Distribution of Mass and Energy in Closed Model of the Universe

Transcription

1 Ieraial Jural f Asrmy ad Asrysics, 05, 5, 9-0 Publised Olie December 05 i SciRes ://wwwscirrg/jural/ijaa ://dxdirg/046/ijaa05540 Disribui f Mass ad Eergy i Clsed Mdel f e Uiverse Fadel A Bukari Dearme f Asrmy, Faculy f Sciece, Kig Abdulaziz Uiversiy, Jedda, Saudi Arabia Received February 05; acceed December 05; ublised 5 December 05 Cyrig 05 by aur ad Scieific Researc Publisig Ic Tis wrk is licesed uder e Creaive Cmms Aribui Ieraial Licese (CC BY) ://creaivecmmsrg/liceses/by/40/ Absrac Te uiverse s riz disace ad vlume are csruced i e clsed csmic mdel Te uiverse riz disace disribui icreases csaly fr < me ad decreases fr > me Hwever, e uiverse s riz vlume sws a sudde reduci i e rage = 05 Gyr me due e cage f e uiverse sace frm fla curved e clsed i e ierval 56 Gyr me O e er ad, is disribui exibis a abru rise i e rage = me due e cage f e uiverse sace frm clsed e curved fla i e ierval Gyr Te mass f radiai, maer ad dark eergy wii e riz vlume f e uiverse are als ivesigaed Tese disribuis reveal similar iceable cages as e uiverse s riz vlume disribui fr e same reass Te mass f radiai dmiaes u = 55 yr, e e mass f maer becmes larger Aferwards, b disribuis f radiai ad maer decrease wile e disribui f dark eergy rises uil = 0007 Gyr, were e mass f dark eergy revails u = me Hece, e disribui f dark eergy reduces uil = 4089 Gyr, were e mass f maer becmes rmie agai A = 5646 Gyr e masses f b maer ad radiai becme areciably ig suc a e iercluser sace will vais ad clusers f galaxies ierfere wi eac er Furermre, ly e iergalacic medium will disaear, bu als galaxies will cllide ad merge wi eac er frm exremely dese ad clse csmlgical bdies Tese very dese bdies will uderg furer successive cllisis ad mergers uder e aci f ceral graviy, were e iersellar medium will vais ad e uiverse wuld devel big cruc a bc = 565 Gyr I is ieresig e a e riz disace f e uiverse i e clsed mdel a = me is i very gd agreeme wi e maximum riz disaces i e five geeral csmic mdels Keywrds Dark Eergy, Radiai, Clsed Csmic Mdel Hw cie is aer: Bukari, FA (05) Disribui f Mass ad Eergy i Clsed Mdel f e Uiverse Ieraial Jural f Asrmy ad Asrysics, 5, 9-0 ://dxdirg/046/ijaa05540

2 F A Bukari Irduci Te disribui f desiy arameers f radiai, maer ad dark eergy i e clsed csmic mdel were ivesigaed i a revius sudy [], were we discvered e mai ecs f e uiverse isry i is mdel I is wry w sudy e disribuis f equivale mass f radiai, mass f maer ad equivale mass f dark eergy wii e riz vlume f e uiverse ge deeer sig f e uiverse evlui i e clsed mdel Te reas fr csiderig e equivale mass f radiai i is sudy is e sigifica value f e radiai desiy arameer i e early uiverse ad befre e big cruc as we ave see i [] Terefre, i is vial devel e disribuis f e riz disace ad riz vlume f e uiverse i e clsed mdel a varius ime rages deedig e bases reseed i [] Descrii f medlgy is illusraed i Seci, wile algrim wuld be sw i Seci Resuls ad discussi are dislayed i Seci 4 Cclusi is give i Seci 5 Medlgy I is bvius frm [] a e riz disace ad riz vlume f e uiverse i clsed csmic mdel a e rese ime are resecively c 4 d( ) = S, ( ), ( ), ( ) d 0 a S m a a S Λ r a a a H Ω + Ω + Ω () a 8π V( ) = d ( ) () were Ω Λ,, Ω m, ad Ω r, are give by Λ, Ω Λ, = () c c, c, m, Ω m, = (4) ρ Ω = r, r, c c, ( ) H c, = (6) 8πG 4 H H ( ) = s Ω Λ, ( a ) + s Ω m, ( a a ) + s Ω r, ( a a ) a H( ) H ( ) were s =, S = H H ( ) Λ ( a ) H H =, m, r, a Ω +Ω +Ω a a (8) Λ, Ω Λ, = (9) c ρ c, c, m, Ω m, = (0) ρ Ω = () r, r, c ρc, (5) (7) 9

