The Flatness Problem as A Natural Cosmological Phenomenon

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1 Inenainal Junal f Pue and Applied Physics ISSN Vlume 4, Numbe (008), pp Reseach India Publicains hp:// The Flaness Pblem as A Naual Csmlgical Phenmenn 1 Maumba G Okey, Mnynk N Manga and 3 Mal J Oien 1 Depamen f Physics, Schl f Physical Sciences, Univesiy f Naibi, P.O. Bx Naibi, Kenya gmaumbaa@yah.cm maumba@unbi.ac.ke Depamen f Physics, Schl f Physical Sciences, Univesiy f Naibi, P.O. Bx Naibi, Kenya nmnynk@unbi.ac.ke 3 Depamen f Physics, Schl f Physical Sciences, Univesiy f Naibi, P.O. Bx Naibi, Kenya jmal@unbi.ac.ke Absac An ineplay beween elemenay paicle ineacins a vey high enegies and he expansin f he Univese is sudied in he cnex f he inflainay mdel find a pssible explanain f an bseved flaness which emains a csmlgical puzzle. As is well knwn fm bsevain, he univese is a leas faily fla implying ha he elaive enegy densiy is alms uniy. This value f enegy is exaplaed fm he mae dminaed and he adiain dminaed phases f he false unbken gand unified vacua sepaaely, he ue vacuum. This pesen elaive enegy densiy f de uniy mus have been f de a he Planck ime f secnds ha leads he gaviy beaking fm sng and weak ineacins. A vey fine uning f ne pa in is needed keep he elaive enegy densiy clse uniy ve a vey lngime ensue ha he univese is vey ld as bseved. This in iself des n ensue ha he cuvaue em in he Fiedman field equains f min is unimpan. These equains ae slved sepaaely, f he pen, clsed and fla univeses cespnding he value f he cuvaue em being minus ne, ze and plus ne. Analysis f hese sluins indicaes ha he pen and clsed sluins cnvege he fla univese sluin. A pwe seies expansin dne f he elaive enegy densiy in he case f pen and clsed univeses shws ha even a fis de cecin is negligible and in his case he hypeblic sluins becme expnenial which is he fla univese esul. The flaness pblem is hen seen as a naual phenmenn in an inflainay

2 16 Maumba G Okey e. al. univese. Such a case-gained sluin becmes he undelying easn f he cuvaue em be idenically ze ahe han assuming he value f eihe minus ne plus ne. Inducin The expeimenal jusificain f he h big bang mdel ess n hee pincipal pieces f evidence 1 : i. The expansin f he univese based n he ecessin f he galaxies and Hubble s law v = Hd, whee v is he ecessin speed, H is he ppinaliy cnsan and d is disance beween w bseves/galaxies. ii. The discvey f he.76 K csmic micwave backgund adiains ha fills he enie univese. iii. The csmic abundance f he vey ligh elemens especially Helium and Deueium. In spie f hese siking chaaceisics, he mdel suffes fm sme seius geneal difficulies he sluin f which demands an impan addiin mdificain he basic mdel 1. These csmlgical puzzles ae badly discussed in he lieaue 1,,3,4,5,6,7. Cuenly he inflainay mdels appea pmising candidaes ha seem ffe he pssibiliy f explaining he fundamenal csmlgical facs. They ae based upn micphysical evens which ccued ealy in he univese bu well afe he Planck epch. In his pape, he Fiedman equain is slved explicily f he hee mdels (k=- 1, k=0 and k=+1). By making he appximain based n he fac ha he agumen f he hypeblic funcins is vey lage, all he hee mdels ae seen educe an expnenial funcin ie bh he k=+1 and k=-1 mdels educe he k=0 mdel in he appximain ha he univese undewen a apid expansin duing is ealy sages f develpmen. I is cncluded ha he flaness pblem is pbably a naual phenmenn in an inflainay univese. The ganizain f he pape is as fllws: In secin II, we biefly eview he inflainay scenai in he simples way while in secin III we fmulae he flaness pblem fm he csmlgical pinciple f hmgeneiy and ispy. In secin IV we give he jusificain f neglecing he enegy densiy f elaivisic paicle adiain duing he apid expansin and in secin V we give u cnclusins and fuhe eseach wk. Inflainay scenai Gand Unified Mdels f paicle physics pedic ha he sae f hemal equilibium f he quanum field will undeg a phase ansiin a a ciical empeaue T c f 15 de f he gand unificain scale, 10 GeV. Hence if he quanum fields in he univese began in an abiay h sae a he big bang singulaiy, as assumed in he sandad csmlgical mdels, hen such a phase ansiin wuld have ccued in

