Extragalactic Astronomy ASTR 5630
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Date Created: 09/21/15
Whittle EXTRAGALACTIC ASTRONOMY Homel Mainl IndexI Toolbox PDFs I Preliminaries Dynamics Star Formation 16 Cosmology Morphology Ellipticals Interactions 17 Structure Growth Surveys Dynamics ll Groups amp Clusters 18 Galaxy Formation Lum Functions Gas amp Dust Nuclei amp BHs 19 Reionization amp IGM Spirals 1O Populations AGNs amp Quasars 20 Dark Matter 4 LUMINOSITY FUNCTIONS Indexl Questionsl Imagesl Referencesl Printl Nextl Top I 1 Introduction Galaxies come in a huge range of luminosity and mass 1O6 MB 75 to 225 This is nicely illustrated by a comparison of M32 amp M87 132 H37 a as seen in the sky and 39 b at the same physical distance 39 f Egilm 11 Two elliptkal galam Nahum appawm aim and hmhtma Em Images am mmmmmmma h 1 k Ian 1 10 x10 ammlln Smlmq Palomar digital sky sun39s mrhwmkna rJnan ital n m in mi n nu inn mum munit The Luminosity Function describes how the relative number of galaxies varies with their luminosity The Luminosity function contains information about o primordial density fluctuations o processes that destroycreate galaxies o processes that change one type of galaxy into another eg mergers stripping o processes that transform mass into light Although this information is badly convolved nevertheless 0 Observed LFs are fundamental observational quantities 0 Successful theories of galaxy formationevolution must reproduce them 2 Brief History 1930 Hubble notes that apparent magnitude correlates tightly with redshift fainter galaxies have higher 2 He concludes galaxies have a narrow Gaussian absolute magnitude distribution ltM Bgt 18 0 O9mag 1942 Zwicky realizes that the Local Group contains many low luminosity galaxies He argues for a rising function for low luminosities As we shall see this disagreement foreshadows two important facts o corrections for sample bias are essential 0 there may be two types of LF one for quotnormalquot galaxies and one for quotdwarfsquot mxtl m Topl 3 The Schechter Function In 1974 Press and Schechter calculated the mass distribution of clumps emerging from the young universe and in 1976 Paul Schechter applied this function to fit the luminosity distribution of galaxies in Abell clusters image The fit turned out to be excellent though the reasons why are still not well understood see sec 7 L 1 L L 2 71 ZIP d 41 Be careful which version of the function is used L per dL which is usually plotted Log 1 vs Log L I M per dM where M is Absolute Magnitude so this is effectively dlogL Plots may sometimes be of cumulative numbers N gt L or N lt M Compare here the Luminosity and Magnitude versions Observationally it is important to specify 0 whether the LP is for specific Hubble Types or integrated over all Types 0 whether the LF if for Field galaxies or Cluster galaxies or whatever the environment is o the value of HO since I varies as h3 while L or M vary as h392 where h HO1OO kmsMpc Mathematically note o The function has two parts o a power law 139 0 3 La dominates at low luminosities LltltL index or 1 so the LF rises as L decreases ie fainter galaxies are more common we use the terms quotsteepquot for or 15 and quotflatquot for or O5 b an exponefmal cutoff GINX e L dofmffates at nfgn lumlnosmes L gt Lr le very lufmffous galaxfes are also very rare Tnere are three parameters a nr normallzatlon can be a numbef a number per umt volume or a probabmty nr o 02 n3 Mpc39a for tne total galaxy average a or steepness of fant end Q39rVO e to 4 3 b Lr lumlnoslty at break between two fegloffs Lr 10 ULBQ n or Mar New 9 7 5Logn lfftegfatfoff over number gwes m NgtL L L dL m I a1rLL 2 n r T For L approachlffg zefo NM mm 1 wmcn ls useful for normalfzatfons Note tnat forotlt 4 tne total number of galaxfes diverges many many dwarfs eallty tne LF must turn over at some lower L to avold thls lfftegfatfoff over luminosity gwes m L0 L39lt1gtL39dL39 nLra2LL 43 L for typfcaf on tne lumlnoslty does not dwerge nor does tne mass lfftegfatmg from zero gwes atotal lumlnoslty denslty of LM nr LrI mn 2 LF a foror 4 tne lumlnoslty denslty lS MOE n L50 Mpc39a WHlCH for NHL 10 gwes a atotal mass denslty of 109 n M0 Mpc39a ooffespoffdfffg to o Nexll Prev Topl 4 Methods of Evaluating Luminosity Functions auster and eld samples fequffe qulte dfferent approacnes a Cluster Samples ance all duster galaxles are at tne same dlstanoe bln galaxfes by apparent magmtude down to some llmlt to get Hm use duster redsnft dfstanoe to get slmply HM Velocmes useful tnougn may s ull be mbfguous dwarfs are too fant to measure o dwarfs except BCDs have low SB while distant background galaxies usually have high SB 0 apply statistical corrections to Nm using field samples A Schechter function is fitted to M by minimizing 31 to obtain M and a b Field Samples In general deriving LFs for the field is more difficult than for clusters o lncompletness is usually found in magnitude limited samples typically o magnitude errors near mum include fainter galaxies 0 often magnitude corrections eg for internal absorption are only applied after the sample is defined In practice a magnitude dependent weighting factor can be applied to all galaxies to compensate for the incompleteness It is possible to check the completeness with the VNmax test e V volume out to object Vmax volume to object if pushed back to mum o for a uniform density of objects not necessarily true l a sample is complete for magnitude m if ltVNmaXgtm 05 o Corrections for Malmquist bias are essential ie survey volume smaller for lower luminosity galaxies 0 Good distances are necessary M from m but peculiar velocities can complicate a simple linear Hubble law Several methods have been developed i Classical Method eg Felten 1977 Form a histogram in absolute magnitude Multiply the number in each bin by 1Vmax This corrects each magnitude bin to the same effective volume Vmax is small for low luminosity galaxies so boosts their number to compensate Unfortunately this method assumes a constant space density This certainly isn39t true eg local group dwarfs are over represented locally ii DifferentialCumulative Ratio Method eg Kirshner et al 1979 This method avoids the previous assumption of uniform density However it does assume that the shape of the LF doesn39t depend on environment ie 0 NM l1JM x Dxyz where Dxyz is a position dependent total galaxy density and 115M is the LF expressed as a probability Because NgtM is the integral of NM then NMNgtM I39M I gtM since Dxyz cancels Consequently ltIJM gtM is independent of Dxyz basically for a iven region if NM is high because of the density then so is NgtM Now QM and gtM are evaluated using the the classical method Either their ratio is fitted with the equivalent ratio of a Schechter function or a smooth function is fitted whose differential is fitted to a Schechter function iii The C method eg LyndenBell 1971 revised by Choloniewski 1987 This method was first devised and applied to quasars It only assumes spherical symmetry but not constant density Consequently it is best applied to pencil beam surveys The method is supposedly simple and elegant but I can39t understand it It involves expressing M and Dr as the sums of weighted delta functions then somehow evaluating the weighting factors using quotCfunctionsquot rNextl im 39m 5 Different LFs for Different Hubble Types Early work showed o Schechter function is a good fit to many galaxy samples but a the parameters L or can vary depending on sample depth cluster or field cluster type Recently things are becoming clearer o it is important to consider the LFs of different galaxy Types 0 it now seems that the LFs of the major galaxy types are 0 different from eachother o relatively independent of environment 0 it is the relative proportions of each galaxy type that vary between cluster and field see next sec on More specifically broken down by type we have the following LFs Spirals Sa Sc Gaussian ltMBgt 168 5ogh 0 14 mag SO galaxies Gaussian ltMBgt 175 5ogh 0 11 mag Ellipticals Skewed Gaussian to bright ltM Bgt 169 5ogh dwarf Ellipticals dEdSph Schechter function M 16 5ogh or 13 dwarf lrregulars dlrr Schechter function M 15 5ogh Q O3 These LFs are illustrated here for the Field and Virgo Clearly full sample LFs a have a steep cutoff due to the Gaussian LP of the luminous Spirals 80s and Ellipticals a have rising faint end due to dEs and to lesser extent dlrr lPrevl 6 Different LFs for Field and Clusters Evaluating LFs for Clusters is reasonably straightforward since the galaxies are all at the same distance In general cluster LFs are well fit by a Schechter function have similar L though a can vary and is often steeper than in the field 13 there can be a dipdrop near MB 16 5ogh there can be an excess at higher luminosities cD galaxies 1OL dont fit and would be considered outliers in any smooth distribution We can now understand much of this See Topic 16 7 for a discussion of the physical origin of the morphologydensity different LFs usually arise from different proportions of Sp 80 E dE and dlrr specifically more E 80 dEs are in clusters while more Spirals and dlrr are in the field this is evidence for a morphological dependence on galaxy density see figure the dip at MB 16 occurs at the changeover from quotnormalquot to quotdwarfquot galaxies cD galaxies have clearly had a different history probably growing by accretion 93quot in dense galactic environments i r relation lNextH Prevu Topl 7 Physical Origin of the Luminosity Function Why does the galaxy luminosity function have the form that it does A complete understanding of this is not yet possible but here are the ingredients Making galaxies involves at least two things dark matter halos must form relatively straightforward baryons must fall in and make stars complex physics Here is a very brief account Cosmological simulations follow cold dark matter from initial slight perturbations to make many halos by hierarchical assembly The mass distribution of these halos follows the Schechter form this was Press and Schechter39s 1974 analytic result Hence one might expect a Schechter function for the galaxy mass distribution see figure However the observed galaxy mass function has completely different u er cutoff and lower slope see figure Specifically there are too many huge and dwarf halos without huge and dwarf galaxies To understand why we need to look at what prevents baryons from making stars within halos of different size see figure a Gas falling into huge halos is too hot to cool This becomes the intercluster medium in galaxy clusters b Gas falling into less massive halos is kept hot by AGN jets c Gas falling into small halos can be easily blown out by supernovae and star winds d Gas cannot fall into tiny halos it is prevented by its own pressure These processes are added to the cosmological dark matter simulations using simple prescriptive formulae to generate socalled quotsemianalytic modelsquot see figure These nicer reproduce many galaxy demographic results including a galaxy mass function that is a much better match to the galaxy luminosity function Ewan T0p M x2272nn512 z m wnm mxsszn mwm Asuammv Whime EXTRAGALAC39HC ASTRONOMV m Toolbox E 16 THE COSMOLOGICAL FRAMEWORK Index Queslions Images References M umeq cunem consuncnon 551 update May 19 2mm Next Top 1 lnlmduclian a The Good News Cosmo ogy s n avuew go den eva new agvea me 390 ean m Some vecth ohsawuonal devdopmaus nave been a me weenny ong unoenamy n HD S new we Ha 7 a me ong se 01 decent s1anaam came 5 also we smaseern 0 mm mu 0 0an amp WMAP nave new menve m OMB specuum amp ansouopy wan ngn aocuvacy a wens amp SDSS nave vecermy mapped oosmo ogwcaw sgnmean oca meme 0 obsevvanons at z NLE ae amos1 vouune and pvobe asgnmean awe m oosrme msmvy n dose pamevmp meole cal devdopmaus and new apweanons ae nae ngw mm a The oosrme consume and new ldanve noponlons 0A 7 rm dak enevgy dommales me wnem umvevse 0W 3K manev s vevy vnponant pammany n s1mctuve Vovmanon 0 an oomponem ME photons vevea an eaw no ph e 34 x m5 ONE neumnos ae pvedmed m nave not yet been obsevved a An ovdmay FRW oosmo ogwca wollu mode nx emevged x bemg oomptete y adequate m x m 53 n UsersmmwmKneshsvaSzmdex mm Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM The FRW parameters have been measured quotThe Concordance Modelquot This yields a reliable framework for charting cosmic history ie we now know tz A number of puzzles are removed by invoking an early period of inflation o A detailed theory exists tracing the average conditions from very early times a fairly full description of t lt 1s exists though it is not yet well tested nucleosynthesis at t N 15 mins nicely recovers the observed light element abundances in fact this measures conditions at t N 1s when the neutronproton ratio was fixed the theory accurately describes recombination at 12Myr and the origin of the CMB o A detailed theory now exists which describes the growth of perturbations Starting from inflation a natural spectrum of fluctations can be followed to z1000 here it matches the CMB anisotropies in great detail it can then be followed to 20 where it accurately matches local structure a After thousands of years of wild speculation the true story of creation is finally emerging we are living through and participating in a historic period of intellectual growth in the future our time will be recalled much like that of Copernicus Newton or Darwin b The NotSoGood News a Before we get too dizzy with euphoria let39s regain some humility by recalling we have no idea what the dark matter or dark energy are actually made of ie 96 although inflation is a promising idea it is far from proven and its cause is unknown the origin of the baryon asymmetry is only guessed at why particular cosmological values are favoured is unknown beyond anthropic arguments why there is something and not nothing is as unknown now as it was in Aristotle39s day 00000 0 Of course these and many other puzzles are not really bad news at all they signify a rich subject in good health with more fascinating work to be done In anycase it would be unseemly to fully understand Creation after only a couple of generations 0 Our Path Through the Subject Understandably quotThe New Cosmologyquot has attracted enormous interest and effort The subject is now mature and sophisticated much is well beyond ourmy range Our aim will be to outline the overall framework while ignoring details 0 Here is our path through the subject 1 Following tradition we begin with a discussion of some global cosmic properties isotropy homogeneity expansion structure composition We need a General Relativistic framework comprising two interdependent parts geometry metrics define the spacetime dynamics contents define coefficients of the metric and hence its time evolution We clarify some confusing aspects of geometry on an expanding coordinate grid various types of distance and horizon We introduce a toolkit for measuring intrinsic properties at cosmological distances these allow us to convert redshift our prime observable to distance time etc Such relations lead to the classic tests of the world model parameters As you may know several deep conundrumspuzzles emerge from the standard picture the socalled flatness horizon structure antimatter problems These are quotexplainedquot by introducing an early period of rapid inflation we note how inflation solves some of these problems and how it might be caused We briefly outline the evolution of the hot early universe the various quoterasquot quark hadron lepton radiation matter dark energy a brief treatment of cosmic nucleosynthesis and recombination ewaww 1 9 fileUsersdmw8fSitesastr553indexhtml Page 2 of 53 wumi ASYRSSZM mimimie Asimmy l2272nns l2 2 m a We see iecoiinoihalioh tileclly x llne cosmic MCloWale Eackglourld OMB ils piihnay gnaacleiislios ae ils lemakable isoliopy and puiily oi glam body low 9 balsa v il pioyiges llne sliohgesl evidence ioi llne hol oi il oohcluges oui ieyiew oi homogeneous cosmolog These pioyige llne lihg poihl ioi oui hexl Topic snucnue loinianon Next Prev Top 2 Glahal Praper es a Large Scale lsolropy To humans llne uhiyeise seeiins inglin anisonopic down is Solld up is llne 9lty wlh llne sh quot100m amp slas in speciic giieclohs eyeh ansncaly slas pieiei llne milky way yiniie uighl galaxies guslei in Ngo and coma only al niucli lainlei levels and much gavel usiances goes isoliopy beng lo eiineige Hele ae soiine hioe exmiples images 2 million iahl m lt 17 galaxies oovel1 s wlh only sighl sliucluie viable BT EIEIEI iagio souioes lypical z 1urlll0lmlyoovellhe hoilheih heiinispineie llne OMB wlh llne galaxy amp dipole ieiinoyeg is isoliopic lo one pal in m5 Al iahl levels 0e large scales llne uniyeise seeiins ianaikahlyisonopic b The Cosmological Principle Eyei shce Copelrllcus We ae loalln lo asigh Ealln aspecial localioh Galrlg one slep when we pioiinole egalilaianisiin lo a Cosmological Pllllclple 1lie uiiveise