Kaluza-Klein Inflationary Universe in General Relativity

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1 November 0 Vol. Issue dhav, K. S., Kaluza-Klen Inflatonary Unverse n General Relatvty 88 rtcle Kaluza-Klen Inflatonary Unverse n General Relatvty Kshor. S. dhav * Deartment of Mathematcs, Sant Gadge Baba mravat Unversty, mravat-44460, Inda bstract Kaluza-Klen nflatonary unverse n general relatvty has been studed. To obtan the determnstc model of the unverse, t has been consdered that the energymomentum tensor of artcles almost vanshes n the course of the exanson of the unverse and thereby total energy-momentum tensor reduces to vacuum stress tensor. Ths assumton leads to () a e, where a s scale factor and H s Hubble constant. () the effectve otental V () constant, where s Hgg s feld. It s observed that nflatonary scenaro s ossble n Kaluza-Klen unverse. Keywords: Kaluza-Klen sace-tme, nflatonary unverse.. Introducton Wenberg(986) studed the unfcaton of the fundamental forces wth gravty whch reveals that the sace-tme should be dfferent from four. Snce the concet of hgher dmensonal sace-tme s not unhyscal, the strng theores are dscussed n ten dmensons or twenty sx dmensons of sace-tme. Because of ths, studes n n- dmensons nsred many researchers to enter nto such feld of study to exlore the hdden knowledge about the unverse. Chodos and Detweller(980), Ibanez and Verdaguer (986), Gleser and Daz(988), Banerjee and Bhu(990), Reddy and VenkateswaraRao (00), Khadekar and Gakwad(00), dhav et al.(008) have studed the multdmensonal cosmologcal models n Ensten s general relatvty theory. The theory of fve dmensons s due to the dea of Kaluza (9) and Klen(96). fve dmensonal [5D] general relatvty s the best outcome of an attemt made by these two by usng one extra dmenson to unfy gravty and electro-magnetsm. Many researchers [Lee (984), elqust et al. (987), Collns et al. (989), Overdn & Wesson (997)] have used ths concet for studyng the models of cosmology and artcle hyscs. ccordng to Wesson (984, 999) and Belln (003), the matter s nduced n 4D by 5D vacuum theory for studyng the cosmology of 5D wth ure geometry n non-comact Kaluza-Klen theory. We are aware of the fact that the outstandng roblems n cosmology lke homogenety, sotroy, horzon, flatness and rmordal monoole roblem n grand unfed feld theory are sgnfcantly solved by nflatonary unverses. The concet of early nflatonary hase n grand unfed feld theores was ntroduced by Guth (98) where symmetry breakng hase transton occurs wth the decrease of temerature at * Corresondence uthor: Kshor. S. dhav, Deartment of Mathematcs, Sant Gadge Baba mravat Unversty, mravat-44460, Inda. E-mal : at_ksadhav@yahoo.co.n ISSN:

2 November 0 Vol. Issue dhav, K. S., Kaluza-Klen Inflatonary Unverse n General Relatvty 89 the very early stages of evoluton of unverse [ Zel dovch et al.(978); Krzhnts et al.(976,977); Lnde (979)]. The lfe tme of suer cooled symmetrc hase has been consdered at = 0, where s Hgg s scalar feld whch breaks the symmetry. In ths case, the energy-momentum tensor of artcles almost vanshes n the course of the exanson of the unverse and thereby total energy-momentum tensor reduces to vacuum stress tensor. Ths stuaton leads to () a e, where a s scale factor and H s Hubble constant at that tme whch s gven by H 8 3M V (0), where M 9 0 G V s the Planck mass [ Tolman,969]. e Zel dovch & Novkov (975) roved that the symmetry breakng hase transton takes lace at low temerature Tc where all vacuum energy V(0) transforms nto 4 thermal energy and the unverse s reheated u to the hgh temerature T V(0) where further evoluton starts. It has been roved by Rothman & Ells ( 986) that the roblem of sotroy can be solved. Sten-Schabes (987) has shown that the nflaton wll take lace f the effectve otental V ( ) has flat regon where Hgg s feld evolves slowly but the unverse exands n an exonental way due to vacuum feld energy. The sgnfcance of nflaton for sotrozaton of the unverse has been exlaned by nnnos et al. (99). The nflatonary scenaro n the large scale structure of the unverse has been studed by Panchaakeshan & Seth (99). Schmdt (993) and Burd (993) dscussed nflatonary scenaro for FRW unverse. Bal & Jan (00) examned nflaton n LRS Banch tye-i sace tme n the resence of mass less scalar feld wth flat otental. Reddy et al. (009) nvestgated nflatonary scenaro n Kantowsk-Sache sace tme. Recently, Bal (0) dscussed nflatonary scenaro n Banch tye-i sace tme by consderng scale factor a e as used by Krzhnts (977) and Krzhnts & Lnde (976). Wth ths motvaton, Kaluza-Klen nflatonary unverse n general relatvty has been studed.. Metrc and Feld Equatons We consder the Kaluza- Klen sace-tme n the form ds dx dy dz B d dt, (.) where and B are functons of t only. The extra dmenson s taken to be sacelke. ISSN:

3 November 0 Vol. Issue dhav, K. S., Kaluza-Klen Inflatonary Unverse n General Relatvty 830 Usng Sten-Schabes (987) aroach, the Lagrangan of gravty mnmally couled wth Hgg s scalar feld havng effectve otental V( ) s gven by S 4 g R g V ( ) d x (.) The varaton of ths acton S wth resect to the dynamcal feld leads to the Ensten feld equatons (Here n geometrcal unts 8 G c. ) R j Rg j T j, (.3) where T j j V g j (.4) and g dv ( g) d (.5) Now, the Ensten feld equatons (.3) for metrc (.) wth the hel of equatons (.4) gve a set of equatons 3 B 3 B V, (.6) B B B B V, (.7) 3 3 V, (.8) Equaton (.5) for scalar feld leads to B 3 dv 0, (.9) B d where dot () ndcates the dervatve wth resect to t. ISSN:

4 November 0 Vol. Issue dhav, K. S., Kaluza-Klen Inflatonary Unverse n General Relatvty Soluton of the Feld Equatons: We assume that regon s flat, so for flat regon the effectve otental s constant. V constant k (say) (3.) Solvng equaton (.9), we get l (3.) 3 B where l s constant of ntegraton. To get a determnstc model, we consder the condton [ Zel dovch et al. (978); Krzhnts (977) ] that the scale factor can be exressed as Ths leads to a e. a B e (3.3) Usng equaton (3.3) n equaton (3.), we have le 4 (3.4) Subtractng equaton (.6) from equaton (.7), we get B B B B 0 Ths gves B B B 3 0 B B B Integratng the above equaton, we get B L L 3 4 B B e, (3.5) where L s constant of ntegraton. Usng equatons (3.3) and (3.5), we get ISSN:

5 November 0 Vol. Issue dhav, K. S., Kaluza-Klen Inflatonary Unverse n General Relatvty 83 Me L 4 ex e (3.6) 6H 3L 4 B Ne ex e 6H, (3.7) where M and N are constants of ntegraton. Usng equatons (3.6) and (3.7) n equaton (.) and after sutable transformaton of co-ordnates wth choce of constants, the Kaluza-Klen nflatonary model s gven by ds dt e L ex e 8H 4 3L 4 dx dy dz e ex e d 8H 4. Physcal Proertes (3.8) Integratng equaton (3.4), we get Hgg s scalar feld l 4H 4 s e, (4.) where s s constant of ntegraton. The average exanson ansotroy arameter s defned as 4 H, 4 H where H H H H H H 3 H 4, B B 3 8 L e (4.) 6H The shear scalar s gven by 4 H 4H H 3 8 L e (4.3) 8 ISSN:

6 November 0 Vol. Issue dhav, K. S., Kaluza-Klen Inflatonary Unverse n General Relatvty 833 The deceleraton arameter q s gven by RR q R (4.4) 5. Dscusson and Concluson () The secal volume s gven by equaton (3.3). It ncreases as tme ncreases. Thus, nflatonary scenaro exsts n Kaluza-Klen unverse. () The equaton (4.) gves Hgg s scalar feld. It decrease slowly as tme ncreases. () The average exanson ansotroy of the unverse s gven by equaton (4.). For sotroy, we need 0. The equaton (4.) leads to L 0. Therefore, Kaluza-Klen unverse sotrozes when L 0. (v) Ths condton L 0 mles that the shear scalar gven by (4.3) vanshes [s equal to zero] for sotroy. (v) From equaton (4.4), we get that the deceleraton arameter s q. Hence, Kaluza-Klen unverse has exonental exanson or de-stter exanson. Thus, the model (3.8) aroaches de-stter unverse. It has been shown that Kaluza-Klen metrc (3.8) s sotrozed under the secal condton as onted out by Rothman & Ells (986). Fnally, one may conclude that the nflatonary scenaro s ossble n Kaluza-Klen unverse. References dhav et al. :Int.J.Theor. Phys.,47, 00 (008):DOI 0.007/s nnnos,p.,matzner,r..,rothman,t., Ryan,M.P.: Phys.Rev. D 43, 38 (99). elqust et al. :Modern Kaluza Klen Theores. ddson-wesley,readng(987). Bal, R. : Int.J.Theor.Phys., 50, 3043 (0). Bal,R.,Jan, V.C.: Pramana (00). Banerjee, S., Bhu, B.: Mon. Not. R. stron. Soc. 47, 57 (990). Belln, M.: Nucl.Phys. B660, 389 (003). Burd,.: Class.Quantum Gravty 495 (993). Chodos,., Detweller, S.: Phys. Rev. D, 67 (980). Collns, et al. : Partcle Physcs & Cosmology.Wley,London (989). Gleser, R.J., Daz, M.C.: Phys. Rev. D 37, 376 (988). Guth,.H.:Phys.Rev.D 3, 347 (98). Ibanez, J., Verdaguer, E.: Phys. Rev. D 34, 0 (986). ISSN:

7 November 0 Vol. Issue dhav, K. S., Kaluza-Klen Inflatonary Unverse n General Relatvty 834 KaluzaT.:Zum Untats Prob der Physk Stz Press.kad.Wss.Phys.Math.K,966(9). Khadekar, G.S., Gakwad, M.: Proc. Ensten Found. Int., 95 (00). Krzhnts, D..,Lnde,.D. :nn.phys.(n.y.) 0, 95 (976). Krzhnts, D..: JETP Lett. 5, 59 (977). Klen,O.:Zets.Phys.37,895(96). Lee,H.C.: n Introducton to Kaluza Klen Theores.World Scentfc,Sngaore(984). Lnde,.D. : Re.Prog.Phys. 4, 38 (979). Overdn, J.M.,Wesson,P.S.:Phys.Re.83, 303 (997). Panchaakeshan,N.,Seth,S.K.: Int.J.Mod.Phys. 7,3769 (99). Reddy, D.R.K., Venkateswara Rao, N.: strohys. Sace Sc. 77, 46 (00). Reddy, DRK.,dhav,K.S., Katore,S.D., Wankhade, K S : Int.J.Ther.Phys. 48, 884 (009). Rothman,T.,Ells,G.F.R.: Phys.Lett. B 80, 9 (986). Schmdt, H.J.: Gen. Relatv. Gravt. 5, 87 (993). Sten-Schabes, J.. : Phys. Rev D 35, 345 (987). Tolman,R.C.: Relatvty Thermodynamcs & Cosmology, Clarendon Press,Oxford (969). Wenberg, S.: Physcs n Hgher Dmensons. World Scentfc, Sngaore (986). Wesson, P.S.:Gen.Reltv.Gravt.6,93(984). Wesson,P.S: Sace-tme Matter Theory.World Scentfc, Sngaore (999). Zel dovch, Ya.B.,Novkov,I.D.: Structure & Evoluton of the Unverse. Nauka, Moscow (975). Zel dovch,ya.b.,khloov,m.yu. : Phys. Lett. B 79, 39 (978). ISSN: