Inerninl Jurnl f Asrnmy nd Asrpysics 05 5 0-7 Publised Online Mrc 05 in SciRes p://wwwscirprg/jurnl/ij p://dxdirg/0436/ij055004 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 34675-5596 yr were i becmes equl e mss f mer Ten e mss f mer dmined unil 9455-0063 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 5 0-7 p://dxdirg/0436/ij055004
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
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 = 05-50 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
Figure Te disribuin f e universe rizn disnce in e generl csmic mdels () up = 05 Gyr; (b) in e rnge = 05-50 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 = 05 50 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 = 34675-5596 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 = 0566-0539 Gyr s seen in Tble 4 Te disribuin f l mss cincides wi f rdiin up 54344 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 576959 yr nwrds Te disribuin f mss nd energy wiin e universe rizn vlume f e universe in ny generl mdel in e rnge = 05-50 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 = 9455-0063 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
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 50 7667 4969 9003 B 40 756 45 949 C 0740 7 470 9074 D 9030 3367 3666 306 E 94370 469 3070 450 n Figure Te disribuin f e universe rizn vlume in e generl csmic mdels () up = 05 Gyr; (b) in e rnge = 05-50 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 300 743 449 575555 B 00 73990 3779 5734 C 37900 7496 4349 577 D 7900 5795 337 53939 E 70300 6077 0504 5634 n 4
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 = 05-50 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 5596 56 49 B 496665 76 07 C 554 59 56 D 3473 333 79566 E 34675 3753 755 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 0539 3053 5540 B 05736 934 55375 C 050 300 55445 D 0566 9407 55350 E 0559 906 556 5
= 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 94650 4507 03 B 96930 4393 0339 C 9455 4594 0937 D 969 4564 07974 E 0063 409 07030 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 0453 3965 43973 B 0434 3963 4375 C 04570 39663 4406 D 04035 39954 439 E 0364 399 433 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 400 5340 47694 B 9797 547 4757 C 7546 547 47779 D 36 463 47706 E 65 5455 475 6
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 944 0 B 96955 0 C 950 0 D 073 0 E 65 0 9057 0 6750 0 9059 0 9034 0 59 0 4564 0 4 4640 0 5034 4667 0 0963 49473 0 034 49099 0 4370 universe up = 34675-5596 yr were i becmes equivlen e mss f mer Aferwrds e mss f mer previled unil = 9455-0063 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