From Primordial Quantum Fluctuations to the Anisotropies of CMBR

147 Pages·2006·0.815 MB·English

by  Straumann N.

Ann. Phys. (Leipzig)15,No.10–11,701–845(2006)/DOI10.1002/andp.200610212
From primordial quantum fluctuations to the anisotropies
∗
of the cosmic microwave background radiation
NorbertStraumann∗∗
InstituteforTheoreticalPhysics,UniversityofZurich,8057Zurich,Switzerland
Received25May2005,accepted17January2006byA.Wipf
Publishedonline22May2006
Keywords Cosmology,inflation,cosmologicalperturbations,CMBR,darkenergy.
PACS 98.80-k,98.70.Vc,98.80.Cq,95.36+x,95.35+d
These lecture notes cover mainly three connected topics. In the first part we give a detailed treatment of
cosmologicalperturbationtheory.Thesecondpartisdevotedtocosmologicalinflationandthegeneration
ofprimordialfluctuations.Inpartthreeitwillbeshownhowtheseinitialperturbationevolveandproduce
the temperature anisotropies of the cosmic microwave background radiation. Comparing the theoretical
predictionfortheangularpowerspectrumwiththeincreasinglyaccurateobservationsprovidesimportant
cosmologicalinformation(cosmologicalparameters,initialconditions).
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Contents
0 EssentialsofFriedmann-Lemaˆıtremodels 702
0.1 Friedmann-Lemaˆıtrespacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702
0.1.1 Spacesofconstantcurvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
0.1.2 CurvatureofFriedmannspacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
0.1.3 EinsteinequationsforFriedmannspacetimes . . . . . . . . . . . . . . . . . . . . . . 704
0.1.4 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
0.1.5 Cosmicdistancemeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
0.2 Luminosity-redshiftrelationforTypeIasupernovas . . . . . . . . . . . . . . . . . . . . . . 708
0.2.1 Theoreticalredshift-luminosityrelation . . . . . . . . . . . . . . . . . . . . . . . . . 708
0.2.2 TypeIasupernovasasstandardcandles . . . . . . . . . . . . . . . . . . . . . . . . . 712
0.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
0.2.4 Systematicuncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
0.3 Thermalhistorybelow100MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
PartI:Cosmologicalperturbationtheory 719
1 Basicequations 720
1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
1.1.1 Decompositionintoscalar,vector,andtensorcontributions . . . . . . . . . . . . . . . 720
1.1.2 Decompositionintosphericalharmonics . . . . . . . . . . . . . . . . . . . . . . . . . 721
1.1.3 Gaugetransformations,gaugeinvariant
amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
1.1.4 Parametrizationofthemetricperturbations . . . . . . . . . . . . . . . . . . . . . . . 722
∗ BasedonlecturesgivenatthePhysik-Combo,inHalle,LeipzigandJena,wintersemester2004/5.
∗∗ E-mail:[email protected]
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702 N.Straumann:Cosmologicalperturbationtheory
1.1.5 Geometricalinterpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
1.1.6 Scalarperturbationsoftheenergy-
momentumtensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
1.2 Explicitformoftheenergy-momentumconservation . . . . . . . . . . . . . . . . . . . . . 727
1.3 Einsteinequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
1.4 Extensiontomulti-componentsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736
1.5 AppendixtoChapter1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
2 Someapplicationsofcosmologicalperturbationtheory 750
2.1 Non-relativisticlimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
2.2 Largescalesolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752
2.3 Solutionof(2.6)fordust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
2.4 Asimplerelativisticexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
PartII:Inflationandgenerationoffluctuations 756
3 Inflationaryscenario 756
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
3.2 Thehorizonproblemandthegeneralideaofinflation . . . . . . . . . . . . . . . . . . . . . 756
3.3 Scalarfieldmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
3.3.1 Power-lawinflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
3.3.2 Slow-rollapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
3.4 Whydidinflationstart? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764
4 Cosmologicalperturbationtheoryforscalarfieldmodels 764
4.1 Basicperturbationequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
4.2 Consequencesandreformulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
5 Quantization,primordialpowerspectra 772
5.1 Powerspectrumoftheinflatonfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
5.1.1 Powerspectrumforpowerlawinflation . . . . . . . . . . . . . . . . . . . . . . . . . 774
5.1.2 Powerspectrumintheslow-rollapproximation . . . . . . . . . . . . . . . . . . . . . 776
5.1.3 Powerspectrumfordensityfluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 778
5.2 Generationofgravitationalwaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
5.2.1 Powerspectrumforpower-lawinflation . . . . . . . . . . . . . . . . . . . . . . . . . 782
5.2.2 Slow-rollapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782
5.2.3 Stochasticgravitationalbackgroundradiation . . . . . . . . . . . . . . . . . . . . . . 783
5.2.4 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
5.3 AppendixtoChapter5:
Einsteintensorfortensorperturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
PartIII:Microwavebackgroundanisotropies 788
6 Tightcouplingphase 790
6.1 Basicequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791
6.2 Analyticalandnumericalanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795
6.2.1 Solutionsforsuper-horizonscales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796
6.2.2 Horizoncrossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796
6.2.3 Sub-horizonevolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799
6.2.4 Transferfunction,numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 800
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Ann. Phys. (Leipzig)15,No.10–11(2006) 703
7 BoltzmannequationinGR 801
7.1 One-particlephasespace,Liouvilleoperatorforgeodesicspray . . . . . . . . . . . . . . . . 802
7.2 ThegeneralrelativisticBoltzmannequation . . . . . . . . . . . . . . . . . . . . . . . . . . 805
7.3 Perturbationtheory(generalities) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806
7.4 Liouvilleoperatorinthelongitudinalgauge . . . . . . . . . . . . . . . . . . . . . . . . . . 808
7.5 Boltzmannequationforphotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811
7.6 TensorcontributionstotheBoltzmannequation . . . . . . . . . . . . . . . . . . . . . . . . 815
8 ThephysicsofCMBanisotropies 816
8.1 Thecompletesystemofperturbationequations. . . . . . . . . . . . . . . . . . . . . . . . . 816
8.2 Acousticoscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817
8.3 Formalsolutionforthemomentsθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822
l
8.4 Angularcorrelationsoftemperature
fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
8.5 Angularpowerspectrumforlargescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
8.6 InfluenceofgravitywavesonCMBanisotropies . . . . . . . . . . . . . . . . . . . . . . . . 827
8.7 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831
8.8 Observationalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834
8.9 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837
Appendices 837
A Randomfields,powerspectra,filtering 837
B CollisionintegralforThomsonscattering 839
C Ergodicityfor(generalized)randomfields 841
Introduction
Cosmologyisgoingthroughafruitfulandexcitingperiod.Someofthedevelopmentsaredefinitelyalsoof
interesttophysicistsoutsidethefieldsofastrophysicsandcosmology.
These lectures cover some particularly fascinating and topical subjects. A central theme will be the
current evidence that the recent ( z < 1) Universe is dominated by an exotic nearly homogeneous dark
energydensitywithnegativepressure.Thesimplestcandidateforthisunknownso-calledDarkEnergyisa
cosmologicalterminEinstein’sfieldequations,apossibilitythathasbeenconsideredduringallthehistory
ofrelativisticcosmology.Independentlyofwhatthisexoticenergydensityis,onethingiscertainsincea
longtime:Theenergydensitybelongingtothecosmologicalconstantisnotlargerthanthecosmological
criticaldensity,andthusincrediblysmallbyparticlephysicsstandards.Thisisaprofoundmystery,since
weexpectthatallsortsofvacuumenergiescontributetotheeffectivecosmologicalconstant.
Sincethisissuchanimportantissueitshouldbeofinteresttoseehowconvincingtheevidenceforthis
findingreallyis,orwhetheroneshouldremainsceptical.Muchofthisisbasedontheobservedtemperature
fluctuationsofthecosmicmicrowavebackgroundradiation(CMB).Adetailedanalysisofthedatarequires
aconsiderableamountoftheoreticalmachinery,thedevelopmentofwhichfillsmostspaceofthesenotes.
Sincethisaudienceconsistsmostlyofdiplomaandgraduatestudents,whosemaininterestsareoutside
astrophysicsandcosmology,Idonotpresupposethatyouhadalreadysomeserioustrainingincosmology.
However,Idoassumethatyouhavesomeworkingknowledgeofgeneralrelativity(GR).Asasource,and
forreferences,Iusuallyquotemyrecenttextbook[1].