looks siaiisiically ilie sanie noni all locanons go lo any galaxy w you wil wmess an isloliopic uhiyeise one oohseguehce oi llnis is llnal llne Uhiyeise has no eoge oi can ioi a Euclidean space llnis aso iineans llne uhiyeise is lll lIlIle hole llnis heeg noi be liue ioi omen dosed geoiineliies see lalei Theie is aso a long cosmologica Piihople cosmologica Piihople a all nines gt in aggilioh lo urlllolmlly il explicilly slales llne uhiyeise ooes noi evolve llnis uhgeipihs llne Sean sale cosmology agyocaleg oy Hoyle Bohgi amp Gold in llne 5D amp 6D The evidence ioi eyoluloh lisl cmne in llne 196D and hx giown sliohgei eyei shce song gt We Worl loorlsldel llne Sleady slale oosiinology iuillnei G Large Scale Homogenein il is e y lo snow llnal isoliopy oosiinological piihciple lioniogeneuy gt all localiohs ae slalislically eguiyalehl e g have llne smne iinean gehsily This is soiinelihnes exlehgeg lo poslulale llnal llne lays oi physics ae aso glooal V iia Homogeneily only sels in on laige enouyi scales sonneyheie oelweeh inn Mpc ampl 3pc images on snnallei scales oi ooulse ohe ehoouhleis iinugn inhomogeneiy see oeloya This leads lo aheal Wamlrlg oohciusioh iii IUsersdmwmXnasISUSSZlridex mi nge z m 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM Right quotnowquot a civilization 1000 Gpc away ie well beyond our horizon would u see a microwave background highz 030s and distant young galaxies o be surrounded by sheets amp voids of mature galaxies o find local galaxies with their stars and planets to be much like ours Far from being bizarrely remote the distant Universe would be remarkably familiar d Small Scale Structure 0 The Universe39s small scale inhomogeneity is much more obvious than its global homogeneity At first sight this is a rather puzzling fact why isn39t the Universe just fully one or the other And why is there a special length scale that marks the boundary between the two a The inhomogeneity appears as a heirarchy of structures stars galaxies clusters tapestry Out of almost perfectly uniform gas comes all these rich forms each with its own character This is a remarkable and profound property of our Universe 0 Our cosmology must explain this origin and development of structure e Expansion i The Hubble Law As soon as galaxy spectra were measured it became clear that most were redshifted In 1929 Hubble found a quotroughly linearquot relation between redshift and distance cz o d as data improved this relation was confirmed and has strengthened ever since image Don39t confuse establishing the linear relation with measuring its gradient It took 75 years to achieve 510 uncertainty in the gradient HO see below This is primarily because calibrating the distance scale is notoriously difficult The current best estimate for H0 is 72 l 5 kmsMpc 72 X 10396 Myr391 in psm units Note it is still customary to quote measurements scaled to h 100 kmsMpc For example quotThe luminosity of M87 is 23 X 1011 h392 Lainquot Topic 13k I Note also that there is always some small scatter few 100 kms in the Hubble relation galaxy redshifts have two components 1 the global quotHubble Flowquot arising from cosmic expansion 2 a small quotpeculiarquot velocity arising as the galaxy responds to the gravity of its neighbors quotpeculiarquot comes from Latin quotpeculiumquot meaning quotprivate propertyquot specific to it alone ii Everyone Sees the Same Law At first glance the Hubble Law seems to violate the Cosmological Principle galaxies appear to move radially away from us suggesting we are somehow central However its linear form ensures that all locations witness the same relation Consider two vector locations k and p and the vector field v H r centered on us at 0 We see p move at vp H p so how does an observer located on k see p move use primes to denote values measured by k image p39pk and v39pvpvkHpHkHpkHp39 Hence for any point p k sees a Hubble Law v39p H p39 if we see v Hr then so does everyone else the cosmological principle still holds true Since everyone witnesses the same Hubble Law we conclude that fileUsersdmw8fSitesastr553indexhtml Page 4 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM the Universe itself is undergoing isotropic expansion with form v H r This is a remarkable and profound result f The Big Bang 0 The linear nature of the Hubble law has a second fascinating consequence Tracing the expansion backwards there was a moment when all galaxies were together This singular event is called quotThe Big Bangquot The term was coined in 1952 by Hoyle to gently mock opponents of his Steady State Cosmology If no forces act to accelerate or decelerate the expansion then vt const In that case the singular event ocurred at a time tHYO W 1HO in the past tHYO Ho391 10 h391 Gyr is called The Current Hubble Time If the average expansion rate isn39t too different from its current value then tHYO x current age For HO 72 kmsMpc tHYO 140 Gyr a ballpark figure for the age of the Universe 9 Cosmic Time 0 Being physics graduates we are often nervous when time is introduced does it depend on our location or motion ie is it different for different observers In this case you can relax the time we39ve introduced is a proper time it is measured by inertial observers and can be agreed upon by everyone a In fact the cosmological principle coupled with global expansion guarantees a universal time fundamental observers may synchronize clocks when the local density reaches some value since these observers are at rest wrt their local frame they measure proper time o If we start the clocks at the Big Bang ie set tBB to 0 at lj39 00 we have cosmic time we can all agree on the age of the universe and the timeage at a given redshift tz h Olbers39 Paradox o Olbers 1820 is credited with first noticing that a dark night sky has cosmological implications Imagine an infinitely large old static homogeneous universe Every sight line intersects a star39s surface whose surface brightness is independent of its distance hence the entire sky should shine with the surface brightness of stars However it doesn39t so one or more of the assumptions must be wrong Absorption doesn39t help dust ultimately heats to reach equilibrium with the radiation field I Let39s use modern numbers to quickly assess how this paradox might constrain cosmology The mean local cosmic luminosity density is P39L 108 LEV3 Mpc393 with ML 10 Topic 43 PL 103932 erg s391 cm393 hf N108 Mpc393 Pi 103931 gm cm393 E Q 001 0 N R There are several ways to cast these numbers all confirming the Universe39s emptiness amp darkness 1 mfp 1 mtr 1018 Mpc a typical sight line terminates at an enormous distance 2 the sky brightness J out to distance D is l lZ39L r2 d0 dr 4397l r2 lvz39LD47T erg s391 cm392 sr391 fileUsersdmw8fSitesastr553indexhtml Page 5 of 53 Wm Ammo Eunqthz Aswaan mums l2 3 m Joy DOME now NAEI an We and J N 1n evg squotm1392 Jo geJ J N Jn evg squotm1392 9quot arm Even PL N n n7 Loo pea Joy Jne MW 9 JJ N24 eased Je vevydaw K new body We need D N Jn Mpc x beJove dJS vequwes D N mg Mpc Jo veaen Obevs buth 9w Oeaw gwen JJs dJnenJ empnne adak 9w onJy Mes ouJ gamma on sJanc Unwevses 3 AJ PL N m evg 5quot m J woqu Jake Jnem JUQB yeas A buth 9w vequwes ouv sJanc umvevse Jo be both nganJJc and JmmeuseJy ad 4 BA vea sJas cannoJ smne maJ Jong anyway nov can mey aeaJe so mudn hth convemng all baryon Jo hth u 7 n4 PM c2 N Jn39 evg m1 WeH beJow ma J needed sJaJed dmeven y n a buth 9w umvevse Jne vadJaJJon aone wqu ooanouJe 9m N m9 J Wu even Jn Theve S oJ oouvse Home souvoe oJ vadJaJJon Jne 3mm pnoJospneve oJ Jne OMB The ommdweonona OMB Js uncanan hke Jne dmc Obevs buth ny excepJ Joy one deJaJ Bedsth km JJs sumace mganess by aJaeJov J z N JEN sec Ed Jn Jne absense oJ vedshm We woqu Jndeed wmess aJeJnaJ 9ltyw1h J N ms Wan quot1392 9quot yougnw equwaent oJ oouvse Jo mung Jne 9w an soJa dJsxs x Obevs ongma y ooncewed Fov Jan hth some Jnen J J emanslon men Keeps Jne 9w dak MJdn oJ Jan boHs dan Jo Jne JaeJ maJ me unveue IS m m mamouynamuc eqnlllhllnm Hadw szpvang snc J ad ans expanson ooupJed Jo gvavJalJona JnsJamJJy Jeads my worn LTE 2 Jan yJest anJgnJy Jnhomogeneous dJanmJJon oJ manev an nuge Jempevaluve vaJanons a JnJevaeJonedmnovanon me as oJJen mudn Jongev man Jne Huome expansJon Jme JJ obeys nad hved Jn Jne st mu m men Jne Unwevse M n LTE meve wqu be no paadox Nextl m Tov 3 Mare an Expansian a Expanding Coordinate Grids uggev cavwvg one can Jmaane mee Kmds oJ monon J PanJs nk don m me papa paJJdpaJe n J monon They ae Jxed on me gnd am pman ae caJed mnuamemal ohsavas mew monon IS me HuooJe Jow As aveady 910 n sec 2eu mey an wmess me me HuooJe Jaw Jn JnJJodchovy Jest Jm Js me expandng 15er dead maogy 2 PanJs an JJ you hke can aso new ova me papev crossing gnd hnes Tan monon Js me pecullal veJocny J J awe space veJomJy a New Jmaane aspeedy 31 men aways Mn und y a1 aoonstawt vale n aszagnJ hne Jm vepvesems a pnoJon aosang me expandng space aways momng Jocaw a1 0 nJJmg Jamwwmumssznmmm m s m 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM c From General Relativity we know that in Cosmology we may need to consider a curved grid For this one imagines our dots and ants on an expanding sphere with longitudelatitude lines Fortunately it turns out that expanding flat graph paper is all that39s needed for our Universe b The Scale Factor at amp Comoving Coordinates How do we treat this kind of coordinate system mathematically Easy we consider the grid and its expansion separately I First notice that the Hubble law preserves shapes patterns of galaxies becomes bigger patterns tor a set of i points cosmic expansion gives rit at ri e tor a tiducial time tO Here at is a universal scalar function of time and is called the scale factor at simply tracks how the separation of galaxies changes over time Finding the form for at is a holy grail in cosmology Sensibly we assign the current time special status t0 now ato 1 and rto r0 Hence the current values r0 provide the coordinate grid and are called comoving coordinates as the grid expands the comoving coordinates do not change at any time the physical coordinate of an object is r at ro by setting ato 1 we ensure that in the past a lt 1 while in the future a gt 1 L 9 quot For example at the time of recombination a H 0001 The comoving distance to protoM87 is still 15 Mpc but its physical distance is only 15 kpc Notice that r and r0 are both proper distances they tell us how many nonexpanding rulers tit endtoend from here to the galaxy Later we introduce several pseudodistances eg luminosity amp angular diameter distance DL DA these are not true proper distances but convenient functions of distance Warning symbol conventions tor physicalcomovingpseudo distances varies greatly You must be careful when reading texts quotis this r or d physical or comovingquot In these notes I will try to be consistent r physical r0 comoving D pseudo c The Hubble Parameter Ht With our new expanding coordinate grid how does H lit in To find out take time derivatives of rt at gtlt rto drdt vt dadt gtlt rto 1a dadt rt But this is simply vt Ht rt with Ht 1a dadt we have found that the Hubble relation applies at all times Ha E Ht 1a dadt and drdt v Hr o In general Ht and at both vary with time For the current epoch we write Hto E HO and it has units of inverse time Our best current estimate for H0 is sec 3g 234 x 103918 Hz or 72 kmsMpc d The VelocityDistance Relation amp The Hubble Law fileUsersdmw8fSitesastr553indexhtml Page 7 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM At this point we need to clarify something there are two apparently similar relations 1 the theoretical quotproperquot velocitydistance relation v H r 2 the observational redshiftdistance quotHubble Lawquot 02 H D These are in fact somewhat different The theoretical relation v H r is globally exact though it is observationally inaccessible v and r are quotproperquot quantities ie as measured in a local rest frame For example for r how many nonexpanding rulers must be laid down between us and the galaxy after 1 second v additional rulers must be laid down where v H r o notice that the values are all measured at the same cosmic time we deal with distant galaxies as they right now not as we see them The Hubble Law is strictly observational o 1 z tobsAem and 02 rather than vz is sometimes taken as a quotDoppler velocityquot u D is usually a luminosity distance which matches the proper distance only at low 2 0 both 2 and D apply to the time when the light set out not the current time during the light39s journey the galaxy moved further away and possibly slowed down At high 2 several factors break linearity indeed this deviation is used to measure go I At low redshift the Hubble Law and the velocitydistance relation look the same Cosmic expansion is best described by v Hr it is exact and holds everywhere at all times the Hubble Law 02 H D only provides imperfect observational access to this cosmic expansion e The Hubble Sphere 0 Let39s push the velocitydistance law to great distances For HO 100 kmsMpc we have atr10Mpc v103 kms atr10Gpc v106 kms at r 1000 Gpc v 108 kms Yes these velocities are faster than light special relativity does not apply to this motion it arises from expansion 2f space not motion through space I Can we see the galaxies which recede faster than light The light they emit moves through space at speed 0 towards us but over time the wavefronts get further from us at speed v c gt 0 we will never see them There is a critical distance rHYO cHO 30 h391 Gpc which is now receeding at v HO rHYO c II rHYO cHO is called the Hubble distance where right now galaxies recede at c For constant rate of expansion we will ultimately see everything inside a sphere of radius rHYO Only if v slows down significantly will we be able see beyond rHYO These and other potentially confusing things should become clearer later sec 7 f The Nature of the Redshift 0 When we first encounter redshift we view it as a Doppler shift due to a galaxy39s motion while this is appropriate for peculiar velocities it is not for expansion quotvelocitiesquot o A more mature view sees the light embedded in space getting stretched on its way to us This kind of redshift is called a Cosmological Redshift image a With this view redshift clearly measures the stretch factor between emission and observation fileUsersdmw8fSitesastr553indexhtml Page 8 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM ll loJle 12 atoate 1ate since the current scale factor ato 1 This is a fundamental relation amp globally exact It tells us the relative change in size since the light set out Some examples at cl 30000 kms z 01 ate 111 0909 m 90 at z 1 a 05 all galaxies were half their current separation 8 x the current density at z 6 a 014 currently the most distant QSOs at z 1000 a 01 at recombination everything is 103x closer 109x denser I Since redshift is so important its interpretation has occasionally been questioned eg quottired lightquot photons simply lose energy during their long and tiring journey to us A nice observation verifies that the redshift is indeed caused by expansion If a distant event has true duration 739 the object is v T further away at the end of the event The light therefore arrives v Tc late and we observe a duration t 5739 vclTc m 1 z639 This is called cosmological time dilation a detailed analysis changes mto It has been observationally verified distant SNla have slower light curves by 1z image L 56 9 Measuring Distances and Ho The struggle to measure reliable astronomical distances and H0 has a long history image Rather than consider that history we shall briefly summarize the current 1995 2005 situation A good review is Mould39s 1997 EAA article quoted HO values are taken from this article To derive HO one must measure both distances and redshifts o Distance measurements take two rather different forms ladder methods which depend on nearby calibration such as parallax amp Cepheids one shot methods which need no calibration and rely only on known physical properties 0 Redshifts while easily measured need correction for peculiar motion our motion includes the sun39s orbit MW motion in local group Virgocentric infall Topic 15 the target galaxy is chosen to be a groupcluster member amp the group39s mean redshift is used i Ladder Method In elementary texts the distance ladder is often presented with many rungs image In practice there are really only three 1 Use the Hipparcos satellite to get trigonometric parallaxes of nearby Cepheids calibrate PeriodLuminosity PL relation 2 Use HST to get Cepheid distances to nearby 25 Mpc galaxies calibrate TullyFisher TF amp FundamentalPlane FP amp other methods 3 Use TF FP amp other distances to groups where peculiar velocities are unimportant group mean redshifts amp distances now give HO Let39s outline these various steps amp techniques a Cepheid variables fileUsersdmw8fSitesastr553indexhtml Page 9 of 53 Whittle ASTR 5534 Extragalactic Astronomy fileUsersdmw8fSitesastr553indexhtml 