Found of Cosmological Thought
Found of Cosmological Thought ASTR 302
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This 104 page Class Notes was uploaded by Khalil Sawayn on Monday September 28, 2015. The Class Notes belongs to ASTR 302 at George Mason University taught by Harold Geller in Fall. Since its upload, it has received 46 views. For similar materials see /class/215138/astr-302-george-mason-university in Astronomy at George Mason University.
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Date Created: 09/28/15
Topics Covered in Chapter 1 Nature of Matter 2 Nuclear Physics 3 Modern Physics 4 Waves 5 Nature of Light 6 Wien39s Law 7 StefanBoltzmann Law 8 Photoelectric Effect 9 Brightness 10Scale Model of Where we are A Subatomic Interlude A Subatomic Interlude Scalein m 39 I Scalein 107mm m mm I 100000000 V 10 000 nucleus i r I a 3 w 39 gt 10 15m men a 1000 I electmn nucleus 39 glo mm quark gleamquot 1 1 sum A Subatomic Interlude III Flavor 39 A ubatomic Interlude V Neutrinos are produced in the Weak Interactionquot for example Neutrinos from the earth natural radioactivity Manmade neutrinos accelerators nuclear power p ants Astrophysical neutrinos Solar neutrinos Atmospheric neutrinos Relic neutrinos left over from the big bang Neutrino Factoids The earth receives about 40 billion neutrinos per second per cm from the sun About 100 times hat amount are passing through us from the big ban This works out to about 330 neutrinos in every cml ofthe universe By comparison there are about 00000005 protons per cm3 in the un39verse Your own body emits about 340 million neutrinos per day from 4UK Neutrinos don t do much when passing through matter Thus it is very dif cult to observe neutrinos Neutrinos reveal information about the Sun s core and have surprises of their own Neutrinos emitted in thermonuclear reactions in the Sun s core were detected but in smaller numbers 13 than expected Recent neutrino experiments explain whythis is so Based u on conversion of electron neutrino to tau neutrino Detecting neutrinos requires a different kind ofa detector Neutrino Detection How we see different size obi ects electron microscope accelerator Neutrinos Y detecting the product of their interaction Neutrino Detection ll Determining the Speed of Light Galileo trie d unsuccessfully to determine the speed of light using an assistant with a lantern on hilltop a distant ttght traveis through empty space at a speed of3 s OiausRamErm d E a m thatthe eltaet rne w mg EcitpsEsUfJupttEr smuuns tmmm depended an the distance quotmumMM DfJupttErtu Earth This happens because rt takesvarytngttmesfurhgrtt tbtravet the vamng distance between Earth and tater Muhammad mmmtwt Using drt wtth a hymn W distance and a measured We gave the speed rate btthe h ht Rom g 2 Stationary mirror lt quotquotmquot 3 1 De ection A aquot939e Lightsource Observer 3r tn tasu mean and Fbueatt atsb expenrnented thh hght by bbunetng tt uff a mtattng rntrrbr and rneasunng ttrne The hght returned tn tts sburee at a shahtiy dtnerent pustttbn because the mtrrur has muve dunng the ttrne tght was travehng drt agatn gave Light is electromagnetic radiation and is characterized by its wavelength A Smmmlillxanxrztn m min bulk tight a tu spew ymnlulvnlywn Magnumqr e m v new man swam munquot hutdmnnuhngilu 1 The Nature of Lig mm a Pvnvigiltnn tttth rnathernattrtan and ph 39hiiit tn ttt Wavelength and Frequency anumn and wavelength or an Cllclmm n hequenn man etertmmaguem WIIVE tin Hzr speed ht tight 2 wmtkngth re um tm mm Electromagnetism Maxwell39s Equations H VE ammo V 313 M VIEU BE x3 1 7 v A may m Wm Because ur us e ectru and magnetu prupemes hghus a su aHed electromagnetic m iation V swb e hght faHs m the 4mm m mm mm ran e Star ga axwes and ether amen emu MW mm m 3H vvave engths Three Temperature Scales calm Fahrenhs mm Sun39s are umpemm Sun39s survm lumvnmmle Bailing pom n1 mm Frankly main or mm Absolm mu An opaque omect ermts e ectromagnehc radwatwon accordmg to us temperature m Em m Ilmvmllhw39nll llmylllmhMh Wen s Law Visible light l Illll l it ill V l I Blackbudy urve ax 5800 K E E Sun39s intensity curve 3 o 500 man 2000 3000 Wavelength nm gt Audlsuncanll Au v 6 Stefan39Bmtzmann Law mums mmm 34 aiarearezeivaxlzwwamol 4 veer lighl pm mm m s 1 The StefanBoltzmann law states that a blackbody radiates electromagnetic waves gt with a total energy ux Edirectly amp 9 quot proportional to the fourth power ofthe Kelvin temperature Tofthe object lamupon Lquammuuahlu E G T4 39 ht Photons and Planck Flam 2r Chemists Observations 1 mical substan e In a ame 2 52nd Ii h from 3 Bright lines in he through a prism light at sped Molecular hydrogen Nenn Lilhlum Imn Barium Calcium The Sun Incandeszanl lam Fluorescent lamp Ki chhoff s Laws 5 ABSOIVI39IDN ma svlnnuu mm m m mm mm um av n1 w u quotI an am y in hmnml u commuoussrsmum m mussawn svmluu hlkay In Hth n u lulu In wow mmluhwlhld mummy nwmugy mum wlvtllnglh x ma my mama In Each chemical element produces its own unique set of spectral lines Wavelenng Intensity Wavelength ii i Warclauth L Astronomers Observations Absorption mum or me Sun Iabamlory For um emission line olimn hero is a corresponding mm Ammanquot line in he xelnr spulmm hens Khu muu be run in Ihe Sun s almosphnre The almnspheve scatters 1 Why the sky looks blue The atmosphere slatters blue light more effectively than red light hence mostly red light reanhes Blue your eye when you look through a thick silte of atmosphere at the setting Sun b Why the setting Sun looks red An atom consists of a small dense nucleus surrounded by electrons Gnld loi seen edgeon An atom has a small dense nucleus composed of protons and neutrons Rutherford s experiments with alpha particles shot at gold foil helped determine the structure Electrons The humhemvmmuhsm ah amm snuc eus sthemnmlc numhenm that pamcmav eTemem The same eTemem mav have m evem numbevs m hemmhs m Ms hueTeus These suehw dmevem kmds unhe same eTemems ave cauemsnmres Thmmihg wholnm Emined phmh A6563nm A amp3nm n m Eleulunhllslmmxhnn 5 mummy 2m onegum hyammgnexinluem g 6561nmphnlun a Nemabwmsassmm nhmmmm mugy uusu nrmmmjumwm muezmn unmlhen somu mm quotmm b nndhhhuwbili y iluuvululvmv vh 4 by 2 Bu meuululummoindu m mmm bu Mghumb unl T x mmmm ammmwm 1 mam he HMMCGW d wulml Ma y 91w mlumemmeewem wan mlmnd Bohr s formula for hydrogen wavelengths Hummer at Wm WM 7 NH Modeling the Universe Chapter 11 HawleyHolcomb Adapted 39om Dr Dennis Papadopoulos UMCP Spectral Lines Doppler summery Rodsth lws vlu Nuhhlnvllwdi r emm lw lll Doppler Examples Doppler Examples Expansion Redshifts p2 three WES zlUi eleven m es Expansion Redshifts it i Expansion Example Current Record Redshift Hubbleology Hubble length DHdH Hubble Sphere Volume enclosed ll l Hubble Universe gh scene it is the limit ofthe observable Universe Everything that could have affece us 339 a o m 2 7 a i 0 Every point has ltS own Hubble sphere Lookrback time Time required for light to travel from emission to observation Gravitational Redshift Interpretation of Hubble law in terms of relativity ew way in look at redshifts observed by Hubble Redsni is not due to Velocity of galaxies r Galaxies appl ximatelv atiunarv in Space 7 Galaxies be unher marl beeauseme Since helwem Ihem Is rhysnzw exmndln as space 2x 7 rbe expansbn b1 space asRm mibe metric Equatiun alsu ailectslhewavelenmh miiebi