3 The Flaness Pblem as A Naual Csmlgical 163 he ealy univese as he fields cled belw T c as a esul f he expansin f he univese. If he gand unified field-hey mdels un u be cec and if he univese behaved accding he sandad csmlgical mdels a hese vey ealy epch, hen his phase ansiin will have ccued and i is f cnsideable impance ha we exple is cnsequences f csmlgy 8. The ms damaic and appealing cnsequence ha has been pediced s fa is he pssible exisence f an ea f inflain. In he iginal scenai 9, i was ppsed ha he ze empeaue penial enegy f a Higgs field Φ culd have a lcal minimum wih enegy densiy (say) abve ha f he abslue minimum. If Φ seled in he lcal minimum ve a sufficienly lage egin f space as he field cled, hen i culd emain apped hee in a measable sae unil i unneled hugh(by nucleain and expansin f bubbles) he ue minimum. While in he false vacuum, he kineic and spaial deivaive enegies f he field wuld be negligible cmpaed wih is penial enegy. Hence he sess-enegy ens Tαβ f he field wuld be dminaed by he em. Thus he effec f he field Φ n he dynamics f he univese via Einsein s equain wuld be jus like he effec f having a lage psiive csmlgical cnsan. In he Rbesn-Walke mdels, his pduces he de-sie sluin which yields an expansin f he univese n an expnenial ime scale. Thus if Φ emained in he false vacuum f many e-flding imes, an enmus inflain f he univese wuld have ccued which culd be f elevance f shedding ligh n such issues as why hee ae elaively few (if any) magneic mnples in u univese, why he univese appeas be nealy csmlgically fla ec 4. Hweve, his mdel suffes fm he seius pblem ha i des n appea allw an exi fm he inflainay ea in such a way as evlve he pesenly bseved univese. The new inflainay scenai 3,9,10 was ppsed pimaily vecme his pblem f baining a gaceful exi fm he inflainay ea. The basic idea f new inflain is as fllws: One cnsides he Higgs field Φ whse ne-lp effecive penial has he fm skeched in figue 1. We assume ha iniially he field is in hemal equilibium a high empeaue and hus is in he V a Φ =0 as in cuve (a). As he field cls due symmeic minimum f T expansin f he univese, i will fall in he measable sae indicaed in cuve (c) f figue 1 by he lcal minimum f V T a Φ =0 f inemediae empeaues. A lw empeaues, he dip in V T ges away and he dynamics f he field is gvened by he classical (ball lling dwn a hill) evluin f Φ in he ze-empeaue penial f cuve (d) wih iniial cndiins Φ 0, Φ 0, whee he magniude f he diffeences fm ze ae esimaed fm quanum flucuains. If V (Φ) is vey fla nea Φ =0, hen i will ake a lng ime f Φ each Φ c. Duing his ime, he sess-enegy ens f Φ will be dminaed by he em, whee is he heigh abve he ue minimum f he fla pa f V (Φ). The univese has an exa enegy densiy a is dispsal which mus have dynamical effecs via he Einsein s equain 4.

4 164 Maumba G Okey e. al. T T c Φ = 0 (a) Symmeic phase (c) Measable T T b T, Φ 0 c (b) Lwe values f T T, Φ 0 b (d) T = 0, Φ 0 Figue 1: shws a skech f he behavi f he ne-lp effecive penial V T in Cleman-Weinbeg mdels. A high empeaues, V has he fm shwn in cuve (a), wih a single minimum a Φ = 0. A lwe values ft, T V T develps side minima as shwn in cuve (b). A sill lwe empeaues V has he fm shwn in cuve (c). Finally cuve (d) epesens he effecive penial at = 0. T Fmulain f he Flaness Pblem The evluin f he univese is descibed by he Rbesn-Walke meic hugh he Fiedmann equain 4 S + kc 8πG = S whee S() is he scale fac, k is he cuvaue paamee and is he enegy densiy. In he k=0 mdel we have a any epch (in he adiain dminaed ea) 1 S () kc S Ω 1 s ha = ( Ω 1) = (3) S S 4 whee Ω = is he csmlgical elaive enegy densiy paamee. F he pesen c kc S epch, we have = ( Ω 1) (4) S S