12272006 1234 PM This class of pulsating stars defines a tight periodluminositycolor relation images measure period to get luminosity and hence distance they are luminous stars M 8 to 12 and hence can be seen to considerable distances 25 Mpc by HST however they are also rare so there are only within pc which tend to be of low luminosity Historically the PL relation was calibrated by Main Sequence fitting to open clusters containing Cepheids Now Hipparcos provides direct trigonometric calibration eg Perryman et al 1997 however this calibration still needs to be improved eg using future astrometric missions SIM GAIA The distance to the LMC plays a very important role and also still needs to be improved it contains enough Cepheids to define the PL relation in m not M hence extragalactic Cepheids yield relative distances to the LMC the current best estimate for the LMC is mM 1850 l 013 E 50 l 32 kpc uses EBV 01 the HST Key Project has now measured Cepheids in galaxies out to 25 Mpc These galaxies were then used to calibrate the following methods TF TullyFisher Relation This is a luminositylinewidth relation for spirals Topic 54b scatter is minimum in the near IR l H hence the method is often referred to as quotIRTFquot about 20 spirals now have Cepheid distances about 25 groupsclusters out to 10000 kms have TF distances HO 71 l 8 eg Sakai et al 1999 FP Fundamental Plane Relation This is a refinement of the luminositylinewidth FaberJackson relation for ellipticals Topic 74b Either DnO isophotal diameterdispersion or surface brightnessradiusdispersion relations since no Cepheids in Es calibration uses Es in groups with Cepheid distances eg Virgo Fornax Leo many groupsclusters out to 10000 kms now have FP distances HO 78 l 10 eg Mould et al 1996 Kelson et al 1999 SNla WD binary thermonuclear detonation These are very luminous so well suited to qO studies high 2 but also useful for H0 lower 2 the light curves aren39t all the same but peak luminosity correlates with fading rate and color image unfortunately very few SNla have ocurred in galaxies with Cepheid distances calibration not ideal Ho 68 l 6 eg Gibson et al 1999 SBF Surface quot 39 39 739 Consider a set of COD pixels recording the light from an E galaxy each one with perfect SN ratio there is still variation between the pixels because of fN fluctuations in stars Although the mean surface brightness is independent of distance the variation is not nearer galaxies have fewer stars per pix larger variation difficulties contamination by globular clusters colorpopulation dependency calibration HST can use this method out to about 7000 kms HO 69 l 7 eg Ferrarese et al 1999 HST Key Proiect combines all these methods plus GC amp PN luminosity function methods HO 72 l 5 km s391 Mpc391 eg Friedman et al 2002 image Page 10 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM ii Direct Methods EPM quot quot Tquot 39 Method originally from BaadeampWesselink This method applies to aH pulsatingexpanding photospheres particularly Type II core collapse SN angular size is derived from flux temperature and emissivity black body 1 linear size is derived from integrating velocity Iinewidth over time distance by comparing angular and linear sizes Fortunately I EPM distances agree statistically with Cepheid distances EPM distances now available for SN ll out to 14000 kms check HO 73 l 11 eg Schmidt et al 1994 o VLBI Masers in Nuclear Gas Disks So far only one good example of this method exists NGC 4258 Miyoshi et al 1995 Topic 134e a compact 1pc molecular disk orbits central black hole VLBI of H20 masers gives Keplerian velocities and proper motions distance by comparing linear and angular velocities l Mpc this method has good potential for future more distant objects eg at z gt 05 it would give HO and go Gravitational Lensing Time Delays 2 030 images have different light paths with different physical lengths this path difference is given by the time delay between QSOs light curved via crosscorrelation the calculated path difference depends on projected mass density and linear scale distance by comparing observed angular scale and calculated linear scale About 10 now done HO 60 65 puzzlingly low o 82 SunyaevZeldovich Effect in Clusters Hot electrons in galaxy cluster lCMs do two things 1 they generate Xrays via bremsstrahlumg LX ac ne2 rc3 TX 2 2 they Compton scatter CMB photons 5TTCMB oc ne rC TX you can solve for rC and compare with 6G to get a distance HO 60 65 eg Birkinshaw 1998 also puzzlingly low 0 Possible Concern Why do the more distant lensing amp SZ methods seem to give systematically low values for Ho Perhaps we live in a void with higher local HO than the global value The answer is quotprobably notquot for several reasons 0 the most distant TF work is now out to 15000 kms 200 Mpc which is hardly local 0 the Hubble relation is linear from 100 to 1000 Mpc o from CMB anisotropies the incidence of voids of size 104 kms is quite rare The local value is probably within a few percent of the global value Why the more distant estimates seem to yield low values is not yet understood I Currently Favoured Value of Ho 72 l 5 km s391 Mpc391 fileUsersdmw8fSitesastr553indexhtml Page 11 of 53 Wm mxsszu Eungahmz Aman x2272nn512 z m Spevge et a guns used an HST Key Med value my new WMAP concovdawce mode Many peode new adopt an x me amemly Vavouved value made 01 6 015 cm 2 also need to say meme wgo ow man oonecnons ae used 01 not Negt l Levl ml 4 campanenls 8 Their Densities 8e Pressures a The Stage and its Contents The amth umvevse oomans we mmk p151 We oomponems Dak Enevgy Dak Manev Ea yomc Manev Photons Neumnos Vou 91mm not mm m mese x meveW oomponems hke Mnmve sung n amom n geneva ve anwy me contents he p to de ne me spacernme n men mey veswde one Mnmve amuany muenoes me shape and sze m me mow To undevs1awd me g oba pvopemes we meve39ove need acompleae mnevay m me oomponems a Mos1 mpona We need to Know mew densmes P a pee me p Fm sexevil components P and p ae mm addmve PM JA and pm Jp Annougn n weeks We We ve ntloducmg Iwo qua mes heve we aen t An eqnanon 01 Male EDS teHs us new pvessuve depends on densuy p pa As me umvevse emanus me densuy and pvessne m eam oomponem changes Fonunalew we Know nowJ and p vay usng oca oonsevuanon m enevgy me EOS AM we need meve39ove s to menve me densuy and pvessne a one nme Obmousy me mos1 convemem nme s me mesa The vest x mey say s mow b cmical Densin amp Spatial Geometry Lev my see nowJ days us me We wu need a ew eduanons men Mp1th Me on see eaL so p151 accept men an new eye s me cosmic elegy nuedmann eqnanon usng scale may av sec 63M dad 2 e EmmaQJ CEHa Tm matches me me mun KE PE Em Mn Em appeanng x aspanal culwlule bound sys1em unbound sys1em Hence We Cm denne acnucal densny A man we d anal geomeuy kEI n lUsersldmwmKnaSxsvaSZmdax mm m 2 m 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM ll 1a2dadt2 H2 awe3C giving PC 3H28iTG li39c 265 X 10397 h2 M pc393 180 X 103929 h2 gm em3 108 h2 mp m393 a Notice lj39c varies with time through Ht or h and we39ll write its current value as r J39CYO However once set k 101 is fixed for all time In particular if k 0 now flat geometry then we always have a flat geometry if 0 l 39cyo now then lquot PC always worth stressing since this seems to be the case 0 With such tiny values it is convenient to express densities relative to lrz39c using HEPl3C o The table lists the measured current relative densities for the various cosmic components values are for the concordence model 10 o uncertainties other functions are given which we39ll come to in a moment p L3 x 31W x 233W 1 3W Component 9 CYO IUD ax a 393 IX sign accel Dark Energy 073 0 a x 91 2 Dark Matter 023 g 0 23 1 Baryons 004 m 0 23 Photons 50 X 105 13 12 Neutrinos 34 X 10395 13 12 o The current best estimate for the total density is 9101 IQ 100 l 002 we live in a universe with quotflatquot spatial geometry This is one of the most important discoveries in recent cosmology I Warning please don39t continue to view 9101 as defining the future of the Universe it doesn39t this was appropriate in the prelambda days A 0 when it was common to state if 9101 gt 1 the Universe will turn around collapse and end in a big crunch if 9101 lt 1 the Universe will expand forever with 0 one cannot infer the future simply from 9m Instead 0101 only fixes the spatial geometry openflatclosed not the future 0 Pressure amp Equations of State 0 Why is pressure relevant Not because quotpressures pushquot they don39t only pressure gradients push It is because pressure is fundamentally momentum flux density it arises from internal motion just as motion of charges make B fields so motion of masses make gravitomagnetic fields if the motion is omnidirectional eg a gas the result is simply an additional gravitational force if the motion is coherent a rotating mass the force has direction frame dragging In slightly more technical terms Einstein39s pr 839Il39Gc2 Tm states spacetime geometry energymomentum distribution fileUsersdmw8fSitesastr553ihdexhtml Page 13 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM the J V indices are 0123 ctxyz and Toyo alone covers the energy including rest mass the other three components are for momentum of which pressures are a form Equations of State EOS tell us how pressure depends on density Here on earth equations of state can be quite complex although gases are simple p lquot kTJ In cosmology we need a relativistic equation of state where lquot contains both rest mass and energy Fortunately the cosmic fluids all have simple forms p W X li39c2 the constant quotwquot is itself often referred to as quotthe equation of statequot w tells us what fraction of the total energy density is in the form of pressure recall the units of pressure are energy density J m393 or erg cm393 It is useful to recall that sound speed cs2 dpd5l w c2 true for relativistic amp nonrel fluids There are some limits on w o w g 1 keeps subluminal sound speed cS lt c w 1 refers to quotstiff matterquot with c8 c c all known matter satisfies the quotdominant energy conditionquot F39 2 p Together these limit plausible values of w to 1 S w S 1 As we shall see there are three relevant values of w II w 0 this is called quotdustquot for historical reasons matter with zero pressure II w w 0 nonrelativistic matter present day baryons CDM II w 13 relativistic matter photons neutrinos II w 1 vacuum energy the cosmological constant is also described by w 1 Do these jive with our prior experience of quotnormalquot equations of state A perfect gas has equation of state p nkT and adiabatic indexI39ll its internal energy density u nkT39l39 1 and its pressure p 39l 1 u nkT recall 39l is closer to 1 when particles themselves store internal rotvib energy eg a monatomic gas has fl 53 with u 32 nkT and p nkT a diatomic gas has 39l39 97 with u 72 nkT and p nkT Let39s quickly recover pV l const and w x 0 for our perfect nonrelativistic Consider U gtuV and adiabatic expansion d0 0 dU p dV so that dU pdV Multiply p l39 1 u by V to get pV l 1 U and then differentiate pdV v dp dUquotlquot 1 pdVquotlquot 1 dpp quotlquotdVv pV l const gas p nkT and 32 kT 12 mltv2gt so using rest density lrz39o we have p lj39Om m ltv2gt3 p lquotoc2 gtlt ltv2gt3c2 so w ltv2gt3c2 m 0 for a nonrelativistic gas For a photon gas Po 0 and rl39l 43 so 1 13 and we find p ll 1 u 13 u 13 lZ39c2 lt39ocz 13 l339c2 w 13 as we had before Finally for the expansion of a vacuum with constant density amp pressure we have dUpdV lquotV02dVpdV gt quotVc w1 Basically you must provide energy to create more vacuum ie do work to increase the volume Normally of course gas does work on the surroundings reducing its internal energy d The Five Components fileUsersdmw8fSitesastr553indexhtml Page 14 of 53 12272006 1234 PM Whittle ASTR 5534 Extragalactic Astronomy Let39s now look briefly at the 5 cosmic components i Dark Energy I This component has had a checkered history 00000 0 0 Famously introduced in 1917 by Einstein to balance matter amp yield a static universe Dumped in 1929 when Hubble discovered cosmic expansion Resurrected in 1930s40s when large HO implies tH lt t loitering model Dumped in 1960s70s when HO revised down giving tH gt t4 Resurrected in 1980s when tH lt tGC amp QM 03 but inflation suggests 9101 1 Discovered in 1998 with accelerating expansion qo Vat2M lt 0 Here to stay 2005 Now 3 independent amp consistent confirmations I Dark Energy is generic term there are three main possibilities A quotcosmological constantquot appearing in Einstein39s equations as an integration constant 11 Quantum mechanical energy of the vacuum Both vacuum energy and have w1 Quintessence a strange new form of matter with 1 lt w lt 13 possibly evolving over time We dont yet know which of these dark energy is Acceleration only demands w lt 13 while current observations constrain w lt 08 vacuum energy is currently favoured which can be treated as a A term a Current theoretical understanding is zero a quotnaturalquot energy density has 1539ch m4hc3 for quotnatural massquot m if m Planck mass hcG12 IZ39ch 3 X 10126 eV cm393 1093 gm cm393 I Observed 7ch 103 eV cm393 or 10123 gtlt smaller maybe the worst guess ever The particle mass which does give the correct value is m N 10393 eV No known candidates Studying DE in the lab is very difficult only energy differences can be measured The Casimir effect for example shows the vacuum is quotlivequot but not by how much ii Dark Matter Introd uced by Zwicky in 1930s to explain high galaxy cluster dispersions Then forgotten Reintroduced in the 1970s to explain spiral rotation curves at large radii Increasingly important on larger scales galaxies binaries clusters large scale flows Necessary to create large scale structure rapidly from CMB baryons alone are inadequate Combined methods yield It39DM 023 I 004 Theoretical understanding zero though some properties are constrained 0 0000 nonbaryonic ie neither protons nor neutrons BBNS rules out high I I39b nonrelativistic after T N 1 MeV cold dark matter CDM clusters efficiently interacts only via weak and gravitational forces dark matter possibly made of quotWIMPSquot Weakly Interacting Massive Particles WIMP candidates not known maybe lightest supersymmetric particle Current laboratory searches underway Nothing yet found iii Baryons In this context baryons refers to neutrons protons and electrons Big Bang NucleoSynthesis BBNS and CMB power spectra together give 9b 0044 i 0005 Primordial HHe abundances are 7525 by mass or 121 by number Since then stars have converted 2 by mass into heavier elements Only 10 of baryons are luminous ie are found in stars The other 90 is mosly in diffuse ionized gas surounding galaxies Page 15 of53 fileUsersdmw8fSitesastr553indexhtm Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM The traction of baryons in neutral form is tiny TBD 0 Mass to light ratios help interconvert lI39m X X MLx in band X Topic 13 For the local universe 3 H 20 X 108 h L133 Mpc393 m 9 Watt AU393 This yields several tiducial values using Hem we have MLCm 1400 h MID353 7 million kgWatt very dark using lquotm 027 quotcm we have MLm 375 close to some clusters using Pb 004 Fem we have MLb 55 significantly more than most galaxies The universe is optically quite dark on average 7000 tonnesWatt iv Photons o The CMB is by far the strongest of all the cosmic background radiation tields images This is true for both number density nquotquot 410 cm393 and energy density uquotquot N 026 eV cm393 lt39s energy density is even comparable to the photon and thermal components of the local ISM o The CMB spectrum spans 50011 to 1 cm and peaks at 10 mm 80 or 19 mm By It is an exceedingly accurate black body with TOMB 2725 i 0002 K 007 image 0 Summarizing the integrated energy and number densities for a BB radiation field we have fileUsersdmw8fSitesastr553indexhtm note here a 8W5 k415h3c3 4 Uc is the radiation constant and NOT the scale factor u a T4 756 X 103915 T4 erg cm393 Ener densit gy y ucmb 417 X 103913 erg cm393 026 eV cm393 J uc47l39 caT447l39 0T4ltl39 180 gtlt10395 T4 erg s391 cm392 sr391 Ener tlux gy 994 X 10394 erg s391 cm392 sr391 Jcmb n aT327kB 203 T3 cm393 Number density 410 3 cm ncmb nc439fl39 484 X 1010 cm392 s391 sr391 978 X 1011 cm392 s391 sr391 N Number tlux Ncmb v Neutrinos Like all particles neutrinos were created out of energy in the early Universe Since they are stable they survive to today in great though undetectable numbers There are three families ye t H VT each with paIticleantipalticle pairs six in all They decoupled at t N 1s 2 1010 kT 1 MeV and have travelled unscattered ever since Cosmic Neutrino Background CNB similar to but much younger than the CMB Since kT gtgt myc2 M 0 at decoupling they were relativistic quothotquot dark matter They behave like a relativistic gas 39l39 43 w 13 similar to the CMB Two tacts make the ONE slightly different from the CMB Shortly after 1239 decoupling the ee39 pairs annihilate at kT 05 MeV this puts