pandsrme WaElenmh expands arm 5 ibere isa reesbm sb ebsmbibgreai redshift is due in ebsmbibgreai Expansion er wavelength br lignt nbube regular Debbier sni frum ibeai mebens Relation between 2 and Rt Using our relativistic interpretation of cosmic redshilts we write Replay Rn Redshi ofa galaxy is de ned by z Hubble Law for nearby zlt01 objects Tnus eze eeutxwm WWW R R Where Hubble s constant l5 defll led by H AR R But also forcomoying coordinates oftvyo galaxles differing by spacentlme interyal dRlwangi naye y Dummy x AWAHdRXAWAI Hence y HRH fortvyo galaxles Witn fixed comoying separation Peculiar velocities peculiaryelocities g motions of galaxies in local gmub l 39 Ell39l leasiill lll dl ninert llllt l39llllllil nil t nllCllgl ll liiie lilll ii attei since random yelocities do not oyerall increase Witn comoying separation but cosmological redsnitt does it is necessary to measure fairly distant galaxies to determine ne Hubble constant accurately Distance determinations further away in modern tlmes Cebneids in tne Virgo galaxy cluster have been measured vvltb Hubble Space av J TeleseubelBMbe avv Vlrgu cluster TullyFisher relation TullyeFlsner relationsnib splral galaxles e correlation between ndtndt eetrun vwm am lmvlnslc iunindsny dteeiexv ampere Mb absevved brientnesstd determine distenoe e Wurksuut to about zuuMpc tnen nydrooen line becumes too nard to measure Hubble time Oncetbe Hubble barameter nas been determined accuvalENi it uiyesyerv useful information about age and size oftne u uniyerse o size oruniy RecallHubblebarameterisratioofrateofenanoeofsizeof 93959 1 AR 1 dR RA RE noniyerseworeexpandinoata cun amrateiwewuuldbave dnaye H uRueconstant and Rtu MARAD tberl Wnul WADkm ie IH7H would be age ofUnlverse slrlie Big Bang WU Modeling the Universe e smol mom a nsmniaq ui emu Sunahrdcasmaiegr lMuul 39quot quot BASIC COSMOLOGICAL ASSUMPTIONS lltli39llr Mi 7 Elnstelnlus t umpieted theory at 3R 7 Expialns aherh us urblt uf Mercury perteetlv Schwarzschlid ls Wurkirlg eh biack huies ete Elhstelrr turns his atterrtlerr te deEiirlg the unlverse as a vmme H39lllv L pl39Dr lEefi rtetr homle orrrlrlot rr ill rl How to make progress 7r item m illilliriiii i lim 7 imaglrle that all matter in unlverse ls smuutned out 7 r e rghere detalis ilke stars and galaxres but deai With a Smuuth distributiun Elf matter V Thei39l make ihw illiillwm imiliiill li 7 Universe rs homogeneous 7 every plaee in the unlverse has the same Eunditiuns as every other plaee on average 7 Universe rs isotropic 7there rs he preferred dire iun in the universe err average ale Eiiiil ill 1lllvcl viuiqt 3 39c n Principle there are he speclai porhts in space Within the Universe The Unlverse has he eehterl are colleetrvelv caiied the Cosmological Principles Key Assumptions Riddles of Conventional Thinking Stability GR vs Newtonian Newtonian Universe mwi mmi Minna Emir sphere Expanding Sphere 2wain my Fates of Expanding Universe Spherical Universe Friedman Universes Einstein s Greatest Blunder 0 smie Unlverses mm m iim Hunt is emit i imi m m m minim iimimwm IHvDIKivu mm with minimumM m in mm magnum ieuiieiw mm An wimmim in a winquot View mumquot im Miami Wm i mm m mm m mi immune mi Wm m amiiniepuugiummimim mm iaue i mm mm mm THE DYNAMICS OF THE UNIVERSE EINSTEIN S MODEL Einstein39s equations of GR G desuiees me space V dESEYibES the meme nme DJWEQJYE intiming its m 5f 11 Unwse peiueme mi nrnE er H i Universe heie39s Mire we ees Mae we tei the equaan that the Univase i5 htrnnumecus and iseunui uiug iii the RW geememes iiiiiggeri iie iiiieeiiiiiiiiigEiiEiiiiiieiiiiuuii wES F i iWEIiiE ieiiiiiiie iiim iii 3 EULiaIiDiiS Eli i i m 39 iiiiciuiiiii iiiriE ii iiimiiiiii e mi ieia siaiic casewurkedas i UNVEYSE WUZCOHSIEHI Univ me spherical a suiuiiuii in his Equaliuns e We Sphere sianeu u siaiici nwuuid iapiuii sian cuiiapsinu since EvaW emacis icuiiapse wasiui me u mere wuqu men he a rlhas nNerse in 513quot UN e n exmnsnn nllnwed by a phase at eniizrse 5u Einstein could nave used thistu predicttnattne universe must be Eitherexpanding urcuntramingi soon a er Hubbie discovered that the buttnis was befo Hubble discovered Expanding I universe more mange everybudvtnuugnttnat quot Verse was expand39 universe Was static neither Expanding nur untracting Einstein called the Cosmological Constant Eiri modified nis GR Equationsl Greates Blunder f My L39fe e Essentially added a repulsive component of gravity but very recent work suggests that he WtE alled Cosmologi al Constant may have been right more later Cadid make nis spherical universe remain static e BUT it was unstable a fine balance of appasng fumes slightest man could make it expand viulently or collapse numb v Sum up Newtonian Universe Newtonian Universe mending acme VJ 7 Send yea x istwcethe Kinetic energV pev unit mass enainind Wneri ine sphere expanded to in nite sze Fates of Expanding Universe Standard Model nnna sPHERE V1ZGM Rk ZGM SM 4 16 R 25 saddened k insan From Nevvtariianta en 3 degraded YheFviedmannEmAatian dz w mhi actlvtne velautvvemmedta expand iaievei RabenmnWalkev an hmvela wtendstazem astandR gatain nnv metric a new 3 a d pastive eneidvaei dnn mass keepsexpanding iayeven reaches in Fnedmann s eddanan a isthe scale ta mvathevthanthe mdius at an in nity wtn an e velautv ta aaaie amian saneye Gravitv at mass and enevdvmine Universe acts an eaadenne scale team 5 6 LEAP gt MD SPHERE WE UWERSE nddn aeine gmv vatmass nede a unitumi Sphere adean deiadide 7quot and Wnai na ens Wneri R7 Lima Eueplaoedhvomvatmeoanstam Yemi retainssgni mnoe as an eriEVEVat 3 in n v but it istiedta ine avemll geametwat eaade Standard Model Simplifications 4 geingppieimi keys2R insulve needievnennennessenedvdensnvenendesmniine s Nawneed ieieiivisiie eddeiien mmassenergv mnseiveiien end eddeiien eisieiei e ppm Nutiuethathere pm lnmnsamhm p e nsieni Whytheenva 7 Maian ppmdns ieii am mBlgrBang Redshmi g dde ie expensen ieddeeseneidv denan pei dnii veidne iesieiinen w eiensdenineni eeiivin Universe are nediidipie mums eispeeeiine mwmuvemmparedvmhmassmdav Nlmadelsdecelevme kn AlsanawdR h e Favall nedeispeu atmnietinie peneiieu Denan Xpansan s iniiniivi end vex ieni nediidipie ei eeiiviines eieei simpli catiari Fate of UniverseStandard Model wnue eenviine independeni atmvvsture iedei ultimate ieie eniieeiv dependeni en veide em sinee nesseneidvierii dedeeses es w Fate didniveise in Newmniari idni depended en veide dis Friedmann Universe ii depends en veide eiediveidie i All nedeis pedin vvin e awe pdi enivine spneiieei ends vvin BANG Wmlethe dineiivid end win e vmripei Theoretical Observables 39Fviedmann eddeiien describesevulmmn eiseeie ieddi immine Rubensuanalkemietm ie dniveise isdimpie end nenddeneeds 39Salutian pi e eneiee m pend v ise nedei dune Universe end dives R vve eenndi epseive Ru diieeiiv vvnei eise an we epseive ie dned vneinei nedei predidiaris m absevva ians7 39Needta nd absewable ddeniiiies deiived quotuni Ru Enter Hubbl e e Since R end iisieie eie mn iansattime H mndien eiiine Nor coNsrANr Canstamarilvatapammlavtime NanivensY lhalHn 19 g H1 R ZnspRtiiei iineeveidiien eddeiienieinii 1 Replacessmle ts mR pv quotGPJL meas abiequammesl i pend R1 speiiei deen ein Aveiede ness densiveiiiei peieneiei w kr zMGH M ni M d eesdene d edive E WT eiieideeensieniv Expidieeddeiidn 2 rieieiieddiieneneieienei 1 24 1 3 KDgtamml denan 3Hi Observing Standard Model snptv universe i0 i nedeiive nvpeipeiie dnivevsei expand tarevev ev 2 p G A Critical Density 39ltHniDDKms Mpv anml s spu 7 Q ensiv is mevdndei id 3H 2 wieden alums peiedpieneiei ei M eee 39Sca es H1 5u miseeMpe dives A denan cdneni veide em dives Emmi denan 3H misee M pd iuzsxdnin muei dives pedndeiv pelvieen epen nvpeipeiie universes end dasedi nnei spneiieei dniveise in e iiei dniveise n iseenseni dineivvse ii enendesvvin msiie hmm w Deoeleration Parameter q DeceierslianPammetev Nann All siendeid nedeis deeeieieie d u Need camal giml mnseniie dnende ii rei sendeid nedeisspeeineeiien did deieniines deeneiiveispeee end ineieidie speeme nedei Summary Definitions sumquot Review How does Rt and H change in time And what is the value ofthe curvature k Need to solve Einstein s equation STAN DAR D COSMOLOGI CAL MODELS in geneiai Ein ein s equation reiaies oeumeirviu dvriamics That means curvature mus1 reiate in evuiuiiun Tums nut that there are three pussibiiiiies x win h xru Important features of standard models All models begin with R0 at a nite time n the past eThis time is known as the BIG BANG 7 Space and time come into existence at this ment there is no time before the big angi rThe big bang happens everwhere in space not at a pointi 39 There Is a connection between the geometry and the dynamics 7 Ciosed xi soidtions foruniverse expand to maximum size then recoiia se 7 Open kei soidtions foruniverse expand forever Fi t x0soidtion for universe expands forever but onividst bareiv aimost grinds to a hait m Hubble time Hubble time for nonuniform i t e Hubbie paiainetei na v2 w iesii iEiaiE iii encetn R ex pan ion steenieeininemmeii nun Beams We n teen ninnenn swim an manning Siupe cit Rn curve is dRdt Recaii Hubbie paiainetei isiatiu eiiate eicnanee ui size at Universetu size uiUnneise lg lg R 41 trunneise wereewandmu at a censtant VatEiWEWuuid haze AWAIconsIanIand Rt wanAu i inenwmiie have H ARAIM n i e iH7H would be a e OfUriiierse since Big Bang n nielime nmienne is ifitnentenim in sing Me at in M an genes We Be W n e We gif iggsm em we age We me i Friedmann Equation Terminology wnaeimmeuiieewnesurmnmnnawsniuiinnscnme han 7 Iillhhiz disanm Dcdis13nc2that iighttvaveisin a Eacktu Eins12in s ee e T HubbietimE rnis gives an appiuximate ieea mine size at r 39 inaunsewamwmveise wnenwe puttnemimetiic in Ein ein SEuuatiun ane Eu we urine universe In the amuunt uicusinic tiine since mm the GR we eenne rneumznn Equamn this is what thebigbanu in arinanmudeis thisisaiwwsiessthan mimimmdvnamicsu rwww theHubbietime GM 1 7 Link me II e tamuuntuicusinictiinetnatpasses 3 R betweenthe Emissiun eiiient w a certain eaiaw ane tne ubservatiun eitnat iient bv us wnat aYE tne tenns imuiv2d7 7 PamL IP lmrlznll a spneie centered untneEannWitn iaeius 7 Si 2 an sumveieimnsanmeiawaiian c yVie the spneieeenneeeitneeisancetnatiientcan 7 nstherateHimaneemthecasmcscaieta m traveisincethe ie ne Thisuwestheeduemtheactuai e ubseNabieUnNerse measAWA inisi mangesintime Pisthe Mai ma evand eneie dens e kisthe geametviccurvatuve eensant Critical density irwt 1i iiite Fiii ii idiiii entiariiiii iiiR2 w iiei What are the obeervabies forfiat soiution 2 k0 EC 7 H2 2 7 Fiiedmann Equatiuntnen gives R 3 R2 Hze Let imiiiiiie ii i eiiiiziiiii ii siiiiie PH iimii 3 7 Sui tnis ase occurs ittne density is Examiy Equai new to tne c Ical density me n p pm we gm 3 62 1 7 Critiear density means fiat suiutiunfura given smmwmesm BMW E WWW vaqu cit Hi mien istne must Easiiy observed e Fiat and whenmi Universes mn aniy anew in presence at parameter enaugh ma ev ln general We can denne tne density Darameter 9 Carl rlovv rewrite Friedmarlrl s equation yet again using tnis We get kc2 0 W Omega in standard models 2 O1 22 HR y Tnus vntnin cuntext ufthe standard rnudel 7 mt Nlcrt tnen universe is nvpemulie and Wlll expand meter 7 21 nlcnr tnen unNerse isllat andWill lus1 manaeetu evpandrurever Hit icot tnen unnerse is Spherlcal andvvlll recullapse i irrreri leltllol if there ls more harl a Banaln amount Elf matter lrl the universe WNW tne attractive nature at gravity vvill ensure tnat tne Universe recullap l Value of critical density Forpl esel lt Destrobserved value ofthe Hubble constant Hn72 kmSMpc critical density is equal to nameltlm HE E H alumsmquot Compare to e an tuna kean 7 pm 25 kglm7at sea level 7 a I k w xt 39 kulm The deceleration parameter q The deceleration parameter measures how quickly the universe is decelerating or accelerating In standard models deceleration occurs use the gravity of matter slows the rate of expansion Forthose comfortable with calculus actual de nition of q is Matteronly standard model ln standard rnudel Mere density istrdrn rest rnass energy at rnatter unly itturns nut nattne value uttne deceleratan parameter is given by Q q z Tnis gives a eunsisteney check for ne standard matterrdumlrlated rnudels vve can attempt tn measure 0 in tva vvays 7 Direct measurement ernuw much mass isintne Universequot re measure massdensitv and eumnaretu crltlcalvalue 7 Use measurement at deceleratlurl parameter 7 easurementutaisanaiueuustu measurement mHubbe Parameterbvuhservln aneein anslunrateasa c ne tn nukathuWH naneesWitnredsniri rurdistanlealaxies Direct observation of q Deceleratlun shows up as a deviatiun trurn Hubble s lavv A very subtle effemi nave to detect deviatiuns trurn Hubble s lavvfur uble s vvlth a large rEdShl iiwnnniqn nisnniannn i7 ihaii inn WWW 7 k i is pusmye Energy universe vynien isvynyii 53 Expand furever Mini 7 km is negative Energy universe mien is miny it Mme recuiiapses at finite time quot M VM39 7 WEI is ZEru Energy universe Wnien is vyny it Expands furever but siuvviy grindstu a naii at infinite time WEEKquot e 71 Expansion rates 2 K R 4 For at we 01 mattei rdomii iated g univerae it turns outthere i5 5 simpie soiutioi i I now Rvai ies Wit L mm x R0 Rani J a tn 0 i 2 knuvvn as ine Einsteinde Sitter suiutiun m in soiutioi is Wiin Qgti expansion i5 siower n5 5 m m R as foH Wed by recoiiapse immmdznmmi inanima dlw in soiutions Wiin Qltiy expansion i5 faster 31min be39Eii mm 535 3139 i MOdlfled ElnSteln S equatlon Modified Friedmann Equation Bi minis mi may gen We an m mmnme mew Wequotequaiunenmnemmimei neneinan g G Enuaiiun EWEmiVVE Evuiu iun u1R changesiu became T 6 c 7H1R 7 pkh 141 nas an aeeiiienai iennwnicniusmepenes en spaceinne 3 3 BeumeWiimes a censianiiacien A ThiscunsiamAGieekie ei Lambdaquotisknuwnasihe DividinngmewcancunsmennemanyMinimian immhwmamnsm ineianm isiinswqiiitmmimisiiir nine 5 A cuiiespundsiuavacuum Enaigv ie aneneigvnm Thaiaimimmmanevai Ii hassubscvim Mi assuciaiedwiiheiiheima eiuiiadiaiiun Twuaddiiiunai densiivpaiameieneimsai naieeennee Acuuidbepusiiiveuinegaive 0 p A M1 7mnyemuieeeasaminenimmienesmneke 3H m 3H1 Uni 22 andissiev p quot my quot 7 Ug iiaiw Qwuyggtvzsandmamw39wsWnicmendsmriiake Mugm WE msmm a W2 W2 Eneveyiennsin cusmuiuw ansineimrn pusmye A ave nuw u en veieneu in as dark clergy 1 QM QA Qk Generalized Friedmann Equation in terms on s The generalized Frledrnal39ll39l e uation governing evolution or RU is written in terrns orthe present 5 densitv pararneterterrns as 2 RE anR f OM OA 5 9 R Rn The cinlvterrns in this equation that vary vvlth tirne are the scale ractcir R and its rate or change 9 nce the constants Hm Mi Al are measured ernpiricallv uslrlg ubservatluns then Whale ruture or the Universe