MODEL-INDEPENDENT PLOTTING OF THE COSMOLOGICAL SCALE FACTOR AS A FUNCTION OF LOOKBACK TIME

Transcription

1 MODE-INDEPENDENT POTTING OF THE COSMOOGICA SCAE FACTOR AS A FUNCTION OF OOKBACK TIME H. I. Ringeracher* and. R. Mead* Dep. of Physics and Asronoy, U. of Souhern Mississippi, Haiesburg, MS 3946, USA ABSTRACT In he presen work we describe a odel-independen ehod of developing a plo of scale facor a () versus lookback ie fro he usual Hubble diagra of odulus daa agains redshif. This is he firs plo of his ype. We follow he odel-independen ehodology of Daly and Djorgovski (4) used for heir radio-galaxy daa. Once he a () daa plo is copleed, any odel can be applied and will display accordingly as described in sandard lieraure. We hen copile an exensive daa se o z = 1.8 by cobining SNe Ia daa fro SNS3 of Conley e al. (11), High-z SNe daa of Riess e al. (4) and radio-galaxy daa of Daly & Djorgovski (4) o be used o validae he new plo. We firs display hese daa on a sandard Hubble diagra o confir he bes fi for CDM cosology and hus validae he joined daa se. The scale facor plo is hen developed fro he daa and he CDM odel is again displayed fro a leas-squares fi. The fi paraeers are in agreeen wih he Hubble diagra fi confiring he validiy of he new plo. Of special ineres is he ransiion-ie of he universe which in he scale facor plo will appear as an inflecion poin in he daa se. Noise is ore visible on his presenaion which is paricularly sensiive o inflecion poins of any odel displayed on he plo unlike on a odulus-z diagra where here are no inflecion poins and he ransiion-z is no a all obvious by inspecion. We obain a lower lii of z.6. I is eviden fro his presenaion ha here is a dearh of SNe daa in he range, z = 1, exacly he range necessary o confir a CDM ransiion-z in he neighborhood of z =.76. We hen copare a Toy Model wherein dark aer is represened as a perfec fluid wih equaion of sae p = (1/ 3) ρ o deonsrae he plo sensiiviy o odel choice. Is densiy varies as 1/ and i eners he Friedann equaions as 3 Ω dark / replacing only he Ωdark / a er. The Toy Model is a close ach o CDM bu separaes fro i on he scale facor plo for siilar CDM densiy paraeers. I is described in an appendix. A ore coplee ransiion ie analysis will be presened in a laer paper. Key words: cosology-dark aer; cosology-disance scale; cosology-heory; * E-ail: ringerha@gail.co and awrence.ead@us.edu 1

2 1. INTRODUCTION Tradiionally, he Hubble diagra plos odulus agains redshif, boh of which are observaional easureens. SNe Ia daa are always seen his way. The chosen cosological odel is hen fied and secondary quaniies such as he deceleraion paraeer, he ransiion redshif, he age of he universe, ec. are exraced by operaions on he fied paraeers. The ransiion redshif of he universe is he redshif value a which he universe ransiions fro deceleraing o acceleraing. The fac ha he universe is acceleraing a all was discovered in 1998 and a Nobel Prize was awarded (Perluer, Schid & Riess 11). The ransiion-z is hus a criical poin ha is highly odel-dependen. Indeed, soe auhors (ia e al. 1) have even suggesed ha he ransiion-z be regarded as a new cosological nuber. The locaion of his poin is no obvious in a sandard Hubble diagra because he disance odulus akes no noiceable changes a ha locaion. In order o obain he ransiion redshif one us evaluae he deceleraion paraeer a he poin where i vanishes. Thus one us ake second derivaives of noisy daa - generally no desirable. Daly and Djorgovski (3) coen on his as a cardinal sin for any epirical scienis, bu auhors do i anyway. An alernaive approach is o uilize he Hubble diagra daa o creae a plo of he scale facor, a ( ), versus lookback ie,. This plo displays he inflecion poin a he ransiion ie visually, unlike a odulus plo where he locaion of his poin is uninuiive. Only one derivaive need be aken on he scale facor plo o locae his poin, hus reducing noise and periing higher sensiiviy o odel discriinaion. Scale facor plos are seen in every cosology exbook bu appear o be underuilized in he lieraure. The reason is apparenly ha i is assued ha a cosological odel us firs be seleced in order o calculae lookback ie. In fac, ha is no necessary. Daly and Djorgovski (3, 4) have developed a odel-independen approach o calculae iporan cosological paraeers, for exaple he expansion paraeer, Ez ( ) and he deceleraion paraeer, qz. ( ) They derive forulas for hese, based upon esiaes of he diensionless coordinae disances of galaxies. We ake his work a sep furher by analyzing lookback ie siilarly. The priary purpose of his paper is o describe and deonsrae a odel-independen approach o develop a scale facor-lookback ie plo. This paper is organized as follows. We firs presen he heory for his approach deonsraing why a odel is no needed, allowing one o plo epirical daa. A red shif daa se is hen seleced for he scale facor plo. In fac we cobine SNS3, 11 SNe Ia daa of Conley, e al. (11) wih he 4 Radio Galaxy daa of Daly and Djorgovski (4) and soe High-z SNe Ia daa of Riess, e al. (4) o provide a baseline o z = 1.8. This daa se is firs validaed on a sandard Hubble diagra of odulus agains red shif by displaying a leas-squares fi of CDM. The sae daa se is hen convered for he scale facor versus lookback ie plo. This process is described in deail. We hen display he sae CDM odel fro a leas-squares fi o he convered daa o validae he ( ) a vs. approach. The wo leas-squares CDM fis o he wo ypes of plos us resul in he sae fiing densiy paraeers in order o insill confidence in he odel-independen approach and