energy amp entropy into the CMB but not the ONE T1quot gt Ty one can show sec 12 that Tilquot 114m Ty giving Ty 194 K x N Neutrinos obey FermiDirac statistics whereas photons obey BoseEinstein statistics Page 16 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM Summing over all six species one can show sec 12 Uyytot 068 Uy l l nmot 911 nilquot 335 cm393 currently 0 The total energy density in 39Il39l39s and U39s alter ee39 annihilation is urel 168 uquotquot 168 a Tquot4 The total relativistic contribution to Q is rel Qquot Up 84 x 10395 today This is the value to use when evaluating the expansion rate during the radiation era What it neutrinos don39t have zero rest mass Currently kTp39 167 X 10394 eV so only neutrino masses larger than this change things For each species one finds a current y myeV 94 h392 Hence a mean mass of 8 eV per species will give 9mm 1 and close the Universe Laboratory limits are presently well above this though structure formation suggests y is small note neutrino oscillation experiments only give mass differences 005eV not masses e Density Changes with Scale Factor 0 As the universe expands how do the densities of the components change The too quick answer is simply Fix a393 QC 1 23 ie densities drop as the volume increases However since pressure also adds to density lF39p pc2 we must include this too Let s consider the expansion of a volume V E a3 it is filled with a component with energy density lAV39c2 and equation of state pc2 Wquot The expansion is clearly adiathermal as much energy enters as exits homogeneity It is also adiabatic the expansion is isentropic because the CMB dominates the total entropy Consider conservation of energy d0 dU pdV 0 since adiabatic dU pdV dl3939c2a3 p da3 c2a3 dli39 sli39c2a3 da 3pa2 da dea 3al39 pcz 31 w lZ39la which has solution l39 lZ39o a3931W 1 z31W where once again we set a0 at0 1 From the above table we find II matter l39ma ljmyo a393 lJ39myo 1 23 as expected by quotconservation of massquot II radiation lI39ra lZ39ryo a394 lZ39ryo 1 24 since nquot lquotac a393 and Effac a391 from redshitt ll vacuum lZ39Va lQ39VYO const space is space Going to earlier times smaller a l539m increases quickly but not as quickly as 1539 radiation dominates over matter at earlier times Since lrz39v is constant while the other densities are increasing the vacuum rapidly becomes irrelevant and one can ignore it for the first Gyr or so More generally the component with most positive w dominates at early times radiation the component with most negative w dominates at late times vacuum images 0 Quantitatively we can identify times when the various component densities are equal fileUsersdmw8fSitesastr553indexhtml Page 17 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM aeq and zeq are easily evaluated while teq requires the integral relations from sec 6c v density match condition a equality z equality t equality l lv l39lm 073 027 a393 072 039 943 Gyr ljv ljlrel 073 84 X 105 a394 0103 83 615 Myr Pb 34quot 004 a393 50 X 10395 a394 125 X 10393 800 620 kyr 939 Pref 027 a393 84 X 10395 a394 311 X 104 3200 57 kyr Pm Pquot 027 a393 50 X 10395 a394 185 X 104 5400 22 kyr Note that here r j39rel refers to the sum of photons and neutrinos relativistic matter Likewise Pm refers to the sum of baryons and CDM nonrelativistic matter f Rates of Expansion k0 single component 0 In general time dependencies are complicated sec 60 However for flat geometries with a single component we find simple forms for at From the cosmic energy equation with k 0 filo IJCYO and Iquot filo a3931W we have dadt2 BWG3a2l l 8WG3a2lg39o a31W HOZIILZ39Opcyo a31W H02 a13W dadt Ho a39lt13Wgt 2 l alt13WV2 da HOldt a 33w2t1H2 lt33Wgt Now as it happens the universe is flat and there are times when just one component dominates ll during the radiation era a 3 112 ll during the matter era a x 23 ll during the vacuum era a 3 e1 from dadtoc a I The function at from zero to the present indeed matches these simple relations see image and sec 6c iii g The Peculiar Role of Negative Pressure 0 In Newtonian gravity Poisson39s equation relates the gravitational field to the matter V2quot 47rGe339 In General Relativity a similar relation gives the acceleration 1ad2adt2 437rG 4339 3pc2 437I39G 4539 1 3w Notice that the source term for gravity is not just 31339 but it includes a pressure term This comes from internal momentum associated with particle motion and it adds to gravity sec 4c 0 Since acceleration is x1 3w then if w lt 13 we find that gravity is repulsive Current observations find an accelerating expansion requiring a component with w 10 l 01 There is presently a strong preference for w 1 vacuum energy or a cosmological constant A fileUsersdmw8fSitesastr553indexhtml Page 18 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM a Note that 15quot is ve and adds to gravity while pV is ve and dominates giving net acceleration a What is ve pressure It is tension eg stretched rubber I pull on it to increase its volume For ve pressure expansion does work on the surroundings internal energy decreases For ve pressure expansion requires work from the surroundings gt internal energy increases The vacuum with constant energy density 539ch behaves the same way To expand a vacuum I must provide the energy for the new volume An extreme example imagine a strange piston with a little quotstrange waterquot in it You pull extremely hard with force F p x A with p 9 x 1020 dyne cm392 1015 atm and A 1 cm2 the piston slowly moves out by d 1 cm you have spent F x d 9 x 1020 erg of energy To your surprise the piston now contains an additional cm3 of water Your 9 x 1020 erg were converted to 9 x 1020c2 1 gm of new water Note since dark energy is 68 x 103930 gm cm393 it only requires 6 x 10399 dyne cm392 tension to create However you can39t verify this experimentally because there is vacuum on both sides of the piston I We can roughly correctly intuit why this leads to accelerated expansion in the following way First forget your tendency to think that quottensionquot should quotpullquot there are no gradients here Instead consider a large volume which is increasing and focus on the difference between the energy required to make more volume because there is now more vacuum and the energy released by the gravitational binding energy Ugrav GM2R which gets more negative For normal matter expansion spreads mass out raising the gravitational energy The loss of energy to the gravitational field comes out of the kinetic energy of expansion it decelerates For radiation the situation is even worse since expansion affects both M and R in the gravitational term But for vacuum the M2 term beats out the R term and the gravitational energy is more negative one detail this only becomes true for regions larger than the Hubble volume The whole thing is not unlike a ball talling downwards by moving down R is smaller so Ugrav is more negative to conserve energy the ball must pick up positive kinetic energy and hence it accelerates For a vacuum dominated universe a larger universe has more negative binding energy G M2R increases Some of this gravitational energy is used to make the additional vacuum Where does the rest go Into the kinetic energy the fluid accelerates Thus a vacuum dominated universe quottalls downwardsquot by expanding and accelerating h Cosmic Cooling Consistent with your intuition cosmic contents cool with cosmic expansion Ultimately this occurs because particles move into receding regions so seem to have lower energy There are two situations to consider photons amp matter strictly relativistic amp nonrelativistic i Photon amp Neutrino Cooling I The relativistic content includes both photons and neutrinos both have a black body spectrum Expansion preserves the black body shape but at a new temperature TZ TO1Z TO a Let39s quickly show that this is the case o Currently we have a 1 T0 2725K amp BB photon number density at frequency V0 nto d yo earc3 yo expht39OkT0 11 d yo 0 At a time when a 1 the corresponding frequency is II V0 a giving d V d V0 a conserving photon numbers gives their density change nl d V nUo a3 d 110 Inserting these into the current BB spectrum we get fileUsersdmw8fSitesastr553indexhtml Page 19 of 53 Whittle ASTR 5534 Extragalactic Astronomy fileUsersdmw8fSitesastr553indexhtml 12272006 1234 PM a3 nll d 1 arrc3 y a2exphVakTo 11 a d 12 giving nb d 11 anc3 V2 exphllkToa391 11 d 11 o This is another black body spectrum with TTOa TO1Z This has been verified using absorption in the Cl line structure lines in damped Lyor systems eg at z 18 the Cl line ratios imply TOMB 76 K as expected ii Matter amp Galaxy Cooling A similar analysis can be applied to the cooling of nonrelativistic matter particles There are two key differences 1 The distribution is MaxwellBoltzmann in which T o lt v2 gt while for BB T xlt 11gt 2 The number density is not set by the temperature but enters as a separate variable First let39s derive the matter equivalent of the redshift relation V V0 a In time dt a peculiar velocity v traverses a distance r v dt at this new location the peculiar velocity is reduced below v by the Hubble flow v Hr H v dt 1adadt v dt vada dvv daa with solution Voca39l or for v vO at a 1 we have v vOa as time passes a increases amp v decreases hence random motions decrease and T drops ievvoa A gas of particles in thermal equilibrium has a MaxwellBoltzmann MB velocity distribution nvodvO 439l39TNO v02 m2739TkTo32 expmvo2 kTo dvO N and n in number per cm3 at time when a 1 we have v voa and dv dvoa particle conservation also requires nvdv nvoa3 dvO and N Noa3 Substituting these into the MB relation we get a3 nvdv 47l39 a3N a2v2 m2 l39l39kTo32 expmv2a2 kTO a dv nvdv 4n N v2 m27l39kToa232 expmv2 kToa2 a dv which is another MB distribution with temperature T Toa2 To1z2 Thus a gas undergoing passive cosmic expansion cools according to T o a392 which is faster than the CMB photons for which T o a391 At 2 1100 both have T H 3000K at z 0 T Ill 2725 K while Tmaner m 248 mK In practice this cooling for the baryonic gas never occurs 0 prior to recombination the gas is coupled to the radiation and follows its temparature c after recombination other processes virialization shocks stars AGN heat the gas However one can consider the peculiar motion of galaxies in a similar manner Cosmic expansion is continually quotcoolingquot their peculiar velocities while gravitational interaction is continually heating them eg cluster virial motions Page 20 of 53 wnm AirksSzo mwm Aslvanmw x2272nn512 z m 5 casmic Geamew RobertsanWalker Metric ax mese pvehmmawes new gene we ae veady to begm am eescnpuen 039 me Unwevse causal danvlry pvenees awamewovk V01 eescnpng both geomeuy amp eynarmcs Enstem s equanens 039 GR can be wmen n adeoepuvew compact Vovm BM28113202 TW 3W amp T mm A A v nggeswyg we mam n 1 Tne ndwoes w mas denote 1 time amp a space ooovdmales e 9 CL x y z 01 1134 G eesc pes me spaceme geome y n noovpovates me memc T eescnpes me dwswbuuon 0V masselagy my my and momauum my 12 s n me sys1em Hence me meus saymg st teHs space home me We space teHs mass how s move The eenvanen and semen 039 me GR equanens s anuge SJqu unto sen Heve we wu exlvact omy mat we need Loosew speakmg ans secnen Week at me geomeuy pat G The next secnen Week at me eynarmca pat I and new ve ales to G a SpaceTime Geometries Mankmd s anempt to Vovma y eescnpe Name pegan wan geome y me pvopemes o39space Not snpnsngm Euchd s eescnpuen aese em 039 cm 003 Mar space waistensed by paa eu hnes nevev meet memo 319 s 039 amang e aim to 1am acme nx aea m2 aspneve nx aea A m2 and vemrne 43 m3 Tm 5 me geomeuy we mst enooumev n gvade schook W was and at gvapn papev y Hm pm enme TRUE Howevev mey ae not We wuuva novms men sewemdem eenecmess mevew veuems ouv paeena expenence weave wan gvea aocuvacy mat a paw We 01 uavevse me Unwevse and mmgs ae dmevem How do we eescnpe omev pessmnnes7 Usng memcs A memc eennes me sepalanon as 039 me neaby pems ev evens n we ndude me i Example 1 Flat 3 omeqry We cax me Euchd space al because n 270 n news We en anat smace F 12D and 370 we nave memes as2 ex 1 av 12 e92 meve as2 as etc as2 ex W e22 m 392 e92 sn29e 2 m 392 W2 pom ev They ae equwaem and eenne me same space vwn mese rnemcs you can vecovev all me vesuHs 0V Eucmm geomeuy Eg nd me snen s1 hnes between nee pems rmrwmze U39ds to oonsuuct amang e gt us memo awg es aim 0 mm and se en n lUsersdmwmXnesxsvaSZmdex mm m 2 m 53 Whittle ASTR 5534 Extragalactic Astronomy fileUsersdmw8fSitesastr553indexhtml 12272006 1234 PM ii Example 2 Uniform Positive Curvature In 2D curved surfaces are everywhere wineglasses chairs lampshades Drawing triangles on them quickly reveals interior angle sums different from 180 In general sums greatersmaller than 180 define positivenegatively curved surfaces table 1 Such surfaces have metrics which are different from those above Anticipating an isotropic universe consider the simplest isotropic surface a sphere of radius R Take coordinates r afrom a pole where r is measured along the surface image ols2 olr2 R2 sin2rR d8 Close to the pole r ltlt R we recover the quotflatquot 2D metric ds2 dr2 r2dl92 Further from the pole curvature reduces the 1192 contribution to dsz Since quotstraightquot lines provide the shortest distance between two points they are all great circles A quotstraight edgedquot triangle is therefore a spherical triangle made of great circles For such a triangle of area A the sum of interior angles is 0 ang I39 AR2 any measured triangle allows you to obtain the radius of curvature R for the sphere all triangles give the same R the space is homogeneous and isotropic the sphere has finite circumference 2l39R and finite surface area 4er2 Now consider extending this metric to 3D giving a unimaginable hypersphere of quotradiusquot R d52 dr2 R2 sin2rR dillquot2 where d39ll39 dI92 sinza dire r is a quotstraightquot line from origin To get a feel for the odd nature of this space imagine holding a laser pointer with visible beam r Turn it through 1 d39ll 1 at r 100m it sweeps sideways by 100 X 1573 174m ds at fixed r R sinrRd39l39quot at r n X 100m it should sweep n X 174m right Wrong as n increases the sweep drops below the predicted linear relation ie R sin rR lt r in fact beyond r R X 1172 the sweep begins to decrease for larger r waving the laser wildly its rays always pass through the remote location r WR A friend located at this quotantipodequot would see a laser quotstarquot zooming around the sky A second mindbending ho ho phenomenon Imagine blowing up a huge balloon starting in one direction it gets bigger and bigger After some time it fills half the volume appearing to extend flat in all directions Keep blowing it begins to bend back behind you walls now approaching getting more bent Finally the balloon walls close in on you from behind and you are inside the balloon To help with this think about drawing ever bigger circles on the surface of a sphere If you explored geometry living in this kind of space you would not recover the Euclidean results In 1840 Gauss actually tried to measure the curvature of space by surveying big triangles image Of course his modestly accurate measurements only recovered the Euclidean value of 180 But in principle he could have discovered the nonEuclidean terrestrial Schwarzschild metric in fact R N 1AU and a 30 km triangle deviates from 180 by 10398 arcsec Like the 2D analog the spherical 3D space also has finite extent IT R and volume 2 7l392Rs iii Example 3 Uniform Negative Curvature Replacing sinrR by sinhrR in the above metrics yields spaces of uniform negative curvature Unlike the 2D sphere it isn39t posible to construct a 2D surface with this property however a saddle does have a region of isotropic negative curvature in its center On this surface quotstraightquot edged triangles have 039 ang 7r AR2 which is less than 180 Furthermore in 3D two points can be infinitely far apart and the total volume is infinite Page 22 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM o This table summarizes the geometrical properties of the three spaces described above image d39t l Z Parallel Triangle Sphere Sphere Global curvature k coefficient Lines Angles Area Volume Form ll Positive 1 R2 sin2rR Converge gt180 lt 4 2 lt431rrr2 Closed ll Flat 0 r2 Never meet 180 432 43 2 Open II Negative 1 R2 sinh2rR Diverge lt180 gt439rrr2 gt43I39r2 Open iv Adding Time I It is straightforward to add an independent Newtonian time t as additional coordinate The 3D 1t metrics for positive flat and negative spacetimes become ds2 c2 on2 olr2 R2 sin2rR 192 sin26dquot 32 ds2 c2 dt2 dr2 r2d 2 sinzadfm Minkowskispacetime ds2 c2 dt2 olr2 R2 sinh2rRd62 sin29dquotfr39quot2 One can also replace dig2 sin2 dquotquotquot2 with dillquot2 the angle between the two events on the sky and group them into a single expression using Skx sinx x sinhx for k 1 0 1 ds2 c2 dt2 olr2 R2 Sk2rR all Note I39ve adopted Peacock39s notation for Sk rather than Ryden39s Notice that