is determlned pv sulvlng this Equatlunl ulutluns howeveri are more cumpllcated than when AE Special solution 39mlll lfillu l ll Sltll39l El 7 Solution vvlth A54er i RZACRZEZ ansicin H0 constanl 7 Nu Exp 7 Clused spherical e or historical interest onlv since Hubble s dlscuvery that Universe is Expandlngl IER2 map3A3e1ir2Rz Rio Effects of A Deceleration pararneterooservaole novv epenos on both rnatter content and A Will discuss rnore later This changes the relation between evolution and geornetrv Depending on value ofA e rinsed kl Universe uuld Expand rcrever e rlat WEI ur hvperpclic that Universe uuld recellapse Consequences of positive A Because A terrn appears vvith positive power or Rin Frledmann equation effects citx increase thh tirne if R e t keeps incr i mtiliu expansiunl i we m wodm m new my mm Aquot Scale Funar R 2 x 4 Time m n 3 Wm a mm A term it ddnl In quotIt hypubnlic mesii model in e mi Ii quotmm mm unalvi mx man mimm with i lt a w DE 5mm UNWERSE d w um Saan m min m many an n nammemanu Agtu mum p Va 21 x 39 Eteam mm 39 g 4 7 Cuns tantexpansmn rate Expm 3 Mane man mm a 1 7 Nu Ehg Bang Hm 39 7 Wed But by Est ung ubsewatmns R R g u an i 2 D mmga m 5m New WWW m m 0 7 p351 dmemum madam ua awes M n CumMWbackgmunmmgaymmm wf y 1 uquot L LSJLZ mu m m c numuns hm hm K w 0mm mmmn mm mm mm mm W 111 sauna quot2315 WE 39 39 W m cum mam i 3 u Lumm n K W a v Now Modeling the Universe Chapter 11 HawleyHolcomb Adapted from Dr Dennis Papadopoulos UMCP Spectral Lines Doppler Spectral lines are the result of the interaction of light photon with gas atoms When a photon has the right energy is will allow an electron in an atom to jump to a different energy level Depending on the geometry of the gas the photon source an t e observer an emission line or an absorption line will be produced If the gas is between the photon source W a ecrease In t e intensity of light in the frequency of the 39 ent p oton wil e seen as t e re emitted photons Will mostly be in directions different than the original one This will be an absorption line If the server sees the gas ut not the original e 0 server WI only the photons reemitted in a narrow frequency range This will be an emission line Emission and absorption lines can be blueshifted galaxy is moving toward us or redshifted galaxy is moving away No Motion Motion gt UNSHIFI39ED REDsHIFrED 39 A BLuEsHIFrED Summary Redshift Laws 1 Doppler Redshift 2 zJr 2 Expansion Redshift z 1 1 3 Gravitational Redshift plus Hubble39s velocitydistance law 11HUgtltd 1 2 m obs Z labs 12m 12m Doppler Examples Spectrum of a typical galaxy at 201234 Spectra are usually t by identifying the calcium H and K lines in conjunction with the strong magnesium an sodium lines at 5178 A and 5891 A Strong absorption in hydrogen and the 6 band can also be seen in this spectrum AV l JmLNM An emissionline galaxy with 200886 The prominent features of such a galaxy are now in emission not a sorption Software fits the redshifts for these galaxies by identifying the various hydrogen nitrogen II sodium ll and oxygen lll lines Doppler Examples The Spectrum of the quasar 3C273 shows Spectra of galaxies showing the redshifted hydrogen emission features displacement of H and absorption Dislmuze i n main Vesonues m Lma 273 gale Is 0 H 30 L ti Ummm 5m a J mum mmumm Z ybm Cumna Bureaus 3C273 273rd radio source in the 3rd Cambridge catalog 1 quasar discovered quot3 quasistellar object an active galaxy or AGN an Scales Q UJ HydrogenBalmer series lines redshifted by 16 z 0159 bLZDEI H F Expansion An electromagnetic wave is continually stretched as it travels through expanding space 20 it r i l WVVWlM As the wave travels the space is stretched b the expansion of the Universe stretching the wave The amount of stretch redshift depends on how far the wave travels Objects further away will have higher redshift z Ro lsrzif R Ra scaling factor Redshifts The expansion redshift expression 1 7 1141 R MR1 i 17 no hen and velocitydistance relationship u HU gtlt d are one of the most important equations in cosmology Note that the expansion redshift is independent of the recession velocity If object is at a redshift of 21 then the Universe is twice as large now compared to when the objects was observed z2 three times 210 eleven times Expansion Redshifts Pulsed signals emitted at 1 sec intervals will be observed at 12 seconds L 1 quot R If something happens at time t at redshift z we observe it to happen at time At 1TT ll Frequencies slow down as we observe R distant objects Things appear to happen slower the further we probe cosmological distances At the frontier of the observable Universe 7 4 Time redshift z a m everything is apparently I I frozen in a static sta e Expansion Example Spectrum of a quasar at z55 Image of a quasar at z55 L I w an The feature marked Lyon is the n2 gt n1 emission feature of hydrogen shifted by 550 from a rest wavelength in the ultraviolet to 3000 A 800 nm The faintness of the spectrum below 8000 A is caused by absorption from intergalactic hydrogen clouds along the lineofsight to the quasar Current Record Redshift Record of the farthest known galaxy z1 0 year 2004 Twordlmensional Spectra ofAhell 1335 mum lVlTANTU l isAAn Record of the fartlhest known galaxy z10 year 2004 Using the ESO s Very Large Telescope and the magnification effect of a gravitational lens a team of French Swiss astronomers has found several faint galaxies believed to be the most remote known Further spectroscopic studies of one of these candidates has prowded a strong case for what is now the new recor holder and by far of the most distant galaxy known in the Universe Named Abell 1835 1R1916 the newly discovered galaxy has a redshift of 210 and is located about 13230 million light years away It is therefore seen at a time when the Universe was merely 470 million years young that is barely 3 percent of its current age Hubbleology Hubble length DHcH Hubble sphere Volume enclosed in Hubble sphere estimates the volume of the Universe that can be in our lightcone it is the limit of the observable Universe Everything that could have affected us Every point has its own Hubble sphere Lookback time Time required for light to travel from emission to observation Gravitational Redshift One of the four tests of General Relativity Light emitted from the surface of a body of an radius observed at large distance has a gravitational redshift of 1 e 1 z a The wavelength of light passing through a graVitational field will be shifted toward Where R5 3 the schwarzschud rad39us re der regions of the spectrum 2G 7395 T Gravitational redshift A g gtlt h c2 A photon 39fights its way out of a gravitational field loses energy and its color reddens Sun 2 7m l 1 123 X 106 observed in 1962 for the first time Interpretation of Hubble law in terms of relativity New way to look at redshifts observed by Hubble Redshift is not due to velocity of galaxies Galaxies are approximately stationary in space Galaxies get further apart because the space between them is physically expanding The expansion of space as Rz in the metric equation also affects the wavelength of light as space expands the wavelength expands and so there is a redshift So cosmological redshift is due to cosmological expansion of wavelength of light not the regular Doppler shift from local motions Relation between 2 and Rt Using our