Holographic Dark Energy in LRS Bianchi Type-II Space Time

Transcription

1 Internatonal Journal Of Matheatcs And Statstcs Inventon (IJMSI E-ISSN: 767 P-ISSN: Volue Issue 09 Septeber. 0 PP-8-6 Holographc Dark Energy n LS Banch Type-II Space Te Gtuan Sara esearch Scholar Departent of Matheatcs Gauhat Unversty ABSTACT: In the present paper, we have dscussed holographc dark energy n LS Banch type-ii space te. The Ensten s feld equaton are solved by usng two assupton. One of the s law off varaton for the Hubble paraeter, whch gves a constant value of deceleraton paraeter and another s a relaton between two etrc co-effcent. The law of varaton generates power law and exponental for of average scale factor n ters of cosc te. The physcal and kneatcal propertes of the odel are also dscussed. The holographc dark energy (DE EOS paraeter behaves lke constant,,e,whch s atheatcally equvalent to cosologcal constant for n 0,whereas the holographc dark energy EOS paraeter behaves lke quntessence for n 0. I. INTODUCTION: ecent cosologcal observatons ndcates that the present unverse s undergong an accelerated expanson. Ths acceleraton of the unverse a well establshed fact that s confred by varous ndependent observatonal data ncludng SNIa, CMB radaton etc. However ths dscovery can be antaned n general relatvty by ntroducng a ysterous knd of energy source called the dark energy that can generate repulsve gravty [,]. It s well known that perfect flud wth a constant equaton of state (EOS paraeter lower than. For soluton of the Ensten s feld equaton one can seek by ntroducng by soe kneatcal ansatz that are consstent wth the observaton kneatcs of the unverse and ay nvestgate the dynacs of the flud as a possble canddate of the DE. For nstance Beran s law [,] for the Hubble paraeter that yelds constant deceleraton paraeter (DP has been wdely consdered for obtanng acceleratng cosologcal odels explctly n the fraework of General relatvty. Astronocal observatons ndcate that our unverse currently conssts of approxately 70% dark energy, 5% dark atter and 5% baryonc atter and radaton. Holographc dark energy s the nature of DE can also be studed accordng to soe basc quantu gravtatonal prncple. Accordng to ths prncple [5], the degrees of freedo n a bounded syste should be fnte and does not scale by t volue but wth ts boundary era. Here s the vacuu energy densty. Usng ths dea n cosology we take as DE densty. The holographc prncple s consdered as another alternatve to the soluton of DE proble. Ths prncple was frst consdered by G. t Hooft[6] n the context of blake hole physcs. In the context of dark energy proble though the holographc prncple proposes a relaton between the holographc dark energy densty and the Hubble paraeter H as H,t does not contrbute to the present accelerated expanson of the unverse. In [7], Granda and Olvers proposed a holographc densty of the for H H,where H s the Hubble paraeter and, are constants whch ust satsfy the condtons posed by the current observatonal data. It s known that Banch unverse ansotropc gve rse to CMB ansotropes dependng on odel type[8].a spatally ellpsodal geoetry of the unverse can be descrbed wth Banch type etrcs. However soe Banch type odels sotropze at late tes even for ordnary atter and the possble ansotropy of the Banch etrces necessarly de away durng the nflatonary era. In fact ths sotropzaton of the Banch etrces s due to the plct assupton that the DE s sotropc n nature. In ths paper, we present general relatvstc cosologcal odels wthn the fraework of spatally hoogeneous but totally ansotropc and non flat LS Banch type-ii space te. We consder the generalzed Beran s law for Hubble paraeter and a relaton between two etrc co-effcent for solvng the Ensten s feld equatons. A slar process was carred out by Sngh and Kuar[9] wthn the fraework of LS Banch type-ii space te. 8 P a g e

2 Holographc Dark Energy n LS Banch Type-II Space Te II. METIC AND FIELD EQUATION: We consder the LS Banch type-ii space te n the for ds dt A ( dx dz B ( dy xdz ( Where A and B are functon of te t only. Ensten s feld equaton s gven by j gj ( Tj Tj ( The energy oentu tensor for atter and the holographc dark energy are defned as T j u u ( j And, T ( p u u g ( j j j Where, are energy denstes of atter and holographc dark energy and holographc dark energy. p s the pressure of Feld equatons are AB A AB A (5 B A A A B B AB B AB A p (6 A A A A B A p (7 For the etrc ( we have the followng for V A B (8 A B V ; A B (9 A B j j A B 6 (0 H ( An observed quantty n cosology s the deceleraton paraeter q whch s defned as q ( H H The sgn of q ndcates whether the odel nflates or not. The postve sgn of q correspond to a standard deceleratng odel,whreas the negatve sgn ndcates nflaton. The average ansotropy paraeter ( s defned as- 9 P a g e