3 F A Bukari were is e csmic ime i Gyr ρ Λ, Λ, = ρ c, ΩΛ, c Ωm, ρλ, m, = ρc, + a c r, Ωr, ρλ, = ρ c, + 4 c a c ρλ, ρλ, 00 = c c c () () (4) (5) H ρ c, = (6) 8πG Te riz disace f e uiverse i e clsed csmic mdel a ay give ime is give by c a 4 ( ) = Ω, ( ), ( ), ( ) d 0 Λ + Ω m + Ωr H a d S a S a a S a a a (7-a) Csequely, e cage i e riz disace f e uiverse i e ime ierval bewee w isas f scale facrs a, a is wrie as c a ( ) 4, ( ) m, ( ) r, ( ) d H a Λ a d = S Ω a + S Ω a a + S Ω a a a (7-b) Te riz vlume f e uiverse i e clsed mdel a ay give ime is exressed as ( ) ( ( ) ) V = f d, k (8) Equai (8) idicaes a e riz vlume f e uiverse a is a fuci f d ( ) ad e curvaure f sace k a Sice is curvaure culd be fla, e ad clsed frm e big bag big cruc as evide frm Table i [] Tus, e lw f V ( ) ca be deermied accrdig e value f k a, as exlaied i e fllwig cases: () Fla sace ( k = 0 ) We ave see i [] a e riz vlume f e uiverse a ime i is case is give by 8π V( ) = d ( ) (9) Terefre, i is bvius frm Table i [] a Equai (9) is used i e ime iervals 0075 < 56 Gyr, 98 < 4075 Gyr, ad 548 < bc () Clsed sace ( k = + ) We recall e equai f rer disace f exragalacic bjec were Ad e vlume f sace wii d ( ) ( ) ( ) ( ) 65 Gyr, d = R f r (0) si r k = + r dr f ( r) = = r 0 0 k = () kr si r k = is exressed as r V R r kr r ( ) ( ) π π = dφ siθdθ d () 9

4 F A Bukari were R( ), r, r, θ ad φ are defied as i [] [4] Fr k = +, Equais (), (0) ad () yield Hece Equai () gives Assume ( ) ( ) si d = R r () r r dr V ( ) = 8π R ( ) (4) 0 r Subsiuig by (5) i (4) we ge Le Subsiuig by (7) i (6) we ave Subsiuig by () i (8) yields Suse r =, ece r r r = si α, d = csαd α, csα = (5) si r ( ) = ( ) ( ) V 4πR cs α d α (6) 0 β = α, dβ = d α (7) r r V( ) = 4πR ( )( si r) ( si r) ( si r ) r r V( ) = 4π d( ) ( ) ( si r si r ) π α = ad Equai (9) becmes (8) (9) 6 V( ) = d( ) (0) π Tus, e riz vlume f e uiverse i e clsed csmic mdel a ime i is case is exressed as 6 V( ) = d ( ) () π I is evide frm Table i [] a Equai () is used i w ime iervals exedig rug 56 < 98 Gyr () Oe sace ( k = ) Equais (), (0) ad () give Assume Subsiuig by (4) i () we ave ( ) ( ) si d = R r () r r dr V ( ) = 8π R ( ) () 0 + r r r r = si ξ, d = csξd ξ, csξ = + (4) si r ( ) = ( ) ( ) V 4πR cs ξ d ξ (5) 0 94