relativistic interpretation of cosmic redshifts we write 1 Rpresent A obs R em emitted Redshift of a galaxy is defined by M So we have Hubble Law for nearby zltO1 objects CAR AR At Thus 6226Atx d gtltH light travel where Hubble s constant is defined by Z R At 2 R dt But also for comoving coordinates of two galaxies differing by spacetime interval dRtchomovng have V Doomwng x ARAtdR xARAt Hence v dxH for two galaxies with fixed comoving separation Peculiar velocities Of course galaxies are not precisely at xed comoving locations in space They have local random motions called peculiar velocities eg motions of galaxies in local group This is the reason that observational Hubble law is not exact straight line but has scatter Since random velocities do not overall increase with comoving separation but cosmological redshift does it is necessary to measure fairly distant galaxies to determine the Hubble constant accurately Distance determinations further away In modern times Cepheids in the Virgo galaxy cluster have been measured with Hubble Space Telescope 16 Mpc away Cepheid Variable in M100 TWFPC2 Virgo cluster TullyFisher relation TullyFisher relationship spiral galaxies Correlation between width of particular emission line of hydrogen Intrinsic luminosity of galaxy So you can measure distance by Measuring width of line in spectrum Using TF relationship to work out intrinsic luminosity of galaxy Compare with observed brightness to determine distance Works out to about 200Mpc then hydrogen line becomes too hard to measure Hubble time Once the Hubble parameter has been determined accurately it gives very useful information about age and size of the expanding Universe Recall Hubble parameter is ratio of rate of change of size of Universe to size of Universe H 1 AR 1 dR R At R dt If Universe were expanding at a constant rate we would have ARAz consz am and Rz z xARAz then would have H ARAtR1t ie tH1H would be age of Universe since Big Bang Rt Modeling the Universe 0 Simple Cosmological Model a The Gravity Riddle 0 Expanding Cosmic Spheres 0 Friedman Universes O Cosmological Constant 9 Standard Cosmological Model we 1915720 We En EAltlng pawn SElnztem publlshss General Relatlvlty uExplalns Mercury s plecesslun llght bendmg pledlcs glavltatlclnal redshl s and lens ng n1 ugh uschwarzs llld makes progress on BHS deling Mme Unlverse eElnszein urns hls anem lon to l e me a who 3 cetc But GR l lergely not well understand because it ls a very complex problem BASIC COSMOLOGICAL ASSUMPTIONS Germany 1915 Einstein just completed theory of GR Explains anomalous orbit of Mercury perfectly Schwarzschild is working on black holes etc Einstein turns his attention to modeling the universe as a whole How to proceed it s a horribly complex problem How to make progress Proceed by ignoring details Imagine that all matter in universe is smoothed out ie ignore details like stars and galaxies but deal with a smooth distribution of matter Then make the following assumptions Universe is homogeneous every place in the universe has the same conditions as every other place on average Universe is isotropic there is no preferred direction in the universe on average There is clearly largescale structure Filaments clumps Voids and bubbles But homogeneous on very largescales So we have the The Generalized Copernican Principle there are no special points in space within the Universe The Universe has no center These ideas are collectively called the Cosmological Principles Key Assumptions Cosmological Principle The Universe is isotropic and Solution to the seeming dilemma homogeneous on iarge scales Make things simple Isotropic The Universe looks the same In all directions Simple assumptions 0 Ignore 39details in the universe such as galaxies stars planets etc 0 Smooth out ali matter in the universe 0 Assume universe is Homogeneous The Universe is unlfarm on large scales a homogeneous there is no preferred t piace in the universes b otropic there is no preferred quotis direction in the universe Riddles of Conventional Thinking The problem Find a curved 4D spacetime continuum which is both homogeneous and isotropic And we have two riddles to solve 1 Why is the nightsky dark holds important clues to the universe a Gravity riddle discussed in the following hasthe same uverall brighlness The night sky should be bright Stability imagine a smooth universe in the shape of a sphere ignore curvature Gravitational effect of stars independent of radius of sphere NM r 1 Fm m r 2 imagine each star on side A has one more atom than stars on side 839 Residual force SideA i SideB GR vs Conclusion isotropy needed for equilibrium of forces Alternative Solution Space is spherically curved but finite Total amount of matter in the universe is finite and all gravitational forces are finite proposed by Rieman Deadlock of cosmic forces Universe is either finite or there will be no stable conditions leading to a collapse Solution to both the nightsky riddle and the gravity riddle light and gravitational forces travel at a finite speed gravity doesn39t act instantaneously Newtonian nbound universe can only be static if it is isotropic Most scientists lt1920 were 39obsessed by the idea that we life in a static universe including Einstein General Relativity opened the door to the concept of a nonstatic universe Attempt to study a nonstatic universe a de Sitter Holland in 1917 u Friedm 39 39 u u u U Hubble USA 1929 However General Relativity is difficult to understand Surprising re elation Newtonian theory and GR theory give same results if universe is uniform Particle with tangential velocity v vm parabolic orbit V lt V elliptical orbit gt v yperboli orbit parabolic hyperbolit Static sphere Strength of gravity depends on the mass M of e sphere and the distance rfrom the 5 here but it is independent of the radius extent of the sphere Fm 6 m M l f Let the sphere expand Surface particle Expanding sphere Expanding Sphere Strength of gravity depends on the mass of e sphere and the distance rfrom e sphere but it is independent of the radius extent of the sphere F9 6 m Mrz Let the sphere expand Surface K particle Expanding sphere in case the radial velocity of the surface is smaller than the escape Velocity the sphere will expand and then collapse in case the radial velocity of the surface is equal or larger han he escape velocity the sphere will expand continually and never collapse Lets consider the energy budget Gravitational energy Egm G Mr Kinetic energy Em V2 V1 Total energy Em Em Egm or Em VzW GMr 12 GMrEm V1 ZGMr2Em V1 ZGMrk Fates of Expanding Universe V1 2 G Mr k k is twice the total energy Three cases if the velocity equals the escape velocityl then the total energy is zero k 0 f the velocity is smaller than the escape velocity then the total energy is negative klt0 f the velocity is larger than the escape velocity then the total energy is pasltlve k Let the expanding sphere represent the entire universe Curvature of the spacetime continuum k k gt 0 k D k lt 0 positive zero negative curvature cunature curvature menu we Spherical Universe 1 Spherical space Scale factor R Universe is closed k1 k1 2 Flat spac RO5 Rl e Universe is open k0 R2 3 Hyperbolic space Universe is open k 1 R1 0 R2 k 0 no curvature stretching of rubber Friedman Universes in modern reiativistic cosmoiogy the Emstem39de Enter unwerse