions () (6), we ge w sin he meric () wih he help o equaions (6) (6) can be wrien as ds d w sin he spaial volume is given by where sin he componen o he ubble parameer w (6) d sin d sin d cos d x (65) w V sin ( ), (66) x, y z y are given by co( ) he Generalized mean ubble parameer is ound o be he expansion scalar is ound o be he deceleraion parameer q is given by he anisoropic parameer o he expansion (67), co( ) z co( ) (68) co( ) (69) q d sec ( ) d (60) is ound o be i i 8 (6) where i i,,, represen he direcional ubble parameers in he direcion o x, y z respecively he Shear scalar is given by co ( ) (6) Using (6) (6) in ()-(), we ge pressure energy densiy as p w cos ec ( ) 7 ( w ) cosec ( ) w w ( ) (6) w ( )( ) (8 ) w cosec ( ) 7 ( w )cosec ( ) w w ( ) w ( )( ) (6) 7 cosec ( ) he ensor o roaion { he voriciy ensor is a measure o he roaion o he local res- rame relaive o he compass o ineria} 0 is idenically zero ence, his universe is non-roaing i, j j, i

8 6 K S dhav e al: ianchi ype-ii, VIII & IX Universe Filled wih We Dark Fluid in ( R, 7 Conclusions ere, we have consruced ianchi ype-ii, ype-viii ype-ix cosmological models in ( arko e al [] ) heory o graviaion wih a new equaion o sae or he dark energy componen o he universe known as We Dark Fluid For all above models o universe [ianchi ype-i, ype-viii & ype-ix], we observe ha (i) Iniially, he proper volume is zero u, laer on, when, i increases & becomes ininiely large (ii) he deceleraion parameer q < -, hence, hese universes are acceleraing which is consisen wih he presen day SNIa observaions lim (iii) he value o limi 0 ence, hese models do no approach isoropy (iv) For hese models, we ge 0 5 {his value is greaer han he presen upper limi (0-5 ) o, Collins e al [7] } his implies ha our models represen early sage o evoluion o universe (v) hese cosmological models o universe are exping, non-roaing acceleraing CKNOWLEDGEMENS uhors are hankul o UGC, New Delhi or inancial assisance hrough M R P REFERENCES [] arko e al: arxiv: 0669v [gr-qc] (0) [] Nojiri,S, Odinsov,SD: arxiv : hep-h/00788(00a) [] Carroll, e al : Phys Rev D 70, 058 (00) [] Nojiri,S, Odinsov,SD: Phys Le 56, 7(00b) [5] Kramer e al: Exac Soluions o Einsein Field Equaions, VE Deuschur V D W, erlin (980) [6] Riess,, e al: sron J 6, 009 (998) [7] Perlmuer, S, e al: srophys J 57, 565 (998) [8] Sahni, V: arxiv: asro-ph/00 (00) [9] ai, PG: he Voyage MS Challenger MSO, London (988),vol, par [0] ayward, J: ra J ppl Phys 8, 965 (967) [] olman, R S Naidu, S : arxiv: sro-phy / 0080 (005) [] Singh,, Chaubey, R: Pramana 7, 7 (008) [] dhav, KS, e al: In J heor Phys 50, 6 (0a) [] dhav, KS, e al: In J heor Phys 50, 9 (0b) [5] Jain, P, e al: In J heor Phys 5, 56 (0) [6] Samana, G C :In J heor Phys: DOI:0007/s (0) [7] Collins, C, awking, SW: srophys J 80, 7 (97)