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AnIntroductionto

ModernCosmology

SecondEdition

Thispageintentionallyleftblank

AnIntroductionTo

ModernCosmology

SecondEdition

AndrewLiddle

UniversityofSussex,UK

WestSussexPOl98SQ,England

Telephone(+44)1243779777

Email(forordersandcustomer

serviceenquiries):cs-books@wile,..co.uk VisitourHomePageonW\lA\'.wileyeufope.comor\\.T·\v\\'.wiley.com

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Tomygrandmothers

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Preface

Constants,conversionfactorsandsymbols

1 A(Very)BriefHistoryofCosmologicalIdeas

2ObservationalOverview

2.1Invisiblelight.. . .

2.2Inotherwavebands.

2.3Homogeneityandisotropy

2.4Theexpansion

oftheUniverse

2.5ParticlesintheUniverse

...

2.5.1Whatparticlesarethere?

2.5.2Thermaldistributionsandtheblack-bodyspectrum

3NewtonianGravity

3.1TheFriedmannequation.....

3.2Onthemeaningoftheexpansion.

3.3Thingsthatgofasterthanlight

3.4Thefluidequation. . . . . . . . .

3.5Theaccelerationequation

.....

3.6Onmass,energyandvanishingfactorsofc

4TheGeometryoftheUniverse

4.1Flatgeometry. . . .

4.2Sphericalgeometry. . . .

4.3Hyperbolicgeometry. . .

4.4InfiniteandobservableUniverses.

4.5WheredidtheBigBanghappen?.

4.6Threevaluesof

k. . . . . . . . .

5SimpleCosmologicalModels

5.1Hubble'slaw.....

5.2Expansionandredshift

5.3Solvingtheequations.

xi xiv 1 3 3 7 8 9 11 11 13 17 18 21
21
22
23
24
25
25
26
28
29
29
30
33
33
34
35
Vlll

5.3.1Matter..

5.3.2Radiation

5.3.3Mixtures

5.4Particlenumberdensities

5.5Evolutionincludingcurvature

36
37
38
39
40

6ObservationalParameters45

6.1TheexpansionrateH

o 45

6.2Thedensityparameterno. . .47

6.3Thedecelerationparameter

qo48

7TheCosmologicalConstant51

7.1IntroducingA . . . . .51

7.2FluiddescriptionofA . . . .52

7.3CosmologicalmodelswithA53

8TheAgeoftheUniverse57

9TheDensity

oftheUniverseandDarkMatter63

9.1WeighingtheUniverse. . . . . . . . .63

9.1.1Countingstars63

9.1.2Nucleosynthesisforeshadowed.64

9.1.3Galaxyrotationcurves. . . .64

9.1.4Galaxyclustercomposition..66

9.1.5Bulkmotions

intheUniverse_67

9.1.6Theformationofstructure. .68

9.1.7ThegeometryoftheUniverseandthebrightnessofsupernovae68

9.1.8

Overview........69

9.2Whatmightthedarkmatterbe?.69

9.3Darkmattersearches. . . . . .

10TheCosmicMicrowaveBackground75

10.1Propertiesofthemicrowavebackground75

10.2Thephotontobaryonratio. . . . . . .77

10.3Theoriginofthemicrowavebackground.78

10.4Theoriginofthemicrowavebackground(advanced)81

11TheEarlyUniverse85

12Nucleosynthesis:TheOriginoftheLightElements91

12.1HydrogenandHelium.. . . . . . . . . . .91

12.2Comparingwithobservations.. . . . . . .94

12.3ContrastingdecouplingandnucIeosynthesis96

13TheInflationaryUniverse

13.1ProblemswiththeHotBigBang

13.I .1Theflatnessproblem. .

13.1.2Thehorizonproblem

13.1.3Relicparticleabundances

13.2Inflationaryexpansion

.....

13.3SolvingtheBigBangproblems.

13.3.1Theflatnessproblem. .

13.3.2Thehorizonproblem

13.3.3Relicparticleabundances

13.4Howmuchinflation?

....

13.5Inflationandparticlephysics

14TheInitialSingularity

15Overview:TheStandardCosmologicalModel

AdvancedTopic1GeneralRelativisticCosmology

1.1Themetricofspace-time. . . .

1.2TheEinsteinequations. . . . .

1.3Aside: TopologyoftheUniverse

AdvancedTopic2ClassicCosmology:DistancesandLuminosities

2.1Lightpropagationandredshift

2.2TheohservableUniverse.

2.3Luminosity

distance.. . .

2.4Angulardiameterdistance

2.5Sourcecounts.

AdvancedTopic3NeutrinoCosmology

3.1Themasslesscase.

3.2Massiveneutrinos.

3.2.1Lightneutrinos.

3.2.2Heavyneutrinos

3.3Neutrinosandstructureformation

AdvancedTopic4Baryogenesis

AdvancedTopic5StructuresintheUniverse

5.1Theobservedstructures..

5.2Gravitationalinstability.

5.3Theclustering

ofgalaxies.

5.4Cosmicmicrowavebackgroundanisotropies

5.4.1Statisticaldescription

ofanisotropies

5.4.2Computingthe

Ct..'...,...

5.4.3Microwavebackgroundobservations.

5.4.4Spatialgeometry.

ix 99
99
99
101
102
103
104
104
105
106
106
107
III U5 119
119
120
122
125
125
128
128
132
134
137
137
139
139
140
140
143
147
147
149
150
152
152
154
155
156
x

5.5Theoriginofstructure

Bibliography

Numericalanswersandhintstoproblems

Index

157
161
163
167

Preface

Thedevelopmentofcosmologywillnodoubtbeseenasoneofthescientifictriumphsof thetwentiethcentury.Atitsbeginning,cosmologyhardlyexistedasascientificdiscipline.

Byitsend,theHotBigBangcosmologystoodsecure

astheaccepteddescriptionofthe

Universe

asawhole.TelescopessuchastheHubbleSpaceTelescopearecapableofseeing lightfromgalaxiessodistantthatthelighthasbeentravellingtowardsusformost ofthe lifetimeoftheUniverse.Thecosmicmicrowavebackground,afossilrelic ofatimewhen theUniversewasbothdenserandhotter, isroutinelydetectedanditspropertiesexamined.

ThatourUniverse

ispresentlyexpandingisestablishedwithoutdoubt. Wearepresentlyinanerawhereunderstandingofcosmologyisshiftingfromthe qualitativetothequantitative, asrapidly-improvingobservationaltechnologydrivesour knowledgeforward.Thetumofthemillenniumsawtheestablishment ofwhathascome tobeknownastheStandardCosmologicalModel,representinganalmostuniversalcon sensusamongstcosmologists astothebestdescriptionofourUniverse.Nevertheless,itis amodelwithamajorsurprise-thebeliefthatourUniverse ispresentlyexperiencingac celeratedexpansion.Add tothatongoingmysteriessuchasthepropertiesoftheso-called darkmatter,whichisbelieved tobethedominantformofmatterintheUniverse,anditis clearthatwehavesomewaytogobeforewe cansaythatafullpictureofthephysicsof theUniverse isinourgrasp.

Suchaboldendeavour

ascosmologyeasilycapturestheimagination,andoverrecent yearstherehasbeenincreasingdemandforcosmologytobetaughtatuniversityinan accessiblemanner.Traditionally,cosmologywastaught, asitwastome,asthetailendof ageneralrelativitycourse,witha derivationofthemetricforanexpandingUniverseand a fewsolutions.Suchacoursefailstocapturetheflavourofmodemcosmology,which takesclassicphysicalscienceslikethermodynamics,atomicphysicsandgravitationand appliesthemonagrandscale. Infact,introductorymodemcosmologycanbetackledinadifferent way,byavoiding generalrelativityaltogether.Byaluckychance,andasubtlebitofcheating,thecor rectequationsdescribing anexpandingUniversecanbeobtainedfromNewtoniangravity.

Fromthisbasis,onecanstudy

allthetriumphsoftheHotBigBangcosmology-theex pansion oftheUniverse,thepredictionofitsage,theexistenceofthecosmicmicrowave background,andtheabundancesoflightelementssuch asheliumanddeuterium-and evengoon todiscussmorespeculativeideassuchastheinflationarycosmology.

Theoriginofthisbook,firstpublishedin1998,

isashortlecturecourseattheUni versityofSussex,around20lectures,taughttostudentsinthefinalyearofabachelor's

XIICONTENTS

degreeorthepenultimateyearofamaster'sdegree.Theprerequisitesareallverystandard physics,andtheemphasisisaimedatphysicalintuitionratherthanmathematicalrigour. Sincethebook'spublicationcosmologyhasmovedonapace,andIhavealsobecome aware oftheneedforasomewhatmoreextensiverangeofmaterial,hencethissecondedi tion. Tosummarizethedifferencesfromthefirstedition,thereismorestuffthanbefore. andthestuffthatwasalreadythereisnowlessout-of-date. Cosmologyisaninterestingcoursetoteach,asitisnotlikemost oftheothersubjects taught inundergraduatephysicscourses.Thereisnoperceivedwisdom,builtupovera centuryormore,whichprovides anunquestionablefoundation,asinthermodynamics. electromagnetism,andevenquantummechanicsandgeneralrelativity.Withinourbroad brushpicturethedetailsoftenremainratherblurred,changingaswelearnmoreaboutthe Universeinwhichwelive.Opportunitiescropupduringthecoursetodiscussnewresults whichimpactoncosmologists'views oftheUniverse,andforthelecturertoimposetheir ownprejudicesontheinterpretation oftheever-changingobservationalsituation.Unless

I'vechangedjobs(inwhichcase

I'msurewww.google.cornwillhuntmedown),you

canfollowmyowncurrentprejudicesbycheckingoutthisbook'sWWWHomePage at http://astronorny.susx.ac.uk/-andrewl/cosbook.htrnl Thereyoucanfindsomeupdatesonobservations,andalsoalistofanyerrorsinthebook thatIamaware of.Ifyouareconfidentyou'vefoundoneyourself,andit'snotonthelist.

