A Closed Model of the Universe

Transcription

1 Inernionl Journl of Asronomy nd Asrophysis hp://dxdoiorg/036/ij0330 Published Online June 03 (hp://wwwsirporg/journl/ij) A Closed Model of he Universe Fdel A Bukhri Deprn of Asronomy Fuly of Siene King Abdulziz Universiy Jeddh Sudi Arbi Emil: flbukhri@homilom Reeived Jnury 3 03; revised Februry 5 03; eped Februry 3 03 Copyrigh 03 Fdel A Bukhri This is n open ess rile disribued under he Creive Commons Aribuion Liense whih permis unresried use disribuion nd reproduion in ny dium provided he originl work is properly ied ABSTRACT A losed model of he universe ws onsrued ording o he ssumpion h very minor frion of he drk energy rnsfers so slowly o r nd rdiion The osmologil prer is no longer fixed bu represens so slowly deresing funion wih i In his model he universe expnds o mximum limi = 6853 Gyr hen i will onr o big runh b = 5365 Gyr Observionl ess o he losed osmi model were illusred Disribuions of he universe expnsion nd onrion speed were esblished in his model whih indied h he expnsion speed in he erly universe is ppreibly high hen i will derese rpidly unil i vnishes However he onrion speed of he universe inreses oninuously unil he i jus before b Disribuions of he universe expnsion nd onrion elerion were perford empirilly whih onfird he previous resul In he losed osmi model he universe hisory n be egorized ino six min sges hese re he firs rdiion epoh he firs r epoh he firs drk energy epoh he ls drk energy epoh he ls r epoh nd he ls rdiion epoh Disribuions of he densiy prers of he rdiion r drk energy nd he ol densiy s well s he disribuions of emperure of he rdiion nd non-relivisi r were ll invesiged in his model ll epohs of he universe Keywords: Cosmologil Prer; Cosmology; Cosmi Dynmis Inroduion In pervious wo riles [] he osmologil prer ws ssud onsn in five generl osmi models However in so osmologil sudies is no ully perfely onsn bu exhibis slow vriion so is ofen desribed s quinessene [3-6] In oher words he drk energy densiy does no remin 8πG onsn wih i This poin of view is in good green wih he Heisenberg s Uneriny Priniple h here is n uneriny in he moun of energy whih n exis This smll uneriny llows non-zero energy E o exis h π for shor inervls of i where h is E Plnk s onsn m Kg s As resul of he equivlene beween r nd energy hese smll energy fluuions n produe virul pirs of r priles (priles nd heir nipriles mus be produed simulneously) whih o ino exisene for shor i nd hen dispper o produe phoons In he presen sudy is ssud o be very slowly deresing funion of he osmi i suh h ny derese in sy should be ompensed by inresing eh of he r densiy nd rdi- r ion densiy by The imporne of his sudy is o know under wh osmologil ondiions he universe n be onring o big runh rher hn expnding for ever s shown in he five generl osmi models invesiged in [] In Seion deiled desripion is given for he hodology Deerminion of is explined in Seion 3 Observionl ess of he losed osmi model re illusred in Seion Resuls nd disussion re presened in Seion 5 Finlly he onlusion is displed in Seion 6 Mehodology From [] we hve seen h he densiies of r rdiion nd drk energy osmi i Copyrigh 03 SiRes

2 90 F A BUKHARI re given by where () m ò () r ò (3) 3H 8πG () 3H 8πG ò (5) H0 m0 3 H H0 r0 H H0 0 H H 0 m0 0 H r0 (6) (7) (8) (9) Subsiuing by () (6) in () we ge 3H 0 m0 3 8πG Or m0 m 0 (0) 3 Similrly we n find r0 0 () 0 0 onsn () Now ssu very smll derese in bou % per Gyr so he derese in in osmi i is expressed s 00 (3) Aording o he onservion lw of mss nd energy he derese in he energy densiy is ompensed by inrese of in eh of r nd Therefore he osmi i he new vlues of r nd re given by 0 0 () m0 0 3 (5) r0 0 (6) The slowly vrying osmologil prer is 8πG (7) Using Equions ()-(5) nd ()-(6) he new vlues of densiy prers in he expnding osmi model i re (8) Le 0 (9) (0) H s hen Equions (6)-(8) n be wrien s H s () 3 m0 m s () r0 r s (3) 0 Subsiuing by ()-(3) in (9) nd using (8)-(0) we ge he Hubble prer in he losed osmi model i or H0 3 H s s m sr Copyrigh 03 SiRes