5 F A Bukari Le Subsiuig by (6) i (5) we ge Subsiuig by () i (7) yields We r =, Equai (8) reduces η = ξ, dη = d ξ (6) ( r) ( si r) ( si r) r cs si V( ) = 4πR ( )( si r) ( r) ( si r) ( si r) r cs si V( ) = 4π d( ) ( ) ( ) (7) (8) V = 98π d (9) Terefre, e riz vlume f e uiverse i clsed csmic mdel a ime i is case is wrie as ( ) ( ) V = 98π d (40) I is clear frm Table i [] a Equai (40) is used i e ime iervals 65< 0075 Gyr, 4075< 548 Gyr Te al desiy f e uiverse i e clsed csmic mdel a ime is Subsiuig by ()-(5) i (4) we ge were r, Λ, ( ) = m, + + (4) c c ( ) = c, Ω m, + c, Ω r, + c, Ω, Λ ( ) ρ ( ) = Ω (4) c, ( ),,, Ω =Ω +Ω +Ω (4) m r Λ Frm Equai (9), (0), (40) ad (4) e al mass f e uiverse wii e riz vlume i clsed csmic mdel a ime is ( ) ( ) ( ) M = V (44) Te masses f maer, radiai ad dark eergy wii e riz vlume f e uiverse i clsed csmic mdel a ime are resecively ( ) m, m, = M M ( ) Ω Ω ( ) r, r, = M M ( ) Ω Ω ( ), Λ, = M M Te ime ierval bewee w isas wi scale facrs a, a durig e uiverse exasi is give by Equai (6) i [4] [5] as Ω Λ Ω ( ) Δ a =, ( a ) m, r, d a a Λ H Ω +Ω +Ω a a (45) (46) (47) (48) 95

6 F A Bukari Hwever, durig e uiverse craci if a < a e mdulus f e rig ad side f (50) suld be ake Algrim I deermiai f e disribuis f ( ), ( ), ( ), m,, r, ad Λ, d V M M M M we use e fllwig ses: () Sage f e uiverse exasi a) Se = 0, d = 0 ad iser e value f amax = 00988, J = 000 fr 05 Gyr, a max = 75587, J = 600 fr 05 < me amax b) Calculae DA = DBLE J ( ) c) Sar geeral DO l I =, J wic icludes e fllwig sub ses: a = DA I, a = DA I d) ( ) e) Cmue ew value f csmic ime umerically usig (48), were = + Δ f) Obai ew value f e uiverse riz disace d umerically usig (7-b), were d = d + Δd g) Deermie e crresdig values f V usig (9), () ad (40), i addii e values f H ( ), c,, Ω m,, Ω r,, Ω Λ,, Ω ( ), ( ), M( ), Mm,, Mr, ad M Λ, usig (7), (6), (4), (5), (), (4), (4), (44), (45), (46) ad (47) resecively ) Ciue e geeral DO l () Sage f e uiverse craci a) Se = me = 6857 Gyr, d = d ( me ) = 8694 Gc, ad iser e values f amax = 75587, J = 600 fr me < *, * = bc 05 Gr y, amax = 00959, J = 90 fr > * amax b) Evaluae DA = DBLE J ( ) c) Sar geeral DO l I =, J wic icludes e fllwig sub ses: J = J I +, a = DA J, a = DA J d) Se ( ) e) Obai ew value f csmic ime umerically usig (48), were = + Δ f) Cmue ew value f e uiverse riz disace d umerically usig (7-b), were d = d Δd Sub ses (g) ad () are similar (g), () meied abve i e sage f e uiverse exasi 4 Resuls ad Discussi Te disribui f e uiverse riz disace i e clsed csmic mdel uil = 05 Gyr is sw i Figure (a) Te disribui icreases quie slwly u = Myr, e e disribui sars raisig raidly Hwever, e disribui f e uiverse riz disace i e rage = 05 Gyr - me icreases very fas uil abu = 577 Gyr Aferwards, i raises gradually as idicaed i Figure (b) Furermre, e disribui f e uiverse riz disace i e rage = me - *, were * = bc - 05 Gyr, decreases quie slwly u = Gyr, ece i decreases relaively fas as reseed i Figure (c) Nevereless, e disribui f e uiverse riz disace i e rage = - * decreases slwly uil = 547 Gyr, = bc Gyr, e i sars reduci sarly wards = bc as dislayed i Figure (d) Te disribui f e uiverse riz vlume i e clsed csmic mdel u = 05 Gyr is sw i Figure (a) Te disribui icreases very slwly uil = Myr Aferwards, e disribui sars raisig areciably Durig is csmic ime rage e sace f uiverse is fla Te disribui f e uiverse riz vlume ciues raisig u = 48 Gyr, ece e disribui suddely decreases uil = Gyr, e i icreases gradually u = me as see is Figure (b) Te sar decrease f e disribui i e rage 48 < < Gyr because e sace f e uiverse cages frm fla curved e clsed i e ierval 56 Gyr me Te disribui f e uiverse riz vlume i e rage = me - * is disclsed i Figure (c) Te disribui decreases uil = Gyr, ece i sws abru raisig u = Gyr, ece i reduces agai uil = * Te abru icrease f e disribui i e rage < < Gyr is due 96

