Chapter 5C Cosmic Geometry: Part 2 Last Update: 12 August 2007

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1 Chaper 5C - Cosmic Geomery: Par Chaper 5C Cosmic Geomery: Par Las Updae: ugus 7. Inroducion We coninue here our ouline of he basic geomerical properies of cosmological spaceimes which we began in Chaper 5B. join Index o boh Chapers is as follows:- Newonian Cosmologies (Ch.5B ); The formulaion of general relaiviy (Ch.5B ); Cosmological models (Friedman-oberson-Walker space-imes) (Ch.5B 4); The surprising observaion ha Newonian cosmologies encompass he general relaivisic cosmologies provided ha pressure and he cosmological consan are ero (Ch.5B & 5); Precisely wha is mean by an expanding universe (Ch.5B 4 + Ch.5C); Precisely wha is mean by an infinie universe, and why an infinie universe would always have been infinie, even a he Big Bang (Ch.5B 4.); Wha is mean by he curvaure of space and how i is relaed o wheher he universe is finie or infinie (Ch.5B 4); Explanaion of why I keep referring o he sie scale of he universe, raher han simply he sie of he universe (Ch.5B 4); Wha is he Cosmological Consan and how does i affec he possible cosmological models (Ch.5B & 5); ow he relaionship beween he age of he universe and he ubble ime depends upon he cosmological model (Ch.5B 6); The curren consensus model (based on observaional daa) (Ch.5B 7); The cosmology of a universe filled wih a pressurised fluid how pressure causes graviy (Ch.5C); Demonsraion ha an expanding universe leads o a redshif, and ha his redshif obeys ubble s Law, a leas o firs order? or is i exac? (Ch.5C); The definiions of several differen lengh measures and derivaion of heir relaion o he redshif for differen cosmological models (Ch.5C); In paricular, he definiion of he lengh scale in ubble s Law v = which permis he velociy v = d/d o exceed c (Ch.5C); The sie of he observable universe and why his is greaer han c, bu depends upon he cosmological model (Ch.5C); Why he horion encompasses a decreasing amoun of maer as we look furher back in ime he orion Problem (Ch.5C); The opology of he universe how a non-rivial opology could solve he finie bu unbounded problem bu wih ero curvaure (Ch.5C).. edshifs and Disances To recap, he mos general meric saisfying he assumpions saed in Secion has he form, ds c d ()dl (.) where dl is a lengh elemen in a spaial sub-space. Since he facor of () carries he unis of lengh, dl is a sor of angular inerval, hence he subscrip. (For k > i Page of 7

2 Chaper 5C - Cosmic Geomery: Par is almos lierally an angular inerval, in he hypoheical 4D Euclidean embedding space). I can be wrien (amongs oher ways) as, dl d S( ) d sin d (.) where, S( ) is, sin or sinh according o wheher space is fla, posiively curved or negaively curved (i.e., k =, + or ).. The edshif We show ha radiaion emied and received by co-moving observers in a oberson-walker space-ime is subjec o a redshif (or a blue shif if he universe is conracing). Consider wo co-moving observers, and. By definiion hese observers are a res in (,, ) co-ordinaes, i.e. observer is a (,, ) a all imes and observer is a (,, ) a all imes. Thus, he angular separaion beween hem, L, is consan. Suppose observer emis a pulse of ligh owards a ime. Thanks o he global ime co-ordinae,, we do no have o be concerned abou he relaiviy of ime beween he wo observers. The assumpion of a global ime co-ordinae makes cosmological kinemaics easier han ha of special relaiviy, surprisingly. Suppose observer receives he pulse of ligh a ime. Imagine he pah of he ligh beam being broken ino a sequence of equal ime periods d. The ligh raverses an angular inerval dl in each period d, bu noe ha he sie of hese dl will vary wih ime. Because ligh ravels on he null cone, i.e. ds =, from (.) we have, ()dl cd (.) Thus, if he universe is expanding monoonically over he period in quesion, he angular inervals dl corresponding o fixed ime incremens, d, become smaller and smaller a laer imes. This is illusraed by, observer observer ( ) ( ) consan angular separaion of observers L Page of 7