3 F A BUKHARI 9 H H s s 0 3 m s r () The riil mss densiy in he losed osmi model i beos 3H (5) 8πG The new densiy prers in he losed osmi model i re (6) m (7) (8) And he ol densiy prer in he losed osmi model i is (9) The speed of he universe dynmis in he losed osmi model is obined from Equion () suh h or H 0 m 3 s r H s s (30) The elerion of he universe dynmis in he losed osmi model is found empirilly s lim (3) 0 The i inervl beween wo insns wih sle fors during he universe expnsion is given by Equion (6) in [] s 0 m0 H 0 r0 d (3) However during he universe onrion if < hen modulus of he righ hnd side of Equion (3) should be ken The redshif lookbk i relion in he losed osmi model is given by Equion (8) in [] In ddiion he disribuions of emperure differen epohs of he universe depend on relions similr o Equions (33) (3) nd (37) in [] 3 Deerminion of The i of he mximum expnsion of he universe in he losed model is evlued by ierive proedure s follows: ) Sr wih 0 0 nd le I I D 0 D ) Clule 000 vlues of nd H0 using Equions (3) (30) The vlue of orrespond- ing o he minimum posiive vlue of is ssud o be nd 3) Sele H0 H0 nd repe he previous wo seps where D 0 Now he vlue of orresponding o he minimum posiive vlue of is supposed o be nd H0 H0 ) Repe his hod severl is using he vlues D nd 0 hen esi he vlues 3 nd 5 6 nd obin he orresponding vlues of nd H0 5) Denoe hese resuls s presened in Tble where i is noieble h he vlues of nd H0 onverge nd beo very lose o zero In oher words he universe sops expending 6 6) From Tble one n esily find h he i of mximum expension of he universe in he losed model is Gyr By similriy he i of big runh is Gyr b Tble Ierive deerminion of he mximum expnsion i of he universe in he losed osmi model H Copyrigh 03 SiRes

4 9 F A BUKHARI Observionl Tess o he Closed Cosmi Model I is onvenien o sr by invesiging he disribuions of he osmologil prer in he losed osmi model vrious epohs ording o Equion (7) Figure () shows no eviden hnge of wih osmi i unil 36 Gyr hen dereses in relively higher re owrds 05 Gyr On he oher hnd exhibis grdul hnge wih i in he i rnge 05 Gyr s seen in Figure where 68 Gyr is he i of mximum expnsion of he universe in he losed osmi model The slow vriion of wih is lso noieble in he i rnges b s displed in Figures () nd (d) respeively where 5363 Gyr is b he i of big runh of he universe in he losed osmi model nd b 05 Gyr Figure () shows h he expnsion disribuion of he universe in he losed osmi model up o 0 is found using Equion (3) This disribuion is in good green wih h of he observed generl osmi model A obined by Equion (6) in [] Moreover 97 Gyr hese wo disribuions beo idenil The redshif look-bk i disribuions in hese wo models up o 0 were esblished nd presened in Figure Boh disribuions re in perfe green The obvious green beween he observed generl osmi model A nd he losed osmi model s seen from Figures () nd srongly rgues in fvour of relibiliy of he losed osmi model () () Figure () The disribuion of he osmologil erm in he losed osmi model up o = 05 Gyr; The disribuion of he osmologil erm in he losed osmi model in he rnge = 05 Gyr ; () The disribuion of he osmologil erm in he losed osmi model in he rnge = ; (d) The disribuion of he osmologil erm in he losed osmi model in he rnge = b (d) Copyrigh 03 SiRes