ds2 E c2d39l392 where d39T is the Lorentz invariant proper time interval Indeed the second is the spacetime of special relativity and is called Minkowski spacetime 0 Notice also that the flat metric involves sums of quadratics of coordinate differentials flat geometry is rooted in the Pythagoraean relation eg triangles have 039 ang 180 Although more general geometries are curved many are locally flat eg in 3D a sphere expanding the metric to first order at any location gives a quadratic form Such geometries are called Reimannian and the spacetimes of GR are all of this kind Why Because locally the Equivalence Principle demands a Minkowski spacetime of Special Relativity As you can see the three metrics above are all of this kind as r 0 they are locally flat Of course the second derivatives do not vanish and it is these that define the curvature Physically these give tidal forces which are apparent across regions of finite size v Geodesics I In plane geometry straight lines have special status Why is this because straight lines are the shortest distance between two points a In curved geometries the shortest path between two points is called a geodesic Eg on the 2D surface of a sphere geodesics are great circles hence quotaphrodisiacsquot are great circles passing through Africa ho ho a In curved 3D spaces how do we find these quotstraight linesquot fileUsersdmw8fSitesastr553indexhtml Page 23 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM 1 One impractical way would be to stretch a thin piece of thread between the two points it39s minimum energy state would occupy the shortest distance 3 Alternatively and more importantly light beams travel in quotstraight linesquot Fermat39s principle ensures they take the path of least time Ethe shortest path this occurs because longer paths interfere destructiver cf Feynman39s little book QED OJ Yet another way would be to follow the path of an object experiencing no external forces Recall Newton39s 1st law quotobjects free from external forces remain at rest or in uniform motion in a straight linequot its wavefunction behaves just like the light waves longer paths interfere destructively o In the curved 4D spacetime of GR the same methods apply Light moves along special null geodesics so named because ds cd39T 0 light never feels time We can also use a freely moving object to define a quotstraight linequot Recall in GR gravity is not a force instead mass curves space and objects quotfollowquot the space Einstein39s version of Newton39s 1st law becomes free falling objects move along quotstraight linesquot geodesics in the curved 4D spacetime eg Earth39s orbit is curved in 3space but follows a geodesic in the 4D spacetime near the sun In FRW cosmology there are two important contexts for geodesics 1 Fundamental observers ie galaxies quotmovequot along geodesics in the expanding spacetime These form a diverging nonintersecting bundle with common origin in the big bang This is known as Weyl39s postulate 1923 and it ensures a legitimate cosmic proper time 2 Light crossing the cosmic spacetime always moves along null geodesics with ds0 In practice this means we only see things which lie on our past light cone a light cone is a conic surface in a 3D spacetime in which the 4th space axis is omitted More generally the path through spacetime an object takes is called its world line World lines need not be geodesics for example if they are acted on by a force you for example held up by the earth39s surface follow a world line not a geodesic Conversely geodesics are all world lines the path through spacetime of a freely falling object b The RobertsonWalker Metric o In the 1930s Robertson and Walker independently considered metrics suitable for the Universe The required metrics need to describe a spacetime which 1 is always isotropic amp homogeneous 2 expands or contracts with time These two conditions limit the possibilities enormously 0 Note that in 1930s the assumption of homogeneity amp isotropy was fairly bold But the algebraic advantages were so great it was the obviousonly option to pursue The degree to which these assumptions are valid depends on scale On large scales gt few Mpc these assumptions are excellent see sec 2ac the RWmetric works well and the treatment is valid On intermediate scales where Killlt13 lt 1 they are still useful the RWmetric can be applied locally with parameters slightly different from the global ones this approach is integral to the analysis of the early stages of galaxy formation On small scales where tillltl gtgt 1 they are poor assumptions fileUsersdmw8fSitesastr553indexhtml Page 24 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM local matter dominates the metric which does not expand planets stars galaxies clusters electrostatic or quantum forces can also overcome the expansion atoms people rulers The form of the RWmetric is warning other equivalent forms exist this one is common II ds2 c2 dt2 at2dr02 R023k2rORO d39l39l 2 I where rO is the comoving proper distance ie as measured today to an object This looks very familiar it has the spatial form introduced above it has a scale factor at which tracks expansion usually atnow is set to 1 The primary parameters of the RWmetric are 1 k 1 0 1 specifies ve flat ve curvature and the corresponding Sk 2 R0 current radius of curvature units of length cf Gaussian curvature HE 1R02 3 at a time varying scale factor a dimensionless function of cosmic time t Much of 20th century cosmology was a struggle to find these three things As the universe expands its radius of curvature increases just like a sphere So why does the RWmetric only include the current radius of curvature R0 and not Rt In fact Rt is present implicitly I 1 one can show 6biii that Rt at R0 and this factor is indeed present in the d39lll quot coefficient 2 r0 is a comoving radius ie as measured today hence roRO in Sk is correct at all times Please don39t think of Rt as quotthe radius of the universequot it is a measure of spatial curvature although for k 1 it yields roughly the correct total volume for k 1 it is negative Also the limiting condition near k 0 with R 00 is well behaved since R sinrR r It is important to recognise that the RWmetric is independent of General Relativity it comes just from requiring isotropy amp homogeneity at all times GR enters by specifying how the curvature and scale factor depend on the universe39s contents We39ll turn to this topic in a moment after briefly getting familiar with the RWmetric 0 Simple Properties of the RWMetric fileUsersdmw8fSitesastr553indexhtml Let39s just check a few simple results which we derived intuitively in sec 3 i Proper Time What time do fundamental observers witness For them drO dquot39quot 0 and so ds c2d39ll392 c2dt2 giving t 39339 As expected fundamental observers experience a proper time on which all can agree ii Proper Distance At time t how far away is a galaxy with current comoving coordiante r0 by quothow far awayquot we mean quotwhat39s the proper distancequot how many rulers would span the gap Between us and it is purely radial so dquotlquot 0 and quotat time tquot means dt 0 so the RWmetric gives ds at drO and the total proper distance is r l ds l at drO at l drO at rO As we suspected the proper distance r is simply the current comoving distance scaled by at Page 25 of53 Wm Ammo Eungmmz Aswaan x2272nns x2 x m iii VelocityDisance Law Let s vecovev ms ve ocuy ms1awas sxpawson ve anon pvopev ve ocuy v 5 ms vale at man 1 naemes am 91 usng v an va We Havs v dvdt damva dadtxva 15dadxv man s on ve ocuy ms1awas aw v H X W Hubb e paametev H 153 dam iv Time Dilmion amp Fledshi A photon emmed wom ams1aw gaaxy a1 12 wave s to us mm W n am as u We Havs u aw 32 a cata dva a fauna faya Two photons as emmed a1 2 31th 65 aw anvs a1 nmss a aw ya WWW Colman 45 a Dmng ms ms 2 65 to a bow photons as m mgm awd so I dta V01 1m News 5 ms sme an V01 ms mu mp fataw aso ms sme so ms ma sawmsn oonmbunons mus1 J gaug nam gt n6 2 amawaue Hang snas am 1 squa Tm teHs us ma ms duvanon 039 31erth We wmsss s amen by a actov agequot A Hsunsua sxuananon 039 1m ouanon srma to anovma Doppbv sweet m gwen m ssa 3e msnnonso meve ms ouanon x ossn sssn m ms engthened ugm wvves 0V mghrz mpemovae Tamng 6 2 am 6 to be ms nms between wavsass1s 0V ugm We Havs ms oosmo ogwca vsasHm 6 6 2 121a A we 12 7 agequot n n s oss1 0 mm 039 vsosHm x Edwer m saas taste dunng ms photon s joumey V Tmal Cosmic Volume mmars ms Ma vomms 0V adosed umvevss At oomovmg ooovdmale ya aw nms L We Havs avama vectov 0V pvopev ength 1 an ya howeven swngmg 1m vamal vectov by of my swsspa om ams1awas ampg1 anonH0 of ms mu avcum evence s 2 7r ampg1 snagRh aw ms mu asa s 4 7r an R02 QuaHo to get ms vomme We mus1 megvale 1m asa ovev dv wom u 0 ms awpo e a1v7r ampg1 v A7Fat2 RffsngoaHg dv 2 an3 R03 x expected ms vomme gvows wm am am 5 dose 0 ms vaus V01 aQVspheve 0V vadms R V01 aw open umvevse ms megva dwevges osaauss 0 ms asa dwevges am my a so Nextl E ml 6 casmic Dynamics lhe Friedmanquot Equa ans Homogeneny souopy aw ms cosmoxogma Pnnaue aons Havs sng ed om ms RWrmemc x ms onw H v m 50 Rm K HWmamwmMUmssymoxHm ngszsmsz Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM These quantities are determined by whatever generates the spacetime geometry Einstein39s General Relativity tells us exactly this Guy 81quotTGc2 Tm spacetime is curved by the distribution of cosmic energy amp momentum It is now time to introduce GR so we can derive expressions for at R0 and k Fortunately for you and me the treatment will be heuristic brief and goal oriented a Gravity39s Field Equation i From Scalars to Vectors to Tensors I There is a helpful progression of field equation analogs scalar gtvector tensor 1 Newton39s theory of gravity describes how a scalar field arises from a distribution of mass V2 4l39G li39 Clearly GR aims to find a more mature quotrelativisticquot version of this iquot A similar equation applies to a static distribution of charge l339q V2 v l539q 60 However it charge moves current then the Lorentz boost yields new magnetic forces Note that the boost does not increase the charge itself We now need a vector potential AH and 4current j to describe this cZVQ 82512 A jH6O here jo is the charge density lZ39q while j1y2y3 are the x y z currents 9 Now return to massgravity When mass moves momentum the Lorentz boost does two things 1 it makes a gravitomagnetic torce 2 it increases the mass density m 39l39mo This second aspect undermines a vector treatment and a tensor treatment is necessary Instead of a 4vector massmomentum current we need a 4 X 4 massmomentum matrix Of course relativistically quotmassquot is superceeded by energy and we speak of pr the quotenergymomentumquot tensor or quotstressenergyquot tensor Here J and U are four 1 time 3 space coordinate indices eg ct x y z or ct r 6 3921quot ii Tm The EnergyMomentum Tensor o Tug has 4 x 4 16 elements arranged in a square For an isotropic tluid all the offdiagonal cross coordinate elements are zero TW is diagonal Toyo E lrz39c2 is the total energy density Tm T22 T33 lt poX gtc2 E is the xmomentum density E pX the xpressure y 2 etc Now pressure is isotropic so pX py pZ p don39t confuse momentum p with pressure p So Tug diag I39339c2 p p p all oftdiagonal elements are zero fileUsersdmw8fSitesastr553indexhtm Page 27 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM iii GIN The Curvature Tensor 0 So much for Type what about Guy On the LHS it provides a description of the spacetime curvature in various quotdirectionsquot In general its components are complex combinations of 2nd order coordinate partial differentials it is afterall replacing V2 in the Newtonian Poisson equation The general curvature is the Reimann tensor of rank 4 RugIa with 44 256 elements Symmetry reduces this to the rank 2 Ricci tensor Rm and its scaler trace the Ricci curvature R Finally the curvature is expressed as Guy pr 12 guy R where gpy are the metric coefficients I There are two very important constraints for the cosmological spacetime 1 Everywhere the spacetime is locally flat the geometry is Reimannian recall this is ultimately rooted in the Equivalence Principle 2 Isotropy ensures all offdiagonal elements are zero GM is diagonal furthermore spatial isotropy demands Gm G22 G33 Evaluating the elements for the RWspacetime one obtains Goyo 3a2 k 02R02 dadt2 Gm G22 G33 1a2 2 a d2adt2 k 02R02 dadt2 iv Two Cosmological Equations 0 Now equate these elements in Guy to the same elements of Tm amp add the proportionality constant Goyo 3a2 k 02R02 dadt2 87FG I339 87rG02 Toyo G 1a2 2 a d2adt2 k 02R02 dadt2 8l39G pc2 awec2 T Combining these we arrive at two fundamentally important cosmic equations II dadt2 87 l39G3l539a2 kc2RO2 The Friedmann Equation or Energy Equation ll d221clt2 47rG3al39 3pm The Acceleration Equation where k 1 0 1 follows the sign of the curvature radius R0 and at is the scale factor Rather than treat these as two simultaneous equations one can effectively reduce them to one differentiate the Friedmann equation substitute for d2adt2 and rearrange dlr39dt 3a dadt l139 3pc2 now divide by dadt insert p wl539c2 and rearrange 131 W X dl39l39 da a which integrates to give lvz39 lj39o a3931 W with 1539 lj39o at a 1 This expression for l39a can now be inserted in the Friedmann equation which then yields at You will recall deriving lr39a in sec 4e using conservation of energy dU pdV Einstein39s equations implicitly contain energy conservation which therefore also provide l3quota Of course it helps to choose an equation of state p wlquotc2 of particularly simple form fileUsersdmw8fSitesastr553indexhtm Page 28 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM v A Newtonian Analog Often a Newtonian analysis is used to give insight into the Friedmann equation So let39s follow tradition and repeat that briefly here Consider a huge sphere uniformly but sparsely filled with rubble technically quotdustquot image Focus on a single rock at radius r0 it feels only interior mass M 43 7rr035390 To follow the rock39s motion define radial coordinate r at rO track using a scale factor Set the whole sphere expanding with drdt r0 dadt for our rock Finally see below assume the whole sphere expands uniformly maintaining uniform density in this case the mass interior to our rock is constant and M 43 rs339 What is the equation of motion for the rock Its total energy per unit mass is the sum of its kinetic and potential energies 12drdt2 GMr E00 where E00 is the rock39s energy at r 00 This is just the equation for throwing a stone vertically upwards if EOQgt 0 then v gt vesc and the rock escapes if EOQlt 0 then v lt vesc and the rock returns Rewriting this equation using the changing scale factor and density we get dadt2 awe3 3539 a2 EOQ r02 which is basically the Friedmann equation with EOO r02 standing in for curvature k c2RO2 Continuing the cosmic acceleration equation is immediately apparent the rock39s acceleration is d2rdt2 GMr2 4TG3 P r incorporating the scale factor d2adt2 4TI39G3 lquot a This would be identical to the GR equation if il39 were replaced by 539 3pc2 Newton obviously misses the fully relativistic energy and pressure contributions to gravity I Let39s now consider all the rocks not just our test rock At tO set them all moving outward with speed v0 HO rO further out moving faster Rocks starting at smaller rO have smaller E00 12vo2 GMrO 12 Ho2 r02 83 39 39G r02 tZ o Inserting E00 back into the Friedmann equation and cancelling the re2 terms we get dadt2 87TG3 r339 a2 12H02 83 7TG 0 which implies the following For all rocks at is independent of re and hence the entire sphere expands uniformly The velocity law v H r is preserved at all times The condition for critical expansion is Ho2 83 7TG iZ39O or critical density iZ39c 3H02 839G The time evolution is identical to an FRW Universe filled with simple pressureless matter Extrapolating backwards we infer a singular quotexplosivequot origin for the expanding sphere macaw x What about the total energy of the system If we set the expansion parameter H by the critical density we have a marginally bound system this should have zero energy when summed over all rocks For PE we just perform the standard integration for the PE of a uniform density sphere fileUsersdmw8fSitesastr553indexhtml Page 29 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM PE G l 43rrr3l339x 47rr2ri339x r391 dr 35 GM2R for R outer radius The KE we also get by integrating over the Hubble law and substituing for H2 87FGl393 KE 12 l 4er l539 Hr2 dr 35 GM2R Hence the total energy of the sphere began and remains zero throughout Later sec 7d we find a similar energy conservation result if we take R to be the horizon distance The Newtonian picture also helps us understand the quotflatness problemquot it shows just how close to Vesc we must launch rocks if they are to quotget up highquot ie make a big universe However let39s postpone this analysis until we specifically look at