3 hey do. Finally, a Toy Model for dark aer is inroduced and displayed on he sae scale facor plo o deonsrae is sensiiviy o odel differeniaion. The Toy Model is described in Appendix A. We will leave a ( ) daa analysis o a laer paper.. THEORY We begin by wriing he FRW eric for he CDM odel: dr ds = d a() + r dθ + r sin θdφ (1) 1 kr We choose a fla 3-space fro curren easureens and se k =. We noe ha r, θ, φ are frozen or cooving coordinaes. However, hey define a posiion for each galaxy observaion iagined o span fro he presen o disan pas hus represening a faily of red shifs and coordinae disances wih an iplici ie dependence. A foral discussion of his poin is presened below. where ookback ie is radiionally calculaed fro he following inegral: z dz ' = H, () (1 + z') E( z') 3 Ez ( ) (1 z) k(1 z) = Ω + +Ω + +Ω (3) is he Hubble paraeer for CDM and he densiy paraeers are Ω for dark plus baryonic aer, Ω k he curvaure paraeer and Ω, he dark energy densiy paraeer. In his paper we se Ω k = for a fla universe. H is he presen Hubble ie, 1/ H. e us exaine his forula in deail. The scale facor is defined by a () = 1/(1 + z). Also we us have, by definiion, a () Ez ( ) = (4) a () where he overdo is he derivaive wih respec o ligh ravel ie (coordinae ie),. Clearly, associaed wih every observed red shif here us be a ligh ravel ie fro ha source. Bu fro he above definiions alone i is clear ha he inegral () is siply z d ' = 1 =τ, (5) z H where z is he ligh ravel ie fro he source a red shif z andτ is he diensionless lookback ie. Here we have noralized ie wih respec o he Hubble ie so ha he presen ie is = 1. Fro he eric, Eq.(1), he ligh ravel inerval along a fixed lineof-sigh is: d = a() dr (6) This ie inerval is inerpreed as he ligh ravel ie inerval beween wo spaially consecuive SNe sighings of a faily of observaions. The space beween he wo observaions expands such ha he su over all observaions of z is he ligh ravel ie, 3