I'dbeverypleasedtohear

ofit. Thestructureofthebookisacentral'spine',themainchaptersfromonetofifteen, whichprovideaself-containedintroductiontomodemcosmologymoreorlessreproduc ingthecoverage ofmySussexcourse.InadditiontherearefiveAdvancedTopicchapters, eachwithprerequisites,whichcanbeaddedtoextendthecourseasdesired.Ordinarily thebesttimetotacklethoseAdvancedTopics isimmediatelyaftertheirprerequisiteshave beenattained,thoughtheycouldalso beincludedatanylaterstage. I'mextremelygratefultothereviewersoftheoriginaldraftmanuscript,namelySteve Eales,CoelHellierandLindaSmith,fornumerousdetailedcommentswhichledtothe firsteditionbeingmuchbetterthanitwouldhaveotherwisebeen.Thanksalsotothose whosentmeusefulcommentsonthefirstedition,inparticularPaddy LeahyandMichael

Rowan-Robinson,and

ofcoursetoalltheWileystaffwhocontributed.MatthewColless.

BrianSchmidtandMichaelTurnerprovidedthree

ofthefigures,andMartinHendry,Mar tinKunzandFranzSchunckhelpedwiththreeothers,whiletwofiguresweregenerated fromNASA's SkyVtewfacility(http://skyview.gsfc.nasa.gov)locatedatthe

NASAGoddardSpaceFlightCenter.Alibrary

ofimages,includingfull-colourversions ofseveralimagesreproducedhereinblackandwhitetokeepproductioncostsdown,can befoundviathebook'sHomePageasgivenabove.

AndrewRLiddle

Brigbton

February2003

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xiv

Somefundamentalconstants

Newton'sconstant

Speed oflight

ReducedPlanckconstant

Boltzmannconstant

Radiationconstant

Electronmass--energy

Protonmass--energy

Neutronmass--energy

Thomsoncross-section

Freeneutronhalf-life

G c

Ii=h/27[

kB o=7[2c 3 mec 2 m p c 2 m n c 2 Ue thalf or or

6.672X10-

11 m 3 kg- 1 sec- 2

2.998x10

8 msec- 1

3.076x10-

Mpcyr-

1.055x10-

34
m 2 kgsec- 1

1.381x10-

23
JK- 1

8.619x1O-

5 eVK- 1

7.565x10-

16 Jm- 3 K- 4

0.511MeV

938.3
MeV

939.6MeV

6.652x10-

29
m 2

614sec

Someconversionfactors

1pc=3.261lightyears=3.086x10

16 m

1yr=3.156x10

7 sec

1eV=1.602x10-

19 J 1M 0 =1.989X10 30
kg

IJ=lkgm

2 sec- 2

1Hz=1sec-

Commonly-usedsymbols

redshiftdefinedonpage9,35

Hubbleconstant9,45

physicaldistance9 vvelocity9 f frequency12

Ttemperature13

kBBoltzmannconstant13

Eenergydensity15

Qradiationconstant15

GNewton'sgravitationalconstant17

pmassdensity18 ascalefactor19 xcomovingdistance19 kcurvature20 ppressure22 (oroccasionallymomentum11)

HHubbleparameter34

n,Nnumberdensity39 hHubbleconstant46 (orPlanck'sconstant12)

00presentdensityparameter47

pccriticaldensity47

0densityparameter48

Okcurvature'densityparameter'48

qodecelerationparameter48

Acosmologicalconstant51

Ot\cosmologicalconstantdensityparameter52

ttime57 topresentage57

0/3baryondensityparameter64

Y4heliumabundance93

d1umluminositydistanceJ29 ddiamangulardiameterdistance132 /::"TIT,Cecosmicmicrowavebackgroundanisotropies152,153 Xy

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Chapter1

ABriefHistory

ofCosmological Ideas Thecornerstoneofmodemcosmologyisthebeliefthattheplacewhichweoccupyinthe

Universeisinnowayspecial.Thisisknown

asthecosmologicalprinciple,anditis anideawhichisbothpowerfulandsimple.Itisintriguing,then,thatforthebulkofthe historyofcivilizationitwasbelievedthatweoccupyaveryspeciallocation,usuallythe centre, intheschemeofthings. TheancientGreeks,inamodelfurtherdevelopedbytheAlexandrianPtolemy,be lievedthattheEarthmustlieatthecentre ofthecosmos.ItwouldbecircledbytheMoon, theSunandtheplanets,andthenthe'fixed'starswouldbeyetfurtheraway.Acomplex combination ofcircularmotions,Ptolemy'sEpicycles,wasdevisedinordertoexplainthe motions oftheplanets,especiallythephenomenonofretrogrademotionwhereplanets appear totemporarilyreversetheirdirectionofmotion.Itwasnotuntiltheearly1500s thatCopernicusstatedforcefullytheview,initiatednearlytwothousandyearsbeforeby Aristarchus,thatoneshouldregardtheEarth,andtheotherplanets,asgoingaroundthe Sun.Byensuringthattheplanetsmovedatdifferentspeeds,retrogrademotioncouldeas ilybeexplainedbythistheory.However,althoughCopernicusiscreditedwithremoving theanthropocentricviewoftheUniverse,whichplacedtheEarthatitscentre,heinfact believedthattheSunwasatthecentre.

Newton'stheoryofgravityputwhathadbeen

anempiricalscience(Kepler'sdiscovery thattheplanetsmoved onellipticalorbits)onasolidfooting,anditappearsthatNewton believedthatthestarswerealsosunsprettymuchlikeourown,distributedevenlythrough outinfinitespace, inastaticconfiguration.HoweveritseemsthatNewtonwasawarethat suchastaticconfigurationisunstable. Overthenexttwohundredyears,itbecameincreasinglyunderstoodthatthenearby starsarenotevenlydistributed,butratherarelocated inadisk-shapedassemblywhichwe nowknow astheMilkyWaygalaxy.TheHerschelswereabletoidentifythediskstructure inthelate1700s,buttheirobservationswerenotperfectandtheywronglyconcludedthat thesolarsystemlayatitscentre.Only intheearly1900swasthisconvincinglyoverturned, byShapley,whorealisedthatwearesometwo-thirdsoftheradiusawayfromthecentre ofthegalaxy.Eventhen,heapparentlystillbelievedourgalaxytobeatthecentreofthe

2ABRIEFmSTORYOFCOSMOLOOICALIDEAS

Universe.Onlyin1952wasitfinallydemonstrated,byBaade,thattheMilkyWayisa fairlytypicalgalaxy,leadingtothemodemview,knownasthe cosmologicalprinciple (orsometimestheCopernicanprinciple)thattheUniverselooksthesamewhoeverand whereveryouare. ltisimportanttostressthatthecosmologicalprincipleisn'texact.Forexample,no onethinksthatsittinginalecturetheatreisexactlythesameassittinginabar,andthe interior oftheSunisaverydifferentenvironmentfromtheinterstellarregions.Rather,it isanapproximationwhichwebelieveholdsbetterandbetterthelargerthelengthscales weconsider.Evenonthescale ofindividualgalaxiesitisnotverygood,butoncewetake verylargeregions(thoughstillmuchsmallerthantheUniverseitself),containingsaya milliongalaxies,weexpecteverysuchregiontolookmoreorlesslikeeveryotherone.

Thecosmologicalprincipleisthereforeaproperty

oftheglobalUniverse,breakingdown ifonelooksatlocalphenomena.

Thecosmologicalprincipleisthebasis

oftheBigBangCosmology.TheBigBangis thebestdescriptionwehave ofourUniverse,andtheaimofthisbookistoexplainwhy.

TheBigBangisapicture

ofourUniverseasanevolvingentity,whichwasverydifferentin thepastascomparedtothepresent.Originally,itwasforcedtocompetewitharivalidea, theSteadyStateUniverse,whichholdsthattheUniversedoesnotevolvebutratherhas lookedthesameforever,withnewmaterialbeingcreatedtofillthegapsastheUniverse expands.However,theobservationsIwilldescribenowsupporttheBigBangsostrongly thattheSteadyStatetheoryisalmostneverconsidered.

Chapter2

ObservationalOverview

Formostofhistory,astronomershavehadtorelyonlightinthevisiblepartofthespec trum inordertostudytheUniverse.Oneofthegreatastronomicalachievementsofthe

20thcenturywastheexploitationofthefullelectromagneticspectrumforastronomical

measurements. Wenowhaveinstrumentscapableofmakingobservationsofradiowaves, microwaves,infraredlight,visiblelight,ultravioletlight,X-raysandgammarays,which allcorrespondtolightwavesofdifferent(inthiscaseincreasing)frequency.