5 The Flaness Pblem as A Naual Csmlgical T Given S( ) T we have f k = ± 1, ( Ω 1) = ( Ω 1)4 H T (5) In he ealy csmlgical chnlgy, we have he fllwing Planck ime 61 ( Ω 1) = h ( Ω 1) 53 GUTs ime ( Ω 1) = h ( Ω 1) Elecweak symmey beaking phase 7 ( Ω 1) = h ( Ω 1) (6) Quak cnfinemen ime ( Ω 1) = Neuin decupling ime ( Ω 1) = h ( Ω h ( Ω 1) 1) I is ineesing ne ha he cefficien n he igh-hand side f equain (5) keeps n gaining i.e. a he Planck ime he appximae value f leads gaviy beaking he sng and elecweak ineacins; he value leads he sng fce beaking fm is cunepa elecweak ineacin and 10-7 leads he elecweak symmey beaking. The abve scenai is n gd f csmlgy, meaning a phase ansiin is bund ake place wheneve he cefficien n he igh hand side f he abve equain aains a paicula value. Hweve if we can ewie equain as 8πG kc = S S (7) hen we can inepe ( kc ) as he al enegy f he univese wih he kineic enegy em epesened by S and he gaviainal penial enegy by he em cnaining. If he al enegy is psiive (k=-1) hen he kineic enegy is gea enugh (he iniial velciy is geae han he escape velciy) and he univese will cninue expand feve i.e. he univese is pen. If he al enegy is negaive (k=+1) he univese will ecllapse i.e. he univese is clsed. In he k=0 mdel, he univese is a he escape velciy and i will expand indefiniely. Since u univese has been expanding 6 a alms exacly he ciical ae avid ecllapse, we expec ha sme mechanism in he ealy hisy f he univese culd have diven he univese is pesen sae. While Guh s 9 iginal mdel was flawed, he new vesin based n slw-llve ansiin ppsed by Linde 3, Albech and Seinhad 10 and A A Sabinsky 11 appeas pmising. I is assciaed wih he evluin f a weakly-cupled scala field which f sme easn was iniially displaced fm he minimum f is penial 1.

6 166 Maumba G Okey e. al. Imagine he univese being cled hugh he ciical empeauet c. As he empeaue T dps belwt c, he sae f lwes enegy shifs in he false vacuum (measable sae) a Φ = 0 unil a sme sage he Φ field unnels acss he V ( Φ) 0 baie and falls dwn he V(0) slpe is ue vacuum as shwn in figue belw. V (Φ) O Φ= σ Φ Figue : Effecive penial f a field Φ f he ype cespnding inflainay mdels wih a slw-ll-ve. If we dene by he diffeence beween he enegies f he w vacua, hen unil he unneling has aken place, he univese has an exa enegy densiy a is dispsal which mus have dynamical effecs in he Einsein s equain f he sandad csmlgy descibed by he Fiedmann equain S + kc 8πG = ( ) + (8) S whee S( ) is he enegy densiy f he elaivisic paicle adiain 1. Since i falls as he univese expands while says cnsan, he vacuum enegy densiy dminaes. Theefe we can assume and slve he equain f min (8) f he k=0, k=+1 and k=-1 mdels. The k=0 mdel S 8πG S 8πG Equain (8) educes = s ha = = H (9a) S s whee H is Hubble s cnsan. We simply inegae equain (9a) ge S exp H (9b)

7 The Flaness Pblem as A Naual Csmlgical 167 The k=+1 mdel S + c 8πG Equain (8) educes = S 8πG S = S c which can be ewien as ds 1 β S s ha S = = S β = 1 = c u 1 d α α β (10a) S whee α =, β = cα and u =. Theefe equain (10a) becmes 8πG β du cd 1 1 =, which can be inegaed bain S = β csh = cα csh (10b) u 1 β α α fm which we aive a The k=-1 mdel 4 c G S = 3 8π csh 8πG and 8πG S csh c (10c) 3 S c 8πG Equain (8) educes = S fm which we have 8πG 1 ds 1 S S = S + c = ( S + β ) and S = = ( S + β ) = c + 1 (11a) α d α β Making use f u pevius subsiuins, we have 4 du 1 c G = d which can be inegaed bain S u + 1 α = 3 8π sinh 8πG (11b) s ha 8πG S sinh (11c) 8πG In he appximain ha is vey lage 13, bh equains (10c) and (11c) educe (9b) which is he fla mdel. This apid expansin in an expnenial fashin cninues unil he unneling akes place and Φ aains is ue vacuum value. kc Ne ha saing wih he cuvaue em in cmpaisn he expansin em S S 58 pi inflain, leads he fme being educed by appximaely S 10 S while he lae alms says cnsan. This lage fac manifess as he fine-uning in he flaness pblem.