4 z, fro he os disan source o he nearby one a a ( ) = a(1) = 1. One us also be cerain ha he inrinsic condiion, a (1) = 1, is also saisfied for a proper plo. I reains o describe he coordinae disance, r, in ers of ie. We shall be working wih several disance easures. Modulus, μ, is a easure of luinosiy disance, D (Mpc) and is defined fro: μ = M = 5ogD + 5 (7) The luinosiy disance is defined fro he cooving disance which is our eric coordinae disance, r : D = r/ a( ) (8) Thus, dr = d [ a( ) D ] Wih disances noralized o he Hubble lengh, and ie o Hubble ie, our coordinae disance r () is he sae as he diensionless coordinae disance, yz, ( ) of Daly and Djorgovski and we ay wrie, adoping heir noaion; D dy = d [ a( ) ], (9) DH where DH = ch. We shall keep Eq. (9) in differenial for because boh a () and D vary wih each SN easureen and we will analyze our daa his way, consisen wih Eq. (6). Finally, fro Eqs. (5) - (9), we can wrie for he epirical diensionless lookback ie, τ ; 1 z τ = a ( ) dy (1) ogeher wih a ( ) = 1/ (1 + z), hus relaing our plo o direc easureens of red shif and luinosiy disance, a nuerical procedure which will becoe clear in he a ( ) plo secion. We nex proceed o selec a daa se. 3. DATA SEECTION AND VAIDATION In selecing daa o validae our plos we desire as high a red shif range as possible. Since we base our approach on he work of Daly and Djorgovski, naurally we are srongly influenced by heir work on radio galaxies as sandard candles (Daly and Djorgovski 1994) We choose o cobine 18 of heir RGs (excluding 3C45 and 3C47.1) ou o z = 1.8 in Daly & Djorgovski (4) wih ore recen 11 SNS3 SNe Ia daa of Conley e al. (11) which alone goes o z = 1.4. To help fill he sparse region beween z = 1 and z = we add High-z SNe Ia daa of Riess, e al. (4) for z 1 also in Union.1 (Kowalski e al. 8). Conley, e al.do no provide a deerinaion of H o peri scaling heir daa wih regard o he esiaed absolue agniude of a Type Ia SN. Daly and Djorgovski do scale heir 4 daa by cobining i wih he SNe daa of Riess, e al. (4). They use heir own esiaed Hubble consan of 66.4 k/s/mpc fro he Riess, e al. daa. They obain his value by exaining he low-z ( z <.1) linear Hubble diagra of Riess, e al. Thus heir RGs are well-scaled o he Riess, e al. SNe. 4

5 We herefore choose o scale he Conley, e al. (11) daa wih respec o he daa of Daly and Djorgovski (4) and he daa of Riess, e al. (4). The Conley, e al. daa are considered very accurae wih uliple correcions described in heir work. We applied heir correcions o obain he correced agniude corr. We hen copare he Conley, e al. and Riess e al. daa o evaluae he supernova absolue agniude noing ha Daly and Djorgovski have successfully scaled o Riess, e al. We selec 8 SNe in coon o he wo daa ses (Table 1). We firs esiae our own Hubble consan fro he low-z Conley, e al. daa. We included daa ou o z =.1 (133 poins) for a low-z ye sufficienly large se for a high confidence Hubble fi. The presen Hubble consan is found fro he coordinae disance, r, given he luinosiy disance, D, and red shif velociy, v, using he relaion: v cz cz(1 + z) H = = = (11) r ad D We find H = 69. k/s/mpc for he Conley, e al. se. This is only for our purposes in scaling calculaions and is no ean o fix heir scale. Conley (1) poined ou ha care us be aken when aking such esiaes due o he exree sensiiviy of he daa o he choice of he SN Ia agniude, M. We laer rescale he Conley, e al. daa by noralizing he coordinae disances o H = 66.4 for consisency. We hen copared he Conley, e al. and he Riess, e al. daa for he 8 SNe seleced and esiaed he leas squares SN Ia absolue agniude, M, necessary o give he wo ses idenical oduli for hose poins. Averaging he 8 values, we find M = ±.13 in agreeen wih he Riess, e al. esiae of M = This absolue agniude is hen subraced fro corr o obain he odulus for he cobined daa. A sligh correcion o M = 19.4, wihin our error, was ade o bes fi our cobined SNe o high z. The joined hree ses are shown in Fig. (1). Also shown in Fig. (1) is he radiional fi of CDM wih leas squares densiy paraeers,.78 Ω = and Ω =.7, essenially he WMAP values, hus confiring he qualiy of he daa se. The cobined Modulus-z daa se is available (CobinedDa13). Table 1: 8 SNe in coon wih he Riess, e al. and Conley, e al. daa source Riess μ, odulus Conley corr M sn1999cc sn1999gp snca sncf sncη sn1ba sn1cn sn1cz

6 Fig. 1. Joined SNe Ia daa ses of Conley, e al., Daly and Djorgovski and Riess, e al. wih leas squares fi of CDM. Modulus is fro Eq. (7). 4. POTTING THE SCAE FACTOR AGAINST OOKBACK TIME We firs calculae he lookback ie. We will follow Eq. (7-1) very closely and will presen a able showing a saple calculaion. We assue we have he redshif, z, and he luinosiy disance in Mpc, D. D is calculaed fro Eq. (7), given ypical odulus daa, μ. Table shows a series of easureens sored by ascending z in colun 1. Shown are a se saring wih he lowes z values followed by a gap juping o around z = 1 in order o show he changes in he running su over colun 5 o ge he lookback ie in colun 6. The labeled coluns are calculaed as follows: Colun 1: z given Colun : a= 1/ (1 + z) Colun 3: D in Mpc fro Eq.(7) given odulus μ Colun 4: D Y = a, DH D H = c/ 66.4 = Mpc ( H = 66.4 k/s/mpc) Colun 5: a delayi = ai ( Yi Yi 1) j Colun 6: ookbacktj = 1 ai ( Yi Yi 1) i = 1 Colun 7: ookbacktcorr = ookbackt.161 j j 6