3 Chaper 5C - Cosmic Geomery: Par The oal angular separaion of he observers is obained by summing over he unequal angular incremens, i.e. from (6.), cd L (.4) () If we consider anoher ligh pulse emied by observer a ime +, and received by observer a ime +, hen clearly, cd L (.5) () Noe ha because he angular separaion of he wo observers is consan, he LS of (6.4) and (6.5) are he same. Now we may imagine he wo consecuive pulses o be he peaks in a sinusoidal wave rain, i.e. he inerval is he periodic ime seen by observer, whils he inerval is he periodic ime seen by observer. Subracing (6.4) and (6.5) gives, cd () cd () (.6) Bu for ime inervals small compared wih he expansion rae of he universe, his gives simply, ( ) ( ) f f ( ) ( ) (.7) where f is he frequency seen by observer and f is he frequency seen by observer, and similarly is he wavelengh of he radiaion. Thus, in an expanding universe, he raio in (.7) is greaer han uniy and he radiaion is red-shifed. We see ha kledge of he universe s sie scale, (), a he wo imes suffices o deermine he red-shif. sronomers defined he red-shif as, ( ) ( ) (.8) In pracical applicaions, we are observer, deecing he radiaion, and is he presen ime ( ). If a cosmological model is assumed, giving he funcion (), an observaion of he red-shif allows he ime of emission o be found from (.8), ha is, ( ) observed (.9) ( ) emission Page of 7

4 Chaper 5C - Cosmic Geomery: Par I follows ha an observed red-shif can be ranslaed ino a ime of emission by inversion of he funcion (), i.e., emission (.9b) Explici examples of (.9b) will be discussed laer for specific cosmologies.. Disance Measures D and D l In Secion. we have alked abou he (consan) angular separaion of wo comoving observers, L. Bu wha is heir acual separaion? his poin we come hard up agains he fac ha here is no unique definiion of he disance beween wo co-moving observers. One obvious measure, since he speed of ligh is supposed o be consan, is he sum of he ligh pah incremens cd ha are required o send a signal from o. Thus, he ligh ransi ime disance, D l, is defined as, D l c (.) This is a very convenien definiion of a disance if we happen o k wha he ligh ransi ime is. owever, if our wo co-moving poins in space are defined by heir (consan),, co-ordinaes (i.e. we are given heir angular separaion, L ), hen i is no immediaely obvious wha he ligh ransi ime is. Since he disance apar of he wo poins a ime is ( ) L and his increases o ( ) L a ime, we have, ( (.) ) L c ( ) L bu wha is c exacly for a given L? Before answering his quesion, we urn o a differen definiion of disance. We have already appealed o i in he above discussion. any ime he insananeous disance beween our wo poins a a given angular spacing, L, is simply ( ) L. Thus if we choose o be he ime, we have he definiion of disance, i.e., D ( ) L (.) D is a very convenien measure of disance if we happen o k he angular disance beween he wo poins in quesion (and we also k he curren sie scale, ( ), of he universe presumably hrough some cosmological model). owever, if he wo poins have been defined by specifying he ligh ransi ime connecing hem, how do we find D in ha case? We see ha his is essenially he problem we had wih D l, bu in reverse. Page 4 of 7

5 Chaper 5C - Cosmic Geomery: Par In boh cases he answer is provided by (.4). This allows us o find he angular inerval L beween poins conneced by a null geodesic (ligh) over he ime inerval [, ]. ence he disance D can be found for a pair of poins conneced by ligh over a specified ime inerval. By invering (6.4) we can also find he ligh ransi ime inerval corresponding o a given angular separaion, L. This is bes illusraed by examples, below.. D and D l for Fla Space (k = ) and = and Maer Dominance s discussed in Chaper 5B Secion 5, his cosmological model has he soluion, () ~ () / (.) where ~ is given by Chaper 5B Equ.(.9), ~ 6 G r. owever, for sufficienly small imes, () behaves like (.) for all cosmological models for which = a a finie ime, assuming maer dominance. ence (.) is a generic behaviour for early imes assuming maer dominance. In (.), is some arbirary ime when he sie scale is. We can carry ou he inegral in (.4) explicily, o give, c L o (.4) We are usually ineresed in he case when he laer ime is ( = ). Choosing he arbirary ime o be also, we ge, c L (.5) Thus, (.5) gives he angular inerval L beween wo poins linked by a ligh ray beween imes and. ence, he disance is, from (.), D c (.5b) special case of (.5) is he maximum angular separaion beween wo poins which are causally linked (). This is he separaion corresponding o ligh ravelling since he Big Bang, obained by puing =, hence, Observable Universe: c L D L c (.6) This is raher a conradicion, since a sufficienly early imes he universe will be radiaion dominaed. Page 5 of 7

