Cosmological Distances in Closed Model of the Universe

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1 Inerninl Jurnl f srnmy n srpysics p://xirg/436/ij333 Publise Online June 3 (p://wwwscirprg/jurnl/ij) Csmlgicl Disnces in Clse el f e Universe Fel Bukri Deprmen f srnmy Fculy f Science King bulziz Universiy Je Sui rbi Emil: flbukri@milcm Receive Jnury 3 3; revise Februry 6 3; ccepe Februry 4 3 Cpyrig 3 Fel Bukri Tis is n pen ccess ricle isribue uner e Creive Cmmns ribuin License wic permis unresrice use isribuin n reprucin in ny meium prvie e riginl wrk is prperly cie BSTRCT Fur Csmlgicl isnces were eermine in e lig f e clse csmic mel wic ws presene in previus suy Ec f ese isnces ws bine in erms f e resif f n exrglcic bjec I is fun e luminsiy isnce f e exrglcic bjec in e clse csmic mel e bserve mel n mel re pprximely e sme up z = 535 wever e luminsiy isnce in e clse csmic mel pprximely grees wi is vlue in e bserve mel up z = 644 Esimins f e rizn isnce f e universe e l mss n e mss f mer wiin e rizn isnce e equivlen numbers f e ilky Wy-like glxies n e Cm-like clusers f glxies e mss f mer were cmpue in e clse csmic mel e presen ime Keywrs: Exrglcic Disnces; Csmlgy; Csmic Dynmics; Csmlgicl Prmeer Inrucin In [] e prper isnce luminsiy isnce ngulr imeer isnce n isnce mulus f e exrglcic bjec were suie in erms f is resif in e lig f five generl csmic mels In is ricle ese csmlgicl isnces will be invesige in e clse csmic mel wic ws inruce in [] Furermre ineresing esimins will be eermine in is mel presen ime Tese re e rizn isnce f e universe e l mss n e mss f mer wiin e rizn isnce e equivlen numbers f e ilky Wy-like glxies n e Cm-like clusers f glxies e mss f mer Descripin f melgy is illusre in Secin wile resuls n iscussin re presene in Secin 3 n cnclusin is given in Secin 4 elgy Te prper isnce f n exrglcic bjec e presen ime is given by p c e () c is e spee f e lig n e e emissin ime f e bjec pn [] Te spee f e universe ynmics in e clse csmic mel is bine by Equin (3) in [] s m 4 s r 3 s s () s (3) Te ubble prmeer in e expning csmic mel ime is m (4) r Te ensiy prmeers in e expning csmic mel ime re (5) c c m m c r r c c (6) (7) Cpyrig 3 SciRes

2 F BUKRI c c c (8) m m c 3 c (9) r r c 4 c c () c c () is e csmic ime in Gyr 3 c () 8πG Subsiuing by () in () we ge c 3 4 s s s (3) p Z m r Te ensiy prmeers in e clse csmic mel ime re c c m m c (4) (5) r r c c (6) 3 c (7) 8πG n e ubble prmeer in e clse csmic mel ime is 3 4 s s m s r (8) using (5) (4) we ve c c subsiuing by () (7) in (9) we ge (9) Le S en Equin () cn be wrien s Similrly we cn ge s () S s m S m () Using (3) s s s () s (3) r S r Subsiuing by ()-(3) in (3) e prper isnce f e exrglcic bjec in e clse csmic mel e presen ime becmes c 3 4 S S S (4) p m r z n is luminsiy isnce ngulr imeer isnce n isnce mulus re respecively c z 3 4 L m r z S S S (5) c 3 4 m r z z S S S (6) c m 5 5lg S S S z 3 4 m r (7) z Cpyrig 3 SciRes