k 0 Nature of space is kRZ with k LCM 1 R A FriedmanLemaitre universe k R Big Bang Time Big Bang Time O O Einstein s Greatest Blunder FriedmanLemaitre universe k 1 Big Bang Time 0 Einstein used all three homogeneous and isotio ic case to solve is equation of General Relativity before the expansion of the Universe was discovered by Hubb e Einstein found that 9 for a static universe only the spherical case worked as a solution for his e s to start with an expanding universe se of expansion Followed by collapse Einstein could have predicted the expansion of the Universe but he took a wrong turn Einstein modified his equations and a repulsive force39 called the Cosmological Constant quotGreatest Blunder of my Lifequot Static Universes Einstein was a firm believer that the Jnlverse l5 static he no evolution static universe has to be a closed Iniverse With positive curvature kgt0 lstatic universe has to be in an quilibrium of forces An unknown force Ieeds to balance gravity ntroduction of the Cosmological Constant s proposed by Einstein in 1 1 Static Einstein de Sitter Universe has a ipherical space closed and finite and ontains a force A which opposes gravity THE DYNAMICS OF THE UNIVERSE EINSTEIN S MODEL Einstein s equations of GR 87Z39G 4 C g G describes the space time curvature including its dependence with time of Universe here s where we plug in the RW geometries IH T describes the matter content of the Universe Here s where we tell the equations that the Universe is homogeneous and isotropic Einstein plugged the three homogeneousisotropic cases of the FRW metric formula into his equations of GR to see what would happen Einstein found That for a static universe Rz consz anz only the spherical case worked as a solution to his equations If the sphere started off static it would rapidly start collapsing since gravity attracts The only way to prevent collapse was for the universe to start off expanding there would then be a phase of expansion followed by a phase of collapse So Einstein could have used this to predict that the universe must be either expanding or contracting but this was before Hubble discovered expanding universe more soon everybody thought that universe was static neither expanding nor contracting So instead Einstein modified his GR equations Essentially added a repulsive component of gravity New term called Cosmological Constant Could make his spherical universe remain static BUT it was unstable a fine balance of opposing forces Slightest push could make it expand violently or collapse horribly Soon after Hubble discovered that the universe was expanding Einstein called the Cosmological Constant Greatest Blunder of My Life but very recent work suggests that he may have been right more later Sum up Newtonian Universe A 39new way to look at redshifts Redshifts are not caused by recession velocities of the galaxies but by the xpansion of s ace wh39le the galaxies are actually nearstationary a 2 u 2 2 o 5 39o a E w Galaxies get further apart because the space between them is h sically expanding DLie tot e expansion of the the wavelengths expand which causes the reds ift m 395 m n m Cosmological redshifts are due to smolo icai expansion of the wavelength niverse started with R0 at a finite time 30 9 in the p35 Bi Ban pace an time of light and has nothing to do With the ome into existence during the Big Bang TEQUlaquot Doppler Shift 39Om galaxy mOtIOHS whi happens v rywhere i c Space is moving apart accordin o Hubble s distancevelocity relationship 1 D 39iH Xd tHFn39il Newtonian Universe Strength of gravity depends on the mass M of the sphere and the distance rfrom e spher but it is independent of the radius extent of the sp ere Fgm 6 m Mrz Let the sphere expand Surface K particle Expanding sphere in case the radial velocity of the surface is smaller than the escape velocity the sphere will expand and then collapse in case the radial velocity of the surface is equal or larger than t e escape veloci y the sphere will expand continually and never collapse Lets consider the energy budget Gravitational energy Egm G Mr Kinetic energy Ekm 12 V1 Total energy Em Em Egm or Em VzW GMr 12WGMi Em W ZGMrZEm lf ZGMrk Send rgt k is twice the kinetic energy per unit mass remaining when the sphere expanded to infinite size Fates of Expanding Universe FINITE SPHERE V22GMsRk 2GM 1e s 2E k 2Ew R Explore Rgt 1 E lt0 negative energy per unit mass expansion stops and recollapses 2 E 0 zero net energy exactly the velocity required to expand forever but velocity tends to zero as t and R go to infinity 3 Egt 0 positive energy per unit mass keeps expanding forever reaches infinity with some velocity to spare BIG LEAP gt CONSIDER SPHERE THE UNIVERSE 1amp5 7IGpR What happens 8 2 V E when Rgt Zg 39GpR 2EOO Standard Model 9 7Z39G0R From Newtonian to GR 3 8 2 Pg 2 E 39GpR ZEOO The Friedmann Equation Egg E GpRz kcz 3 Robertson Walker RW metric kO 1 1 ln Friedmann s equation R is the scale factor rather than the radius of an arbitrary sphere Gravity of mass and energy of the Universe acts on space time scale factor much as the gravity of mass inside a uniform sphere acts on its radius and E replaced by curvature constant Term retains significance as an energy at infinity but it is tied to the overall geometry of space Standard Model Simplifications 4 F 7erRZ kc2 SE 713011 To solve we need to know how massenergy density changes with time If only mass pR3constant Now need relativistic equation of massenergy conservation and equation of state ie pE fpm Notice that here me3 constant but pER4constant Why the extra R Mainly photons left out of BigBang Redshifting due to expansion reduces energy density per unit volume faster than 1R3 Photons dominant early in Universe are negligible source of space time curvature compared with mass to day All models decelerate 118k 0 Also now dRdtgtO expansion For all models RO at some time RO at t0 Density gt infinity and kc2 term negligible at early times Great simplification Fate of UniverseStandard Model While early time independent of curvature factor ultimate fate critically dependant on value of k since massenergy term decreases as 1R Fate of Universe in Newtonian form depended on value of E In Friedmann Universe it depends on value of curvature k All models begin with a BANG but only the spherical ends with BANG while the other two end with a whimper Emma saw my k z u Friedmannrumaiue imam k z a Theoretical Observables Friedmann equation describes evolution of scale factor Rt in the RobertsonWalker metric ie universe isotropic and homogeneous Solution for a choice of p and k is a model of the Universe and gives Rt We cannot observe Rt directly What else can we observe to check whether model predictions fit observations Need to find observable quantities derived from Rt Enter Hubble H V1 154R Since R and its rate are functions of time H function of time NOT CONSTANT Constant only at a particular time Now given symbol HO F 7erR2 kc2 Time evolution equation for Ht g 2 Replaces scale factor R by 2 H2 E G ki measurable quantities H p and R2 3 R2 spatial geometry Observing Standard Model Average mass density critical parameter why kc2 2 R02 2 H0 Explore equation 1 Empty universe p20 k negative hyperbolic universe expand forever 1 Measurement of HO and p0 give 3 H 02 curvature constant k 2 Flat or require matter or energy 