Bianchi Type V Magnetized Anisotropic Dark Energy Models with Constant Deceleration Parameter

Transcription

1 Th fican Rviw of Physics (0 9: ianchi Typ V Magntizd nisotopic ak Engy Mods with Constant cation Paamt S Kato,* and Y Shaikh patmnt of Mathmatics, SG mavati Univsity, mavati-60, India patmnt of Mathmatics, N Cog of Engg & Tch, Yavatma-500, India In this pap, w hav studid th soutions of ianchi typ V univs with vaiab ω in th psnc and absnc of magntic fid of ngy dnsity spcia aw of vaiation fo Hubb s paamt poposd by man [9] has bn utiizd to sov th fid quations Som physica and kinmatica poptis of th mod a aso discussd Intoduction fw yas ago two goups of sachs (th Supnova Cosmoogy Pojct and th High-Z Supnova Tam psntd som vidnc that th pansion of th univs is accating (Ganavich t a [,]; Pmutt t a [-5]; Riss t a [6]; Schmidt t a [7] ak ngy is a mystious substanc with ngativ pssu and accounts fo nay 70% of tota matt-ngy of univs, but has no ca panation ak ngy has bn convntionay chaactizd by th quation of stat (EoS paamt ω p, which is not ncssaiy constant Th simpst dak ngy candidat is th vacuum ngy ( ω, which is mathmaticay quivant to cosmoogica constant Λ Th oth convntiona atnativs a quintssnc ( ω >, phantom ngy ( ω < and quintom as vovd and hav tim dpndnt EoS paamt so, som oth imits obtaind fom obsvationa suts coming fom SN Ia data (Knop t a [8] and SN Ia data coaboatd with CMR anisotopy and gaay custing statistics (Tgmak ta [9] a -67< ω <-06 and - <ω <-079, spctivy Sva mods hav bn poposd to pain dak ngy [0-8] Howv, it is not at a obigatoy to us a constant vau of ω man [9] poposd a spcia aw of vaiation of Hubb paamt in FRW spac-tim, which yids a constant vau of dcation paamt (P Such a aw of vaiation fo Hubb paamt is not inconsistnt with th obsvations and is aso appoimaty vaid fo sowy timvaying P mod Th aw povids picit foms *katosd@diffmaicom of sca facto govning th FRW univs and faciitats to dscib accating as w as dcating mods of voution of th univs Nw dak ngy mods in anisotopic ianchi typ-i (-I spac-tim with vaiab EoS paamt and constant dcation paamt hav bn invstigatd Padhan t a [0] Locay otationay symmtic ianchi-i cosmoogica mods with dynamicay anisotopic dak ngy (E and pfct fuid hav bn constuctd in Gna Rativity by kasu t a [] Thy assumd that th dak ngy (E is minimay intacting, has dynamica ngy dnsity and anisotopic quation of stat paamt (EoS ianchi typ III anisotopic dak ngy mods with constant dcation paamt hav bn invstigatd by Yadav and Yadav [], ianchi typ III magntizd anisotopic dak ngy mods with constant dcation paamt by Tad t a [] and dak ngy mods with anisotopic fuid in ianchi typ- VI 0 spac-tim with tim dpndnt dcation paamt hav bn invstigatd by Padhan t a [] ijan Saha [5] has studid th voution of th univs fid with dak ngy within th scop of a ianchi typ-v mod Eact soutions to fid quations a obtaind using th popotionaity condition and vaiationa aw of Hubb paamt Shi Ram t a [6] psntd act soutions of Einstin s fid quations fo a ianchi-typ VI 0 spac-tim fid with pfct fuid satisfying th baotopic quation of stat und th assumption that th pansion scaa is popotiona to sha scaa ijan Shah [7] studid th voution of th univs fid with dak ngy within th scop of a ianchi typ-vi mod In this pap, w hav studid ianchi typ V univs with vaiab ω in th psnc and