the flatness problem in sec 10a i 0 Finally the Newtonian framework is extremely useful in studying the growth of perturbations denser patches expand more slowly than average so the density contrast increases these ultimately act as mini closed universes and collapse to make stars amp galaxies We39ll return to this analysis in Topic 19 0 Overall then the parallel with GR is is really quite close though it does lack important details he sphere is bounded and hence not isotropic there is a special location the center Making the sphere arbitrarily large still requires an even larger space in which to embed it Since expansion is truer kinematic velocities approach and exceed c at large radii At very early times small size one finds v N c and black hole conditions Gravity as seen as a force which misses the link to the spacetime in which expansion occurs Nevertheless the analogy can be helpful as a conceptual starting point b Cosmological Parameters Let39s now draw together some useful cosmological parameter relations i Hubble Parameter H Scale factor a 1 z 1 a1 now Hubble Parameter H 1a dadt varies with time Hubble Constant HO dadtO 72 X 10396 Myr391 today39s value psm units ii Density Parameters 113 Q 0 Critical density l539c 3H287I39G varies with time Today39s value lI39CYO 3H0287TG 137 X 10397 M13 pc393 I Density parameters 1 2 lrz39 lj39c 87l39G3H 2 li39 vary with time Example of today39s value Qmp lVZ39mYO INCO 87rG3H02 ri39myo o Summing gives the total density 5391 lj39m lrz39r lZ39V Alternatively at 9m Q R all thesevary with time Including their dependence on scale factor sec 4e and inserting today39s known values gives II l39ltal39lco mo as ro a4 Qvo This is an exceedingly important relation fileUsersdmw8fSitesastr553indexhtml Page 30 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM it gives the full density evolution as a function of a or 2 using today39s measured values 39 39 391 warning l139ta l139CYO is not the evolving density parameter Ma E l ta lZ39c see sec chii iii Curvature Parameters R0 amp 2k What about an expression for the curvature radius R0 Start with the Friedmann equation and divide by a2 1a2dadt2 H2 awe3 1539 kc2a2R02 substitutetorlri39 SHz tBWG to get H2 H291 kc2a2Fto2 which yields R0 cHo k Qtyo 1 V2 notice k Qtyo 1 is always positive This is as expected R0 00 for Um 1 Current estimates put 910 1 lt 002 so R0 gt 7 cHO 30 Gpc At earlier times R a R0 is smaller I It is also useful to express the curvature term as an effective density k E k02R2H2 kc2a2R02H2 and Qkyo E kc2F102HO2 Inserting intothe Friedmann equation we find H2 H291 HZQK 1 Q 9k which is simply our Newtonian energy identig normalized to KE KE PE E00 Notice Gk and k have opposite sign ve k means open geometry Using 9k allows us to recast the Friedmann equation in compact and soluble form sec 6ci 1a2 dadt2 H02 l339tat39co Qkyo a2 H02meoa393 Lea4 QVYO kyoa39Z II where nkyo is simply1 910 1 meo 910 QVYO Finally we can reexpress R0 in terms of 2K0 and the Hubble distance rHYO c HO R0 rHYO i Qkyol V2 iv Deceleration Parameter q What about a parameter for accelerationdeceleration A dimensionless parameter is q a d2adt2 dadt2 d2adt2 aH2 To find an expression for this start with the acceleration equation amp divide by a 1ad2adt2 H2q 47rG3I5393pc2 4l39G3 i39 1 3w q 47rG3H2 3H287l39G 01 3w 12 Q 1 3w 0 In practice we should sum over the various components each with their own w q 12nm 29 29m Vanm 51 9V atanytime or fileUsersdmw8fSitesastr553indexhtm Page 31 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM ll q0 Va mp ryo QVYO today 0 Notice that q is defined to be we tor deceleration In our pre v cosmology days 1970s90s this was always ve with q0 12 no no is negligible But with a vacuum term q can be either ve or ve depending on um and Currently with Qm 026 and RV 073 we have q0 06 we are accelerating v Cosmological Constant A I What about the cosmological parameter A Recent tasion amp these notes treat this term by simply adding a vacum component Historically however it first appeared as a possible term in Guy Gulf Rpl 39 VER gulf Aggy I This propagates into the Friedmann amp Acceleration equations dadt2 87rG3li339a2 kc2R02 13Aa2 d2adt2 47l G3al39 Spcz 13Aa Since the new term appears with the same power of a we can simply incorporate it in l339 and p this is done by defining A in terms of lquotV and discovering that w 1 ll 11 arrGlF39V 3H4 units ottime392 and 35quot pV02 ie w1 c The Many FRW World Models 0 Let39s now use the Friedmann equation to study the time development of the scale factor at basically derive those expansioncontraction graphs we39ve seen since high school The form of at depends on two things 1 the curvature closed critical open which depends on Rt 2 the form of the density relation lrquota which depends on the fluid A specific solution is called an FRW world model FRW Friedmann Robertson Walker Let s begin with the general case then look at some special cases i The General Case a Start with the Friedmann equation and reexpress the density amp curvature using Q39s sec 6biii 1a2dadt2 87FG3 33339 1a2 kc2R02 H02 awe3H0 l339 1a2 H02 Qkyo l39lo2 ll39zll39jcp Qkp a2 l l39lo2 lnmp as ro a4 vo kp a2 l HO2 E2a which gives ll 1a dadt Ha HO Ea tor the evolution of the Hubble parameter fileUsersdmw8fSitesastr553indexhtml Page 32 of 53 Whittle ASTR 5534 Extragalactic Astronomy fileUsersdmw8fSitesastr553indexhtml 12272006 1234 PM where we use Peeble39s notation for Ea or EZ and the curvature quotdensityquot kyo ll Ea E ln39mp as ro a4 nvp Qkp a2 l 2 Hm 1z3 0ro1z4 9W kp 122 1 2 1 39Qtp 1 39 0mg 39 nrp39 mo ll EZ These are exceedingly important functions and are central to all cosmological calculations Notice that they involve current hence observable values of Q and so can be evaluated directly We are now set to evaluate at the time evolution of the scale factor by simple integration dadt HO a Ea giving l daa Ea HO tl dt Hot datrom 0 to a dt from 0 tot In general this and related integrals need to be done numerically Since Ea and its integral are often needed it39s good to have a working subroutine handy IDL Mathematica Matlab all have ready integrators as does Numerical Recipes eg qromb Notice we have three related variables here cosmic time t scale factor a and redshift 2 you can move between them with the following important relations between their ditterentials u where tHYO 1 H0 is the Hubble time one can also use the Hubble radius c dt rHYO da aEa etc For example the relation for current cosmic age is simply tHYO da aEa tHyo dz 1zEz along with da dz1z2 a2dz 39 dt 0 to now tHYO l dz 1zEz 0010 0 tHYO l da aEa o to 1 Let39s now look at the time evolution of some specific FRW world models ii The Concordance Model The real Universe has several components with current densities II vyo 073 meo 027 9 84 gtlt1O395 9m 100 kyo 1 Hm 0 Integrating the general relation gives a current age 0988 tHYO and at curve shown here image In practice each term is dominant over a certain range in a giving three eras during the radiation era Ea H ryolz a392 during the matter era Ea m mo as2 during the dark energy era Ea w QVYOVZ Taken individually these yield reasonable approximations for at over most of cosmic history This image shows the approximations and their errors while the table gives the functional forms The Toolbox includes a more extensive set including Hubble radius particle and event horizons Page 33 of53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM Approximations to the Concordance model Radiation era Ea g Qrvon a2 t tHYOJ daa Ea o to a m 121HY0 ryo39Vz a2 m 741 a2 h72391 Gyr 234x1019 a2 h72391 a m 116x10396h7212tyr12 m 209x103910 h72 2 18V Matter era Ea m myoyz a3932 t tHYOJ daa Ea o to a x 23 tHYO lmV2 a32 x 174 a32 h72391 Gyr a m 149x10397 h7223tyr23 Dark Energy Era Ea m vol2 ttnow 61 tHyoal daaEa 1 to a w tHYO QVYO39VZIMa m 159 na h72391 Gyr a x exp6tGyh72159 tnOW0988tHYO Two details Formally the exponential term has infinite past so it is sensible to integrate from a1 the current time The matter solution is improved slightly by adding 39000 years to correct the integral over the radiation era iii Velocity Histories An alternative and possibly more illuminating diagram than at is the velocity history vt In our Newtonian analog sec Gav how does the rock39s speed change after the initial throw To find this we simply use the Hubble relation v H r H a r0 dadt r0 HO a Ea rO By choosing r0 1 Mpc we get the velocity history for a galaxy now at 1Mpc The full velocity history for the concordance model is shown here image notice the minimum at a m 065 when YO W VYO and we change from de to acceleration notice also the ever increasing velocity as t 0 a feature of all nonzero m or 1 models The initial expansion is essentially infinitely fast with at curves all starting out vertical This parallels the Newtonian case where v Hvesc GMr 2 00 as R 0 For the universe va increases even faster since radiation dominates giving Va 21391 as a 0 0 Don39t confuse va with Ha though they are related Ha va a 1 z va Ha gives the velocity at 1 Mpc real distance not 1 Mpc comoving distance Thus Ha increases even more quickly as a 0 iv Flat Models Single Component The special case of flat geometries with a single component are good pedagogical models They also illustrate the behaviour of the various quoterasquot in the multicomponent models 0 We use t tHYO l da aEa from 0 to a with Ea containing just one term fileUsersdmw8fSitesastr553indexhtml Page 34 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM 1 Pure matter 9m 1 a Ea a391 2 t 23 tHYO a32 a 3211H912 3 2 Pure radiation ryo 1 a g a21V2 r 121HYO a2 a 2r1HY0112 3 Pure vacuum QVYO 1 a Ea 312 2 t t0 tHYO loge a a elO toIH o 4 Pure curvature kyo 1 a Ea kyofZ t tHO a a t NHO These recover the forms from sec 4f a 0 t23 t 2 e1 for matter radiation vacuum The pure curvature model is open not flat with simple linear expansion at all times Pure matter amp curvature models are also called Einsteinde Sitter amp Milne sec ecvi Note the coefficients are different from the concordance approximations because the current Q39s aren39t unity For a general equation of state parameter w we have sec 4f a Ea a39lt13Wgt 2 t 233w tHYO alt33Wgt2 a 3 3w2 t tHYO 12 lt33Wgt you can quickly check this gives 1 amp 2 above for w 0 and 13 Note also that the flat model with w 13 behaves like an open pure curvature Milne model It is easy to find the current age of all these models just set a 1 and solve for t Here I chose to use tH in place of tHYO since the relation for tage is true at all times As we suspected from the outset tage m tH to within factors unity The exception is pure vacuum with exponential expansion and infinite age t 00 as a 0 tage 23 3wtH 23 tH 12 tH 00 tH for1 4 above Note that w 13 separates acldecelerating models and hence tage greaterless than tHYO For example for flat matter Einsteinde Sitter w 0 we have tage 23 tHyo a famous result By including a component with w lt 13 eg vacuumlambda then tage can get longer than tHYO This was a key motivation for including Ato solve tHO ltt1 1930s or tHYO lt tGC 1990s 0 Some more at curves for various models are given here images v Flat Models Matter Vacuum As it happens it is possible to derive an analytic form for a flat universe with both nm and 9V Apart from a brief 50 kyr radiation period this is basically the Universe we live in As an excercise in integration follow the analysis through to derive the following result at myo vyo 3 sinh2332QVY012ttHYO 0712xsinh23128xttHYO tage 23tHJOQVYO3912 sinh39l QVYO nyo 078gtlt127tH 09931HY0 which recovers an age conveniently close to tHYO E Ho391 ie the decelerating amp accelerating portions quotaverage outquot to approximate a constant expansion the numbers here use VYO 1 me 073 vi Curved Models Matter Only For a long time 1960s90s curved matter models were favoured justifying their inclusion here open with infinite future closed ending in a big crunch or quotcriticalquot balanced between the two fileUsersdmw8fSitesastr553indexhtml Page 35 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM There is however a more compelling reason they are crucial in theories of galaxy formation while the real Universe seems to be globally tlat this is not true locally voids form from open regions while starsgalaxiesclusters form from closed regions FRW curved matter models apply since local evolution is independent of the surroundings o Forasingle component 1339s lrz39syo a3931W and a Ea 9810 a3913W 1 nsyo1 2 For w 0 matter we have ta tHYO l meo a391 1 meo 3912 da from 0 to a This has a parametric solution which uses a quotdevelopment anglequot ll The form of the solution depends on whether 9va gt 1 closed or meo lt 1 open For closed models with ll 0 to 27f the Universe expands halts and recollapses in a cycloid aquotl39 12 amax 1 cos ll ll tquoti39 121H amaXquoti39 sin li39 myo 1 2 which turns at a7T amax nmvo mp 1 and collapses at t2l39 39l l39tHO amax vao 1 2 for example it meo 15 we have amax 3 and tcrunch 67 tHYO For open models the Universe expands forever with similar solution with ll 0 to 00 aquotl39 12 apar cosh 7 1 II tl i39 121HY0 apasinhquotquot vquoti391limo 2 where apar meo 1 nmyo plays the role of amax Graphs of these classic solutions for meo 09 00 11 are shown here BR 61 vii Specific amp Named World Models This figure illustrates most classes of FRW world models The two key parameters are 1 curvature closed k 1 0k ve tlat k 0 0k 0 open k 1 k ve 2 lambda RV ve odd 0 simplest ve what seems to be true a Full histories in linear time usually consider pressureless matter along with vacuum This is because radiation quickly becomes irrelevant when lquot lt lquotm at t N 50 kyr I Referring to the figures the overall forms are negative unusual simply adds to matter all models recollapse zero A most discussed until recently all decelerate tuture depends only on curvature closed recollapses tlat halts 00 open expands torever Examples for Hm 09 10 11 are shown here L76 BR 61 positive A early matter dominates decelerates late vacuum dominates accelerates For a closed geometry additional possibilities include see below fileUsersdmw8fSitesastr553indexhtml Page 36 of 53 Wmvnve mum Eunngxuvz Mmmv x2272nns x2 x m F010Vlt V E bom oovvabse and bounce sovuvvons ebsv Fov Ovz v g vacuuvn vebuvsvon and dosed geomeuy neavv baance Mong vatvc bevvods Ages and mules vov av mess vnodevs can be nvoevv ssen vn mess bvovs vmage Some ov mess vnodevs go by me Have ov moss vmo advocatedsvudved mevn a Ensbemde sue buve manev ML cvvnca d sca modev bevween open amp dosed vvorvvca y nevmev ansvevn nov de Snev advocated mvs a be sue buve vacuum vat emonamal emanslon axe vnvnvve pm Ha oonsv Thvs vnodev nx venewed vmponawce snce vv vs bevveved vo oocuv duvvng Manon vv aso descvvbes me sveadv svave modev vmvcn needs Ha oonsv a ansbem Hvs ovvgvna modev wvvaluvemanevvacuum av balance vo gvve nan unvvevse bveedaved Hubbve s dvscovevv ov expansorv doesn t m above anavss snce Ha u vrvstead ssv dadd adv u vo vnd baance vmen Ow vev 32 WWWa The vnodev vs uns1abve vo smav devanons vvovn baance a sbbmgoonvemame Two addvvvona sovuvvons vo Enstevrv s vnodev vnvvnvvevv vong svanc nvs1ovv bevove s1avvng ex son bvg bang vn vevnove on men expanson oovnes vo gvadua nav a1vnvnvve vuvuve uemavue dose vo Ensvevn modem vvm 0V 0quot svgnvvv below above a xbansongt Iomeungv gt exponenvva ex an The vnonvanon vov vovvevvng m me voo snovv Unoonecl Hubbve n The age can be vrvaemed bv vnvoducvng Levnavve bvecvenvvv aso waned a b me avvvca vaues v me va a sse mvs exmvpve vvnages g bang vo make evevnenvs a Mlne buve wvvaluve empty nov even vacuuvn enevgv k n operv k 71 a v gt no use acroevevalvorv vvnea expanson a v vav vwe vH vnonvanon mowvg gaaxves SJqu onvv vo specva vevalvvvty HT mv m Mvne s gnoonecv Next Prevl Top 7 Dismnces 8c Horizans Dscussng dvsvances on an expandvng bossvbvv wvved ooovdvrvale gvvd can be mcky Ea vvsv vers dvs1vnguvsn bevween mvee dvvvevenv vltvnds ov dvsvance menvemem 1 Hope nuances v as ue dvsvances 7 me nuvnbev ov nonrexpandvrvg vuvevs bevween obvecvs v vnaemes due o expanson adcovdvng vo me vevocvtyrdv awce vevalvorv v dvvdv v me cunem bvobev dv awce may vo an obvecv vs aso caved vvs comovmg nuance va me bvobev dvsvances av mange vvm me scae vacvov 1 ad van ad va 2 pseudo vnv venn nuances D as devvved vvovn oevvan menvemems e vuvnvnosvv DL ov mguval dvmvetev va as n spacernme weve sac and Eucvvdean mev as not vea dvs1mces so vnudn x oorvvervvem mncnons 01 nance mev as dosevv ned vo obssvvanons and bay aauoa vove vn es1abvvsnvng me wovvd vnodev a veashm z mougn nov an exbvvcvvdvsvance vv vs ouv bvvvnav obssvvabve vv mundmwmmevxsvaSKvndex mv V19 7 m 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM it is derived from spectra and gives the scale factor when the light set out ate 1 21 Given a world model it is relatively easy to derive all the other distances using 2 a This section deals mainly with proper distances r The