6 Chaper 5C - Cosmic Geomery: Par Thus, we have he resul ha, for hese cosmological models, he sie of he observable universe (in erms of he disance measure D ) is jus hree imes c, where is he curren age of he universe. Noe ha he facor of hree originaes from he dependence of () on a power of ime in Equ.(.). One may reasonably ask how he boundary of he observable universe can be a a disance of c given ha ligh ravels a speed c and has been doing so for he ime. The answer is ha i depends upon wha disance measure you choose o employ. I we choose he ligh ransi ime based disance measure, D l, hen by definiion he observable universe is of sie D l c. owever, i is ofen more useful o k he sie of he observable universe, ha is he sie measure D. The reason he laer is larger is ha, in addiion o he disance covered by he ligh, space has also expanded during he ime ha he ligh was in ransi. s a check we can derive he key equaion, (.4), in anoher way. Le he ligh sar a ime and be observed a ime, as above. Consider some inermediae ime. In he inerval d he ligh ravels cd. owever, beween ime and when he ligh is observed he universe will expand from ( ) o ( ). ence, he disance cd ravelled by he ligh a around will be sreched o a disance [( )/( )]cd by he ime of he observaion. To ge he oal disance we mus add all hese sreched conribuions, i.e., ( ) D cd (.7) ( ) Bu his is jus he same as (.4), by virue of he definiion of D, see (.). The Varying Causal orion (.6) shows ha he disance D o he boundary of he observable universe increases linearly wih ime. Bu, of course, he universe iself is expanding nonlinearly wih ime, as given by (.). Consequenly, he observable universe will encompass a changing amoun of maerial as he expansion proceeds. The bes way of seeing his is from (.6) which shows ha he angular disance o he boundary of he observable universe a any ime is, c c Causally Conneced egion: L () ~ ~c (.8) If he universe had been expanding linearly wih ime, hen his angular disance o he observable boundary would have been independen of ime. In his case, a given angular inerval would always have conained exacly he same maer. owever, for any cosmological model exhibiing a Big Bang he acual behaviour is as given by (.8) for sufficienly early imes (and for all imes in a fla universe wih =, assuming maer dominance). Thus, he observable universe is described by an angular inerval which increases wih ime, and hence encompasses an increasing amoun of maerial. Page 6 of 7

7 Chaper 5C - Cosmic Geomery: Par Of course, he observable universe is jus anoher way of alking abou he maximum exen of he region which is causally conneced o is cenral poin (us) a a given ime. Since he angular inerval (.8) shrinks o ero a he Big Bang i follows ha each paricle of maer is causally conneced o less and less surrounding maer as we approach he Big Bang (going backwards in ime). Is his anoher way of looking a reacion freee-ou by cosmic expansion rae? Check. Finally, we have sill no found explicily he D l disance for a given angular inerval, L. From (.5) and (.) wih = + we find, ~ L (.9) c ence, he disance D l will in general depend upon he absolue ime as well as upon he angular inerval, L, raversed. If = hen we ge, ~ D l c L (.) c which is jus he inverse of (.8). I is raher surprising ha he ligh ransi ime disance beween wo poins depends upon he cube of heir angular separaion. The elaionship of D and Dl o edshif I is convenien o use our scaling facor a() = ()/ in his secion. The red-shif can hus be wrien, using (.9), ( () ) a a (.) nd from (.) we have, a() which also yields he useful ime-red-shif relaion: and hence (.) (.b) Noe ha (.) and (.b) apply only for maer dominance and fla space, k =, wih =, or for sufficienly early imes if k or and non-ero. cually, (.) and (.b) are raher inaccurae. The reason is ha he erm in he numeraor canno really be inerpreed as he presen age of he universe. This sems from (.). This power law expression does no generally hold up unil he presen ime. more accurae ime-red-shif relaion is derived as (.) below. Noe ha (.b) shows ha signals from he Big Bang ( = ), i.e. poins on he horion, will have a divergen red-shif,. We can derive D in erms of direcly from (.5b), i.e., Page 7 of 7