Weareeven

entering anepochwherewecangobeyondtheelectromagneticspectrumandreceivein formation ofothertypes.Aremarkablefeatureofobservationsofanearbysupernovain

1987wasthatitwasalsoseenthroughdetection

ofneutrinos,anextraordinarilyweakly interactingtypeofparticlenormallyassociatedwithradioactivedecay.

Veryhighenergy

cosmicrays,consisting ofhighly-relativisticelementaryparticles,arenowroutinelyde tected,though asyetthereisnoclearunderstandingoftheirastronomicalorigin.Andas Iwrite,experimentsarestartingupwiththeaimofdetectinggravitationalwaves,ripples inspace-timeitself,andultimatelyofusingthemtoobserveastronomicaleventssuch as collidingstars.

Theadvent

oflargeground-basedandsatellite-basedtelescopesoperatinginallparts oftheelectromagneticspectrumhasrevolutionizedourpictureoftheUniverse.While thereareprobablygapsinourknowledge,some ofwhichmaybeimportantforallwe know,wedoseem tohaveaconsistentpicture,basedonthecosmologicalprinciple,of howmaterialisdistributed intheUniverse.Mydiscussionhereisbrief;foramuchmore detaileddiscussionoftheobservedUniverse,seeRowan-Robinson'sbook'Cosmology' (fullreferenceintheBibliography).Aset ofimages,includingfull-colourversionsofthe figuresinthischapter,canbefoundviathebook'sHomePageasgiveninthePreface.

2.1Invisiblelight

Historically,ourpictureoftheUniversewasbuiltupthroughevermorecarefulobserva tionsusingvisiblelight. Stars:ThemainsourceofvisiblelightintheUniverseisnuclearfusionwithinstars.The

Sunisafairlytypicalstar,withamassofabout2 x10

30
kilograms.Thisisknown asasolarmass,indicatedM G, andisaconvenientunitformeasuringmasses.The

4OBSERVATIONALOVERVIEW

fl'igure2.1Ifviewedfromabovethedisk,ourownMilkyWaygalaxywouldprobably bletheM100galaxy.imagedhereby{heHubbleSpacetelescope.[FigurecourtesyNASAr neareststarstousarcafewlightyearsaway.alightyearbeingthedistance(about 10 16 metres)thailightcantravelinayear.Forhistoricalreasons.analternative unit.knownastheparsecanddenoted'pc',Iismorecommonlyusedincosmology. Aparsecequals3.261lightyears.Incosmology,oncseldomconsidersindividual stars.insteadpreferring toadoptasthesmallestconsideredunittheconglomerations ofSlarsknownas... Galaxies:Oursolarsystemliessomewayoff-centreinagiantdiskstructureknownas theMilkyWaygalaxy.Itcontainsastaggeringhundredlhousandmillion(lOll)or sostars,withmassesrangingfromaboutatcnthlhatofourSuntolensoflimes larger. Itconsistsofacentralbulge.plusadiskofradius12.5kiloparsccs(kpc. equalto10 3 pc)andathicknessofonlyabout0.3kpc.Wearelocatedinthedisk about8kpcfromthe centre.Thediskrotatesslowly(andalsodifferentially,with theouteredgesmovingmoreslowly.justasmoredistanlplanets inthesolarsystem orbilmoreslowly). AIourradius.thegalaxyrotateswithaperiodof200million years.Becausewearewithinil.wecan 'Igetanimageofourowngalaxy.bUIit probablylooksnotunliketheM100galaxyshowninFigure2.1.

Ourgalaxyissurroundedbysmallercollections

ofstars.knownasglobularcluslers. Thesearedislributedmoreorlesssymmetricallyaboulthe bulge.aldistancesof5- IAparsecisdefinedasthedisfanccalwhichthemeandistancebefwe<'ntheEarthandSunsublendsasecond ofarc.1lw:meanEanh-5undislance(calledanASlronomicalVniUis1.496xlOllm.anddi,idingthaIby tan(13Tcscc)l1ivesIpc'"3.086X10 16 m.

2./.INVISIBLHLIGHT

Figure2.2AmapofgalaxypositionsinanarrowsliceoftheUniverse,identifiedby theCfA(CenterforAstrophysics)redshiftsurvey.Ourgalaxyislocatedattheapex,andthe isaround200Mpc.Thegalaxypositionswereobtainedbymeasuremenloftheshiftof spectrallines,asdescribedinSection2.4.Whilemoremodernandextensivegalaxyn.-dsrnft surveysexist.thissurveystillgivesoneofthebeStimpressionsofstructureinthe [FigurecourtesyLarsChristensen]

30kpc.Typicallytheycontainamillionstars,andarethoughttoberemnanlsofthe

fonnation ofthegalaxy.Asweshalldiscusslater,itisbelievedthattheentiredisk andglobularclustersystemmay beembeddedinalargersphericalstructureknown asthegalactichalo. Galaxiesarethemostvisuallystrikingandbeautifulastronomicalobjects inthe Universe,exhibitingawiderangeofproperties.However,incosmologythedetailed structure ofagalaxyisusuaJlyirrelevant,andgalaxiesarenonnallythoughtofas point-likeobjectsemittinglight,oftenbrokenimosub-classesaccordingtocolours. luminositiesandmorphologies. Thelocalgroup:Ourgalaxyresideswithinasmallconcentratedgroupofgalaxiesknown asthelocalgroup.ThenearestgalaxyisasmallirregulargalaxyknownastheLarge MagellanicCloud(LMC),whichis50kpcawayfromtheSun.Thenearestgalaxy ofsimilarsizetoourownistheAndromedaGalaxy,atadistanceof770kpc.The Milky Wayisoneofthelargestgalaxiesinthelocalgroup.Atypicalgalaxygroup occupiesavolume ofafewcubicmegaparsecs.Themegaparsec,denotedMpc andequal toamillion parsecs,isthecosmologist'sfavouriteunitformeasuring distances,becauseitisroughlythesepardtionbetweenneighbouringgalaxies.It equals3.086x10 22
metres.

Clusters

ofgalaxies,superclustersandvoids:SurveyinglargerregionsoftheUniverse, sayonascaleof100Mpc,oneseesavarietyoflarge-scalestructures,asshown inFigure2.2.ThisfigureisnOIaphotograph,butratheracarefullyconstructed map ofthencarbyregionofourUniverse,onascaleofabout1:10 27
!Insome

6OBSERVATIONALOVERVIEW

Figul't'2.3ImagesoflheComaclusterofgalaxiesinvisiblelighl(Iefl)andinX-rays(righI), onthesamescale.Colourversionscanhefoundonthebook'sWWWsite.[Figure..coor1e'iY oftheDigitizedSkySurvey,ROSATand1 placesgalaxiesareclearlygroupedintoclustersofgalaxies;afamousexampleis theComaclusterofgalaxies.IIisabout100Mpcawayfromourowngalaxy,and appearsinFigure2.2asthedenseregion inthecentreofthemap.Theleftpanelof Figure2.3showsanopticaltelescopeimageofComa;althoughtheimageresembles astarfield.eachpointisadistinctgalaxy.Comacontainsperhaps10000galaxies. mostlytoofainllOshow inthisimage.omittingintheircommongravitationalfield. However,mostgalaxies.sometimescalledfieldgalaxies.arenotpart ofacluster. Galaxyclustersarethelargestgmvitationally-collapsedobjeclS intheUniverse.and theythemselvesaregroupedintosuperclusters.perhapsjoinedby filamentsand walls ofgalaxies.Inbetweenthis'foamlike'structurelielargevoids,someaslarge as50Mpcacross.StructuresintheUniversewillbefurtherdescribedinAdvanced

Topic5.

Large-scalesmoothness:Onlyoncewegettoevenlargerscales,hundredsofmega parsecsormore.doesthe

Universebegintoappearsmooth.Recentextremelylarge

galaxysurveys,the2dFgalaxyredshiftsurveyandtheSloanDigitalSkySurvey. havesurveyedvolumesaroundonehundredtimesthe sizeoftheefAsurvey.each containinghundreds ofthousandsofgalaxies.Suchsurveysdonotfindanyhuge structuresonscalesgreaterthan thoseseenintheefAsurvey:thegalaxysuperclus tersandvoidsjustdiscussedarelikelytobethebiggeststructuresinthepresent

Universe.

Thebeliefthatthe

Universedocsindeedbecomesmoothonthelargestscales.the cosmologicalprinciple, istheunderpinningofmodemcosmology.Itisinteresting thatwhilethesmoothness ofthematterdistributiononlargescaleshasbeenakey assumption ofcosmologyfordecadesnow.itisonlyfairlyrecentlythatithasbeen po<;sible10provideaconvincingobservationaldemonstration.

2.2.TNOTHERWAVEBANDS7

2015

Errorbars

multipliedby400 10

Wavespercentimetre

5 olL_-L__ o 100
400
300
Figure2.4ThecosmicmicrowavebackgroundspectrumasmeasuredbytheFIRASexperi mentontheCOBEsatellite.Theerrorbarsaresosmallthattheyhavebeenmultipliedby400 tomakethemvisibleonthisplot,andthebest-fitblack-bodyspectrumat

T=2.725Kelvin,

shown bytheline,isanexcellentfit.