following two sections consider among other things the pseudo distances D 0 Before we start let39s recall two usefulsensible units of time and distance for cosmology II Hubbletime tHYO Ho391 100 h391 Gyr 139 h72391Gyr ll Hubble distance rHYO cHO ctHyO 139 h72391Glyr 426 h72391Gpc they are units comparable to the current age and visible size of the Universe a Three Proper Distances Because the Universe is expanding the proper distance to an object depends on time rt In fact there are three distances one might consider image 1 The proper distance to the object when the light set out rte e emit if the light has travelled for a long time rte could be quite small but certainly less than 2 The proper distance to the object when the light arrives rto o observed now recall rto r0 is also called the comoving distance Since te lt tO then rte lt rto or using the scale factor rte ate gtlt rto ate gtlt rO The distance light travelled during its journey rlbt ctO te lbt lookbacktime expansion guarantees that rlbt is intermediate in length between rte and rto the light travel time tO te is also the lookback time as invoked in quotpopularquot statements quotThat galaxy is 5 billion light years away hence we see it as it was 5 billion years agoquot 9 o What is rto as a function of 2 Here are two approaches Imagine a photon at time t on its way to us In interval dt it crosses c dt which then expands to become c dt at at time t0 tnOW We can also use the various forms for dt given in sec 6ci to give several equivalent integrals ll rto cldtat 1e toto rHYOldaaz Ea ato1 rHYOldzEz zto 0 Alternatively from the RW metric photons move on radial null geodesics so dlrll ds 0 giving c2 dt2 a2t dro2 cl dt at l dr rto te to t0 and 0 to re as before 0 What about the lookback time Again using the alternatives sec eci for dt II LBT to 1e l dt 1e to to tHyol daa Ea a to 1 tHYOJ dz1z Ez zto 0 Which are all straightforward to evaluate numerically I Examples of rto rte and rlbt ctO te are shown here images fileUsersdmw8fSitesastr553indexhtml Page 38 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM A different presentation using quotThe Astronomer39s Universequot is shown here image i Two Examples 2 6 amp 1000 Let39s do a couple of examples using the Einsteinde Sitter Universe vao 1 MO nkyo 0 In this case Ea a3932 and Ez 1 z32 1 For a z 6 030 what are r00 rtea amp 0 t0 te There are several ways to treat this let39s use the z relations r00 rHyol 1 z393 2 dz 2 to 0 2 rHYO 1 712 124 rHYO rte rto1 z 124 rHYO 7 018 rHYO to 1e tHJOlI1 z395 2 dz 2 to 0 23 tHYO 1 732 063 tHJO The 030 was 018 rHYO when the light set out it is now 124 rHYO the light travelled for 063 tHYO 2 Pushing further consider the CMB at z 1000 we have r00 2 rHYO 1 10001 2 194 rHYO rte 194 rHYO 1001 00019 rHYO or only 8 Mpc We are seeing the CMB when it was closer than the Virgo cluster That is the main reason physical scales appear so large on the CMB sec 8b b Horizons a Current estimates suggest 9101 10 we are in a spatially infinite Universe Can we see all of this universe No for three somewhat overlapping reasons 1 Right now galaxies beyond rHYO cHO 426 Gpc are receding faster than light As a result light is actually getting further from us not closer 2 The universe is only 14 Gyr old light cannot travel infinitely far in that time 3 Our view of the universe is inevitably restricted to events on our past light cone this is a 3d slice within a 4d spacetime like a sheet in space There is much in the Universe not on that sheet I On earth our view is limited by the horizon Hence some of these cosmic limitations are referred to as horizons i The Hubble Sphere rH where expansion velocity c 0 Consider point 1 the velocity distance relation v Hr defines a critical radius rH cH c tH Beyond this distance objects recede faster than light and wavefronts actually get further from us If the expansion remained constant then ultimately we cancannot see objects insideoutside rH Of course the rate of expansion changes and so therefore does the size of the Hubble sphere For anytime c Ha rHa HO Ea rHa giving II rHa rHoEa rHoEz fileUsersdmw8fSitesastr553indexhtml Page 39 of 53 Whittle ASTR 5534 Extragalactic Astronomy fileUsersdmw8fSitesastr553indexhtml 12272006 1234 PM or reexpressing rHa in its larger current comoving size royHa we have In physical coordinates rH increases from zero at t a 0 z 00 and continues to increase In comoving coordinates rOYH growsshrinks in deaccelerating universes rOYHa rHaa rHYOa Ea rHYO 1 zEz For the concordance model ignoring the early period of inflation we find for comoving rOYH the Hubble sphere initially grew outwards eg at z 9 a 01 rOYH 061 rHYO rOYH then grew to a maximum size when 9m R at a 072 when rOYH 115 rHYO N N N N 49 Gpc Since then rOYH has been shrinking and is currently at rOYHa1 rHYO 426 h72391 Gpc It is currently shrinking at a rate drOYHda gtlt dadt HO drHda 018 MpcMyr or about 06 c Objects currently at the Hubble sphere are not of course visible to us nor will they ever be ii The Particle Horizon rm r furthest currently visible Consider point 2 above and ask what is the furthest object currently visible in the sky lts light has travelled nonstop since the big bang so that t6 0 and LBT t0 te tage We want the comoving version of this distance rto From the above sec 7a relations just take limits t 0 to to or a 0 to 1 or z 00 to 0 ll rphorto cldtat 01010 rHYOldaa2Ea 0101 rHYO ldzEz 0010 0 For nonzero 9m or 9 the integral is finite and there is indeed a most distant object visible For example take the Einsteinde Sitter case flat matter only Ez 1 z32 giving rphoto rHYo I 1 z393 2 dz 0010 0 2 rHyo so our three proper distances are rte 0 rlbt c t0 23 rHyo rto rphorto 2 rHYO Using the concordance values in Ez gives rphor 255 rHYO 109 h72391 Gpc 355 Gly It seems puzzling that light cannot cross the quotentire universequot when it has quotno sizequot at the BB But at t 0 for nonzero m or r all locations expand with infinite speed sec 6cii Hence at t 0 all points are profoundly isolated even the closest location is outside the horizon Only after some deceleration does light make progress and crosses comoving coordinate space Only if the initial expansion is zero can light cross everywhere in the first instant This occurs in two FRW cases empty open Milne linear expansion and pure vacuum de Sitter exponential expansion In these cases there is a period when the entire universe is causally connected If the conditions are right it can establish coherent global properties during this time As we shall see sec 11 an early period of inflation may have done just this in our universe The particle horizon is important in understanding structure formation By defining the region visible at any time it also defines the region in causal contact Even when optically opaque gravity still travels at light speed and sets the sphere of influence Thus the particle horizon defines the largest possible selfinteracting region Everything on larger quotsuperhorizonquot scales is fundamentally disconnected Page 40 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM I So how does the particle horizon size grow with time Simply change the upper limit to t a z In comoving coordinates we have for the particle horizon at anytime II rphot lcdtat Otot rHyoldaa2Ea Otoa rHyoldzEz 0010 z Giving 2 a 2 rHYO for Einsteinde Sitter a rHYO for flat radiation 00 for flat vacuum de Sitter I The time defined by rphor c goes by another name conformal time quotF39t dtat o to t daa2Ea o to a E lldzEz 0010 z This is a useful surrogate for time in part because it gives the horizon size at a given t a 2 It is the time variable of choice for studying growth of perturbations Spacetime diagrams see below can also look much simpler when plotted using conformal time iii The Event Horizon remor furthest ultimately visible If an event happens now in a distant galaxy eg a supernova when will we actually see it If the answer is quotneverquot then the galaxy is said to lie beyond our event horizon Like the other horizons it can change with time and even be infinite ln comoving coordinates for anytime it is given by Which is the same as rphor but with different complementary integration ranges Not surprisingly if an FRW model has finite rphor it often has infinite rehor and visaversa Einsteinde Sitter amp flat radiation have rehor 00 while flat vacuum de Sitter has rHYO 2a2 Vacuum in the concordance model ensures we will never see remote parts of the Universe Indeed as time passes we will see less and less as the event horizon shrinks in comoving radius rehort cldtat t to 00 rHJOldaa2Ea a to 00 rHYOJ dzEz zto1 I The event horizon is important during vacuum driven inflation sec 11 A small causally connected region expands and is pushed outside the shrinking event horizon After inflation we have a huge smooth superhorizon region previously causally conntected As normal expansion resumes the horizon moves out and the region reenters the horizon It also plays a crucial role generating and amplifying quantum fluctations Ultimately these fluctations provide the seeds for future galaxy formation sec 11 o The three horizons rH rphor and rehor for the concordance model are shown here image 0 SpaceTime Diagrams I In special relativity spacetime diagrams are often useful in clarifying a situation They usually plot ct vertically and a spatial coordinate horizontally so light rays move at 45 Such diagrams are more complex on curved expanding spacetimes in cosmological GR a locally light cones are quottippedquot to align with tilted worldlines o globally light rays can follow curved paths as they move across expanding coordinate grids It is however often possible to clean up these diagrams amp reestablish 45 light paths fileUsersdmw8fSitesastr553indexhtml Page 41 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM a change the spatial axis to comoving distance so objects have vertical world lines u change the time axis from ct to c39I t conformal time 0 These images show spacetime diagrams with light cones for the concordance model There are three of them a normal distance vs normal time light cone appears like an avocado seed 0 comoving distance vs normal time light cone appears like a spikey shield volcano o comoving distance vs conformal time light cone appears like a 45 degree cone They are quite instructive take a little time to figure them out d Energy within the Hubble Sphere 0 Although we used local energy conservation to legitimately derive l39 No a3931W discussion of global cosmological energy is much more trickyimpossible not least because the Universe is infinite in extent there is no global inertial frame expansion kinetic energy is illdefined for r gt rH where v gt c gravitational potential energy is illdefined with no zerolevel at infinity Ignoring these concerns I we proceed to estimate the total energy within a Hubble sphere rH c H Our aim is merely to suggest that the Universe39s total energy might actually be ZERO Let39s push the simple Newtonian analysis sec 6a v a little further for a critically expanding sphere of radius R density lquot 3 H2 8710 and velocity law v H r PE 35GM2R 320R5 H4 G KE 12 47rr2drl539H r2 320 R5 H4G and we recover the Newtonian result E101 PE KE E00 0 for all R o Relativistically however we mustn39t forget the positive rest mass energy RE Me2 43rR3l339c2 12R3 H2 02G 12R3rH2 H4G Hence IPEI RE N R r2 so that on small scales rest mass utterly dominates cosmic energy However when R rH the ve gravitational energy grows to match the we rest mass energy eg for R rHYO we have RE IPEI rHYO5 HO4G c5Ho G x 1076 erg E 1022 MEIIE 1o11 Mgal This crudely illustrates how on cosmic scales the total energy including rest mass could be zero a A more professional analysis of inflation does indeed suggest the Universe might have zero net energy ll Everything may have arisen from Nothing This is surely one of the most astonishing of Nature39s possible properties All matter in the Universe is borrowed against the negative gravitational energy of spacetime A similar notion arises in the context of black holes the mass of a black hole can be found entirely in the energy of its gravitational field In the cosmic context the Hubble radius provides a kind of Schwarzchlidgravitational radius fileUsersdmw8fSitesastr553indexhtml Page 42 of 53 h IUsersdmwmXnesISUSSZThdax hm WhmTa mxsszb Eunqthz Asuahnmy x2272nnsx23wM RS 2GMH02 2GA TvHaJCQ GAfiTcaH33H2 Tarec cH 1H ythh Ts yhy The HubbTe sbheTe TaTs oh The madame The Th amassTaaTus aTagTam Tmage Nexll Prevl 8 observables vs Redsh A Taal 1 h The UhTyeTse have quotmum bmbeTTTes Tummos y sze veTocTty space aehsTTy me Ta Ta away mess maven bmbeTTTes ux mgTe pvopev monoh eTc The TeTaTTohsths beTweeh These Two ae sTTaghTToTwaa Th anauc anclmean space Howeyeh The Teal UhTyeTse hx aw exbahaThg bossTbTy wvved spacernme how does ThTs aTTecT The smbTe Euchdeaw TeTaTTohsthe7 WeH ThThgs geT a bT move obmbTex As usua we need To Keep 0M eye oh Iwo hew pe s T The TaeT ThaT The geomehy nghT be an 2 The TaeT ThaT auhhg TTghTsTouThey Tom ve obyecT To us The UhTyeTse hx emanue th These h we we cm aehye dose 31509 To The Euchdeaw TeTaTTohsths Tndeed ohe usJa y expvesses Them h anclmean Tom ushg apeamoduance D h pTaoe on The mos1 Tmmus oT These ae TumThosTy dT a Ce Db 31d g M tly To Keep The obhyehTToh oT TabeTTThg pseudordmawces nyh c aw uTaI aTmTeTeT dT a Ce D a Luminosin Distance 9 393 We ve ohTy used The ma baT oT The RWrmemc Now we need To use TTs angular baT Th asTanc Euchaeaw spacey vaghThess T W m39Q sheTT nyh Teams equal To Th e obsevyevbbyecT de1 T aehyea by sbveamhg TumThosTy L W oyev aspheucal awe a m T L TAM MOTmg To The RWrmemc we need To make Ion moamcatbhs T The appmpuale aTSTawce T van 2 The sunaee aea Ts Teauoea Unae ed a The eheTgyoTeam bhoToh Ts Tea A The Vale a1 ythh bhoTohs an m z T Ez 2 To u The cunem pvopev aTSTawce TeTaTTye To we Toy dosed open geomemes need by aTaeToT T 2 ye Ts also manned by aTae by T z cohsTaeT bbThTsT amp 2 The RWrmemc Toy THENBIS oh The obmoyThg sbheTe T a d5 R02 860mg a92 ah29 W2 ThTegTaThg we 9 31d Tov The ToTa Spheucal sheTT sea We geT AU W R02 SKQUDHo W DVD2 WWe Don Ho SAMFm DUB Ts acomovlng nance measure 31d Ts 0M TvsT pseudordmawce ThThTlt oT a X D x gTWTg The collecl am we pTaoed phySTCaT aea all at pvopev aTSTawce a X YD H am gtlt DU 12 a0 TT Ts maTev nagev Than The pvopev dT a Ce ax Tm Toy adosed open geomeTvy mu m 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM for flat k 0 geometry Dro r0 and spherical shells have Euclidean area A 41ll39D2 o For flux we set a 1 and include the redshift factors points 3 amp 4 above H t L439ll39D2gtlt11Z2 L4l39l39DL2 whereDLE1zD DL is the luminosity distance and is our second pseudo distance it gives the correct bolometric luminosity using the Euclidean formula 0 Making the equations more explicit Dro R0 SkrORo effective angular comoving distance roz rHYO l dz Ez from z to 0 true comoving proper distance Skx sinx x sinhx k 1 0 1 corrects for curvature R0 rHO 9K0 V2 the curvature radius sec 6b iii 9kg 1 ntp the curvature parameter Notice that for k 0 D r0 and apart from the 1 22 term we recover the Euclidean relation Likewise for rO ltlt R0 and 2 ltlt 1 we recover the full Euclidean relation Here is a figure showing DLZ for several world models image 0 There is an important detail we39ve ignored the above relation works for bolometric fluxes In practice observations are usually in a spectral band which introduces two additional effects 1 the bandwidth is stretched r 39 in 39 g 39 5 by a factor 1 z 2 the rest frame spectral region is bluer than the bandpass by a facor 1 2 K correction which lead to the following modifed relations f L4l39 BL X Ll1 z391 Lt X 1 21 units ergscm2A f1 Ly 47F DL2 gtlt LyU1 z LyV X 1 2 units ergscm2Hz lfA A0 z391 Lt1 z391 47r of units ergscm2 lify V1 z LyV1 2 MW of units ergscm2 Usually continuum fluxes require these relations while emission lines are bolometric b Angular Diameter Distance I In a static Euclidean space an object at distance d of size ds subtends and angle d39ltl l ds d In moving to the RWmetric we have two modifications 1 the object was closer when the light set out 2 curvature affects the linear distance swept out by an angle a All rays we see have been travelling on radial null geodesics d5 d39flI dt 0 for a proper transverse length ds which subtends angle d39lt quot the RWmetric gives fileUsersdmw8fSitesastr553indexl39itml Page 44 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM ds ateRo mooRemitquot ateDrodquot 1 quot DrO1 2 d39l fquot note ate is included since the light ray geodesics start at emission time te image RampT 61 Thus we have for the angular diameter II dquot r quotdsateDdsDA whereDA E D1z DL1z2 DA is the angular diameter distance and is our third pseudo distance it gives the correct angular diameter using the Euclidean tormula DAz curves are shown here image BR 74 they are famous for turning over place galaxies