8 168 Maumba G Okey e. al. Fine-Tuning in he Flaness Pblem Le he al enegy densiy be + ) (. Then since have S. We can expand he enegy densiy as 4 T and 3 4 n ( x ) = a + a1x + a x + a3x + a4 x an x whee x = S. afe finding he cefficiens, we have ( x) = + S 4S + 0S 10S +... Nice ha even fis-de, S is negligible. 1 T S, we Cnclusin The naue and age f he univese cnsain he value f he elaive enegy densiy Ω be n diffeen fm uniy. Had he enegy densiy been subsanially diffeen fm he ciical enegy densiy c, he heavy elemens, he sas and he galaxies culd n have fmed. F fa much less han c, he univese wuld have expanded apidly f sas galaxies fm while f fa much geae han c, he univese wuld have ecllapsed befe sa ( galaxy) fmain culd have been achieved. The fine-uning equied in mainaining Ω nealy uniy ve he equied peid is s exemely delicae ha he inflainay mechanism pbably seems have played a significan le. The cnveninal esul ha Ω is nealy uniy day means ha in he fis mmens f he big bang i was pecisely uniy. The naual infeence is ha he value is and has always been exacly uniy. An impan implicain f his esul is ha he value Ω = 1 is a naual phenmenn in an inflainay scenai. Alenaively, u esul wuld als imply ha hee is a lage amun f unseen mae in he univese ha is cnibuing indiecly he enegy densiy s ha he value Ω = 1. This pssibiliy pvides a windw f fuhe eseach wk. Acknwledgemens We wuld like expess u deep gaiude Musapha A O f many enlighening discussins, encuagemens and advice. The fis auh is exemely gaeful he Highe Educain Lans Bad (Kenya) f he Reseach Gan N. HELB/LD/PSA/VOL.1/15 006/07 S/NO.731 Refeences [1] [1] Hughes, I S, 1991, Elemenay Paicles, Cambidge Univesiy Pess, Cambidge UK, Chap 15. [] Cughlan, G.D., and D, J.E, 1991, Ideas f Paicle Physics, Cambidge Univesiy Pess, Cambidge, UK, Chap.4.

9 The Flaness Pblem as A Naual Csmlgical 169 [3] Linde, A.D., 198, A new inflainay Univese scenai: A pssible sluin f he Hizn, flaness, hmgeneiy, ispy and pimdial mnple pblems, Phys Le 108, N.6, 389. [4] Jayan, V.N, 1993, inducin Csmlgy, Cambidge Univesiy Pess, Cambidge, U K, Chap. 6. [5] Maumba, G.O, Csmlgical puzzles, Univesiy f Naibi Pepin NRB- TP [6] Hawking, S.W, and Mss, I.G, 198, SupecledPhase ansiins in he vey ealy univese, Phys.Le 110B, N.1, 35. [7] Wikipedia, 006, Big Bang, The fee encyclpedia, p1. [8] Gene, F.M, e al, 1985, Des a phase ansiin in he ealy univese pduce he cndiins needed f inflain? Phys.Rev D31, N.4, 73. [9] Guh, A.H, 1981, Inflainay Univese: A pssible sluin he hizn and flaness pblems, Phys.Rev D 3, N., 347. [10] Andeas, A., and Paul, J.S, 198, Csmlgy f Gand Unified Theies wih adiaively induced symmey beaking, Phys. Rev Le 48, N. 17, 10. [11] Sabinsky, A.A, 198, Dynamics f phase ansiin in he new inflainay univese scenai and geneain f peubains, Phys. Le 117B, N. 3, [1] Rbe, J.S, and Michael S.T, 1985, Decaying paicles d n hea up he univese Phys.Rev D 31, N. 4, 681. [13] Thmas and Finney, 1993, Calculus and Analyic Gemey, Nasa Publishing Huse, New Delhi, India, Chap. 9. Lis f symbls and abbeviains c-speed f ligh G-gaviainal cnsan GeV-giga elecnvl h-accuns f he unceainy in Hubble s cnsan H-Hubble s ppinaliy cnsan k-he cuvaue paamee S()-csmic scale fac T c -ciical empeaue T -Enegy-Mmenum ens αβ v -speed f ecessin V T -empeaue dependen effecive penial -al enegy densiy c -ciical enegy densiy -vacuum enegy densiy Φ -Higgs scala field Ω -elaive csmlgical enegy densiy paamee Ω -pesen elaive csmlgical enegy densiy paamee.

2. The units in which the rate of a chemical reaction in solution is measured are (could be); 4rate. sec L.sec

2. The units in which the rate of a chemical reaction in solution is measured are (could be); 4rate. sec L.sec Kineic Pblem Fm Ramnd F. X. Williams. Accding he equain, NO(g + B (g NOB(g In a ceain eacin miue he ae f fmain f NOB(g was fund be 4.50 0-4 ml L - s -. Wha is he ae f cnsumpin f B (g, als in ml L - s -?