2.2Inotherwavebands

Observationsusingvisiblelightprovideuswithagoodpictureofwhat'sgoingoninthe present-dayUniverse.However,manyotherwavebandsmakevitalcontributionstoour understanding. Microwaves:Forcosmology,thisisbyfarthemostimportantwaveband.Penzias&Wil· son'saccidentaldiscoveryin1965thattheEarthisbathedinmicrowaveradiation, withablack-bodyspectrumatatemperature ofaround3Kelvin,wasandisoneof themostpowerfulpiecesofinformationinsupportoftheBigBangtheory,around whichcosmologyisnowbased.ObservationsbytheFIRAS(FarInfraRedAbso luteSpectrometer)experimentonboardtheCOBE(COsmicBackgroundExplorer) satellitehaveconfirmedthattheradiationisextremelyclosetotheblack-bodyform atatemperature2.725 ±0.001Kelvin.ThisdataisshowninFigure2.4.Further more,thetemperaturecomingfromdifferentparts oftheskyisastonishinglyuni form,andthispresentsthebestevidencethatwe canusethecosmologicalprinciple asthefoundation ofcosmology.Infact,ithasrecentlybeenpossibletoidentify tinyvariations,onlyonepartinahundredthousand,betweentheintensities ofthe microwavescomingfromdifferentdirections.

Itisbelievedthattheseareintimately

relatedtotheorigin ofstructureintheUniverse.Thisfascinatingtopicisrevolu tionizingcosmology,andwillbeexploredfurtherinAdvancedTopic5.

8OBSERVATIONALOVERVIEW

Radiowaves:Apowerfulwayofgaininghigh-resolutionmapsofverydistantgalaxiesis bymappingintheradiopartofthespectrum.Manyofthefurthestgalaxiesknown weredetected inthisway. Infrared:Carryingoutsurveysintheinfraredpartofthespectrum,aswasdonebythe highly-successfullRAS(InfraRedAstronomicalSatellite) inthe1980s,isanexcel lentway ofspottingyounggalaxies,inwhichstarformationisatanearlystage.

Infraredsurveyspickupasomewhatdifferentpopulation

ofgalaxiestosurveyscar riedout inopticallight,thoughobviouslythebrightestgalaxiesareseeninboth.

Infrared

isparticularlygoodforlookingthroughthedustinourowngalaxytosee distantobjects,asit isabsorbedandscatteredmuchlessstronglythanvisiblera diation.Accordingly,it isbestforstudyingtheregionclosetoourgalacticplane. whereobscurationbydust isstrongest.

X-rays:Theseareavitalprobe

ofclustersofgalaxies;inbetweenthegalaxiesliesgasso hotthatitemitsintheX-raypart ofthespectrum,correspondingtoatemperature oftensofmillionsofKelvin.Thisgasisthoughttoberemnantmaterialfromthe formation ofthegalaxies,whichfailedtocollapsetoformstars.X-rayemission fromtheComagalaxycluster isshownintherightpanelofFigure2.3.Theindivid ualgalaxiesseen inthevisiblelightimageintheleftpanelarealmostallinvisible inX-rays,withthebrightdiffuseX-rayemissionfromthehotgasdominatingthe image.

2.3Homogeneityandisotropy

TheevidencethattheUniversebecomessmoothonlargescalessupportstheuseofthe cosmologicalprinciple. Itisthereforebelievedthatourlarge-scaleUniversepossesses twoimportantproperties, homogeneityandisotropy.Homogeneityisthestatementthat theUniverselooksthesameateachpoint,whileisotropystatesthattheUniverselooksthe same inalldirections. Thesedonotautomaticallyimplyoneanother.Forexample,aUniversewithauni formmagneticfieldishomogeneous, asallpointsarethesame,butitfailstobeisotropic becausedirectionsalongthefieldlinescan bedistinguishedfromthoseperpendicularto them.Alternatively,aspherically-symmetricdistribution,viewedfromitscentralpoint,is isotropicbutnotnecessarilyhomogeneous.However,ifwerequirethatadistributionis isotropicabouteverypoint,thenthatdoesenforcehomogeneityaswell. Asmentionedearlier,thecosmologicalprincipleisnotexact,andsoourUniverse doesnotrespectexacthomogeneityandisotropy.Indeed,thestudy ofdeparturesfrom homogeneityiscurrentlythemostprominentresearchtopicincosmology.I'llintroduce thisinAdvancedTopic5,but inthemainbodyofthisbookIamconcernedonlywiththe behaviour oftheUniverseasawhole,andsowillbeassuminglarge-scalehomogeneity andisotropy.

2.4.THEEXPANSIONOFTHEUNIVERSE

2.4TheexpansionoftheUniverse

Akeypiece

ofobservationalevidenceincosmologyisthatalmosteverythingintheUni verseappearstobemovingawayfromus,andthefurtherawaysomethingis,themore rapiditsrecessionappearstobe.Thesevelocitiesaremeasuredviathe redshift,which isbasicallytheDopplereffectappliedtolightwaves.Galaxieshaveaset ofabsorption andemissionlinesidentifiableintheirspectra,whosecharacteristicfrequenciesarewell known.However,ifagalaxyismovingtowardsus,thelightwavesgetcrowdedtogether, raisingthefrequency.Becausebluelightisatthehigh-frequencyend ofthevisiblespec trum,this isknownasablueshift.Ifthegalaxyisreceding,thecharacteristiclinesmove towardstheredend ofthespectrumandtheeffectisknownasaredshift.Thistech niquewas firstusedtomeasureagalaxy'svelocitybyVestoSlipheraround1912,and wasappliedsystematicallybyone ofthemostfamouscosmologists,EdwinHubble,inthe followingdecades. Itturnsoutthatalmostallgalaxiesarerecedingfromus,sothestandardterminology isredshift z,definedby (2.1) whereA em andAobsarethewavelengthsoflightatthepointsofemission(thegalaxy)and observation(us). Ifanearbyobjectisrecedingataspeedv,thenitsredshiftis v z==-, c (2.2) wherecisthespeed oflight. 2 Figure2.5showsvelocityagainstdistance,aplotknownas theHubblediagram,forasampleof1355galaxies.

Hubblerealisedthathisobservations,whichwere

ofcoursemuchless extensivethan thoseavailabletousnow,showedthatthevelocity ofrecessionwasproportionaltothe distanceofanobjectfromus:

11=HoT.(2.3)

ThisisknownasHubble'slaw,andtheconstantofproportionalityH o isknownasHub ble'sconstant.Hubble'slawisn'texact,asthecosmologicalprincipledoesn'tholdper fectlyfornearbygalaxies,whichtypicallypossesssomerandommotionsknownaspe culiarvelocities.Butitdoesdescribetheaveragebehaviour ofgalaxiesextremelywell.

Hubble'slawgivesthepicture

ofourUniverseillustratedinFigure2.6,wherethenearby galaxieshavethesmallestvelocityrelativetoours.Overtheyearsmanyattempt"have beenmadetofindaccuratevaluesfortheproportionalityconstant,but,aswewillseein

2Thisformulaignoresspecialrelativityandsoisvalidonlyforspeedsv.z:c.Ifyou'reinterested,thespecial

relativityresult, ofwhichthisisanexpansionforsmallvIc.is l+z=

1-+vic

1 -vic

However.fordistantobjectsincosmologytherearefurtherconsiderations,concerningthepropagationtimcof thelightandhowtherelativevelocitymightchangeduringit,andsothisexpressionshouldnotbeused.

10OBSERVATIONALOVERVIEW

100
. . o . . 00 .. 50
.. -..-... • 0r.'" •I••••-,. ....-..,..-.....-.-.: ...-.. •'\0'1.-:--' ....:.-. .-,.. "I'"'. :;I..,..• -to.,:•0 .0• oL....:_"""'-_......._---'-_----''--_.L.._"'''''-_......._---'-_----''--_.L.._"'''''-_...J o ........ 'j !II

Distance(Mpc)

Figure2.5Aplotofvelocityversusestimateddistanceforasetof1355galaxies.Astraight linerelationimpliesHubble's law.Theconsiderablescatterisduetoobservationaluncertain tiesandrandomgalaxymotions,butthebest-fitlineaccuratelygivesHubble's law.[The x-axisscaleassumesaparticularvalue ofHo.]

Chapter6,aconsensusisonlynowbeingreached.

Atfirstsight,itseemsthatthecosmologicalprinciplemustbeviolated ifweobserve everythingtobemovingawayfromus,sincethatapparentlyplacesusatthecentre ofthe Universe.However,nothingcouldbefurtherfromthetruth.Infact, everyobserverseesall objectsrushingawayfromthemwithvelocityproportionaltodistance.