InanopeningchapterthosepartsoftheStandardModelofcosmologywillbetreatedthatareneededfor
themainpartsofthelectures.Moreonthiscanbefoundatmanyplaces,forinstanceintherecenttextbooks
oncosmology[2–6].
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704 N.Straumann:Cosmologicalperturbationtheory
InPartIwewilldevelopthesomewhatinvolvedcosmologicalperturbationtheory.Theformalismwill
laterbeappliedtotwomaintopics:(1)Thegenerationofprimordialfluctuationsduringaninflationaryera.
(2)Theevolutionoftheseperturbationsduringthelinearregime.AmaingoalwillbetodeterminetheCMB
powerspectrum.
0 EssentialsofFriedmann-Lemaˆıtremodels
ForreasonsexplainedintheIntroductionItreatinthisopeningchaptersomestandardmaterialthatwillbe
neededinthemainpartsofthesenotes.Inaddition,animportanttopicalsubjectwillbediscussedinsome
detail, namely the Hubble diagram forType Ia supernovas that gave the first evidence for an accelerated
expansionofthe‘recent’andfutureuniverse.MostreaderscandirectlygotoSect.0.2,wherethisistreated.
0.1 Friedmann-Lemaˆıtrespacetimes
Thereisnowgoodevidencethatthe(recentaswellastheearly)Universe1is–onlargescales–surprisingly
homogeneousandisotropic.Themostimpressivesupportforthiscomesfromextendedredshiftsurveysof
galaxiesandfromthetrulyremarkableisotropyofthecosmicmicrowavebackground(CMB).IntheTwo
DegreeField(2dF)GalaxyRedshiftSurvey,2completedin2003,theredshiftsofabout250’000galaxies
have been measured. The distribution of galaxies out to 4 billion light years shows that there are huge
clusters,longfilaments,andemptyvoidsmeasuringover100millionlightyearsacross.Butthemapalso
showsthattherearenolargerstructures.ThemoreextendedSloanDigitalSkySurvey(SDSS)hasalready
producedverysimilarresults,andwillintheendhavespectraofaboutamilliongalaxies3.
OnearrivesattheFriedmann(Lemaˆıtre-Robertson-Walker)spacetimesbypostulatingthatforeachob-
server,movingalonganintegralcurveofadistinguishedfour-velocityfieldu,theUniverselooksspatially
isotropic. Mathematically, this means the following: Let Iso (M) be the group of local isometries of a
x
Lorentzmanifold(M,g),withfixedpointx∈M,andletSO3(ux)bethegroupofalllineartransforma-
tions of the tangent space T (M) which leave the 4-velocity u invariant and induce special orthogonal
x x
transformationsinthesubspaceorthogonaltou ,then
x
{Txφ: φ∈Isox(M), φ(cid:1)u=u}⊇SO3(ux)
(φ denotesthepush-forwardbelongingtoφ;see[1,p.550]).In[7]itisshownthatthisrequirementimplies
(cid:1)
that(M,g)isaFriedmannspacetime,whosestructurewenowrecall.Notethat(M,g)isthenautomatically
homogeneous.
AFriedmannspacetime(M,g)isawarpedproductoftheformM = I ×Σ,whereI isanintervalof
R,andthemetricgisoftheform
g =−dt2+a2(t)γ, (1)
such that (Σ,γ) is a Riemannian space of constant curvature k = 0,±1.The distinguished time t is the
cosmictime,anda(t)isthescalefactor(itplaystheroleofthewarpfactor(seeAppendixBof[1])).Instead
oftweoftenusetheconformaltimeη,definedbydη =dt/a(t).Thevelocityfieldisperpendiculartothe
slicesofconstantcosmictime,u=∂/∂t.
1 ByUniverseIalwaysmeanthatpartoftheworldarounduswhichisinprincipleaccessibletoobservations.Inmyopinionthe
‘Universeasawhole’isnotascientificconcept.Whentalkingaboutmodeluniverses,wedeveloponpaperorwiththehelp
ofcomputers,Itendtouselowercaseletters.Inthisdomainweare,ofcourse,freetomakeextrapolationsandventureinto
speculations,butoneshouldalwaysbeawarethatthereisthedangertobedriftedintoakindof‘cosmo-mythology’.