5 F A BUKHARI 93 () dereses bruply up o bou 06 Gyr Aferwrds fluues grdully wih unil 0 On he oher hnd Figure exhibis he disribuion of he universe onrion speed in he losed model in he rnge I is ler h he inrese of wih is grdul up o 5 Gyr hen rpidly inreses wih unil The disribuion of he universe expnsion elerion in he losed model in he rnge 05 Gyr is dedued from Equion (3) nd exhibied in Figure 5() Abrup inrese in wih is obvious up o 0 Gyr Then hnges very slighly wih unil 5 Gyr where srs deresing grdully up o 673 Gyr Aferwrds dereses () Figure () The expnsion of he universe in he generl osmi model A nd he losed model up o = 0 ; Redshif look bk i relion in he generl osmi model A nd he losed osmi model up o = 0 5 Resuls nd Disussion The expnsion of he universe in he losed osmi model up o is obined by Equion (3) nd presened in Figure 3() I is noieble h he inrese of wih is oninuous s liner relion unil bou 63 Gyr hen inreses relively slow wih Neverheless he onrion of he universe in he losed model in he i rnge b is illusred in Figure 3 I is obvious h lmos linerly dereses wih However redues relively slow wih jus before b The disribuion of he universe expnsion speed in he losed model in he rnge 05 Gyr is perford using Equion (30) nd displed in Figure () The vlue of is high in he erly universe hen i Figure 3 () Expnsion of he universe in he losed osmi model up o = ; Conrion of he universe in he losed osmi model in he rnge = b Copyrigh 03 SiRes

6 9 F A BUKHARI () () Figure () The disribuion of he universe expnsion speed in he losed osmi model in he rnge = 05 Gyr ; The disribuion of he universe onrion speed in he losed osmi model in he rnge = rpidly owrds he mximum expnsion i I is ler h in he rnge Gyr 0 Km s M p Furhermore Figure 5 shows he disribuion of he universe onrion elerion in he losed model in he rnge I is noieble h suddenly redues up o 690 Gyr hen redues grdully unil 393 Gyr where 06 Aferwrds rises grdully up o 593 Gyr where srs inresing quie r- pidly owrds 0Kms M p in he inervl Gyr I is remrkble o noe h he disribuions of nd in he losed osmi model in he rnges 05 Gyr b will be invesiged in deils in sepre sudy sine in hese wo i rnges he pressure of he osmi fluid is signifin nd n no be negleed Figure 5 () The disribuion of he universe expnsion elerion in he losed osmi model in he rnge = 05 Gyr ; The disribuion of he universe onrion elerion in he losed osmi model in he rnge = The disribuion of he densiy prers in he losed osmi model up o 05 Gyr is dislosed in Figure 6() I is prominen h he disribuion of he rdiion densiy prer oinides on he disribuion of he ol densiy prer up o 0663 yr However he disribuion of he r densiy prer oinides on he disribuion of 6656 Myr I is lso obvious h he disribuions of he drk energy densiy prer nd he disribuion of re inresing while he disribuion of remins lmos fixed he vlue up o 0663 yr hen i srs deresing Neverheless he disribuion of sys lmos onsn he vlue in his epoh of he universe Thus 05 rm 5595 yr wheres r m Copyrigh 03 SiRes