2 Th fican Rviw of Physics (0 9: absnc of magntic fid of ngy dnsity togth with constant dcation paamt Som physica and kinmatica poptis of th mod a aso discussd Th out in of th pap is as foows In sction, th mod and fid quations a dscibd and th soution of fid quations a psntd in sction whi sction concuds th findings Mod and Fid Equations W consid spatiay homognous ianchi typ- V mtic in th fom ds m m dt d dy C dz ( Wh,, and C a th functions of t ony and m is constant Th simpst gnaization of EoS paamt of pfct fuid may b to dtmin th EoS paamt spaaty on ach spatia ais by psving th diagona fom of th ngy momntum tnso in a consistnt way with th considd mtic H, w a daing with a magntizd anisotopic dak ngy fuid whos ngy momntum tnso is [, T, T, T T ] ν T µ diag (, Thn, on may paamtiz it as foows T [ p p p ] ν µ diag, y, z, diag [ ω ω ω ], y, z,, [ ω ( ω δ ( ω γ ] diag (,,,, Wh, is th ngy dnsity of th fuid; is th ngy dnsity of magntic fid, p, p y, p z a pssus and ω, ω y and ω z a dictiona EoS paamts fo fuid on, y and z as, spctivy, and ω is th dviation f EoS paamt of th fuid W hav paamtizd th dviation fom isotopy by stting ω ω and thn intoducing th skwnss paamts δ and γ that a th dviations fomω, spctivy, on y ais and z ais Th Einstin fid quations, in natua imits ( 8 π G, c a C C C m C m C C m C C C 0 C ( ω δ ( ω γ (6 (7 (8 (9 G µν Rµν Rgµν Tµν ( Wh, th ovhad dot dnots ( divativ with spct to cosmic tim t µ Wh, u ν g, µ (,0,0,0 µν u u is th fouvocity vcto, R µν is Ricci tnso, R is Ricci scaa, and T µν is th ngy-momntum tnso In commoving coodinat systm, Einstin s fid quations (, fo spatiay homognous ianchi typ V mtic (, in cas of (, ads to Soution of th Fid Equations Th fid quations (5 (9 a a systm of fiv quations with ight unknown paamts, C,,, ω, γ, δ Th systm is thus initiay, undtmind and w nd additiona constaints to cos th systm Intgating Eqn (9, w hav C C m C C ω (5 C (0

3 Th fican Rviw of Physics (0 9:005 7 ν Th ngy consvation quation: T 0, µ ; ν which ads to two quations fo th anisotopic fuid and magntic fid & & C& & C& & ( ω δ γ 0 ( C C and ( Finay, w constain th systm of quations with a aw of vaiation fo th avag Hubb s paamt that yids a constant vau of dcation paamt Such a typ of ation has aady bn considd by man and Gomid [8] fo soving FRW mods Th avag sca facto of ianchi-typ V mtic is givn m ( C R ( Th dictiona Hubb paamt in th diction of, y and z as, spctivy, fo th ianchityp V mtic a H &, H y &, H z Th man Hubb paamt H is givn by ( H H H H Th pop voum V is dfind by y m ( C z C& ( C (5 V ( g (6 Fom Eqns (-(6, w obtain V H V R C R C (7 In od to sov th fid quations, w us a physica condition that th pansion scaa is popotiona to sha scaa, that is n C (8 Wh, n is popotionaity constant ccoding to Thon [9], th obsvations of th vocity d-shift ation fo tagaactic soucs suggst that Hubb pansion of th univs is isotopic within about 0% ang appoimaty (Kantowski and Sachs [0], Kistian and Sacks [] and d shift studis pac th imit σ 0, on th atio of sha σ to Hubb H in H th nighbohood of ou gaay today Coin [] discussd th physica significanc of this condition fo pfct fuid and baotopic EoS in a mo gna cas In many paps (Saif and Zubai [], Yadav and Yadav [], this condition is poposd to find th act soutions of cosmoogica mods W hav th in mnt (Eqn ( is compty chaactizd by Hubb s paamt H Thfo, t us consid that man Hubb paamt H is atd to avag sca facto R by th ation H Wh, k ( and ( 0 s k R (9 > 0 s a constants n impotant obsvationa quantity is th dcation paamt q, which is dfind as RR&& q R& (0 Fom Eqns (7 and (9, w hav n R & kr ( n ( s R R & k ( Using Eqns (0, ( (, w gt constant vaus fo th dcation paamt fo th man sca facto as: q s fo s 0 ( q fo s 0 ( Th sign of q indicats whth th mod accats o not Th positiv sign of q (i, s > cosponds to standad dcating mods, whas th ngativ sign of q < 0 fo 0 s < indicats accation and q 0 fo s cosponds to pansion with constant vocity It is makab to mntion h that