Itisperhapseasiest

toconvinceyourself ofthisbysettingupasquaregridwithrecessionvelocityproportional todistancefromthecentralgrid-point.Thentransformtheframe ofreferencetoanearby grid-point,andyou'llfindthattheHubblelawstillholdsaboutthenew'centre'.Thisonly worksbecause ofthelinearrelationshipbetweenvelocityanddistance;anyotherlawand itwouldn'twork. So,althoughexpanding,theUniverselooksjustthesamewhichevergalaxywechoose toimagineourselveswithin.Acommonanalogyistoimaginebakingacakewithraisins init,orblowingupaballoonwithdotsonitssurface.Asthecakerises(ortheballoonis inflated),theraisin(ordots)moveapart.Fromeachone, itseemsthatalltheothersare receding,andthefurtherawaytheyarethefasterthatrecessionis. Becauseeverythingisflyingawayfromeverythingelse,weconcludethat inthedistant pasteverything intheUniversewasmuchclosertogether.Indeed,tracethehistoryback farenoughandeverythingcomestogether.TheinitialexplosionisknownastheBigBang. andamodel oftheevolutionoftheUniversefromsuchabeginningisknownastheBig BangCosmology.Lateron,wewillfindoutwhyitiscommonlycalledtheHotBigBang.

2.5.PARTICLESINTHEUNIVERSE

1 de II / \ Figure2.6AccordingtoHubble'slaw.thefurtherawayfromusagalaxyis.thefasteritis receding.

2.5ParticlesintheUniverse

2.5.1Whatparticlesarethere?

EverythingintheUniverseismadeupoffundamentalparticles.andthebehaviourofthe

Universeasawhole

depends011thepropertiesoftheseparticles.

Onecrucialquestioll

iswhetheraparticleismovingrelativisticallyornot.Anyparticle hastwocontributions LOitsenergy,onebeingthekineticenergyandtheotherbeingthe mass--cnergy,whichcombinetogive £ 22422
total=mc+pc, (2.4) whereTnisthepaniclerestmassandpthepaniclemomentum.Ifthemass-energydom inates,theparticle willbemovingatmuchlessthanthespeedoflight,andwesayitis non-relativistic.Inthatlimitwecancarryoutanexpansion (2.5)

WerecognizethefirsttennasEinstein'sfamousE=mc

2•

knownastherestmass--energy asitistheenergyoftheparticlewhenitisstationary.Thesecondtennistheusualkinetic energy (p=mvinthenon-relativisticlimit).Ifthemass-energydoesnotdominate.the paniclewill bemovingatasubstantialfractionofthespeedoflightandsoisrelativistic. Inpanicular,anyparticlewithzerorestmassisalwaysrelativisticandmovesatthespeed oflight,thesimplestexamplebeinglightitself. Let'sreviewthenatureoftheparticleswhicharehelievedtoexistinourUniverse. 12

Baryons

OBSERVATIONALOVERVIEW

Weourselvesarebuiltfromatoms,thebulkofwhosemassisattributabletotheprotons andneutronsintheatomicnuclei.Protonsandneutronsarebelievedtobemadeup of morefundamentalparticlesknownasquarks,aprotonbeingmadeoftwoupquarksand adownquark,whileaneutronisanupandtwodowns.Ageneraltermforparticles madeup ofthreequarksisbaryons.Ofallthepossiblebaryons,onlytheprotonand neutroncanbestable,3andsothesearethoughttobetheonlytypes ofbaryonicparticle significantlyrepresentedintheUniverse.

Yetanotherpieceofterminology,nucleon,refers

to justprotonsandneutrons,butI'llfollowthestandardpracticeofusingthetermbaryon.

Inparticlephysicsunits,themass-energies

ofaprotonandaneutronare938.3MeVand

939.6MeVrespectively,where

'MeV'isaMega-electronvolt,aunitofenergyequaltoa millionelectronvolts(eV)andrathermoreconvenientthanaJoule. Althoughelectronsarecertainlynotmadefromquarks,theyaretraditionallyalsoin cludedunderthetitlebaryonbycosmologists(totheannoyance ofparticlephysicists).A crucialproperty oftheUniverseisthatitischargeneutral,sotheremustbeoneelectron foreveryproton.Weighinginatapuny0.511MeV,wellunderathousandth ofaproton mass,thecontribution ofelectronstothetotalmassisatinyfraction,notmeritingseparate discussion. InthepresentUniverse,baryonsaretypicallymovingnon-relativistically,meaningthat theirkineticenergy ismuchlessthantheirmass-energy.

Radiation

OurvisualperceptionoftheUniversecomesfromelectromagneticradiation,andsuch radiation,atalargevariety offrequencies,pervadestheUniverse.Inthequantumme chanicalview oflight,itcanbethoughtofasmadeupofindividualparticles-like packets ofenergy-knownasphotonsandusuallyindicatedbythesymbol').Photons propagate,naturallyenough,atthespeed oflight;sincetheyhavezerorestmasstheirtotal energyisalwaysgivenbytheirkineticenergy,and isrelatedtotheirfrequencyfby

E=hf,(2.6)

wherehisPlanck'sconstant. Photonscaninteractwiththebaryonsandelectrons;forexample,ahigh-energyphoton canknockanelectronout ofanatom(aprocessknownasionization),orcanscatteroff afreeelectron(knownasThomsonscattering inthenon-relativisticcasehf"m e c

2•

otherwiseComptonscattering).Themoreenergeticthephotonsare,themoredevastating theireffectsonotherparticles.

3Theprotonlifetimeisknowntobeeitherinfinite,correspondingtotheprotonbeingabsolutelystable.or

muchlongerthantheage oftheUniversesothattheyareeffectivelystable.Isolatedneutronsareunstable (decayingintoaproton.anelectronand anantineutrinol.butthoseboundinnucleimaybestahle:thiswillprow crucialinChapter12.

2.5.PARTICLESINTHEUNIVERSE

Neutrinos

Neutrinosareextremelyweaklyinteractingparticles,producedforexampleinradioactive decay.Thereisnowsignificantexperimentalevidencethattheypossessanon-zerorest mass,butit isunclearwhetherthismassmightbelargeenoughtohavecosmological effects,anditremainsaworkingassumptionincosmologytotreatthemaseffectively massless.Iwilladoptthatassumptionforthemainbody ofthisbook,andinthatcasethey, likephotons,arealwaysrelativistic.Thecombination ofphotonsandneutrinosmakesup therelativisticmaterialin ourUniverse.Confusingly,sometimestheterm'radiation'is usedtorefertoalltherelativisticmaterial.

Therearethreetypes

ofneutrino,theelectronneutrino,muonneutrino andtauneu trino,and iftheyareindeedallmasslesstheyshouldallexistinourUniverse.Unfortu nately,theirinteractionsare soweakthatfornowthereisnohopeofdetectingcosmologi calneutrinosdirectly.Originallytheirpresencewasinferredonpurelytheoreticalgrounds, thoughwewillseethattheexistence ofthecosmicneutrinobackgroundmaybeinferred indirectlybysomecosmologicalobservations. Becausetheyaresoweaklyinteracting,theexperimentallimitsontheneutrinomasses, especially ofthelattertwotypes,arequiteweak,anditisinfactperfectlypossiblethat theyaremassiveenoughtobenon-relativistic.Thepossibleeffects ofneutrinomasses,Ire exploredinAdvancedTopic3.

Darkmatter

Inthisbookwe'llencounteronefurtherkindofparticlethatmayexistinourUniverse, whichisnotpart oftheStandardModelofparticletheory.Itisknownasdarkmatter,and itspropertiesarehighlyuncertainandamatter ofconstant debateamongstcosmologists.

We'llreturntoitinChapter

2.5.2Thermaldistributionsandtheblack-bodyspectrum

Iendthissectionwithsomediscussionofthephysicsofradiation.Ifthisisunfamiliarto you,thedetails aren'tallthatcrucial,thoughsomeoftheresultswillbeusedlaterinthe book. Ifparticlesarefrequentlyinteractingwithoneanother,thenthedistributionoftheir energiescanbedescribedbyequilibriumthermodynamics.Inathermaldistribution,in teractionsarefrequent,butabalancehasbeenreachedsothatallinteractionsproceed equallyfrequently inboththeforwardandbackwarddirections,sothattheoveralldistri bution ofparticlenumbersandenergiesremainsfixed.Thenumberofparticlesofagiven energythendependsonlyonthetemperature. Theprecisedistributiondependsonwhethertheparticlesconsideredarefermions. whichobeythePauliexclusionprinciple, orbosons,whichdonot.Inthisbookthemost interestingcaseisthat ofphotons,whicharebosons,andtheircharacteristicdistribution attemperature TisthePlanckorblack-bodyspectrum.Photonshavetwopossiblepolar izations,andeachhas anoccupationnumberpermodeNgivenbythePlanckfunction N=1 exp(hf/kBT)-1 ' (2.7)

14OBSERVATIONALOVERVIEW

0r--,.--T'"'"""--,.---,...---.....,....--........---r---r--...,

00 0.5 hf/IBoltzmannconstant,whosevalueis1.381x10- 23
JK- 1 =8.619X10- 5 eVK- 1. Tointerpretthisequation,rememberthathfisthephotonenergy.Thepurposeofthe Boltzmannconstantistoconverttemperatureintoacharacteristicenergy k B

T.Belowthis

characteristicenergy, hf"kBT,itiseasytomakephotonsandtheoccupationnumberis large(asphotonsarebosons,thePauliexclusionprinciple doesn'tapplyandtheremaybe arbitrarilymanyphotonsinagivenmode).Abovethecharacteristicenergy, hf»kBT,it isenergeticallyunfavourabletomakephotonsandthenumber isexponentiallysuppressed, asshowninFigure2.7.