further away they first look smaller then stay the same then look bigger the reason they look bigger is not due to curvature but their proximity when the light set out o The above analysis applies to an object of specific physical size ds How does this change for a large comoving size dS which expands with the universe For example how big would a 100 Mpc SDSS supercluster appear at z 02 5 1000 A comoving ie current size dS was smaller at redshift 2 ds dS1 2 Using this in the relation dquot3939quot ds DA we get II dillquot dS1zD1z dSD dSDEA where DEA E D DEA D is the angular diameter distance for an object expanding with the Hubble flow Let39s do our example of 100 Mpc at z 02 5 1000 choosing Einsteinde Sitter tlat matter for this we have D r0 rHYO1l dz 1 z3 2 2 dHYO 1 1 zquot2 017 118 194 X rHYO Using rHyo 426 Gpc we have 11391quot 790 1140 0690 for z 02 5 1000 These angular sizes don39t get bigger at high 2 because our object was smaller back then Notice that our supercluster is 80 in the SDSS it is still 10 at z 5 and 07 on the CMB 0 Proper Motion Distance Consider an object with transverse velocity v1 the object takes time dt39 to travel ds v1 dt39 but we witness this as time dt dt391 z Combining the angular diameter and time dilation relations the observed proper motion is II d39rr39y39th dsDAdt39 12 vtD vtDM where DM E D DM D is the proper motion distance and is seen to be our original pseudo distance D it gives the correct transverse velocity from a proper motion assuming the Euclidean relation This is the appropriate distance to use when measuring projected jet speeds in radio galaxies note David Hogg39s classic quotcheat sheetquot uses DM DO and DH for my D rO and rHJO d Surface Brightness o In a static Euclidean space surface brightness is famously independent of distance fileUsersdmw8fSitesastr553indexhtm Page 45 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM SB fd39lll quot2 L47l39d2sd2 L47rs2 independent of d Moving to the RWmetric f decreases 39x 1 z392 and dlrll2 increases a 1 z2 II SB fd39lrl392 L439lTDL2sDA2 L439l39l39s21z394 This is the almost equally famous 1 z 4 rapid drop in surface brightness with redshift note it is independent of curvature and of course is Euclidean at lowz o The relations for continuum fluxes are 38 o 1 z393 and SBA a 1 z395 I the steep drop in surface brightness with redshift is a mixed blessing it severely hinders observation of high redshift galaxies however without it we would all be dead 1 z 4 takes the utterly lethal black body radiation at recombination and renders it harmless e Cosmic Volumes 0 As we look out to a given redshift the volume witnessed increases Let39s first consider the comoving volume ie the volume it will expand to today How does this depend on redshift The proper comoving area at comoving distance rO is Aro d9 D2 where D ROSkrORO The proper comoving volume of a shell of depth drO is dVC Aro drO d9 D2 drO Now since r0 rHO l dzEz then we have drO myO dzEz which gives dVC d rHYO D2 dzEz which can be integrated between redshifts ll Vcz1 to 22 erHyo Ll Dz2 dzEz from Z1 to 22 Notice that Dz is itself an integral since D ROSkrORO and r0 rHYO l dzEz z to 0 The total d9 439Ill39 comoving volume out to redshift 2 turns out to be VCZ 2 rHo3 ko l DrHo1 mommph I kp I39Vz Ask kp DrHo 1 where ASkx arcsinx amp arcsinhx for k 1 amp 1 and VCZ 43 7r D3 for k 0 Let39s now consider the actual not comoving volume which is much less as 2 increases Repeating the analysis but keeping noncomoving quantities we have Ara all a2 D2 dro becomes a dro a rHyo dzEz giving dV all rHyo 02 1 z393 dz Ez ch 1 z393 which can also be integrated to yield a significantly smaller volume f Summary of Relations 0 Let39s gather many of these relations together Imagine observing an object at redshift z with bolometric luminosity L and physical diameter S fileUsersdmw8fSitesastr553indexhtml Page 46 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM First some auxiliary functions amp definitions we39ll need an mo1z3 Qro1z4 The kp 1 321quot2 kg 1 39 Qmp Qro Qvo 1 39 910 D R0 SkrORO with Skx sinx x sinhx for k1 0 1 R0 rHYOIQKYOIVZ with rHYO cHo ctHYO drO cdta rHO dzEz myO daa2Ea 0 Proper distance at time of observing r00 r0 cl dta 1e to to rHYOllRdzEz z to 0 rHyodaa2Ea a to1 Proper distance at time of emission r1e a e r00 r00 1 Z Lookback time LBT to 1e l dt 1e to to tHYOJ dz 1 zEz z to 0 tHYOJ daa Ea a to 1 o The physical and comoving Hubble spheres at redshift z rHz cHz rHYOEz rOYHz caHz rHYO1zEz o The comoving particle horizon at time redshift scale factor t z a rphor c dt a 0 to t rHyo dz Ez 0010 z rHyo da a2Ea o to a I The comoving event horizon at time redshift scale factor t z a rem c dt a t to 00 rHYO dz Ez z to 1 rHYO da a2Ea a to 00 0 Observe a bolometric flux W m392 f L 4 l7l39D2 DL D1 z is the luminosity distance I Observe an angular diameter radians l S DA DA D1 z is the angular diameter distance 39lll llc SC D for an object expanding with the universe with comoving size SC 0 Bolometric surface brightness SB W m392 sr39l SB L32x1z394 o For transverse velocity v1 km s39l we observe a proper motion PM radians s39l fileUsersdmw8fSitesastr553indexhtml Page 47 of 53 wnm Ammo mwm Aswanan x2272nn512 z m PM v DM DM D 5 me pvopev monon dame Comovmg pvopev vomme dvD m3 n amen norn z to z dz n 9on 319 s d0 dvD d0 D2 dva d0 D2 MD dz Ez d dvD x 1 z393 5 me physical nonroomovmg vomrne n me smwe sneu 9 Useful Charts w n m Dev 039 Wm and gvapns men snow ooncovdance mode oosrmc pvopemes nu an H n pauw al mey an use Imam nme snce s s doses1 to on expenence To oovev evevylmng we need nee epocns a Fuumsmvy Pame evs and Events awd aapns a ThemstGyv Pame evs and Events 0 Themst kyv Pame evs and Events Ves mese dwd me me a ong me to make Nexl Prevl Top 9 Measuring casmalagical Parameters Ea ygeo ogy m pvenocuped Mn mennng me sze shape oomposuon and age loe Eam 9o 00 Mn eaw ne vecerm wsmoogy n ng housed on mennng me om 005mm paametevs ndeed adme n women m m wsmoogy m the hum V01 mo numbevs HD and a Today we vecogruze seveval move Hm W W a W WV pm a ew omevs Em me 199u72mu penod x no ess concemed Mn mennng mese Mndmwema paametevs How does one go aoom mennng mem7 Theve s an ovevan suategy The ve anonswps denned n me pvemous secnon ndude many 039 me om paametevs ence one needs to compare quotmum and unsaved pvopemes 0V dam objects The mam dummy 5 Knowng me nmnsc pvopemes gt one need good 5 31de cawd es 01 5 31de Mesz x we swan see Some useM guves Wages The Pumay cosmoxogma Pame evs Mennng Mean Component Densmes Num Dev urns Lurmnosuy Ds1ance Meawernen s An ma Dme ev Meawernen s e ev Next Prev Top 10 Prahlems Mlh lhe Smndard Madel n HA5wdmwmKnaSxsvaSZmdax nwnw mg m 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM Some useful figures images The Flatness Problem The Horizon Problem The Structure Problem The Antimatter Problem The Entropy Problem The Existance Problem The Standard Hot Big Bang model described so far seems remarkably cogent Recall its basic assumptions are c The Cosmological Principle large scale homogeneity amp isotropy yielding the RW metric General Relativity relates contents to dynamics and geometry Five components with equations of state radiation neutrinos baryonic matter dark matter dark energy An initial perturbation spectrum close to quotscale invariantquot Adiabatic cooling from a hot early phase 0 The Standard Model of particle physics 00 O Pk o k 0 QED QCD etc This framwork manages to account for an extremely wide range of observations It also provides several independent estimates of its basic parameters Apart from the unknown nature of dark matter amp energy the framwork seems quite robust However there exist a number of problems with this standard picture quotProblemsquot here does not refer to an error or mismatch to data Rather they are deeper concerns about the seemingly unlikely nature of the initial conditions why was the expansion launched exactly the way it needed to be to yield our current universe This section summarizes these cosmic peculiarities 7 of them The section following shows how an early burst of inflation can explain how most of them come about a The Flatness Problem fileUsersdmw8fSitesastr553indexhtm The flatness problem starts our by noting the Universe is close to a rather special condition a The universe39s geometry is within a few o of flat zero curvature 0 its density is within a few of the critical density 0 its expansion speed is within a few of the Newtonian escape speed As we39ll see the standard model rapidly evolves away from this special state not towards it Having expanded over many factors of 10 its early state must have been exceedingly close to the special e In the absence of an explanation for this fine tuning we seem to have a quotproblemquot i Newtonian Escape Speed Let39s return to our Newtonian sphere of expanding rocks to gain an intuitive picture review sec 6a v Imagine a large sparse spherically expanding ensemble of rocks and focus on an outer rock moving upwards recall that no rocks pass eachother whatever its radius this rock always has the same mass M beneath it We will imagine that the whole ensemble was once at one location and was quotlaunchedquot in an explosive manner Since M is constant near the origin our rock must initially move very close to vesc to get quotup highquot If it starts at rm moving at vin and turns around high up at rtum then we have for vin and vesc 12 Vin2 GMrin E00 GMrtum and vexm GMrin Page 49 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM Hencewv1n2GM1rm1rlum giving VinVesc21rin391rturn1rin w 139rinrturn So in order to reach rtum it must start within Av of vesc where Av vesc m 12 rinrturn As rm gets smaller and smaller the initial velocity must get closer and closer to vesc This is the Newtonian analog of the quotflatness problemquot in order to get so BIG the Universe must have been launched incredibly close to vesc Let39s briefly illustrate with the Earth radius 6000 km where vesc 11 kms Imagine starting when R 1 km where vesc 850 kms lt55 m 106 tonnes cm393 To reach 6000 km it must be launched within 007 km s391 10394 of 850 km s39l From 1 m it must be launched within 2 m s391 10398 of 27000 km s391 to reach 6000 km These times correspond to redshifts of 6000 and 6 million 20 kyr amp 1 wk not all that remote In this terrestrial context such precision would require a bizarrly controlled initial explosion In the cosmological context of course inflation provides precisely this kind of quotexplosive launchquot ii CurvatureOmega Evolution This topic is usually cast in terms of curvature which is of course related to density Measurements suggest the current total density 910 is within 5 of unity equivalently Qkyo lt 005 Now although an exactly flat universe is always flat what if the current Universe is in fact slightly curved Let39s derive how the curvature changes as the Universe evolves does it get more or less curved I The current total density parameter is 9m PLO l 39cyO 87TGF391YO 3H02 and its deviation from 1 specifies the spatial curvature 0m 1 Qkyo kc2Fl02HO2 see sec 6b iii At epochs other than the present Ha 87TGl39t 3H2 where H and lZ39t are now both functions of a Let39s substitute Ha HOEa and 15391 IPCJO Emma39s eroa394 QVYO E2a Qkyoa392 111a 81161500 53911339co3H02E2a E2a 9koa3921 IE2a 1 9m I a2E2a which is the relation we need More specifically since a2E2a Qmoa391 eroa39z 04an kyo then for any currently nonflat universe kjo 0 we find going backwards in time ie a 0 the geometry was more flat at 1 9k 0 as long as either matter or radiation dominates There is a more transparent way to express this using a2E2a a2H2a Ho2 va Ivo2 This is the normalized velocity history discussed in sec 60 ill with its image shown here Summarizing II ma 1 Qkoa2E2a 1 kovaVO12 giving a simple rule 0 Decelerating expansion matterradiation makes the Universe less flat 0 Accelerating expansion vacuum makes the Universe more flat Hence for the standard model the Universe was flatter in the past and will get flatter in the future Let39s look at the past more quantitatively In the radiation era we have sec 6c ii fileUsersdmw8fSitesastr553indexhtml Page 50 of 53 Whittle ASTR 5534 Extragalactic Astronomy 12272006 1234 PM a2Ea2 w Elma2 and t tHYOJ daa Ea o to a x 121HY0 Gm a2 m 233 x1019 h7239 l 2 a sec so the curvature becomes ll 9km Qkyo azQnO 104 a2 9K0 50x103916tsec h72391 kyo At the time of the CMB a N 1039s the curvature 9k is only 1 of the current value Qkyo at He synthesis a 10399 9k 103914 Qkyo and at the GUT era a 10 9k m 103952 QKYO Clearly the fact that we are within 5 of flat today implies the early universe must have been extremely flat Of course inflation39s early acceleration can generate just this kind of extreme flatness One final potentially confusing point although the Universe was flatter in the past ie Qk 0 amp Qt 1 its curvature radius R a RO was smaller in the past and indeed R 0 as a 0 How do we understand this apparent contradiction Easy R and UK depend differently on a Recall sec 6biii the definitions of 2k and 9K0 which use R and R0 Gk kc2R2H2 kc2 a2R02 H02E2a QKYOa2E2a So R2H2 follows a2E2a and although R 0 as a 0 R X H increases and drives UK to 0 Of course a X Ha o va the velocity history our previous result and we have v 00 as a 0 b The Horizon Problem 0 The Structure Problem d The Antimatter Problem e The Monopole Problem f The Entropy Problem 9 The Existance Problem 0 Look around you the objects the trees the earth moon sun and stars Where did all that matter come from As physicists we are triply perplexed a more than most people we appreciate just how MUCH matter exists in the universe b we know that matter is concentrated energy multiply by c2 and its a HUGE amount of energy c we have deep respect for conservation of energy again where did it all come from I From sec 7d we get half the answer appearances are misleading integrated over a Hubble sphere the positive massenergy is balanced by an equal negative gravitational energy The total energy is ZERO In a sense the Universe quotSUMS TO NOTHINGquot Your matter and everything around you is quoton loanquot borrowed against a huge intergalactic gravitational debt Our puzzle doesn39t vanish however it changes to something a little more tractable what mechanism can start with nothing and create arbitrary amounts of matter and gravity You guessed it inflation can read on fileUsersdmw8fSitesastr553indexhtml Page 51 of 53 wnni Ammo mimimie Aswaan l2272nns l2 2 w some useiui iguies images i Next i Prev Topl 11 lnllalian amp lls Salulians As pioiniseq We now iuin to exmiine iniation and its iinpiications ioi Soiving ine above piooieins imianon neie ieiei any quotameva inat inquqes an eaiy peiioq oi accdaabe emanslon q s to oes no iepiaee oi oontiaqict ine standard not big bang modei uno s wtues ai ieinan vdes a Way to launch ine standard modei it Fiatnev iniation is inseiteq at me beginning and pm i As you ae piooaqy awae it add me ioiiovnng newieaiuies in in oaeivaqe univeise m onoe WeH insqe EHOHZOH inis aioue nine to estaqisn noinogeneity anq isoimpy which we now obsevve on ialge seaes e expanson at exaciiy ine escape veiocity x aiowng ine univeise to become lame it qiove ine geomeivy exceedingiy qose to iat Euqiqian it qiove ine iota density exceedingiy qose to me qitica density a it miqiieq quantum iuciuatons into qmca peviuvbalionsi aiowng subsequent stiuciuie ionnanon a attei in ation Vehealing piouiqes an exceedingiyhoi ieiatinsncaiyqoininateq eaiy 6 a it qiiutes to NZeYO any ieiic patiqu ieit ovei iioin ine eaiiest nines eg magnetic monopoies a it iniiaqtiousty comens naming into amiaiiy ialge mounts oi some1 mg amuaiy exaciiy equa mounts oi posiuve mas enevgy anq neganve giaviia onai ieiq enevgy This ieiq is veiy sopnisticateq anq mos1 is beyond oui ieaq my expeitise 9o We ii appioaqn inis at asuitaqy unsopnisncateq ievei Weii stat win ine iinpiications oi aoceievanon and men iooilt at mat inignt be dHWig ine aoceievanon a Virtues of Accelermed Expansion i Preliminaries Fonurialeiy we nave aieaqy enoounteieq aoeeieiateq expanson in me context oi dank elegy see Topics Adii 4quot 4n We sinpiy neeq aqoininant component win equation oi state paametei wlt 46 ie p lt rPc2a Tnisioiiows qiiecity won me Fiieqinann aoceieianon equation ctQactt2 twee aJ Epic ioi Weill q am is positive giaviiy is Vepuisive anq expanson academies VevieW sec 4n1ioi an iriMiHe qeseiiption oi my inis nappens Foi aiiat geomeivy We nave see so iii P Pa a39m W and me Fiieqinann enevgy equation gives qaqi MG 5 a melona a 0W Ha a 0W which nx soiutions wgtrt an 3 3M2 mqiw m pouei iawexpanson inqex gt1 aeoeieiation ioi 71 lt w 13 lt71 W 391 a emuquotN puie exponentia expansori win erioiding nine tH Hquot W 71 W2 r imWMHF G M qg iip aaooi at i iWW 2m i mu The iast two soiutions nave no qea big bang ie agtu oniy x i LOO wih wwri Ne IIUsersldmwmKneSxsvaSZindex mi V19 52 m 53
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