in cosmology The annotated bibliography at the end of the text provides a selection of recommended cosmology books, at the popular, intermediate, and advanced levels Many people (too many to name individually) helped in the making of this book; I thank them all I owe particular thanks to the students who
Cosmology is the science about the structure and evolution of the Universe on the large scale, its past, present and future This is a very broad and rapidly developing science It is based on modern fundamental Physics, Astronomy and employs a variety of mathematical methods In our lectures, we will not be able to cover
Alain Blanchard Cosmology: Basics Outline Introduction Theory of Observations in RW space Dynamics and Solutions Cosmological parameters estimations Successes and
Part I: Cosmology Basics [Ryden chap 2 to 4] Miguel Quartin Instituto de Física, UFRJ Astrofísica, Relativ e Cosmologia (ARCOS) Curso de Cosmologia Pós – 2019/1 2
years there has been increasing demand for cosmology to be taught at university in an accessible manner Traditionally, cosmology was taught, as it was to me, as the tail end of a general relativity course, with a derivation ofthe metric for an expanding Universe and a few solutions Such a course fails to capture the flavour of modem cosmology
Cosmology (from the Greek: kosmos, universe, world, order, and logos, word, theory) is probably the most ancient body of knowledge, dating from as far back as the predictions of seasons by early civiliza-tions Yet, until recently, we could only answer to some of its more basic questions with an order of mag-nitude estimate
65446_7Book_Astro__Liddle_Introduction_to_Modern_Cosmology.pdf
AnIntroductionto
ModernCosmology
SecondEdition
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AnIntroductionTo
ModernCosmology
SecondEdition
AndrewLiddle
UniversityofSussex,UK
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Tomygrandmothers
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Contents
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
2
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
CONTENTS
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. . . . . .
72
10TheCosmicMicrowaveBackground75
10.1Propertiesofthemicrowavebackground75
10.2Thephotontobaryonratio. . . . . . .77
10.3Theoriginofthemicrowavebackground.78
10.4Theoriginofthemicrowavebackground(advanced)81
11TheEarlyUniverse85
12Nucleosynthesis:TheOriginoftheLightElements91
12.1HydrogenandHelium.. . . . . . . . . . .91
12.2Comparingwithobservations.. . . . . . .94
12.3ContrastingdecouplingandnucIeosynthesis96
CONTENTS
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
CONTENTS
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-
7
Mpcyr-
1
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-
1
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
9
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
9.
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/I
Boltzmannconstant,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
B 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.
16 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
of 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