Distribution of Mass and Energy in Five General Cosmic Models
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1 Inerninl Jurnl f Asrnmy nd Asrpysics Publised Online Mrc 05 in SciRes p://wwwscirprg/jurnl/ij p://dxdirg/0436/ij Disribuin f Mss nd Energy in Five Generl Csmic Mdels Fdel A Bukri Deprmen f Asrnmy Fculy f Science King Abdulziz Universiy Jedd KSA Emil: fdbukri@gmilcm Received 3 Mrc 05; cceped 7 Mrc 05; publised 0 Mrc 05 Cpyrig 05 by ur nd Scienific Reserc Publising Inc Tis wrk is licensed under e Creive Cmmns Aribuin Inerninl License (CC BY) p://creivecmmnsrg/licenses/by/40/ Absrc Disribuins f e universe rizn disnce nd universe rizn vlume were invesiged in e lig f five generl csmic mdels wic were cnsruced in previus sudy B disribuins increse s slwly up 5444 Myr en ey sr rising very fs up 60 Gyr Aferwrds ey increse gin very slwly unil 4 Gyr Disribuins f mss f rdiin mer nd drk energy wiin e rizn vlume f e universe were ls sudied in e five generl csmic mdels Te msses f b rdiin nd mer decrese grdully wi ime wile e mss f drk energy increses Te mss f rdiin previled in e erly universe up yr were i becmes equl e mss f mer Ten e mss f mer dmined unil Gyr were i becmes equl e mss f drk energy Tencefrwrd e mss f drk energy previls e universe Te csmic spce becmes pprximely mer empy in e s fr fuure f e universe Keywrds Generl Csmic Mdels Disribuin f Mss nd Energy Inrducin In previus sudy [] e disribuin f densiy prmeers f rdiin mer nd drk energy were invesiged in deils in five generl csmic mdels Hence i wuld be ineresing sudy e disribuins f equivlen mss f rdiin mss f mer nd equivlen mss f drk energy wiin e rizn vlume f e universe in e generl mdels Terefre i is necessry sr is sudy by invesiging e disribuins f e rizn disnce nd rizn vlume f e universe in e generl mdels differen ime inervls depending n e bses discussed in [] Descripin f medlgy is given in Secin wile lgrim wuld be illusred in Secin 3 Resuls Hw cie is pper: Bukri FA (05) Disribuin f Mss nd Energy in Five Generl Csmic Mdels Inerninl Jurnl f Asrnmy nd Asrpysics p://dxdirg/0436/ij055004
2 nd discussin re presened in Secin 4 Cnclusin is swn in Secin 5 Medlgy We ve seen in [] e rizn disnce nd rizn vlume f e universe e presen ime re respecively were r m Λ is c d( ) = Λ ( ) 0 m r d H + + () π 3 V( ) = d ( ) () 3 ch re ll defined s in [] Tus e rizn disnce f e universe ny given ime c d( ) = Λ ( ) 0 m r d H + + (3) Cnsequenly e cnge in e rizn disnce f e universe in e ime inervl beween w insns f scle fcrs is wrien s c Δd( ) = Λ ( ) m r d H + + (3b) Te rizn vlume f e universe ny given ime is I is ls bvius frm [] e l densiy f e universe is given by were π 3 V( ) = d ( ) (4) 3 ρ ( ) ρ ( ) = c (5) ( ) ρ c ( m r Λ ) = + + (6) 3H ρ c = (7) πg ρm H 0 m m = = 3 ρ c H () ρr H 0 r r = = 4 ρ c H ρλ H0 Λ = = Λ ρ c H (9) (0) H = Λ m r + + ( ) H Hence e l mss wiin e rizn vlume f e universe ny given ime is expressed s ρ () M = V () Te mss f mer M m e equivlen mss f rdiin M r nd e equivlen mss f drk energy