2 ConsulttheHomePage:http://www.mso.anu.edu.au/2dFGRS.
3 Foradescriptionandpictures,seetheHomePage:http://www.sdss.org/sdss.html.
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Ann. Phys. (Leipzig)15,No.10–11(2006) 705
0.1.1 Spacesofconstantcurvature
Forthespace(Σ,γ)ofconstantcurvature4thecurvatureisgivenby
R(3)(X,Y)Z =k[γ(Z,Y)X −γ(Z,X)Y]; (2)
incomponents:
R(3) =k(γ γ −γ γ ). (3)
ijkl ik jl il jk
Hence,theRiccitensorandthescalarcurvatureare
(3) (3)
R =2kγ , R =6k. (4)
jl jl
Forthecurvaturetwo-formsweobtainfrom(3)relativetoanorthonormaltriad{θi}
Ω(3) = 1R(3) θk∧θl =kθ ∧θ (5)
ij 2 ijkl i j
(θ =γ θk).Thesimplyconnectedconstantcurvaturespacesareinndimensionsthe(n+1)-sphereSn+1
i ik
(k = 1),theEuclideanspace(k = 0),andthepseudo-sphere(k = −1).Non-simplyconnectedconstant
curvature spaces are obtained from these by forming quotients with respect to discrete isometry groups.
(Fordetailedderivations,see[8].)
0.1.2 CurvatureofFriedmannspacetimes
Let{θ¯i}beanyorthonormaltriadon(Σ,γ).OnthisRiemannianspacethefirststructureequationsread
(weusethenotationin[1];quantitiesreferringtothis3-dim.spaceareindicatedbybars)
dθ¯i+ω¯i ∧θ¯j =0. (6)
j
On(M,g)weintroducethefollowingorthonormaltetrad:
θ0 =dt, θi =a(t)θ¯i. (7)
Fromthisand(6)weget
a˙
dθ0 =0, dθi = θ0∧θi−aω¯i ∧θ¯j. (8)
a j
ComparingthiswiththefirststructureequationfortheFriedmannmanifoldimplies
a˙
ω0i∧θi =0, ωi0∧θ0+ωij ∧θj = aθi∧θ0+aω¯ij ∧θ¯j, (9)
whence
a˙
ω0 = θi, ωi =ω¯i . (10)
i a j j
The worldlines of comoving observers are integral curves of the four-velocity field u = ∂ .We claim
t
thatthesearegeodesics,i.e.,that
∇ u=0. (11)
u
Toshowthis(andforotherpurposes)weintroducethebasis{eµ}ofvectorfieldsdualto(7).Sinceu=e0
wehave,usingtheconnectionforms(10),
∇uu=∇e0e0 =ωλ0(e0)eλ =ωi0(e0)ei =0.
4 ForadetaileddiscussionofthesespacesIrefer–forreadersknowingGerman–to[8]or[9].
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706 N.Straumann:Cosmologicalperturbationtheory
0.1.3 EinsteinequationsforFriedmannspacetimes
Inserting the connection forms (10) into the second structure equations we readily find for the curvature
2-formsΩµ :
ν
a¨ k+a˙2
Ω0 = θ0∧θi, Ωi = θi∧θj. (12)
i a j a2
AroutinecalculationleadstothefollowingcomponentsoftheEinsteintensorrelativetothebasis(7)
(cid:1) (cid:2)
a˙2 k
G00 =3 a2 + a2 , (13)
a¨ a˙2 k
G11 =G22 =G33 =−2a − a2 − a2, (14)
G =0(µ(cid:6)=ν). (15)
µν
Inordertosatisfythefieldequations,thesymmetriesofG implythattheenergy-momentumtensor
µν
musthavetheperfectfluidform(see[1,Sect.1.4.2]):
Tµν =(ρ+p)uµuν +pgµν, (16)
whereuisthecomovingvelocityfieldintroducedabove.
Now,wecanwritedownthefieldequations(includingthecosmologicalterm):
(cid:1) (cid:2)
a˙2 k
3 + =8πGρ+Λ, (17)
a2 a2
a¨ a˙2 k
−2 − − =8πGp−Λ. (18)
a a2 a2
Although the ‘energy-momentum conservation’does not provide an independent equation, it is useful
to work this out. As expected, the momentum ‘conservation’is automatically satisfied. For the ‘energy
conservation’weusethegeneralform(see(1.37)in[1])
∇ ρ=−(ρ+p)∇·u. (19)
u
Inourcasewehavefortheexpansionrate
∇·u=ωλ0(eλ)u0 =ωi0(ei),
thuswith(10)
a˙
∇·u=3 . (20)
a
Therefore,Eq.(19)becomes
a˙
ρ˙+3 (ρ+p)=0. (21)
a
Foragivenequationofstate,p=p(ρ),wecanuse(21)intheform
d
(ρa3)=−3pa2 (22)
da
todetermineρasafunctionofthescalefactora.Examples:1.Forfreemasslessparticles(radiation)we
havep=ρ/3,thusρ∝a−4.2.Fordust(p=0)wegetρ∝a−3.
