BIANCHI TYPE I HYPERBOLIC MODEL DARK ENERGY IN BIMETRIC THEORY OF GRAVITATION

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1 May 0 Issue-, olume-ii 7-57X BINCHI TYPE I HYPERBOLIC MODEL DRK ENERGY IN BIMETRIC THEORY OF GRITTION ay Lepse Sene College, Paun-90, Inda bstrat: In ths paper, we presented the solutons of Banh Type-I spae -tme wth dark energy (untessene and Chaplygn gas by solng them Rosen s feld euaton n Bmetr theory of grataton. The hyperbol geometr ew pont of the models wll be helpful to the people who use obseratonal data to searh for suh types of geometry. Keywords : Banh type-i, bometr, gratatonal theory, osmology, hyperbol geometr, sotropze geometry MSC 00 No: 8D05;8F05. Introduton Seeral theores of grataton hae been proposed as alternate to Ensten s theory of grataton n order to explan the osm aeleraton and exstene of dark energy and dark matter n the unerse. One among them s Rosen s Bmetr theory of grataton. The latest dsoery of modern osmology s that the urrent unerse s expandng and aeleratng. The data base lke Cosm Mrowae Bakground Radaton (CMBR, the experment of Tegmark et al. (00 and suh as Type Ia supernoa (S Ne Ia obseratonal data of Esensten et al.(005, ster et al. (00 and Spergel et al. (007 ndate that the unerse ontanng only % ordnary baryon matter, % dark matter and remanng 7% dark energy. The dark matter s unknown form of matter whh has lusterng propertes of ordnary matter and has not been yet deteted. The dark energy s the term represents for an unknown form of energy whh has not also been deteted. The euaton of state parameter (t has an mportant role n dark energy models. The onstant alues of represents auum, 0, and flud, dust, radaton and stff domnated unerse. The arable (t of tme or red shft s onsdered and the untessene model, (explanaton of obseratons of aeleratng unerse nolng salar feld and phantom model (expanson of unerse nreases to nfnte degree n fnte tme ge rse tme dependant parameter (t dsussed by Turner et al. (997, Jmenez (00, Das et al. (005. arous forms of parameter (t hae been used for dark energy models and a bnary mxture of perfet flud and dark energy hae been onsdered by many researhers lke Caldwell et al. (998, Lddle et al. (999, Stenhardt et al. (999, Saha (005, 00, Rahaman et al. (00, 009, Mukhopadhyay et al. (008, Sngh et al. (009 Ray et al. (00, karsu et al. (00, Sharf et al. (00, Yada et al. (0 and Pradhan et al. (0 to nestgate the osmologal models of the unerse and dedued the dfferent geometral and physal aspets of the models. Metr and Feld Euatons We onsder the Banh Type I metr n the form ds where (. dt dx B dy, B and C are the funtons of t only. The flat metr orrespondng to metr (. s d (. dt dx The energy momentum tensor T dy dz C dz T of the soure wth perfet flud and dark energy s gen by ( p p (. wth (. Here p and are the pressure and energy densty of matter respetely and The uantty s the salar of expanson whh s gen by s the flow etor.

2 May 0 Issue-, olume-ii 7-57X (.5 We assume the oordnates to be o-mong, so that 0, The euaton (.. of energy momentum tensor yeld ts omponents as T T T p and T (. Our am s to sole Rosen s feld euatons (. and (. for the metr (. and (.. For ths purpose, frst we are gong to alulate the omponents of Rosen s R tensor,,,,. For (. for euaton (. s N s (g g s fter solng, we get the followng euatons, the omponent N of Rosen s R tensor B C B C BC p B C B C B C B C B C B C BC p B C B C B C B C BC p B C B C B C B C BC From euatons (.7 and (.8, we wrte B and from euatons (.7 and (.9, we get C Thus we hae (. B C ddng the euatons (.7 and (.0, we get 8 BC ( p The olume s the funton of t and t s defned as BC (. From the aboe euatons we wrte.e., By straghtforward alulatons, we get (. B C (. ( p N from euatons (.7 (.8 (.9 (.0 (.5 N, from