7 F A BUKHARI 95 () () Figure 6 () The disribuion of he densiy prers in he losed osmi model up o = 05 Gyr; The disribuion of he densiy prers in he losed osmi model in he rnge = 05 Gyr ; () The disribuion of he densiy prers in he losed osmi model in he rnge = ; (d) The disribuion of he densiy prers in he losed osmi model in he rnge = b (d) Myr Figure 6 shows he disribuion of he densiy prers in he losed osmi model in he rnge 05 Gyr I is eviden h he disribuion of displys rpid inrese unil he i m 00 Gyr where 0593 hen i rises grdully up o 6770 Gyr where i exhibis brup inrese gin The disribuions of beo lose ogeher from 63 Gyr o The vlue of is lmos 0 in he i inervls Gyr Gyr The disribuions of nd hnge quie slowly up o 6770 Gyr where hey lso rise up suddenly They ge lose ogeher from 3536 Gyr o The disribuion of he densiy prers in he losed osmi model in he rnge is presened in Figure 6() All disribuions revel seep derese up o Gyr Disribuions of re djen o eh oher unil 377 Gyr hen hey diverge pr nd derese slowly In ddiion he disribuions of nd re lso ner eh oher up o 860 Gyr Aferwrds hese wo disribuions redue grdully nd ge wy from eh oher Neverheless fer he i Gyr he disribuions of nd redue quie rpidly nd inerse wih eh oher 5579 Gyr where However he disribuions of nd inerse m 07 Gyr where 08 The disribuions of nd ge lose o eh oher m Copyrigh 03 SiRes

8 96 F A BUKHARI from 5650 Gyr unil Figure 6(d) illusres he disribuion of densiy prers in he losed osmi model in he rnge b I is ler h he disribuions of nd lmos oinide on eh oher up o bou Gyr hen he disribuion of srs deresing slighly bu sill lose o h of unil 5368 Gyr while kes he vlues beween 090 hroughou he inervl b However he disribuion of rises grdully nd inerses wih he disribuion of rm Gyr In ddiion he disribuion of ges loser o he disribuion of b Finlly he disribuion of indies slow derese unil bou Gyr hen i exposes quie rpid derese owrds he i of big Crunh I is essenil o relize h he universe hisory hs six min sges in he losed model hese re ) The firs rdiion epoh in he rnge rm ) The firs r epoh in he rnge rm m 3) The firs drk energy epoh in he rnge m ) The ls drk energy epoh in he rnge m 5) The ls r epoh in he rnge m rm 6) The ls rdiion epoh in he rnge rm b These epohs of he universe wih heir relevn densiy prers re ll summrized in Tble Forhermore he geory of spe hroughou he universe hisory in he losed osmi model is presened in deils in Tble 3 One n see in Tble 3 h he spe of he universe is fl jus fer he big bng up o 653 Gyr where he ol densiy prer lies in he rnge Aferwrds he spe of he universe beos open unil 0075 Gyr sine 095 Then he universe spe reurns o fl up o 56 Gyr s Aferwrd he universe spe ges urved hen losed unil 6 beuse Hene he universe spe remins being losed hen urved up o Gyr sine Aferwrd he universe spe evolves ino fl unil 075 Gyr s Then he universe spe develops ino open up o 538 Gyr owing o 095 Evenull he spe of he universe os bk o fl unil he i jus before he big rnh by he reson of The disribuion of he universe emperure Tu in he losed osmi model in he firs rdiion epoh is obined using Equion(3) in [] nd displyed in Figure 7() I is eviden h Tu redues oninuously in liner mnner during his er The emperures of he rdiion Tr nd non relivisi r Tm re deermined from Equions (33) (37) in [] respeively The disribuions of Tr nd Tm in he firs r nd drk energy ers re presened in Figure 7 I is prominen h Tr Tm K rm hen he disribuions of Tr Tm derese shrply up o = 0070 Gyr However boh disribuions redue grdully ferwrds The disribuion of Tr is bove h of T hroughou hese wo epohs A m Tble Epohs of he universe hisory in he losed osmi model Epoh Ti inervl of epoh Firs rdiion er rm 5595 yr Firs r er rm m Firs drk energy re m Ls drk energy er m Ls r er m rm Ls rdiion er rm b Relevn densiy prer Gyr Gyr Gyr 08 m m rm Tble 3 Geory of spe hroughou he universe hisory in he losed osmi model Ti inervl/gyr Tol densiy prer Geory of spe in he universe Relevn epohs Fl spe Firs rdiion er nd firs r er 653 < Open spe Firs r er 0075 < Fl spe Firs r er nd firs drk energy er 6 56 < Curved hen losed spe Firs drk energy er 7 < Closed hen urved spe Ls drk energy er 3938 < Fl spe Ls drk energy er nd ls r er 075 < Open spe Ls r er 538 < < b Fl spe Ls r er nd ls rdiion er Copyrigh 03 SiRes