8 Chaper 5C - Cosmic Geomery: Par D c c (.) Thus, for a fla cosmological model (or whenever () is believed o be a good approximaion) Equ.(.) can be used by asronomers o find he disance of an objec from is measured red-shif,. For example, objecs wih a red-shif of will be halfway o he horion (for his simple cosmological model wih k = and = ). general means of finding such relaions exiss which does no require explici inegraion for D analogous o (.5). This is obained by noing ha, cd cda D (.4) a() aa where he limis of he a inegral have been obained from (.). We do need o k a in erms of a. owever his is provided direcly by he Friedman equaion, Chaper 5B Equ.(5.), wihou he need for inegraion:- 8 kc c a G a a (.4b) In his case, k = =, and wih maer dominance (consan ) his becomes, a a (.5) so ha (.4) readily reproduces (.). In like manner we have, in general, cda D l cd (.6) a which, in his case, using (.5), gives, D l c (.7) consisen wih (.b)..4 Oher Disance Measures: eal-ngle Disance and Luminosiy Disance Page 8 of 7

9 Chaper 5C - Cosmic Geomery: Par <<<<< TO BE DDED >>>>>. Numerical Soluions for Curved Space (k ), Wih and Wihou, (P = ) In Chaper 5B we described he qualiaive feaures of he soluions o he Friedman equaion for a pressureless fluid. owever we did no aemp o find closed form soluions. In his Secion we shall find numerical soluions of he Friedmann equaion for arbirary k, arbirary cosmological consan, and arbirary mass-energy densiy. We shall coninue o assume maer dominance in his Secion. These numerical soluions yield, or he red-shif,, as funcions of ime. Working wihin hese numerical soluions i is lile exra work o evaluae he inegrals which yield he disance measures D l, D, ec. I is convenien o work in he red-shif, so we firs re-cas he Friedmann equaion in erms of. ecall ha he Friedmann equaion is [Chaper 5B, equ.()]:- 8 8 G The densiy parameer is c G kc (.) 8 G we can wrie is value as in (.) is hus, cri cri,. This parameer can vary wih ime, bu. Using he firs erm 8 G 8 G (.) The second erm in (.) can be wrien, using he definiion of he curvaure parameer given in Chaper 5B, kc (.) where he second from follows from he fac ha k is a consan. The las erm in (.) can be wrien in erms of he dimensionless dark-energy parameer as follows, c (.4) where he second form follows because is a consan. Noe ha (.) involves he sie scale,, whereas (.4) involves he ime dependen, and (.) involves boh. dding (.), (.) and (.4) and dividing by gives, This means ha we need only deal wih he firs order Friedmann equaion, as given in Chaper 5B Equ.(5.), since he second order equaion, Chaper 5B Equ.(5.), is implied by he fluid equaion which, for P = reduces o he maer dominance condiion ha is consan. Page 9 of 7

10 Chaper 5C - Cosmic Geomery: Par (.5) Noe ha whils, and are all ime dependen, he quaniies occurring in Equ.(.5) are heir values, and hence are consans. The same is rue of. From he definiion of he red-shif, (.), we can replace / in (.5) by +, giving, (.6) Moreover, by differeniaing (.) we find ha, and (.7) So ha (.6) becomes, (.8) Noe ha, when he universe is expanding, we will need o ake he negaive squareroo of (.8) o find. Thus, we find a firs order differenial equaion for he redshif, d d 5 4 (.9) The hree erms, arising respecively from mass-energy, curvaure, and he cosmological consan, are disinguished in (.9) by he differen orders of he corresponding, or, dependence. Before looking a numerical soluions of (.9) we derive a commonly used relaion beween ime and red-shif. If we confine aenion o sufficienly early imes, i.e. o sufficienly large red-shis, hen he firs erm in (.9) is dominan giving, d d 5 (.) We inegrae (.) from ime o, when he red-shif drops from gives, o, which Bu no so early ha he assumpion of maer dominance (ero pressure) breaks down. Page of 7