7 Table : Saple calculaion of lookback ie z a D, Mpc Y a*delay ookbackt ookbacktcorr * * * * * * * There are several poins o noe in order o properly calculae lookback ie. Colun 5 clearly shows he presence of noise. This is effecively soohed by he inegraion in colun 6 bu lookback ie neverheless carries he noise. More iporanly, wo crieria us be saisfied: [A]; a (1) = 1 and [B]; a (1) = 1. An inspecion of he able a row shows ha a ookbackt. Tha is because his a is no he one for he presen ie, bu raher for he neares easured z. So here is an apparen ie gap of.161. This is subraced fro ookbackt o generae ookbacktcorr in colun 7. In effec his gap is an aoun Δa () z by virue of he definiion of a () and is considered an inegraion consan. Condiion [A] is hen saisfied and he daa is cenered on he presen ie. ookbacktcorr is used in he final plo bu will be referred o as lookback ie. The slope a ie 1 is deerined by adjusen of he Hubble consan. In he presen case he slope is approxiaely.98 and hus saisfies [B]. For he final plo, Table is sored again by he correced lookback ie. Thus he rando noise presen in he lookback ie is ransfored o scaer in a (). 7

8 Fig.. Plo of scale facor agains lookback ie for he cobined daa se. The blue curve is he leassquares fi for CDM. The red curve is he Toy Model for Planck densiy paraeers. Figure shows he cobined daa se ploed as scale facor agains correced lookback ie. Also shown on he plo is he leas-squares fi of CDM wih resuling densiy paraeers: Ω =.735and Ω =.65. These values are exreely close o hose fro he Modulus plo, Fig.1, hus supporing he validiy of he lookback ie calculaion. The R-squared goodness of fi for CDM in Fig. is.98. Also shown on he plo is a Toy Model (Appendix A) wherein dark aer is represened as a perfec fluid wih equaion of sae p = 1/3 ρ. Is densiy varies as consan / and i eners he 3 Friedann equaions as Ω dark / replacing only he Ωdark / a er in CDM. Oherwise i is calculaed in exacly he sae way as he CDM scale facor. Wih his replaceen, a new soluion for he Toy a ( ) can be found. The Toy odel is no a leas-squares fi in order o deonsrae he separaion on he plo. The Planck paraeers were used for he Toy Model: Ω =.68, Ω =.5 and Ω dark =.7. The Toy Model is a close ach o CDM. A Toy leas-squares fi would be indisinguishable fro CDM and he fi paraeers would be: Ω =.61, Ω =.5 and Ω dark =.34. Boh curves lie well wihin he daa scaer for heir curren paraeers. 8

9 5. TRANSITION-TIME TO AN ACCEERATING UNIVERSE A full analysis of he a ( ) daa will be presened in a laer paper. However, a siple inspecion of he daa suggess he a ( ) inflecion poin, or ransiion-ie, lies conservaively a.6 ( z.57 ). A laer ies he slope is acceleraing. A siple quadraic leas-squares daa fi aches he wo closely spaced odel curvaures. A earlier ies he inflecion region is very broad and he daa us evenually urn over oward he origin. The CDM ransiion-ie for Ω =.735 and Ω =.65 is expeced a.514corresponding o z =.77. Riess, e al. have saed a value of +.7 z =.46 ±.13 (4) and z = (7). Daly and Djorgovski have +.8 independenly found z.45 (4) and, wih an expanded daa se, z =.78.7 (8). ia, e al. (8) also checked he Riess, e al. (4) daa and confired heir esiae wihin error. Cunha and ia (8) exained Asier, e al. (6) SNS daa and found z =.61. In he sae paper hey also exained he daa of Davis, e al. (7) and found z =.6. They separaely exained Union daa (Kowalski, e al. 8) and +.14 found z = Transiion ies end o be clusered around z =.45 and z = The Daly and Djorgovski (8) value, z =.78.7, agrees wih CDM bu has exreely wide error bars. The wide variaion in ransiion ies would indicae a proble, or as ia, e al. (8) have pu i, his could be seen o raise soe ild flags wih he sandard CDM odel. Clearly he daa is noisy and siply insufficien o deerine his nuber precisely a he presen ie. More daa in he range 1< z < would be helpful. 6. CONCUSIONS We describe a novel odel-independen approach o plo he cosological scale facor agains lookback ie. This is a new way of ploing epirical sandard candle daa as opposed o he usual Hubble diagra. We seleced and joined wo SNe daa ses ogeher wih Radio-Galaxy observaions o creae a sandard candle baseline o z = 1.8 o be uilized in validaing he new plo. The daa was firs ploed in he usual for of odulus agains red shif and he CDM odel was seen o presen a classic fi hrough he daa, hus validaing he joined daa se. The a () plo was hen consruced and he sae CDM odel was found o fi well, hus validaing he new plo. A Toy Model was also consruced and superposed on he scale facor plo using Planck paraeers o copare agains CDM. The ach was surprisingly good well wihin he plo scaer bu he new plo successfully discriinaed he suble difference. I is clear fro inspecion of he a () plo ha here is a dearh of daa beween z = 1and z = hus resuling in a wide range of esiaes of he ransiion-z and apparenly spanning he enire range of z =.45 o z =.78 biased in general oward he lower values. This ay siply be noisy daa or i igh sugges ension wih he CDM odel. 9