9 F A BUKHARI 97 () () Figure 7 () The disribuion of he universe emperure in he losed osmi model up o = rm ; The disribuion of emperure of he rdiion nd non-relivisi r in he losed osmi model in he rnge = rm () The disribuion of emperure of he rdiion nd non-relivisi r in he losed osmi model in he rnge = rm ; (d) The disribuion of he universe emperure in he losed osmi model in he rnge = rm b (d) = T r = 7 K T m = K The disribuion of Tr nd Tm in he ls drk energy nd ls r epohs re exposed in Figure 7() Boh disribuions inrese slowly up o = Gyr hen hey sr rising rpidly unil hey join ogeher rm where T r = T m = K Evenully Figure 7(d) indies he disribuion of he universe emperure in he ls rdiion epoh This disribuion rises slowly up o yr before b hen i inreses rpidly o he vlue T u = K = 365 yr before b Furher ineresing physil properies of he universe in he losed osmi model would be invesiged in sepre sudies in omprison wih he orresponding properies of he universe in he five generl osmi models Copyrigh 03 SiRes 6 Conlusion In his sudy losed model of he universe ws developed depending on he ssumpion h very slow rnsfer of he drk energy o r nd rdiion is llowed Thus he osmologil prer is no longer onsn bu so slowly deresing funion of i In he ligh of his model he universe expnds o mximum limi 6853 Gyr hen i will reollpe o big runh b 5365 Gyr Observionl ess o his model were presened The disribuions of he universe expension nd onrion speed were invesiged in he losed model whih dislosed h he expnsion speed in he erly universe is very high hen i will redue rpidly unil i vnishes Neverheless he onrion speed of he universe rises oninuously unil he i jus before b The disribuion of he universe expn-

10 98 F A BUKHARI sion nd onrion elerion were rried ou empirilly whih suppored he previous resul In his model he universe hisory is lssified in o six min ers hese re he firs rdiion epoh he firs r epoh he firs drk energy epoh he ls drk energy epoh he ls r epoh nd he ls rdiion epoh The disribuions of he densiy prers of he rdiion r drk energy nd ol densiy in ddiion o he disribuions of emperures of he rdiion nd nonrelivisi r were ll deermined nd disussed in his model in he vrious ers of he universe 7 Aknowledgens This pper ws funded by he Denship of Sienifi Reserh (DSR) King Abdulziz Universiy Jeddh The uhor herefore knowledges wih hnks DSR ehnil nd finnil suppor REFERENCES [] F A Bukhri Five Generl Cosmi Models Journl of King Abdulziz Universiy: Siene Vol 5 No 03 [] F A Bukhri Cosmologil Disnes in Five Generl Cosmi Models Inernionl Journl of Asronomy nd Asrophysis 03 [3] J V Chunh J A S Lim nd N Pires Deflionry () Cosmology: Observionl Expressions Asronomy nd Asrophysis Vol 390 No 3 00 pp doi:005/ :00076 [] J A S Lim nd M Trodden Physil Reviews D Vol p 80 [5] S A Bludmn nd M Roos Quinessene Cosmology nd he Cosmi Coinidene Physil Reviews D Vol 65 No 00 Arile ID: doi:003/physrevd [6] I Zlev L Wng nd P J Seinhrd Quinessene Cosmi Coinidene nd he Cosmologil Consn Physil Review Leers Vol 8 No pp doi:003/physrevle8896 Copyrigh 03 SiRes