3 May 0 Issue-, olume-ii 7-57X We are gong to sole ths dfferental euaton (..7 for the olume and then usng the euaton (.., we are alulatng the sale fators, B and C and then we plan to study three dfferent models n related wth untessene and haplygn gas dark energy Model wth dark energy Quntessene model Let us onsder the model wth dark energy n whh the dark energy s gen by untessene whh obeys the euaton of state p, 0 (. n whh s euaton of state parameter orresponds to untessene model. The onseraton law of energy momentum tensor, euaton n the form of dark energy gen by untessene, an be wrtten as ( where s an ntegraton onstant. Usng the alue of energy densty p (, we wrte the pressure p as Thus we hae the pressure p p and energy densty of the perfet flud as, ( (. ( where s the onstant of ntegraton. d 5 8 ( where 5 and are onstants of ntegraton. In partular, 0, d t 8 ( 5 t (. we wrte we dsuss the soluton wth the partular ases of 0, and. Case ( For 0 (een though t s a ase of dusty unerse, t s onsdered n untessene (. euaton of state, the ntegral euaton has no soluton and hene the model does not exst. Case ( For. The ntegral euaton has a soluton (there s no need to repeat the alulaton part, as t s smlar to the ase of perfet flud for (.5 (snh( mt n

4 May 0 Issue-, olume-ii 7-57X where m and n are non zero poste onstants gen by k s onstant. From euatons, we hae the sale fators 9 m and n m k B C (. Thus the reured metr s ds (snh( mt n dt ( dx dy dz (.7 (snh( mt n, n whh Ths s Banh type I osmologal model wth untessene dark energy n Bmetr theory of grataton for, whh represents dark energy star. Ths model has hyperbol geometry and ts olume and sale fators, B and C are n nerse of hyperbol sne funtons. From the Graph-, t s obsered that, at t 0, the olume of the model attan the maxmum alue and t s gradually dereasng as t nreasng and approahes to zero alue, when t. Ths shows that the model starts wth maxmum olume and the olume slowng down and approahes to zero at later stage. From euatons the pressure p and the energy densty, orrespondng to untessene model are p (snh( mt n (.8 Graph-: s t The nature of densty Graph-: follows s t the graph of hyperbol sne funton. Graph-: t t P 0 s, t the densty attan the non zero poste alue and goes on nreasng, as tme t nreasng and reahes to nfnty, when

5 May 0 Issue-, olume-ii 7-57X t as shown n Graph-. Ths shows that n the begnnng, the model has matter and densty of the matter goes on nreasng wth tme t and t s nfnty at later stage. It s shown n Graph- that the nature of the pressure p n the model s negate and dereasng n nature and tends to mnus nfnty at later stage. The Hubble parameter H and ts dretonal s H H H H moth( mt n (..9 Graph- shows the nature of the Hubble Graph-: parameters H s H and t ts dretonal s H, H and H and ts graph s n nerse of hyperbol tangent funton and the graph les n I th uadrant. Ths shows that there s a negate rate of expanson n the model, whh ndate ontratng model. In the untessene dark energy model for hae been alulated as moth( mt n, the physal parameters,, and (.0 (. 0 (. seh ( mt n. The salar expanson n the model obeys the same nature as that of Hubble parameter and t s always negate supportng the ontraton of the model as shown n Graph-5. Graph-5: s t Graph-: s t The deeleraton parameter s always negate and ts graph shown n Graph-. Thus ths untessene dark energy model s represents dark energy star and t s ontratng startng wth maxmum olume. 5