10 APPENDIX A TOY MODE FOR DARK MATTER E. Kolb, in 1989, describes a coasing universe wih a doinan for of aer ha he refers o as K-aer (Kolb, 1989). K-aer derives fro he Friedann equaion for he FRW eric, Eq.(1), for a aer densiy ha varies as 1/ a ( ). Any densiy of his for, in effec, eners he Friedann equaion as a siple consan curvaure conribuion, k hence his nae, K-aer. Kolb found ha he equaion of sae of his fluid is p = 1/ 3 ρ wih he resul ha he scale facor acceleraion vanishes since we have: a () 4π G = ( ρ + 3p) = (A1) a () 3 He goes on o describe various universes dependen upon he curvaure and properies of hose universes such as effecs on red shif, ec. The Concordance odel did no exis. Today, a coasing universe odel based on K-aer has been rejeced since he curvaure of he universe has been easured as fla ( k = ). However, we now ake a soewha differen view. We know ha he FRW spaial curvaure vanishes wih high confidence. K-aer in he for described by Kolb would add nohing o his scenario. However, we igh consider a for of K-aer as a new ype of ie-dependen aer, replacing dark aer. This is essenially coasing dark aer. Kolb does no discuss his direc consequence. The appropriae aer densiy is 3 insered ino he Friedann equaion for eric (1) as Ω dark / replacing Ωdark / a in CDM. The baryonic aer and dark energy are lef inac. For his new densiy, we solve he Friedann equaion nuerically, valid for all ies. This new soluion reains consisen wih he consrain k = ; Ω k = over all ies as found observaionally. We call his alernaive odel our Toy Model. A leas squares fi of our Toy Model akes i indisinguishable fro CDM wihin he widh of he plo line using densiy paraeers; Ω =.61, Ω b =.5 and Ω dark =.34. Figure siply displays he Toy Model as a conrasing odel for a choice of Planck densiy paraeers. Furher pursui of his odel is ouside he scope of our paper. REFERENCES Asier, P. e al. 6, A&A 447, 31 (asro-ph/51447) Conley e al. 11, ApJS, 19,1 and Guy e al.1, A&A, 53, 7 Conley, A.J. 1 (e-ail discussions) Cunha, J.V. and ia, J.A.S. 8, MNRAS 39, 1 (arxiv:85.161v [asro-ph]) Cunha, J.V. 8, Phys. Rev. D 79, 4731 (arxiv: v1 [asro-ph]) Davis e al. 7, ApJ, 666, 716 Daly, R.A. 1994, ApJ, 46, 38 1

11 Daly, R.A. and Djorgovski, S.G. 3, ApJ, 597, 9 Daly, R.A. and Djorgovski, S.G. 4, ApJ, 61, 65 Daly, R.A. and Djorgovski, S.G. 8, ApJ, 677, 1 Kolb, E.W. 1989, ApJ, 344, 543 Kowalski, M. e al(the Supernova Cosology Projec), ApJ.,8 ia, J.A.S. e al. 8, arxiv: v [asro-ph.co] Perluer, Saul, Brian P. Schid, Ada G. Riess, Nobel Prize in Physics, 11 Riess, A.G. e al. 4, ApJ, 67, 665 Riess, A.G. e al. 7, ApJ, 659, 98 CobinedDa13 is available a 11

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13 Scale facor vs ookback ie Conley e al Daly e al Riess e al CDM Model Toy Model Planck daa a()