Moreinterestingthanthenumber

ofphotonsinamodeisthedistributionofenergy amongstthemodes. Wefocusontheenergyperunitvolume,knownastheenergyden sityE.Becausethereareveryfewphotonswithhf»kBTthereisn'tmuchenergyat highfrequencies.But,despitetheirlargenumber,therealsoisn'tmuchtotalenergyatlow frequencies hf"kBT,bothbecausethosephotonshavelessenergyeach(E=hj),and becausetheirwavelengthislongerandsoeachphotonoccupiesagreatervolume.Witha considerableamount ofwork,theenergydensityinafrequencyintervaldfaboutfcanbe showntobe

81rhf3df

E(J)df=7exp(hJlkBT)-1'

(2.8) whichtellsushowtheenergyisdistributedamongstthedifferentfrequencies.Weseein

Figure

2.8thatthepeakofthedistributionisat!peak:::2.8k

T/h.correspondingtoan

energyE peak =hfpeak:::2.8kBT.Thatistosay.thetotalenergyintheradiationis

2.5.PARTICLESINTHEUNIVERSE

IS J5 ;;--. .r:. .., o * ¥o

246810

Figure2.8Theenergydensitydistributionofablack-bodyspectrum,givenbyequation (2.8).Mostoftheenergyiscontributedbyphotonsofenergyhf'"kBT. dominatedbyphotonswithenergiesoforderkBT.Indeed,themeanenergyofaphoton inthisdistribution isE mean 3kBT. WhenwestudytheearlyhistoryoftheUniverse,animportantquestionwillbehow thistypicalenergycomparestoatomicandnuclearbindingenergies.

Afurtherquantity

ofinterestwillbethetotalenergydensityoftheblack-bodyradi ation,obtainedbyintegratingequation(2.8)overallfrequencies.Setting y=hfjkBT quicklyleadsto (2.9) Theintegralisnotparticularlyeasytocompute,butyoumightliketotryitasachallenge.

Theansweris

7l'4j15,givingaradiationenergydensity

(2.10) wheretheradiationconstant0:isdefinedas (2.11)

Here1i=hj27l'isthereducedPlanckconstant.

Problems

OBSERVATIONALOVERVIEW

2.1.SupposethattheMilkyWaygalaxyisatypicalsize.containingsay1011stars,and

thatgalaxiesaretypicallyseparatedbyadistance ofonemegaparsec.Estimatethe density oftheUniverseinSIunits.Howdoesthiscomparewiththedensityofthe

Earth?

1M 8 2X10 30
kg,1parsec3 x10 16 m.

2.2.IntherealUniversetheexpansionisnotcompletelyuniform.Rather,galaxiesex

hibitsomerandommotionrelativetotheoverallHubbleexpansion,knownastheir pecuLiarveLocityandcausedbythegravitationalpulloftheirnearneighbours.Sup posingthatatypical(egrootmeansquare)galaxypeculiarvelocityis600kIn S-1, howfarawaywouldagalaxyhavetobebeforeitcouldbeusedtodeterminethe

Hubbleconstanttotenpercentaccuracy,supposing

(a)Thetruevalue oftheHubbleconstantis100kms-1Mpc-I? (b)ThetruevalueoftheHubbleconstantis50kInS-1Mpc-1') Assumeinyourcalculationthatthegalaxydistanceandredshiftcouldbemeasured exactly.Unfortunately,thatisnottrue ofrealobservations.

2.3.Whatevidencecanyouthink

oftosupporttheassertionthattheUniverseischarge neutral,andhencecontainsanequalnumber ofprotonsandelectrons?

2.4.Thebindingenergy

oftheelectroninahydrogenatomis13.6eV.Whatisthefre quency ofaphotonwiththisenergy?Atwhattemperaturedoesthemeanphoton energyequalthisenergy?

2.5.Thepeak

ofthe energydensitydistributionofablack-bodyat!peak2.8k s T/h impliesthat!peak/Tisaconstant.EvaluatethisconstantinSIunits(seepagexiv forusefulnumbers).TheSunradiatesapproximatelyasablack-bodywith T sun

5800K.Compute!peakforsolarradiation.Whereintheelectromagnetic

spectrumdoesthepeakemissionlie?

2.6.Thecosmicmicrowavebackgroundhasablack-bodyspectrumatatemperature

2.725K.RepeatProblem2.5tofindthepeakfrequencyofitsemission,andalso

findthecorrespondingwavelengthandcomparetoFigure2.4.Confirmthatthe peakemissionliesinthemicrowavepart oftheelectromagneticspectrum.Finally. computethetotal energydensity ofthemicrowavebackground.

Chapter3

NewtonianGravity

Itisperfectlypossibletodiscusscosmologywithouthaving alreadylearnedgeneralrel ativity. Infact,themostcrucialequation,theFriedmannequationwhichdescribesthe expansion oftheUniverse,turnsouttobethesamewhenderivedfromNewton'stheory ofgravityasitiswhenderivedfromtheequationsofgeneralrelativity.TheNewtonian derivationis,however,somewayfrombeingcompletelyrigorous,andgeneralrelativity is requiredtofullypatchitup,adetailthatneednotconcernusatthisstage. InNewtoniangravityallmatterattracts,withtheforceexertedby anobjectofmassM ononeofmassmgivenbythefamousrelationship (3.1) whereristhedistancebetweentheobjectsandGisNewton'sgravitationalconstant.

Thatis,gravityobeysaninversesquare

law.Becausetheaccelerationofanobjectisalso proportionaltoitsmass,via

F=ma,theaccelerationanobjectfeelsundergravityis

independent ofitsmass. Theforceexertedmeansthereisagravitationalpotentialenergy V __ GMm -, r (3.2) withtheforceexertedbeinginthedirectionwhichdecreasesthepotentialenergythe fastest.Liketheelectricpotential oftwooppositecharges,thegravitationalpotentialis negative,favouringthetwoobjectsbeingclosetogether.Butwithgravitythereisno analogue oftherepulsionoflikecharges.Gravityalwaysattracts.

Thederivation

oftheFriedmannequationrequiresafamousresultdueoriginallyto Newton,whichIwon'tattempttoprovehere.Thisresultstatesthatinaspherically symmetricdistribution ofmatter,aparticlefeelsnoforceatallfromthematerialatgreater radii,andthematerialatsmallerradiigivesexactlytheforcewhichonewouldget ifall thematerialwasconcentratedatthecentralpoint.Thispropertyarisesfromtheinverse squarelaw;thesameresultsexistforelectromagnetism.Oneexample ofitsuseisthat thegravitational(orelectromagnetic)forceoutsideasphericalobject ofunknowndensity profiledependsonlyonthetotalmass(charge).Anotheristhatan'astronaut'insidea

18NEWTONIANGRAVITY

Contributing

mass

Centralpoinl

Figure3.1Theparticlealradiusronlyfeelsgravitationalattractionfromtheshadedregion. AnygravilationalatlractionfromthematerialO\Itsideout,accordingtoNewton's lheorem. sphericalshellfeelsnogravitationalforce,notonlyif(heyareatthecenlrebutiftheyare atanypositioninsidetheshell.

3.1TheFriedmannequation

TheFriedmannequationdescribestheexpansionoftheUniverse,andisthereforethemost importantequationincosmology.Oneoftheroutinetasksforaworkingcosmologist issolving Ihisequationunderdifferentassumptionsconcerningthematerialcontentof theUniverse.ToderivemeFriedmannequation,weneed(0computelhegravitational potentialenergyandthekineticenergyofalestpanicle(itdoesn',matterwhichone,since everywhereintheUniverseisthesameaccordingtothecosmologicalprinciple),andthcn usecnergyconservation.

Let'sconsider

anobserverinauniformexpandingmedium,withmassdensityp,the massdensitybeingthemassperunitvolume,WebeginbyrealiZingthatbecausethe

Universelooksthesamefromanywhere.wecanconsiderany

p'inttobeitscentre.Now considerapanicleadistancerawaywithmassm,asshowninFigure3.1,[By'particle', Jreallymeanasmallvolumecontainingthemassm.1DuctoNewton'stheorem,this particleonlyfeelsaforcefrom

Ihematerialalsmallerradii,shownastheshadedregion_

3.1.THEFRIEDMANNEQUATION

ThismaterialhastotalmassgivenbyM=4npr

3 /3,contributingaforce

GMm4nGpnn

F=-----=--------

r231 andourparticlehasagravitationalpotentialenergy v= _GMm= _4nGpr 2 m r3 19

0·3)

(3.4) Thekineticenergyiseasy;thevelocityoftheparticleisr(I'llalwaysusedotstomean timederivatives)giving (3.5) Theequationdescribinghowtheseparationrchangescannowbederivedfromenergy conservationforthatparticle,namely

U=T+ll,(3.6)

whereUisaconstant.NotethatUneednotbethesameconstantforparticlesseparated bydifferentdistances.Substitutinggives

1:24n2

U=-mr 2 3 (3.7) Thisequationgivestheevolutionoftheseparationrbetweenthetwoparticles. Wenowmakeacrucialstepinthisderivation,whichistorealizethatthisargument appliestoanytwoparticles,becausetheUniverseishomogeneous.Thisallows usto changetoadifferentcoordinatesystem,knownas comovingcoordinates.Theseare coordinateswhicharecarriedalongwiththeexpansion.Becausetheexpansionisuniform, therelationshipbetweenrealdistance randthecomovingdistance,whichwecancallX, canbewritten r=a(t)x,(3.8) wherethehomogeneitypropertyhasbeenusedtoensurethataisafunctionoftimealone. Notethatthesedistanceshavebeenwrittenasvectordistances.Whatyoushouldthink ofwhenstudyingthisequationisacoordinategridwhichexpandswithtime,asshownin