8 7 G 0 0 Q 1 2 M 3 kO gt critical density 3 H 0 3 H 02 PC 8 7 G 0 Q M 0 Critical Density 87er0 3H 3H 87239G Q amp M PC I00 EQM If H0 100 km s1 Mpc1 critical density is 2x1O3926 kgm3 or 10 Hydrogen atoms per cubic meter of space Scales as H2 50 kmsec Mpc gives 11 density Current value of 72 kmsec Mpc gives critical density 1026 kgm3 Q M1 gives boundary between open hyperbolic universes and closed finite spherical universe In a flat universe 9 is constant otherwise it changes with cosmic time Deceleration Parameter q A HgoM 5 M R 185 Deceleration Parameter Now qO All q E 2 standard models decelerate qO Need RH cosmological constant to change it q 2 19 For standard models specification of q determines M 2 geometry of space and therefore specific model The deceleration parameter 1 g ves a measure of haw as the Universe 5 deceieratin I q 7 m2 mrnw Hm I l 1lltth mm mm sma me mg 5qu 1quot case we Th2 quotmm denim or many come Hf uuwcrsc I5 expandhu at onsmt rain the Un verse an be measured either R diremy by determlnlng how much mass is in me Universe or by measuring the deteleratjon parameter q mm 1 m TI 4 Wm D mnte an e m Th2 nae n u New Is a wavs less than the who u39rre In standard mats Summary Definitions 9 standard Cosmological Model Homogeneous and isotropic universe that started in the Big Bang and expands A o a critical Density poi Average density of the universe needed to rna e the universe flat 9 Density Parameter as Q 9 mm determines ifthe ensity has been reached a Cosmological Constant A Acts as repulsive force to gravity set to zero in most scenarios 9 Deceleration parameter 1 Defined as q v 9 gives a measure of how fast the universe is deceiefating a Hubble Constant Hi Defined as HD veiocicy distance a Hubble Time tn Defined as t 1 since the Big Ben De ned criticai d Ha is the cosmic time 9 summary k 7 71 q C v1 2 lt 1 open hyperbolic universe k E q 12 s1 1 open fiat universe k 1 q gt 5 1 gt 1 closed spherical universe RA a Big Bang Time STANDARD COSMOLOGICAL MODELS In general Einstein s equation relates geometry to dynamics That means curvature must relate to evolution Turns out that there are three possibilities Open hyperbolic EXpands forever k1 Open flat Just manages to expand forever kO Closed spherical Recollapses k1 Cosmic time Important features of standard models All models begin with RO at a finite time in the past This time is known as the BIG BANG Space and time come into existence at this moment there is no time before the big bang The big bang happens everywhere in space not at a point There is a connection between the geometry and the dynamics Closed k1 solutions for universe expand to maximum size then recollapse Open k1 solutions for universe expand forever Flat k0 solution for universe expands forever but only just barely almost grinds to a halt Hubble time We can relate this to observations Once the Hubble parameter has been determined accurately from observations it gives very useful information about age and size of the expanding Universe Recall Hubble parameter is ratio of rate of change of size of Universe to size of Universe 1 AR 1 dR R At R dt If Universe were expanding at a constant rate we would have ARAz consz am and Rz z xARAz then would have H ARAtR1t ie tH1H would be a e of Universe since Big Bang Rt Hubble time for nonuniform expansion Slope of Rt curve is dRdt Bang Hubble time time Hubble time is tH1HRdRdt Since rate of expansion varies tH1H gives an estimate of the age of the Universe This tends to overestimate the actual age of the Universe since the Big Bang Terminology Hubble distance DctH distance that light travels in a Hubble time This gives an approximate idea of the size of the observable Universe Age of the Universe tage the amount of cosmic time since the big bang ln standard models this is always less than the Hubble time Lookback time tb amount of cosmic time that passes between the emission of light by a certain galaxy and the observation of that light by us Particle horizon a sphere centered on the Earth with radius ctage ie the sphere defined by the distance that light can travel since the big bang This gives the edge of the actual observable Universe Friedmann Equation Where do the three types of evolutionary solutions come from Back to Einstein s eq 9 872461 C When we put the RW metric in Einstein s equation and go though the GR we get the Friedmann Equation this is what determines the dynamics of the Universe zzg 39G ZZZGM T M2 kc kc2 What are the terms involved G is Newton s universal constant of gravitation iis the rate of change of the cosmic scale factor same as ARAt for small changes in time p is the total matter and energy density k is the geometric curvature constant If we divide Friedmann equation by R2 we get 2H287zG kc2 R 3 R2 Let s examine this equation H2 must be positive so the RHS of this equation must also be positive Suppose density is zero p0 Then we must have negative k ie k1 So empty universes are open and expand forever Flat and spherical Universes can only occur in presence of enough matter Critical density What are the observables for flat solution k0 Friedmann equationg tjlziceyn gives 2 H Tquot So this case occurs if the density is exactly equal to the critical density 3H2 p pcrit Critical density means flat solution for a given value of H which is the most easily observed parameter 87Z39G kc2 H2 3 p R2 In general we can define the density parameter A 871G Q 2 pcrit Can now rewrite Friedmann s equation yet again using this we get Q1H2R2 Omega in standard models kc2 Hm Cosmic lime Thus within context of the standard model Qlt1 if k1 then universe is hyperbolic and will expand forever Q1 if k0 then universe is flat and will just manage to expand forever Qgt1 if k1 then universe is spherical and will recollapse Physical interpretation If there is more than a certain amount of matter in the universe pgtpcritical the attractive nature of gravity will ensure that the Universe recollapses Value of critical density For present bestobserved value of the Hubble constant H072 kmsMpc critical density is equal to pcritical103926 kgm3 ie 6 H atomsm3 Compare to pwater 1000 kgm3 pair125 kgm3 at sea level 2 x 103921 kgm3 pinterstellar gas The deceleration parameter q The deceleration parameter measures how quickly the universe is decelerating or accelerating ln standard models deceleration occurs because the gravity of matter slows the rate of expansion For those comfortable with calculus actual definition of q is ISL F Matteronly standard model In standard model where density is from rest mass energy of matter only it turns out that the value of the deceleration parameter is given by Q q 2 This gives a consistency check for the standard matterdominated models we can attempt to measure Q in two ways Direct measurement of how much mass is in the Universe ie measure mass density and compare to critical value Use measurement of deceleration parameter Measurement of q is analogous to measurement of Hubble parameter by observing change in expansion rate as a function of time need to look at how H changes with redshift for distant galaxies Direct observation of q Deceleration shows up as a deviation from Hubble s law Velocity distance A very subtle effect have to detect deviations from Hubble s law for objects with a large redshift Newtonian interpretation is therefore