4 Th fican Rviw of Physics (0 9:005 7 though th cunt obsvations of SN Ia [5,6 ] and CMR favo accating mods i, q < 0 Using Eqn (, w obtain th aw of avag sca facto as Eqns ( and (, ad to σ ( n k Θ k and ( t c s R fo s 0 (5 n k k ( fo s 0 (6 kt R c Wh, c and c a constants of intgation Cas (i: Mod fo 0 s ( q Fom Eqns (0, (7, (8 and (5, w gt th foowing act pssion fo th sca functions n ( t ( t c (7 n ( t ( t c (8 ( t ( t c (9 Wh, c 0 is th constant of intgation and n c0 n, and ( Thfo, th mod (Eqn ( bcoms ds dt ( t c ( t c n m n dy d ( t c m dz (0 Th pssion fo kinmatica paamts i, th Hubb s paamt H, th scaa pansion Θ, sha scaa σ fo mod (0 a givn by σ k H ( ( t c k Θ H ( ( t c n k ( t c ( n k k ( Using Eqn (7 in Eqn (, w obtain ngy dnsity fo magntic fid as (5 ( n ( t c Using Eqns (8, (7, (8, (9,(5, w obtain ngy dnsity fo fuid as ( n n ( t c ( t c ( n ( t c m n (6 It is obsvd that th Hubb paamt H, th scaa pansion Θ, sha scaaσ, magntizd dak ngy dnsity and ngy dnsity is dcasing function of tim and appoachs 0 as σ t Sinc im constant, th mod is not t Θ isotopic fo ag vaus of t Using Eqns (5, (7, (8, (9, (5 and (6, th quation of stat paamt ω is obtaind as ω ( n n n ( t c m n ( n ( t c s ( t c (7 It is obsvd that th quation of stat ω is tim dpndnt, it can b function of d shift z o sca facto R as w

5 Engy nsity Th fican Rviw of Physics (0 9: Tim Fig: Th pot of ngy dnsity ( vs tim (t Using Eqns (6, (7, (8, (9, (5, (6 and (7, th skwnss paamt, δ (i, dviation fom ω aong y ais, is givn by δ ( n n ( t c (8 ( n ( t c Using Eqns (7, (7, (8, (9, (5, (6, (7 and (8, th skwnss paamt, γ (i, dviation fom ω aong z ais, is givn by γ (n n ( t c ( n ( t c (9 pansion Θ, sha scaa σ mains as it is and th ngy dnsity fo magntic fid, th ngy dnsity fo fuid, th EoS paamt ω, th skwnss paamts δ and γ a givn by ω 0 ( n n m n ( t c ( t c ( n n n t c s n ( t c ( ( n n δ ( t c γ (n n m ( t c In th absnc of magntic fid i, 0 th vaus of Hubb s paamt H, th scaa

6 Th fican Rviw of Physics (0 9:005 7 Cas (ii: Whn 0 s ( q Fom Eqns (0, (7, (8 and (6, w gt th foowing act pssion fo th sca function ( n k t ( L t (0 nk t ( t L ( k t ( t L C ( Wh, 0 n n L c n c, L L and L LL Thfo, th mod ( bcoms ( n n k m (9 L ( n k t ( n kt L It is obsvd that th Hubb paamt H, th scaa pansion Θ and sha scaa σ has th σ constant vaus and im constant, th mod is t Θ not isotopic fo ag vaus of t Magntizd dak ngy dnsity is dcasing function of tim hnc appoachs 0 and ngy dnsity appoachs constant vau as t Using Eqns (5, (, (, (, (8 and (9, th EoS paamt ω is obtaind as ds dt L L kt ( n kt m dz d L nkt m dy ( ω m ( n n k ( n k t ( n L L t k (50 Th pssion fo kinmatica paamts i, th Hubb s paamt H, th scaa pansion Θ, sha scaa σ fo mod ( a givn by σ ( n k H k ( Θ H k (5 k Eqns (5 and (6 giv σ Θ k ( nk k ( n k k ( k ( nk k ( k k } k (6 (7 Using Eqn (0 in Eqn (, w obtain ngy dnsity fo magntic fid as (8 ( n kt L Using Eqns (8, (, (, ( and (8, w obtain ngy dnsity fo fuid as Using Eqns (6, (, (, (, (8, (9 and (50, th skwnss paamt δ (i, dviation fom ω aong y ais is givn by ( n k δ ( n (5 k L t Using Eqns (7, (, (, (, (8, (9, (50 and (5, th skwnss paamt γ (i, dviation fom ω aong z ais is givn by (n k γ ( n (5 k L t In th absnc of magntic fid i, 0 th vaus of Hubb s paamt H, th scaa pansion Θ, th sha scaaσ, th skw nss paamts δ and γ as it is and th ngy dnsity fo magntic fid, th ngy dnsity fo fuid, and th EoS paamt, ω a givn by 0 ( n n k L m ( n kt m ω ( n n k ( n k L t