Figure3.2.Thegalaxiesremainatfixedlocationsinthe

xcoordinatesystem.Theoriginal rcoordinatesystem,whichdoesnotexpand,isusuallyknownasphysicalcoordinates. Thequantitya(t)isacrucialone,andisknownasthescalefactoroftheUniverse. Itmeasurestheuniversalexpansionrate.Itisafunctionoftimealone,andittellsus howphysicalseparationsaregrowingwithtime,sincethecoordinatedistances ifareby definitionfixed.Forexample,if,betweentimestlandt2,thescalefactordoublesinvalue, thattells usthattheUniversehasexpandedinsizebyafactortwo,anditwilltakeustwice aslongtogetfromonegalaxytoanother. Wecanusethescalefactortorewriteequation(3.7)fortheexpansion.Substituting

20NEWTONIANGRAVITY

Time Figure3.2Thecomovingcoordinatesystemiscarriedalongwiththeexpansion,sothatany objectsremainatfixedcoordinatevalues. equation(3.8)intoit,rememberingi;=0bydefinitionasobjectsarefixedincomoving coordinates,gives

Multiplyingeachsideby

2/ma 2 x 2 andrearrangingthetermsthengives (

2=8nGp_kc

2, a3a 2 (3.9) (3.10) wherekc 2 =-2U/mx 2.

ThisisthestandardformoftheFriedmannequation,andit

willappearfrequentlythroughoutthisbook.Inthisexpression kmustbeindependentofx sincealltheothertermsintheequationare,otherwisehomogeneitywillnotbemaintained. Soinfactwelearnthathomogeneityrequiresthatthequantity

U.whileconstantfora

givenparticle,doesindeedchange ifwelookatdifferentseparationsx.withUIXx 2.

Finally,sincek=-2U/m2'x

2 whichistimeindependent(asthetotalenergyUis conserved,andthecomovingseparation xisfixed),welearnthatkisjustaconstant,un changingwitheitherspace ortime.Ithastheunitsof(lengthJ- 2.

AnexpandingUniverse

hasauniquevalue ofk,whichitretainsthroughoutitsevolution.InChapter4wewillsee that ktellsusaboutthegeometryoftheUniverse,anditisoftencalledthecurvature.

3.2.ONTHEMEANTNGOFTHEEXPANSION

3.2Onthemeaningoftheexpansion

21
SowhatdoestheexpansionoftheUniversemean?Well,let'sstartwithwhatitdoesnot mean.Itdoesnotmeanthatyourbodyisgraduallygoingtogetbiggerwithtime(and certainly isn'tanexcuseifitdoes).ItdoesnotmeanthattheEarth'sorbitisgoingtoget furtherfromtheSun. Itdoesn'tevenmeanthatthestarswithinourgalaxyaregoingto becomemorewidelyseparatedwithtime. Butitdoesmeanthatdistantgalaxiesaregettingfurtherapart.

Thedistinctioniswhetherornotthemotion

ofobjectsisgovernedbythecumulative gravitationaleffect ofahomogeneousdistributionofmatterbetweenthem,asshownin Figure3.1.Theatomsinyourbodyarenot;theirseparation isdictatedbythestrength ofchemicalbonds,withgravityplayingnosignificantrole.Somolecularstructureswill notbeaffected bytheexpansion.Likewise,theEarth'smotioninitsorbitiscompletely dominatedbythegravitationalattraction oftheSun(withaminorcontributionfromthe otherplanets).Andeventhestarsinourgalaxyareorbitinginthecommongravitational potentialwellwhichtheythemselvescreate,andarenotmovingapartrelativetoonean other.Thecommonfeature oftheseenvironmentsisthattheyareonesofconsiderable ex.cessdensity,verydifferentfromthesmoothdistribution ofmatterweusedtoderivethe

Friedmannequation.

But ifwegotolargeenoughscales,inpracticetensofmegaparsecs,theUniversedoes becomeeffectivelyhomogeneousandisotropic,withthegalaxiesflyingapartfromone anotherinaccordancewiththeFriedmannequation.Itisontheselargescalesthatthe expansion oftheUniverseisfelt,andonwhichthecosmologicalprincipleapplies.

3.3Thingsthatgofasterthanlight

Acommonquestionthatconcernspeopleiswhetherfarawaygalax.iesarerecedingfrom usfasterthanthespeed oflight.Thatistosay,ifvelocityisproportionaltodistance,then ifweconsidergalaxiesfarenoughawaycan wenotmakethevelocityaslargeaswelike, inviolationofspecialrelativity?

Theanswer

isthatindeedinourtheoreticalpredictionsdistantobjectscanappearto moveawayfasterthanthespeed oflight.However,itisspaceitselfwhichisexpanding. There isnoviolationofcausality,becausenosignalcanbesentbetweensuchgalaxies.

Further,specialrelativity

isnotviolated,becauseitreferstotherelativespeedsofobjects passingeachother,andcannotbeusedtocomparetherelativespeeds ofdistantobjects.

Onewaytothink

ofthisistoimagineacolonyofantsonaballoon.Supposethatthe fastesttheantscanmoveisacentimetrepersecond.

Ifanytwoantshappentopasseach

other,theirfastestrelativespeedwouldbetwocentimetrespersecond, iftheyhappened tobemovinginoppositedirections.Nowstarttoblowtheballoonup.Althoughtheants wanderingaroundthesurfacestillcannotexceedonecentimetre persecond,theballoon isnowexpandingunderthem,andantswhicharefarapartontheballooncouldeasily bemovingapartatfasterthantwocentimetrespersecondiftheballoonisblownupfast enough.But iftheyare,theywillnevergettotelleachotheraboutit,becausetheballoon ispullingthemapartfasterthantheycanmovetogether,evenatfullspeed.Anyants thatstartcloseenoughtobeabletopassoneanothermustdosoatnomorethantwo centimetrespersecondeven iftheUniverseisexpanding. 22
it.NEWTONIANGRAVITY Theexpansionofspaceisjustlikethatoftheballoon,andpullsthegalaxiesalongwith

3.4Thefluidequation

Fundamentalthoughitis,theFriedmannequationisofnousewithoutanequationto describehowthedensity pofmaterialintheUniverseisevolvingwithtime.Thisinvolves thepressure pofthematerial,andiscalledthefluidequation.[Unfortunatelythestandard symbol pforpressureisthesameasformomentum,whichwe'vealreadyused.Almost alwaysinthisbook, pwillbepressure.]Aswe'llshortlysee,thedifferenttypesofmaterial whichmightexistinourUniversehavedifferentpressures,andleadtodifferentevolution ofthedensityp. Wecanderivethefluidequationbyconsideringthefirstlawofthennodynamics dE+pdV=TdS,(3.11) appliedtoanexpandingvolumeVofunitcomovingradius.IThisisexactlythesameas applyingthennodynamicstoagasinapiston.Thevolumehasphysicalradius a,sothe energyisgiven,using E=mc 2, by (3.12)

Thechange

ofenergyinatimedt,usingthechainrule,is whilethechangeinvolumeis dV_42da dt-1radt' (3.13) (3.14)

Assumingareversibleexpansion

dS=0,putting theseintoequation(3.11)andrearrang inggives .a(p) p+P+c 2 =0, (3.15) whereasalwaysdotsareshorthandfortimederivatives.Thisisthe fluidequation.As wesee,therearetwotennscontributingtothechangeinthedensity.Thefirst tennin thebracketscorrespondstothedilutioninthedensitybecausethevolumehasincreased, whilethesecondcorrespondstotheloss ofenergybecausethepressureofthematerialhas doneworkastheUniverse'svolumeincreased.Thisenergyhasnotdisappearedentirely ofcourse;energyisalwaysconserved.Theenergylostfromthefluidviatheworkdone hasgoneintogravitationalpotentialenergy. IDon'lconfuseVforvolumewithVforgravitationalpolentialenergy.

3.5.THEACCELERATIONEQUATION23

LetmestressthattherearenopressureforcesinahomogeneousUniverse,becausethe densityandpressureareeverywherethesame.Apressure gradientisrequiredtosupply aforce.Sopressuredoesnotcontributeaforcehelpingtheexpansionalong;itseffectis solelythroughtheworkdoneastheUniverseexpands. Wearestillnotinapositiontosolvetheequations,becausenowweonlyknowwhat Pisdoingifweknowwhatthepressurepis.Itisinspecifyingthepressurethatweare sayingwhatkind ofmaterialourmodelUniverseisfilledwith.Theusualassumptionin cosmologyisthatthereisauniquepressureassociatedwitheachdensity,sothat p==p(p). Sucharelationshipisknownastheequationofstate,andwe'llseetwodifferentexamples inChapter5.Thesimplestpossibilityisthatthereisnopressureatall,andthatparticular case isknownas(non-relativistic)matter.