11 Chaper 5C - Cosmic Geomery: Par (.) Noe ha an inaccuracy has crep ino (.) since, a he earlies imes he assumpion of maer dominance (ero pressure) upon which (.9) is based, will no longer be correc. Because we inegrae from = o derive (.), his will affec he ime-redshif relaionship. More imporanly, (.) only applies for large red-shifs ( >> ), because we have negleced he second and hird erms in (.9). Consequenly i is no valid o pu = in (..) and falsely conclude ha. The correc derivaion of can be found in Chaper 5B Secion 6. I depends also upon, and possibly also upon if his is non-ero. Numerical Soluions for and / au E-9.E-9.E-9.E-9.E-9 8.7E- 5.9E- 4.69E-8 4.8E-8.96E-8.96E-8.96E-8.97E-8.E-8.49E-6.6E-6.6E-6.6E-6.6E-6 9.4E E E-6.85E-6.56E-6.56E-6.56E-6.66E-6.88E-6 4.7E-5 4.E-5.98E-5.98E-5.98E-5.98E-5.E-5 5.E-4.E-4.E-4.E-4.E-4 8.4E E-5.47E-.4E-.4E-.4E-.4E- 9.9E E E-.74E-.46E-.46E-.46E-.59E-.8E-.55E-.4E-.E-.E-.E- 9.8E- 6.9E- 5.E-.E-.97E-.97E-.97E-.47E-.4E- 4.8E-.7E-.45E-.46E-.45E-.58E-.8E E- 4.E-.98E-.99E-.97E-.98E-.E E- 5.4E- 4.66E- 4.67E- 4.65E-.49E-.47E E- 6.E- 5.56E- 5.58E- 5.55E- 4.7E-.95E- 6 8.E- 7.4E- 6.79E- 6.8E- 6.77E- 5.9E-.6E- 5.E- 9.4E- 8.56E- 8.59E- 8.5E- 6.4E- 4.54E- 4.E-.E-.E-.E-.E- 8.4E- 5.96E-.84E-.69E-.56E-.57E-.56E-.8E- 8.E-.8E-.57E-.9E-.4E-.7E-.8E-.8E-.5.6E-.4E-.E-.E-.8E-.6E-.69E- 4.9E- 4.54E- 4.4E- 4.8E- 4.E-.7E-.6E E- 5.E- 4.89E- 4.94E- 4.85E-.8E-.76E E- 6.7E- 5.7E- 5.76E- 5.66E- 4.49E-.9E E- 7.E- 6.75E- 6.8E- 6.69E- 5.9E- 4.E-. 8.6E- 7.77E- 7.7E- 7.4E- 7.E- 5.96E- 4.5E E- 8.49E- 8.8E- 8.4E- 8.E- 6.6E- 5.7E-. 9.8E- 9.E- 8.89E- 8.96E- 8.8E- 7.9E- 5.78E-.8E+.E+ 9.8E- 9.89E- 9.76E- 8.E- 6.67E- Page of 7

12 Chaper 5C - Cosmic Geomery: Par The numerical soluions presened above are valid all he way up o he presen ime (=), since proper accoun is aken of all hree erms in (.9). owever, hey sill suffer from he inaccuracy caused by neglecing pressure in he early universe..e+.e- ed-shif, +.E-.E- Time in ubble Unis,.E-4.E-5.E-6.E-7.,.8.4,.76.8,.7.8,.74.8,.7.5,.5,.E-8.E-9.E- Numerical Inegraion of Pressureless FW Friedmann Equaion: Time for a Given ed-shif, ed-shif, +.E Numerical Inegraion of Pressureless FW Friedmann Equaion: Time for a Given ed- Shif, Time in ubble Unis,.E-.,.8.4,.76.8,.7.8,.74.8,.7.5,.5,.E- The key o he above graphs gives,, and follows from. The bes curren values for he cosmological parameers are, roughly, consisen wih ero and probably lying ~.4 o.8; ~.7 o.76; beween -. and +.. The cases which consider non-er values for, namely +/-., are indisinguishable in he above graph. The difference beween =.4 and.8 is discernable bu small. The graph for = was generaed by he Page of 7