3 M Λ wiin e rizn vlume f e universe ny given ime re given by ρm Mm = M( ) ρ Te csmic ime is given by Equin (6) in [] s m Mm = M (3) ( ) r Mr = M (4) ( ) Λ MΛ = M (5) ( ) Λ ( ) 0 m r d (6) = + + H Tus e ime inervl beween w insns wi scle fcrs during e universe expnsin is expressed s 3 Algrim Λ m r Δ = d H + + (6b) d V M M M nd M we use e fllwing seps: i) Se = 0 d = 0 K = K = 000 en inser e vlue f mx = 009 fr 05 Gyr mx = 0 fr 05 < 50 Gyr nd mx = 000 fr 50 < 4 Gyr mx ii) Cmpue DA = DBLE K In deerminin f e disribuins f m r Λ iii) Sr generl DO lp I = K K wic includes e fllwing sub seps: iv) = DA( I ) = DAI v) Clcule new vlue f csmic ime numericlly using(6-b) were = + Δ vi) Deermine new vlue f e universe rizn disnce d numericlly using (3-b) were d = d + Δd vii) Obin e crrespnding vlues f V H ρc m r Λ ( ) ρ ( ) M( ) Mm Mr MΛ using (4) () (7) () (9) (0) (6) (5) () (3) (4) nd (5) respecively viii) Cninue e generl DO lp 4 Resuls nd Discussin Te disribuin f e universe rizn disnce in e generl mdels up = 05 Gyr is swn in Figure () Te disribuins f ll mdels cincide n ec er unil 5444 Myr Ten e disribuins f mdels A B nd C cincide n ec er nd ge upper n e cincided disribuins f mdels D nd E Te universe rizn disnce increses quie slwly wi ime in ll generl mdels up 5444 Myr ence i ss rising very fs Te disribuin f e universe rizn disnce in e generl mdels in e rnge = Gyr is illusred in Figure (b) Te disribuins f ll mdels cincide n ec er up Gyr Aferwrds e disribuins f mdels A B nd C cincide n ec er nd becme iger n e cincided disribuins f mdels D nd E In ll generl mdels e universe rizn disnce disribuins increse very fs wi ime Te disribuin f e universe rizn disnce in e generl mdels in e rnge = 50-4 Gyr is presened in Figure (c) Te disribuins f mdels A B nd C re clse ec er nd lie upper n e disribuins f mdels E nd D Te increse in e universe rizn disnce ges very smll wi incresing ime in ll generl mdels
4 Figure Te disribuin f e universe rizn disnce in e generl csmic mdels () up = 05 Gyr; (b) in e rnge = Gyr; (c) in e rnge = 50-4 Gyr Tble sws e universe rizn disnces in e generl mdels specil imes Tese imes re e ime f rdiin-mer mss equivlence rm e ime f mer-drk energy mss equivlence mλ e presen ime = 37 ± 0 Gyr nd e ime n = 4 Gyr Te resuls illusred in Figures ()-(c) re suppred by se displyed in Figures ()-(c) wic sw e disribuins f e universe rizn vlume in e generl mdels in e rnges up = 05 Gyr = Gyr nd = 50-4 Gyr respecively Tble presens e universe rizn vlumes in e generl mdels e specil imes rm mλ nd n Te disribuin f mss nd energy wiin e universe rizn vlume f e universe in ny generl mdel up = 05 Gyr is exibied in Figure 3() Te disribuins f b rdiin nd mer decrese grdully wi ime nd inersec e ime rm = yr s swn in Tble 3 On e er nd e disribuin f drk energy increses grdully unil i inersecs wi e rdiin disribuin e ime rλ = Gyr s seen in Tble 4 Te disribuin f l mss cincides wi f rdiin up yr Aferwrds e w disribuins diverge frm ec er Hwever e disribuin f e l mss cincides n e disribuin f mer frm e ime yr nwrds Te disribuin f mss nd energy wiin e universe rizn vlume f e universe in ny generl mdel in e rnge = Gyr is displyed in Figure 3(b) I is bvius e disribuins f mer nd rdiin decrese grdully wi ime nd e frmer lies bve e ler Te disribuin f drk energy increses wi ime nd inersecs wi e disribuin f mer = Gyr s illusred in Tble 5 Te disribuin f e l mss cincides n e disribuin f e mer up = 4574 Gyr en ey diverge frm ec er Furermre e disribuin f e l mss cincides n e disribuin f e drk energy frm = 57 Gyr nwrds Msses f rdiin mer nd drk energy wiin e universe rizn vlume in e generl mdels e presen ime re illusred in Tble 6 Te disribuin f mss nd energy wiin e universe rizn vlume in ny generl mdel in e rnge 3