Oncetheequation

ofstateisspecified,theFriedmannandfluidequationsareallwe needtodescribetheevolutionoftheUniverse.However,beforediscussingthisevolution, Iamgoingtospendsometimeexploringsomegeneralproperties oftheequations,as wellasdevotingChapter4toconsiderationofthemeaningoftheconstantk.Ifyou prefertoimmediatelyseehowtosolvetheseequations,feelfreeto jumpstraightawayto Sections5.3to5.5, andcomebacktotheinterveningmateriallater.Ontheway,youmight wanttoglanceatSection3.6tofindoutwhyafactor ofc 2 mysteriouslyvanishesfromthe

Friedmannequationbetweenhereandthere.

3.5Theaccelerationequation

TheFriedmannandfluidequationscanbeusedtoderiveathirdequation,notindepen dent ofthefirsttwoofcourse,whichdescribestheaccelerationofthescalefactor.By differentiatingequation(3.10)withrespecttotimeweobtain aali-0,28nG.kc 2 a

2----=---p+2---.

a a 2 3a 3 (3.16)

Substitutinginfor

pfromequation(3.15)andcancellingthefactor20,/aineachtermgives ..(.)2a ap --- =-4nG(p+-:-)+-, a ac 2 a 2 (3.17) andfinally,usingequation(3.10)again,wearriveatanimportantequationknownasthe accelerationequation =_(p+3P) . a3c 2 (3.18)

Noticethat

ifthematerialhasanypressure,thisincreasesthegravitationalforce,andso furtherdeceleratestheexpansion.Iremindyouthattherearenoforcesassociatedwith pressurein anisotropicUniverse,astherearenopressuregradients.

Theaccelerationequationdoesnotfeaturetheconstant

kwhichappearsintheFried mannequation;itcancelledoutinthederivation.

24NEWTONIANGRAVITY

3.6Onmass,energyandvanishingfactorsofc

2 Youshouldbeawarethatcosmologistshaveahabitofusingmassdensitypanden ergydensity finterchangeably.TheyarerelatedviaEinstein'smostfamousequationas f=pc 2, andifonechoosesso-called'naturalunits'inwhichcissetequaltoone,thetwo becomethesame.Forclarity,however,Iwillbecarefultomaintainthedistinction.Note thatthephrase'massdensity'isusedinEinstein'ssense-itincludesthecontributions tothemassfromtheenergy ofthevariousparticles,aswellasanyrestmasstheymight have.

Thehabit

ofsettingc=1meansthattheFriedmannequationisnormallywritten withoutthec2inthefinalterm,sothatitreads a3a 2 (3.19) Theconstantkthenappearstohaveunits[time]-2-settingc=1makestimeandlength unitsinterchangeable.Sincethepractice ofomittingthec2intheFriedmannequationis widelyadoptedinothercosmologytextbooks,Iwilldropitfortheremainder ofthisbook too.Inpractice,itisararesituationindeedwhereonehasto becarefulaboutthis.

Chapter4

TheGeometry

oftheUniverse Wenowconsiderthe realmeaningoftheconstantkwhichappearsintheFriedmann equation

2=81l"GP_.

a3a 2 (4.1) WhiletheNewtonianderivationinthelastchapterintroducedthisasameasureofthe energyperparticle,thetrueinterpretation,apparentinthecontext ofgeneralrelativity,is thatitmeasuresthecurvatureofspace.Generalrelativitytellsusthatgravityisduetothe curvature offour-dimensionalspace-time,andafullanalysiscanbefoundinanygeneral relativitytextbook.HereIwillbepurelydescriptive,andfocusontheinterpretationof kasmeasuringthecurvatureofthethreespatialdimensions.Furtherdetailsofgeneral relativisticcosmologycanbefoundinAdvancedTopic 1. WehavedemandedthatourmodelUniversesbebothhomogeneousandisotropic.The simplesttype ofgeometrywhichcanhavethispropertyiswhatiscalledaflatgeometry, inwhichthenormalrules ofEuclideangeometryapply.However,itturnsoutthatthe assumption ofisotropyisnotenoughtodemandthatastheonlychoice.Instead,thereare threepossiblegeometriesfortheUniverse,andtheycorrespondto kbeingzero,positive ornegative.

4.1Flatgeometry

Euclideangeometryisbasedon asetofsimpleaxioms(e.g.astraightlineistheshort estdistancebetweentwopoints),plusonemorecomplexaxiomwhichsaysthatparallel straightlinesremainafixeddistanceapart.Thesearethebasisforthestandardlaws of geometry,andleadtothefollowingconclusions: •Theangles ofatriangleaddupto180

0•

•Thecircumferenceofacircleofradiusris21l"r. SuchageometrymightwellapplytoourownUniverse.Ifthatisthecase,thenthe

Universemustbeinfiniteinextent,because

ifitcametoadefiniteedgethenthatwould

26THEGEOMETRYOFTHEUNIVERSE

clearlyviolatetheprinciplethattheUniverseshouldlookthesamefromallpoints.I AUniversewiththisgeometryisoftencalledaflatUniverse.

4.2Sphericalgeometry

Euclidalwayshopedthatthemoreartificialfinalaxiomcouldbeprovenfromtheothers. Itwasn'tuntilthe19thcenturythatRiemanndemonstratedthatEuclid'sfinalaxiomwas anarbitrarychoice,andthatonecouldmakeotherassumptions.Indoingso,hefounded thesubject ofnon-Euclideangeometry,whichformsthemathematicalfoundationforEin stein'stheory ofgeneralrelativity.

Thesimplestkind

ofnon-Euclideangeometryisactuallyveryfamiliartous;itisthe sphericalgeometrywhichweuse,forinstance,tonavigatearoundtheEarth.Beforewor ryingabouttheUniversehavingthreedimensions,let'sexaminetheproperties ofthetwo dimensional surface oftheEarth,showninFigure4.1. First ofall,weknowthataperfectspherelooksthesamefromallpointsonitssurface, sothecondition ofisotropyissatisfied(e.g.ifsomeonehandsyouasnookerballandasks whichwayupitis,you'renotgoingtobeabletotellthem).But,unlikethecase ofaflat geometry,thesphericalsurfaceisperfectlyfiniteinextent,itsareabeinggivenby 41rr
2. Yetthereisnoboundary,no'edge'tothesurfaceoftheEarth.Soitisperfectlypossible tohaveafinitesurfacewhichneverthelesshasnoboundary. IfwedrawparallellinesonthesurfaceoftheEarth,thentheyviolateEuclid'sfinal axiom.Thedefinition ofastraightlineistheshortestdistancebetweentwopoints,which meansthatthestraightlinesinasphericalgeometryaresegments ofgreatcircles,suchas theequator orthelinesoflongitude. 2

Thelinesoflongitudeareanexcellentexampleof

thefailureofEuclid'saxiom;astheycrosstheequatortheyareallparalleltooneanother. butratherthanremainingaconstantdistanceaparttheymeetatbothpoles. Ifwedrawatriangleonasphere,wefindthattheanglesdonotaddupto180 0 degrees either.TheeasiestexampletothinkaboutistostartattheNorthPole.Drawtwostraight linesdowntotheequator,ninetydegreesapart,andthenjointhemwithalineonthe equator.

Youhavedrawnatriangleinwhichallthreeanglesare90

0, showninFigure4.1.

Thecircumference

ofacirclealsofailstoobeythenormallaw.Supposewedrawa circleataradius rfromtheNorthPole,andwe'llchoosersothatourcircleistheequator.

Thatradius,measuredonthesurface

ofthesphere,correspondstoaquarterofacomplete circlearoundtheEarth,sor =1rR/2whereRistheradiusoftheEarth.However,the circumferencecisgivenby

21rR,soinsteadoftheusualrelationonehasc=4rfora

circledrawnattheequator.Thecircumferenceislessthan

21rr.Problem4.1looksatthe

generalcase;youmayfindithelpfultoglanceatthefigureonpage

31now.

AlthoughIhaveonlyconsideredspecificcaseswherethealgebra iseasy.itistruethat whatevertriangle orcircleisdrawn,you'llalwaysfind •Theangles ofatriangleadduptomorethan180 0. •Thecircumferenceofacircleislessthan21rr. IWell.that'salmosttrue.SeeAdvancedTopic1.3forawaytobypassthaIconclusion.

2NoteIhal.

apartfromtheequator,linesoflatitudearenotstraightlines;thisiswhyaeroplanesdonolfollow lines oflatitudewhenflying,becauseIheyarenottheshorteslway10go'

4.2.SPHERICALGEOMETRY

Figure4.1Asketchofasphericalsurface,representingpositivek.Atriangleisshown whichhasthreerightangles! IfyoumakethetrianglesorcirclesmuchsmallerthanthesizeoftheEarth,thenthe Euclideanlawsstarttobecomeagoodapproximation;certainlywedon'thavetoworry aboutEuclideanlawsbeingbrokeninoureverydayexistence(thoughtheappreciation thatthe Earthissphericalisvitalfortheplanningoflongdistancejourneys).Soasmall triangledrawnon aspherewillhavethesum ofitsanglesonlymarginallylargerthan180

0•

ThispropertymakesitratherhardtomeasurethegeometryofourownUniverse,because theneighbouringregionwhichwe canmeasureaccurately isonlyasmallfractionofthe sizeoftheUniverseandsowillobeynearlyEuclideanlawswhatevertheoverallgeometry. One ofthemostimportantconceptualpoints thatyouneedtograsp

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