6 May 0 Issue-, olume-ii 7-57X Case ( For whh has a soluton e d the ntegral euaton beomes t 5 t The sale fators are B C e where 5 Thus, the reured metr s ds a t a s a poste onstant. dt e at ( dx dy dz Ths s the Banh type I osmologal model wth dark energy (untessene n Bmetr theory of grataton for whh has olumetr exponental expanson. The olume and sale fators, B and C are n exponental law. t t 0, exponental, as t nreasng and reahes to nfnty, when. they admt alue one and nreasng t Ths shows that the model starts eolng wth metr of speal relatty, wth non zero olume and non zero sale fators and the olume and the sale fators are exponental nreasng as t nreases and admt nfnte alues at later stage. The Hubble parameter H from euaton and ts dretonal s H, H, H are H H The physal parameters H H.,, and admts the alues a a, 0,. It s seen that the model has onstant rate of expanson foreer. The densty and pressure the matter for 5 p of s onstant rght from the begnnng. The salar expanson admts onstant alue whh supports the unform expanson. nsotrop parameter and shear admt zero alues shows that the model s sotropze n all dretons wthout shear. The deeleraton parameter s shows model has aeleratng phase foreer and there s no deeleratng phase. Conluson Three dfferent Banh type I osmologal models wth untessene dark energy, n Bmetr theory of grataton hae been dedued.. The dark energy untessene model, for s n ontratng n nature and for, the model s expandng.. The hyperbol geometr ew pont of the models wll helpful to the peoples of obseratonal data to searh suh type of geometry. 5. REFERENCES karsu, O. and Kln, C. B. (00. LRS banh type I models wth ansotrop dark energy and onstant deeleraton parameter, General Relatty and Grataton, ol., No., pp. 9. karsu, O. and Kln, C. B. (00. Banh type III models wth ansotrop dark energy, General Relatty and Grataton, ol., No., pp. 7. karsu, O. and Kln, C.B. (00. De-stter expanson wth ansotrop flud n banh

7 May 0 Issue-, olume-ii 7-57X type I spae tme, strophyss and Spae Sene, ol., No., pp. 5. Bal, R. and Dae, S. (00. Banh type-ix strng osmologal models wth bulk sous flud n general relatty, strophyss and Spae Sene, ol. 88, pp. 50. Bal, R. and Upadhaya, R.D. (00. LRS banh type I strng dust magnetzed osmologal models, strophyss and Spae Sene, ol. 8, pp. 97. Bal, R. and Sngh, D. (005. Banh type- bulk sous flud strng dust osmologal model n general relatty, strophyss and Spae Sene, ol. 00, pp. 87. Bal, R. and nal (00. Banh type I magnetzed strng osmologal model n general relatty, strophyss and Spae Sene, ol. 0, pp. 0. Bal, R. and Pareek, U. K. (007. Banh type I strng dust osmologal model wth magnet feld n general relatty, strophyss and Spae Sene, ol., pp. 05. Bennett, C. L., Hll R. S., Hnshaw G., Nolta M. R., Odegard N., Page L., Spergel D. N., Weland J. L., Wrght E. L., Halpern M., Jarosk N., Kogut., Lmon M., Meyer S. S., Tuker G. S. and Wollak E., (00. Frst year Wlknson ansotropy probe obseratons: Foreground Emsson, The strophysal Journal Supplement Seres, ol. 8, pp.97. Borkar, M. S. and Charan, S. S. (00. Banh type I magnetzed osmologal model n bmetr theory of grataton, pplaton and ppled Mathemats (M, ol. 5, No., pp. 5. Borkar, M. S. and Charan, S. S. (00. Banh type I bulk sous flud strng dust osmologal model wth magnet feld n bmetr theory of grataton, pplatons and ppled Mathemats (M, ol. 5, No. 9, pp. 9. Gakwad, N. P. and Borkar, M. S. (0. Banh type I masse strng magnetzed baratrop perfet flud osmologal model n bmetr theory of grataton, Chnese Physs Letter, ol. 8, N. 8, pp Borkar, M. S., Charan, S. S. and Lepse,.. (0. Ealuaton of banh type I 0 osmologal models wth a bnary mxture of perfet flud and dark energy n bmetr theory of grataton, Internatonal Journal of ppled Mathemats, ol. 8, No., pp. 05. Borkar, M. S. and Gayakwad, P.. (0. Ealuaton of Shwarz hld s exteror and nteror solutons n bmetr theory of grataton, pplatons and ppled Mathemats (M, ol. 9, No., pp.0. 7