7 F A Bukari (a) (b) (c) Figure (a) Te disribui f e uiverse riz disace i e clsed csmic mdel u = 05 Gyr; (b) Te disribui f e uiverse riz disace i e clsed csmic mdel i e rage = 05 Gyr - me ; (c) Te disribui f e uiverse riz disace i e clsed csmic mdel i e rage = me - * ; (d) Te disribui f e uiverse riz disace i e clsed csmic mdel i e rage = * - (d) (a) (b) 97

8 F A Bukari (c) Figure (a) Te disribui f e uiverse riz vlume i e clsed csmic mdel u = 05 Gyr; (b) Te disribui f e uiverse riz vlume i e clsed csmic mdel i e rage = 05 Gyr - me ; (c) Te disribui f e uiverse riz vlume i e clsed csmic mdel i e rage = me - * ; (d) Te disribui f e uiverse riz vlume i e clsed csmic mdel i e rage = * - e fac a e uiverse sace cages frm clsed e curved fla i e ierval Gyr Te disribui f e uiverse riz vlume i e rage = - * is reseed i Figure (d), wic is reducig gradually wards = bc Te disribui f mass ad eergy wii e riz vlume f e uiverse i e clsed csmic mdel uil = 05 Gyr is sw i Figure (a) Te disribuis f radiai ad al mass decrease very slwly u = 4596 yr ad = 606 yr resecively, e ey reduce mre raidly Hwever, e disribui f maer icreases s slwly uil = 86 yr ad i decreases raidly aferwards Tis disribui iersecs wi e disribui f radiai a = 55 yr ad cicides wi e disribui f al mass a = 89 Myr Te disribui f dark eergy icreases ciuusly wards = 05 Gyr Te disribui f mass ad eergy wii e riz vlume f e uiverse i e clsed mdel i e rage = 05 Gyr - me is illusraed i Figure (b) I is iceable a e disribui f radiai iersecs e disribui f dark eergy a = 0668 Gyr, ece i reduces very fas u = 047 Gyr Aferwards, i raises gradually uil = 4965 Gyr, e i decreases abruly u = 697Gyr, were i sars icreasig agai wards me Te disribui f maer decreases u = 794 Gyr, ece i raises very slwly wards me Tis disribui cicides wi e disribui f al mass u = 985 Gyr, ad iersecs wi e dark eergy disribui a = 0007 Gyr Te maer disribui idicaes bvius decrease i e rage = Gyr Te disribui f dark maer icreases uil = 4965 Gyr, ece i als sws marked decrease u = 697 Gyr Aferwards, i raises sligly wards me Te al mass disribui reduces u = 4965 Gyr, e i decreases abruly uil = 697 Gyr Hece, is disribui icreases gradually wards me Te iceable decrease f all fur disribuis i e rage abu 495 < < 697 Gyr is due e cage f e uiverse sace frm fla curved e clsed i e rage 56 me Te disribui f mass ad eergy wii e riz vlume f e uiverse i e clsed mdel i e rage = me - * is illusraed i Figure (c) Te disribui f radiai reduces very slwly uil = 7989 Gyr, e i raises suddely u = 4040 Gyr Hece, is disribui decreases agai uil = Gyr were i iersecs wi e disribui f dark eergy, e i sars raisig wards * Te maer disribui icreases s slwly u = 7989 Gyr, ece i exibis marked raise uil = 4040 Gyr ad iersecs wi e dark eergy disribui a = 4089 Gyr Aferwards, e disribui f maer sws subsaial icrease wards * ad cicides wi e al mass disribui a = 590 Gyr ece fr Te dark eergy disribui reduces gradually uil = 7989 Gyr were i dislays usadig raise u = 4040 Gyr, ece i decreases agai wards * Te al mass disribui reduces als gradually uil = 7989 Gyr, e i exibis bvius icrease u = 4040 Gyr Aferwards, i raises very fas as e maer disribui Te rmie icrease f e fur disribuis i e ierval abu 7989 < < 4040 Gyr is wig e cage f e uiverse sace frm clsed e curved (d) 98