3 Holographc Dark Energy n LS Banch Type-II Space Te H H ( Where H H H, (,, The holographc dark energy densty are gven by H H ( The contnuty equaton can be obtaned as H ( p 0 (5 The contnuty equaton of the atter s H 0 (6 The contnuty equaton of the holographc dark energy s H( p 0 (7 The barotropc equaton of state p (8 Usng ( and (8 n (7, H HH H H H (9 III. SOLUTION OF FIELD EQUATION: There are three equaton (5-(7 wth fve unknown A, B,,, and p. Therefore to solve the feld equaton we need two condton. For any physcally relevant odel, the Hubble paraeter H and deceleraton paraeter q are ost portant observatonal quanttes n cosology. Hence to solve these equaton we frstly assue a law of varaton for the generalzed Hubble paraeter n the ansotropc odel whch was orgnally proposed by Beran[] and Beran and Gode[] for FW odel. Later on several authors such as Saha[], Maharaj and Nadoo[], Kuar [], Pradhan et al.[], Arhasch et al.[], Baghel and Sngh[5], have studed FW and Banch type odels by usng the law of varaton for Hubble paraeter. Accordng to the law of varaton of generalzed Hubble paraeter H s related to the average scale factor of LS Banch type-ii space te as H n n l l( A B (0 Where l( 0 and ( 0 n are constant. Fro equaton (8 and (0, n l ( n ( l ( n 0 P a g e

4 Holographc Dark Energy n LS Banch Type-II Space Te Usng equaton ( and ( n (, q n ( We see that the relaton ( gves q as a constant. Agan let us assue A ( n kb, where n s constant. Fro ( we obtan the average scale factor as n (ln t c for n 0 (5 exp{ l( t c} for 0 Fro equaton (6 we have got n (6 c, where c s a constant of ntegraton (7 Usng (8 and (, n A (8 n(n B (9 Where, k ( n c n k c k where c s a constant of ntegraton. Usng ( and ( n (, ( (0 Fro equaton (8, p L M N 6(n N (n ( Where P a g e

5 Holographc Dark Energy n LS Banch Type-II Space Te ( n L n(n M ( n { n(n } 9(n n (n n ( N k.-behavour of the odel for n 0 For n 0 n,we have got (ln t c Equaton (8,(9 gves n IV. SPECIAL CASES: n(n A (ln t c ( n (n B (ln t c ( Equaton (7,(0 and ( gves l ( n ( l (ln t c ( p 6(n n( n L (ln t c L(ln t c (5 Where L M ( n N ( n n L ( k n, M M, M n l { n(n } n(n, N l n And, n (ln t c c (6.:Behavour of the Model for n 0 For n 0 we have got exp{ l( t c} P a g e

6 Holographc Dark Energy n LS Banch Type-II Space Te Equaton (8 and (9 gves exp{ l( t c ( n A } (7 exp{ l( t c } n(n B (8 Agan equaton (7,(0 and ( gves exp{ l( t c c (9 } l (0 exp{ ( 6(n l t c n(n p } W N ( Where W ( L M l Equaton (0 shows that holographc dark energy densty s constant for n 0. V. SOME PHYSICAL POPETIES OF THE MODEL: The physcal and kneatcal paraeter of the odel have the followng expressons n V ( ( n Equaton ( shows that for n 0 the expanson scalar s constant whereas for n 0 l n(ln t c (ln t c ( l 50n n 6 where 6n (n H l (5 ln t c elaton (5 shows that for n 0, Hubble paraeter s constant. q n,0 n (6 P a g e