7 Th fican Rviw of Physics (0 9: ( n k δ, (n γ k Concusion i In this pap, w studid th ianchi typ V cosmoogica mod in th psnc of magntizd anisotopic dak ngy ii W consid th ngy momntum-tns consist of anisotopic fuid with anisotopic EoS p ω and a unifom magntic fid of ngy dnsity iii Th aw of vaiation fo Hubb s paamt dfind in (7 fo ianchi typ V spac-tim mod givs two typs of cosmoogis wh th EoS paamt ω is a function of tim Fist, fo s 0, shows th soution fo positiv vau of dcation paamt indicating th pow aw pansion of th univs Scond, fo s 0, shows th soution fo ngativ vau of dcation paamt, which indicats th ponntia pansion of th univs iv W aso discuss th paamts in th absnc of magntic fid in both cass v It is obsvd that in both cass, EoS paamt ω is vaiab function of tim, which has bn suppotd by cnt obsvations Knop, t a [8] and Tgmak, t a [9] viin both cass, it is obsvd that in ay stag th EoS paamt ω is positiv i, th univs was matt dominatd in ay stag vii It is intsting to not that ou suts smb th suts obtaind by Yadav t a [5] in th absnc of magntic fid This study wi thow som ight on th stuctu fomation of th univs, which has astophysica significanc Rfncs [] P M Ganavich t a, stophys J 9, L5 (998 [] P M Ganavich t a, stophys J 509, 7 (998 [] S Pmutt t a, stophys J 8, 565 (997 [] S Pmutt t a, Natu 9, 5 (998 [5] S Pmutt t a, stophys J 57, 5 (999 [6] G Riss t a, ston J 6, 009 (998 [7] P Schmidt t a, stophys J 507, 6 (998 [8] R Knop t a, stophys J 598, 0 (00 [9] M Tgmak t a, (SSS coobaation stophys J Supp 69, 050 (00 [0] P J E Pbs and Rata: Rviws of Modn Physics 75, No, 559 (00 doi:00/rvmodphys75559 [] T Padmanabhan, Phys Rp 80, Nos5-6, 5 (00 doi:006/s070-57( [] E Totoa and M mianski, stonomy & stophysics, No, 7 (005 doi:005/000-66:00508 [] V F Cadon t a, stonomy & stophysics 9, No, 9 (005 doi:005/000-66:00 [] R R Cadw, Phys Ltt 55, Nos-, (00 doi:006/s070-69( [5] P J E Pbs and Ratha, stophys J Pat, Ltts 5, No, L7 (988 doi:0086/8500 [6] Ratha and P J E Pbs, Phys Rv 7, No, 06 (988 doi:00/physrv706 [7] V Sahni and Staobinsky, Int J Mod Phys 9, No, 7 (000 doi:0/s [8] Y-Z Ma, Nuc Phys 80, Nos-, 6 (008 doi:006/jnucphysb [9] M S man, Nuovo Cimnto 7, 8 (98 [0] niudh Padhan, H mihashchi and ijan Saha, Int J Tho Phys 50, 9 (0 OI 0007/s z, axiv:00v [g-qc] [] Özgü kasu and Can atta Kıın, Gn R Gav, Issu, 9 (00, [axiv: v [g-qc]] [] ni Kuma Yadav and Laan Yadav, Int J Tho Phys 50, Issu, 8 (0 [] S Tad and M M Sambh, Pspactim J, No5 (0 [] Padhan, R Jaiswa, K Jotania and R K Kha, stophysics and Spac Scinc 7, Issu, 0 (0 [5] ijan Saha, axiv:0969v [physicsgn-ph] (0 [6] S Chand Piyanka, M K Singh and Shi Ram, G J Scinc Fonti Rsach Mathmatics and cision Scincs, Issu, Vsion 0, (0

8 Th fican Rviw of Physics (0 9: [7] ijan Shah: axiv:09609v [g-qc] (0 [8] M S man and F M Gomid, Gn R Gav 0, 9 (988 [9] K S Thon, stophys J 8, 50 (967 [0] R Kantowski and R K Sachs, J Math Phys 7, (966 [] J Kistian and R K Sachs, stophys J, 79 (966 [] C Coins, E N Gass and Wikisons, Gn R Gav, 805 (980 [] M Shaif and M Zubai, stophys Spac Sci doi 0007/s y (00 [] K Yadav and L Yadav, axiv:007v [g-qc] (00 [5] K Yadav, F Rahman and S Ray, Int J Tho Phys 50, 87 (0 Rcivd: 5 Jun,0 ccptd: 7 Octob, 0