3 F BUKRI Te rizn isnce f e universe in e clse csmic mel e presen ime is given by Equin (8) I is bvius frm Tble (3) in [] e spce f e universe is fl e presen ime ence s expline in eils in [] e vlume f spere f rius is expresse s 8π 3 V (9) 3 Te l ensiy f e universe in e clse csmic mel ime is r m (3) c c Subsiuing by (4)-(6) in (3) we ge c m c r c c (3) m r e presen ime we ve c (3) ence e l mss n e mss f mer wiin in e clse csmic mel e presen ime re respecively expresse s V (33) m m (34) Te equivlen numbers f e ilky Wy-like glxies n e Cm-like clusers m re respecively wrien s m NW (35) N W m CO (36) CO W CO re e msses f e ilky Wy glxy n e Cm cluser f glxies respecively 3 Resuls n Discussin Resuls f e clse csmic mel wul be cmpre wi se f e generl csmic mel wic bes represens bservins [3] Te prper isnce resif relin (4) in e clse csmic mel n e crrespning relin in e generl csmic mel [] up z = 6 re ple in Figure I is fun b isribuins increse wi z n ey cincie n ec er fr z 98 bu ey grully iverge frm ec er ferwrs suc e isribuin f e generl mel becmes sligly upper e isribuin f e clse mel Tis inices fr z > 98 e prper isnce f exrglcic bjec in e clse csmic mel is sligly less n is vlue in e generl csmic mel is becuse e effec f slw rnsfer f rk energy ensiy in mer n riin ensiies wic becmes prminen fr z > 98 Te luminsiy isnce resif relin (5) in e clse csmic mel n e crrespning relins in e generl csmic mel n e = mel [] up z = re represene in Figure Te ree isribuins increse wi z n cincie n ec er up z = 535 wever e isribuin f = mel srs iverging frm e isribuins f e firs w mels fr z > 535 suc i becmes lwer em Furermre z = 644 e isribuin f e clse mel grully iverges frm f e generl mel n becmes sligly lwer i Tis is ls becuse e effec f minr rnsfer f e rk energy ensiy in mer n riin ensiies wic ges niceble fr z > 644 Te ngulr imeer isnce resif relin (6) in e clse csmic mel n e crrespning relin in e generl csmic mel n [] up z = 6 re illusre in Figure 3 B isribuins isply seep increse wi z up z = 9857 en e clse mel isribuin reces is mximum pc z = 683 wile e mx generl mel isribuin ges is mximum pc z = 644 s given in [] mx Te isribuins cincie n ec er up z = 6857 wever e isribuin f e clse mel grully eflecs frm f e generl mel fr z 6857 n remins lwer i gin e nly resn fr is resul is e slig rnsfer f rk energy ensiy in mer n riin ensiies wic becmes bvius fr z > 6857 Te isnce mulus-resif relin (7) in e clse csmic mel n e crrespning relin in e generl csmic mel [] up z = re swn in Figure 4 Te isribuins cincie n ec er wiu ny significn iversin s slig ifferences in e luminsiy isnces f e w mels given vlue f z will ispper in e lgrims f e isnce mulus relins Furermre in e clse csmic mel relins (8) (33)-(36) give e fllwing csmlgicl esimins respecively e presen ime: c 3 4 S S S (8) m r Cpyrig 3 SciRes

4 F BUKRI Figure Te prper isnce resif relin in e clse csmic mel n e generl csmic mel up z = 6 Figure Te luminsiy isnce resif relin in e clse csmic mel e generl csmic mel n e Λ = mel up z = Figure 3 Te ngulr imeer isnce resif relin in e clse csmic mel n e generl csmic mel up z = 6 Figure 4 Te isnce mulus resif relin in e clse csmic mel n e generl csmic mel up z = ) Te rizn isnce f e universe 444 Gpc ) Te l mss wiin is 3) Te mss f mer in wiin is 7388 m ) Number f e ilky Wy-like glxies equivlen e mss f mer wiin is N W ) Number f e Cm-like cluser f glxies equivlen e mss f mer wiin is N CO Errrs in ese esimins were bine by similr wy s in [] 4 Cnclusin In is pper isribuins f e prper isnce luminsiy isnce ngulr imeer isnce n isnce mulus were invesige in erms f e resif f n exrglcic bjec in e lig f e clse csmic mel wic ws cnsruce in previus pper I is fun ll f ese csmlgicl isnces increse cninuusly wi resif excep e ngulr imeer isnce wic seeply increses wi resif wrs mximum vlue pc z mx mx 683 en i ecreses grully wi incresing z I is ls fun e luminsiy isnce f e exrglcic bjec in e clse csmic mel bserve mel n = mel re pprximely similr up z = 535 wever e luminsiy isnce in e clse csmic mel pprximely grees wi is vlue in e bserve m- Cpyrig 3 SciRes

5 F BUKRI 3 el up z = 644 Evluins f e rizn isnce f e universe e l mss n e mss f mer wiin e rizn isnce e equivlen numbers f e ilky Wy-like glxies n e Cm-like clusers f glxies e mss f mer were clcule in e clse csmic mel e presen ime 5 cknwlegemens Tis pper ws fune by e Densip f Scienific Reserc (DSR) King bulziz Universiy Je Te ur erefre cknwleges wi nks DSR ec- nicl n finncil suppr REFERENCES [] F Bukri Csmlgicl Disnces in Five Generl Csmic els Inerninl Jurnl f srnmy n srpysics 3 [] F Bukri Clse el f e Universe Inerninl Jurnl f srnmy n srpysics 3 [3] F Bukri Five Generl Csmic els Jurnl f King bulziz Universiy: Science 3 Cpyrig 3 SciRes