k1 is positive energy universe which is why it expands forever k1 is negative energy universe which is why it recollapses at finite time k0 is zero energy universe which is why it expands forever but slowly grinds to a halt at infinite time open hyperbolic Expands forever Qlt1 k1 Open flat Just manages to expand forever 91 k0 Closed spherical Recollapses Qgt1 k1 Cosmic time Expansion rates For flat k0 91 matterdominated universe it turns out there is a simple solution to how R varies with t 120 Room 0 This is known as the Einsteinde Sitter solution ln solutions with Qgt1 expansion is slower followed by recollapse ln solutions with Qlt1 expansion is faster Scale Factor R 0 0 1 Time Fig 115 The scale factor R as a function of time for the Einsteinde Sit ter standard model This geometrically at model begins with a bang at time t 0 Modified Einstein s equation But Einstein s equations most generally also can include an extra constant term ie the T term in 8726 g T 4 C has an additional term which just depends on spacetime geometry times a constant factor A This constant A Greek letter Lambda is known as the cosmological constant A corresponds to a vacuum energy ie an energy not associated with either matter or radiation A could be positive or negative Positive A would act as a repulsive force which tends to make Universe expand faster Negative A would act as an attractive force which tends to make Universe expand slower Energy terms in cosmology arising from positive A are now often referred to as dark energy M Modified Friedmann Equation When Einstein equation is modified to include A the Friedmann equation governing evolution of Rt changes to become 2 2H p i m2 Dividing by HR we can consider the relative contributions of the various terms evaluated at the present time to The term from matter at to has subscript M Two additional Q density parameter terms at to are defined m sz k pcrit 3H0287ZG 3 102 ROZILIO2 QkE A Altogether at the present time to we have 1QMQAQk Generalized Friedmann Equation in terms of Q s The generalized Friedmann equation governing evolution of Rt is written in terms of the present Q s density parameter terms as 22 a 52 HORO QMLRQA R0 9 The only terms in this equation that vary with time are the scale factor R and its rate of change Once the constants H0 QM QA Qkare measured empirically using observations then whole future of the Universe is determined by solving this equation Solutions however are more complicated than when AO Special solution Static model Einstein s Solution with AC47po kACR2c2 No expansion H0 Rconstant Closed spherical Of historical interest only since Hubble s discovery that Universe is expanding rS R2 87sz3A3 kc2 R2 1amp0 Effects of A Deceleration parameter observable now depends on both matter content and A will discuss more later This changes the relation between evolution and geometry Depending on value of A closed k1 Universe could expand forever flat k0 or hyperbolic k1 Universe could recollapse Consequences of positive A Because A term appears with positive power of R in Friedmann equation effects of A increase with time if R keeps increasing szRzzngszlkcz 3 Positive A can create accelerating expansion CDSITH TIrrE Emma Trrc A positive A can produce an accelerat ing universe 4 Agt0 Scale Factor R N O w p Time Fig 118 When a positive A term is added to the hyperbolic standard model A 0 R expands more rapidly and soon accelerates with q lt 4 Scale Factor R N gt ll 0 2 Time Fig 119 When a positive A term is added to the at standard model A 0 R eventually begins accelerat ing with q lt 0 although at a later time compared with the hyperbolic model 4 A AgtAc E E 22 3 0ltAltAc 00 1 3 4 Time Fig 1110 The Lemaitre model has spherical geometry an a A value slightly greater than the Einstein crit ical value Ac This model features a hovering period during which the scale factor remains nearly constant over a lengthy time interval Following the hovering period expansion continues at an accelerating rate with q lt 0 R k1 A0 AltO 39 t V A quot R f k0 I I I Alt0 r t IAzo R I I I 39 AltO I t Figure 6 The addilion of a negative uumcuvc A force 0 any of III smndard models rugunllcss of geometry resulls inevimbly in rcCDlIapse The evnlminn wilhum th mm is shown as a dashed In for 1 ucc THE DE SITTER UNIVERSE de Sitterquot model SOIUtion With Qk0 at Space1 QM The de Sitter universe 0 no matter and A gt0 5 Hubble parameter is constant 4 Expansion is exponential 3 2 Scale Factor R R Rothto 0o 1 2 Time Fig 116 The scale factor R as a func tion of time for a de Sitter universe The exponential curve of the de Sitter model never goes to R 0 so there is no big bang this model is in nitely old Steady solution Constant expansion rate Matter constantly created No Big Bang Ruled out by existing observations Distant galaxies seen as they were light travel time in the past differ from modern galaxies Cosmic microwave background implies earlier state with uniform hot conditions big bang Observed deceleration parameter differs from what would be required for steady model q 1 1 I q0 I I I I qlt12 I I R q12 1 quot39 qgt12 Now Time Igt Figure 115 The behavior of the scale factor for a variety of models all constrained to pass through the time now with the same slope The value of the deceleration constant determines both the model s future and the age of the universe The larger the value of q the shorter the time back to the big bang The exponentially expanding de Sitter model with q 71 never intersects R O Cosmological models Model Geometry A q Fate Einstein Spherical Ac 0 Unstable collapse or expand de Sitter Flat gt 0 1 Exponential expansion Steady state Flat gt 0 1 Exponential expansion Lemaitre Spherical gt AC lt 0 after hover Expand hover expand Closed Spherical 0 gt 12 Big crunch Einstein de Sitter Flat 0 12 Expand forever Open Hyperbolic 0 O lt q lt 12 Expand forever Negative A Any lt 0 gt 0 Big crunch 4 E A gt Ac It The de Sitter unwersc ZesseS rent a 4 my of Z 2 lane 5 3 R e 39 verse g 0 lt A lt Ac stat 30 2 could le n g vould 00 3 4 r A m l els is Time fr 00 1 2 Time Fig 1110 The Lemaitre model has spherical geometry and a A value slightly greater than the Einstein crit ical value Ac This model features a hovering period during which the scale factor remains nearly constant over a lengthy time interval Following the hovering period expansion continues at an accelerating rate with q lt 0 Fig 116 The scale factor B as a funce tion of time for a de Sitter universe The exponential curve of the cle Sitter model never goes to R 0 so there is no big bang this model is in nitely old R k1 A0 AltO 39 t V A quot R f k0 I I I Alt0 r t IAzo R I I I 39 AltO I t Figure 6 The addilion of a negative uumcuvc A force 0 any of III smndard models rugunllcss of geometry resulls inevimbly in rcCDlIapse The evnlminn wilhum th mm is shown as a dashed In for 1 ucc Fig 117 The behavior of the scale factor for a variety of models All are constrained to pass through the present time arbitrarily set to to 1 with the same slope The value of the deceleration parameter deter mines both the model s future and the age of the universe The larger the de celeration the shorter the time back to the big bang The exponentially expanding de Sitter model never in tersects R 0 Scale Factor R Time
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