9 F A Bukari (a) (b) (c) Figure (a) Te disribui f mass ad eergy i e clsed csmic mdel u = 05 Gyr; (b) Te disribui f mass ad eergy i e clsed csmic mdel i e rage = 05 Gyr - me ; (c) Te disribui f mass ad eergy i e clsed csmic mdel i e rage = me - * ; (d) Te disribui f e mass ad eergy i e clsed csmic mdel i e rage = * - fla i e rage Gyr Te disribui f mass ad eergy wii e riz vlume f e uiverse i e clsed mdel i e rage = * is dislayed i Figure (d) Te radiai disribui icreases quie slwly uil = 550 Gyr, ece i sars raisig areciably fas Te disribui f b maer ad al mass cicide eac er ad lie ver e radiai disribui Te w disribuis icrease gradually u = 5574 Gyr, e ey raise u Hwever, e dark eergy disribui decreases s slwly uil = 5574 Gyr, aferwards i reduces subsaially Esimais f d( ), V( ), Mr( ), Mm( ), M () ad e equivale umber f e Cma-like clusers e mass f maer wii e uiverse riz vlume NCOMA ( ) i e clsed csmic mdel a secial imes are reseed i Table I is ieresig e a a = = bc Gyr, e riz vlume f e uiverse V ( ) 0000( 0 Gc) = 6 Sice e radius f e Cma cluser is r COMA = 6 Mc [5], e V ( ) = VCOMA, were V COMA is e Cma cluser vlume Hwever, e mass f maer wii e riz vlume f e uiverse a = is Mm ( ) = MCOMA Tis idicaes very clearly a e iercluser medium will disaear a = ad galaxy clusers will ierfere wi eac er Furermre, e radius f e Milky Way galaxy is r Mw = 50 Kc [6] Tus, V ( ) = VMW, were V MW is e Milky Way galaxy vlume Nevereless, i is fud a Mm ( ) = MMW Terefre, ly e iergalacic saces will vais a 6 =, bu als galaxies will cllide ad merge wi eac er frm exremely dese ad clse csmlgical bdies Tese very dese bdies will uderg furer successive cllisis ad mergers uder e aci f ceral graviy, were e iersellar medium will vais ad e uiverse wuld devel big cruc a bc = 565 Gyr I is als ieresig e frm Table a e riz disace f e uiverse a maximum exasi is d ( me ) = 869 Gc Tis riz disace is i very gd agreeme wi e maximum value f e uiverse riz disace i e bserved geeral csmic mdel A, d ( ) = 900 Gc, were = 4 Gyr (d) 99