13 Chaper 5C - Cosmic Geomery: Par inegraion procedure and was found o be idenical o Equ.(.). The value of () follows, of course, from () = / ( + ). For =.8, =.7 we find,.e+ 9.E- versus for Omerga=.8, Lambda=.7 8.E- 7.E- 6.E- / 5.E- 4.E-.E-.8,.7 pproximaion: Equ.(.b) pproximaion: Equ.(.).E-.E- ime,.e Noe ha he acceleraing expansion rae of he universe due o he posiive value for is easily discernable in he above graph for imes beyond abou./, where he numerical inegraion diverges from he approximae resul of Equ.(.). The approximaion of Equ.(./.b) is bound o be righ a he presen ime, bu is in error a all earlier imes. Numerical soluions for he disances D l and D follow Page of 7

14 Chaper 5C - Cosmic Geomery: Par Numerical Soluions for he Disances D l and D ( =.8; =.76) D l D / au numerical Equ.(.7) Numerical Equ.(.).E E E E E E E E Page 4 of 7

15 Chaper 5C - Cosmic Geomery: Par 4..5 Inegraed Ligh Transi Time Disance and Disance "Now" Compared for Omega =.8, Lambda =.7 lso Comparison wih Equs.(.4) and (.7). Disance (unis of c/).5..5 Dl (inegraed) Dl (Equ..7) D (inegraed) D (Equ..)..5 ed-shif,....5 Inegraed Ligh Transi Time Disance and Disance "Now" Compared for Omega =.8, Lambda =.7 lso Comparison wih Equs.(.4) and (.7). Disance (unis of c/).5. Dl (inegraed) Dl (Equ..7) D (inegraed) D (Equ..).5 ed-shif, The above graphs show he disance measures D l and D for a source which has he red-shif,, ploed as he abscissa. Noe ha for he case = ; = i was confirmed ha he numerical inegraion exacly reproduced he analyic formulae given in Equs.(.4,.7). Page 5 of 7

16 Chaper 5C - Cosmic Geomery: Par Disance, unis of c/ Inegraed Ligh Transi Time Disance and Disance Now Compared for Omega =.8 and Lambda =.7: Ploed gains Time When Signal Was Emied Dl (inegraed) Dl (Equ..7) D (inegraed) D (Equ..) Time, in unis of /.5 Inegraed Ligh Transi Time Disance and Disance Now Compared for Omega =.8 and Lambda =.7: Ploed gains Time When Signal Was Emied..5 Disance, unis of c/..5 Dl (inegraed) Dl (Equ..7) D (inegraed) D (Equ..) Time, in unis of / The above graphs give he disance measures of a source which emis he received signal a cosmic ime. Thus, when / is small, he ligh-ransi-ime disance is necessarily close o c. owever, he physical disance o he horion is given by he disance measure D for = (or equivalenly for a divergen red-shif). ence, he horion is a a disance of.4c in our universe (assuming =.8 and =.76). Noe ha, wih hese parameers, =.98 so ha he horion is a a physical disance D =.4c =.46c. Page 6 of 7

17 Chaper 5C - Cosmic Geomery: Par 4. ubble s Law ubble s Law is formulaed in erms of he disance measure D. The velociy in quesion is he rae of change of his quaniy, i.e. how rapidly he wo co-moving poins (galaxy clusers) appear o be moving apar. We have herefore, D ( ) L and V ( ) L (6.?) d ence, he angular separaion cancels beween he disance and velociy measures, giving, dd ( ) V ()D where () (6.?) ( ) The essence of he ubble relaion is ha he parameer does no depend upon spaial posiion, only upon ime. The origin of his is clear from he cosmological geomery provided he righ disance and velociy measures are used! Noe ha hese correc measures are such ha he velociy may exceed c (and is c a he boundary of he observable universe, for fla space-ime a leas). Page 7 of 7

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