7 Holographc Dark Energy n LS Banch Type-II Space Te Fro equaton (6 we observed that when n,deceleraton paraeter s postve whch shows the deceleratng behavour of the cosologcal odel. It s worthwhle to enton the work of Vshwakara[], where he has shown that the deceleratng odel s also consstent wth recent CMB observatons odel by WNAP, as well as wth the hgh redshft supernova Ia. But for n 0, deceleratng paraeter s -. So we have exaned that n ust be les between 0 and for deceleratng odel. The atter densty paraeter ( and holographc dark energy densty paraeter are gven by H and H (7 Usng ( n (5, (ln t c (ln t c 6(n n (n 9n( n l ( Where, n (n k Now, n l {n6n n } n ( (ln t c for n 0 (9 n Equaton (9 gves the su of the energy densty paraeter approaches to at late te. So at late te the l unverse becoes flat. At large te, ths odel gve that the ansotropy of the unverse wll dapout and unverse wll becoe sotropc. Ths result also shows that n the very early stage of the unverse,e durng the radaton and atter donated era the unverse was ansotropc and the unverse s approaches to sotropy as dark energy starts to donate the energy densty of the unverse. Usng (5 and (6 n ( l ( n (ln t c for n 0 (50 Fro equaton (50 we have observed that for n 0the holographc dark energy densty s constant. Usng (5 and (6 n (9,we obtan I l Q { ( n }(ln t c (5 Where P l ( n( n, Q l ( n l 6 ( n I { ( n } l P { ( n } P a g e

8 Holographc Dark Energy n LS Banch Type-II Space Te Fro (5 we have observed that for n 0, the holographc dark energy EOS paraeter constant.,e n n 0 n t Fg-The plot of holographc energy densty vs. te n. H n n t Fg- The plot of holographc dark energy EOS paraeter vs. te 8 6 n. n 0 n t Fg-The plot of Hubble paraeter vs.te VI. DISSCUSION: The exact soluton of the Ensten s feld equatonshave been obtaned by assung the law of varaton for the generalzed Hubble paraeter. There s a cosologcal vew that the unverse ght be nhoogeneous 5 P a g e

9 Holographc Dark Energy n LS Banch Type-II Space Te and ansotropc n the very early stage of the unverse. Observatonal data also suggest that dark energy s responsble for gearng up the unverse soe fve bllon years ago. But at the te the unverse need not to be sotropc. So we have assued the unverse to be ansotropc and consder the LS Banch type-ii unverse flled wth atter and holographc dark energy. In ths paper we have exaned the present unverse s acceleratng under certan condton. We have also observed that the behavour of the odel for n 0and n 0. For n 0, the EOS paraeter of the holographc dark energy s constant.,e,whch n atheatcally equvalent to cosologcal constant(. EFEENCES: []. Copeland,E.J.,Sa,M.,Tsujkawa,S.:Int.J.Mod.Phys. D 5, 75-95(006. []. L,M., et al.: Coun. Theor.Phys (0 []. Beran, M.S.: Nuovo Cento B 7,8-86(98 []. Beran,M.S.,Gode,F.M.: Gen. elatv.grav. 0,9-98(988 [5]. Sussknd,L.:J.Math.Phys. 6, 677(995 [6]. t Hooft,G.:gr-qc9006 [7]. Granda,L.N.,Olveros,A.:Phys.Lett. B 669,75(008 [8]. G.F.. Ells: Modern Cosology, Insttute of Physcs Publshng, Brstol, pp8-9(00 [9]. Sngh,C.P.,Kuar,S.:Int.J.Mod.Phys. D 5,9-8(006 [0]. Pradhan, A., Arhasch, H.: arxv: 0.09v [physcs.gen-ph].5 oct.0 []. Debnath,U.:Class.Quantu Gravty 5,0509(008 []. Huang,J.Z.,et al.:astrophyscs.space sc.5,75(008 []. Shao,Y.,Gu,Y.:Phys.Lett.A,65(008 []. Pradhan,A.,et al.:int.j.theor.phys.(0.do:0.007s z.arxv:00.[gr-qc] [5]. Perlutter, S., et al.:nature(london 9,5(998 [6]. ess,a.g.,et al.:astron.j.6,009(998 [7]. Sarkar, S., Mahanta, C..: Int.J.Theor.Phys.5, 8 (0 [8]. Saanta, G.C.: Int.J.Theor.Phys.5; 89(0 [9]. Adhav, K.S.:Eur.Phys.J.Plus 6, 5(0 [0]. Adhav,K.S.:Astrophys.Space sc.5,6(0 []. B.Saha,Asrophys.Space sc.0,8(006. []. S.D.Maharaj,.Nadoo,Astrophys.space sc.08,6(99 []. S.Kuar,Astrophys.space sc.,9(0 []. A.Pradhan,H.Arhasch,B.Saha,Int.J.Theor.Phys.50,9(0 [5]. P.S.Baghel,J.P.Sngh,esearch n Astron.Astrophys,57(0 6 P a g e