10 F A Bukari Table Esimais f e riz disace, riz vlume, mass f radiai, mass f maer, mass f dark eergy ad e equivale umber f e Cma-like clusers e mass f maer wii e uiverse riz vlume i e clsed csmic mdel a secial imes d ( ) Gc V ( ) lg ( Mr M ) lg ( Mm M ) lg ( M M ) N ( ) COMA rm (55) yr u u m (0007) Gyr (7) Gyr me (685) Gyr u u u u u u 4 m (4089) Gyr (5646) Gyr u u u u were = Gyr, u = (Gc), u = (0 Gc), bc u = 0 ad 8 u = 0 4 Te value f d ( ) is als i ig agreeme wi e values f d ( ) mdels as sw frm Table i [7] 5 Cclusis me i e er fur geeral csmic I is aer we ave ivesigaed e disribuis f e uiverse riz disace ad e uiverse riz vlume i e clsed csmic mdel I is fud a e uiverse riz disace disribui icreases csaly fr < me ad decreases fr > me Hwever, e uiverse riz vlume disribui sws sudde reduci i e rage = 05 Gyr - me due e cage f e uiverse sace frm fla curved e clsed i e ierval 56 Gyr me O e er ad, is disribui exibis abru raise i e rage = me - * because f e cage f e uiverse sace frm clsed e curved fla i e ierval Gyr Disribuis f mass f radiai, maer ad dark eergy wii e riz vlume f e uiverse were als ivesigaed i e clsed csmic mdel Tese disribuis reveal similar iceable cages as e uiverse riz vlume disribui fr e same reass Te mass f radiai dmiaes u = 55 yr, e e mass f maer becmes larger Aferwards, b disribuis f radiai ad maer decrease wile e disribui f dark eergy rises uil = 0007 Gyr, were e mass f dark eergy revails u = me Hece, e disribui f dark eergy reduces uil = 4089 Gyr were e mass f maer becmes rmie agai A = 5646 Gyr e masses f b maer ad radiai becme areciably ig suc a e iercluser sace will vais ad clusers f galaxies will ierfere wi eac er Furermre, ly e iergalacic medium will disaear, bu als galaxies will cllide ad merge wi eac er frm exremely dese ad clse csmlgical bdies Tese very dese bdies will uderg furer successive cllisis ad mergers uder e aci f ceral graviy, were e iersellar medium will vais ad e uiverse wuld devel big cruc a = 565Gyr Refereces bc [] Bukari, FA (0) A Clsed Mdel f e Uiverse Ieraial Jural f Asrmy ad Asrysics,, ://dxdirg/046/ijaa00 [] Bukari, FA (0) Csmlgical Disaces i Clsed Mdel f e Uiverse Ieraial Jural f Asrmy ad Asrysics,, 99-0 ://dxdirg/046/ijaa00 [] Bukari, FA (0) Csmlgical Disaces i Five Geeral Csmic Mdels Ieraial Jural f Asrmy 00

11 F A Bukari ad Asrysics,, 8-88 ://dxdirg/046/ijaa00 [4] Bukari, FA (0) Five Geeral Csmic Mdels Jural f Kig Abdulaziz Uiversiy: Sciece, 5 [5] Ryde, B (00) Irduci Csmlgy Addis & Wesley, Bs [6] Sceider, P (00) Exragalacic Asrmy ad Csmlgy Sriger, New Yrk [7] Bukari, FA (05) Disribui f Mass ad Eergy i Five Geeral Csmic Mdels Ieraial Jural f Asrmy ad Asrysics, 5, 0-7 ://dxdirg/046/ijaa