5 Tble Hrizn disnces f e universe in e generl csmic mdels specil imes Mdel d ( ) / Mpc d ( ) d ( ) / Gpc d ( ) / Gpc rm mλ / Gpc A B C D E n Figure Te disribuin f e universe rizn vlume in e generl csmic mdels () up = 05 Gyr; (b) in e rnge = Gyr; (c) in e rnge = 50-4 Gyr Tble Hrizn vlumes f e universe in e generl csmic mdels specil imes 3 Mdel V ( ) / Mpc V ( ) / ( 0 Gpc) 3 V ( ) / ( 0 Gpc) 3 V ( ) / ( 0 Gpc) 3 rm mλ A B C D E n 4
6 Figure 3 Te disribuin f mss nd energy wiin e universe rizn vlume in ny generl csmic mdel () up = 05 Gyr; (b) in e rnge = Gyr; (c) in e rnge = 50-4 Gyr Tble 3 Csmic imes wic M r ( ) = M m ( ) wiin e universe rizn vlume in e generl csmic mdels Mdel rm / 03 yr Lg ( M rm M ) Lg ( M Λ M ) A B C D E Tble 4 Csmic imes wic M r ( ) = M Λ ( ) wiin e universe rizn vlume in e generl csmic mdels Mdel rλ / Gyr Lg ( M rλ M ) Lg ( M m M ) A B C D E
7 = 50-4 Gyr is exibied in Figure 3(c) Agin e disribuin f b mer mss nd rdiin mss decrese wi ime nd e frmer is iger n e ler Te disribuins f drk energy mss nd l mss cincide n ec er Msses f rdiin mer nd drk energy wiin e universe rizn vlume in e generl mdels n re given in Tble 7 Tble sws e equivlen number f e Cm-like clusers e mss f mer wiin e universe rizn vlume NCOMA ( ) in e generl mdels e specil imes rm m Λ nd n I is bvius is cnen f mer srngly decreses wi ime suc e csmic spce becmes lms mer empy in e fr fuure f e universe 5 Cnclusin In is ricle disribuins f e universe rizn disnce nd universe rizn vlume were deermined in e five generl csmic mdels wic were esblised previusly Te w disribuins were fund incresing slwly up 5444 Myr ence ey rise pprecibly fs up = 60 Gyr en ey increse gin s slwly unil = 4 Gyr Disribuins f mss f rdiin mer nd drk energy wiin e universe rizn vlume were ls invesiged in e five generl mdels Te msses f rdiin nd mer re decresing wi ime lug e mss f drk energy is incresing Te mss f rdiin ws dminn in e erly Tble 5 Csmic imes wic M ( ) M ( ) m = wiin e universe rizn vlume in e generl csmic mdels Λ Mdel Lg ( M M mλ ) Lg ( M M r ) / Gyr mλ A B C D E Tble 6 Msses f rdiin mer nd drk energy wiin e universe rizn vlume in e generl csmic mdels = Mdel Lg ( M M r ) Lg ( M M m ) Lg ( M M Λ ) A B C D E Tble 7 Msses f rdiin mer nd drk energy wiin e universe rizn vlume in e generl csmic mdels = n Mdel Lg ( M M r ) Lg ( M M m ) Lg ( M M Λ ) A B C D E
8 Tble Equivlen number f e Cm-like clusers e mss f mer wiin e universe rizn vlume in e generl csmic mdels specil imes Mdel rm mλ n A B C D E universe up = yr were i becmes equivlen e mss f mer Aferwrds e mss f mer previled unil = Gyr were i becmes equl e mss f drk energy Frm is ime nwrds e mss f drk energy dmines e universe Te csmic spce ges pprximely mer empy in e very reme fuure f e universe References [] Bukri FA (03) Five Generl Csmic Mdels Jurnl f King Abdulziz Universiy: Science 5 [] Bukri FA (03) Csmlgicl Disnces in Five Generl Csmic Mdels Inerninl Jurnl f Asrnmy nd Asrpysics 3 3-7
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