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Ann. Phys. (Leipzig)15,No.10–11(2006) 707
WiththisknowledgetheFriedmannequation(17)determinesthetimeevolutionofa(t).
Exercise.Showthat(18)followsfrom(17)and(21).
Asanimportantconsequenceof(17)and(18)weobtainfortheaccelerationoftheexpansion
4πG 1
a¨=− (ρ+3p)a+ Λa. (23)
3 3
Thisshowsthataslongasρ+3pispositive,thefirsttermin(23)isdecelerating,whileapositivecosmological
constantisrepulsive.Thisbecomesunderstandableifonewritesthefieldequationas
Λ
G =κ(T +T ) (κ=8πG), (24)
µν µν µν
with
Λ
TΛ =− g . (25)
µν 8πG µν
Thisvacuumcontributionhastheformoftheenergy-momentumtensorofanidealfluid,withenergydensity
ρΛ = Λ/8πG and pressure pΛ = −ρΛ. Hence the combination ρΛ +3pΛ is equal to −2ρΛ, and is thus
negative.Inwhatfollowsweshalloftenincludeinρandpthevacuumpieces.
0.1.4 Redshift
AsaresultoftheexpansionoftheUniversethelightofdistantsourcesappearsredshifted.Theamountof
redshiftcanbesimplyexpressedintermsofthescalefactora(t).
Considertwointegralcurvesoftheaveragevelocityfieldu.Weimaginethatonedescribestheworldline
of a distant comoving source and the other that of an observer at a telescope (see Fig. 1). Since light is
propagatingalongnullgeodesics,weconcludefrom(1)thatalongtheworldlineofalightraydt=a(t)dσ,
wheredσisthelineelementonthe3-dimensionalspace(Σ,γ)ofconstantcurvaturek =0,±1.Hencethe
integralontheleftof
(cid:3) (cid:3)
to dt obs.
= dσ, (26)
a(t)
te source
between the time of emission (t ) and the arrival time at the observer (t ), is independent of t and t .
e o e o
Therefore,ifweconsiderasecondlightraythatisemittedatthetimet +∆t andisreceivedatthetime
e e
t +∆t ,weobtainfromthelastequation
o o
(cid:3) (cid:3)
to+∆to dt to dt
= . (27)
a(t) a(t)
te+∆te te
Forasmall∆t thisgives
e
∆t ∆t
o = e .
a(t ) a(t )
o e
Theobservedandtheemittedfrequencesν andν ,respectively,arethusrelatedaccordingto
o e
ν ∆t a(t )
o = e = e . (28)
ν ∆t a(t )
e o o
Theredshiftparameterzisdefinedby
ν −ν
z := e o, (29)
ν
o
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708 N.Straumann:Cosmologicalperturbationtheory
Observer (t )
o
σ
d
a(t)
=
dt Integral curve of uµ
Source (t )
e
Fig.1 RedshiftforFriedmannmodels.
andisgivenbythekeyequation
a(t )
1+z = o . (30)
a(t )
e
Onecanalsoexpressthisbytheequationν·a=constalonganullgeodesic.
0.1.5 Cosmicdistancemeasures
Wenowintroduceafurtherimportanttool,namelyoperationaldefinitionsofthreedifferentdistancemea-
sures,andshowthattheyarerelatedbysimpleredshiftfactors.
IfDisthephysical(proper)extensionofadistantobject,andδisitsanglesubtended,thentheangular
diameterdistanceD isdefinedby
A
D :=D/δ. (31)
A
IftheobjectismovingwiththepropertransversalvelocityV⊥andwithanapparentangularmotiondδ/dt0,
thentheproper-motiondistanceisbydefinition
V⊥
D := . (32)
M dδ/dt0
Finally,iftheobjecthastheintrinsicluminosityLandF isthereceivedenergyfluxthentheluminosity
distanceisnaturallydefinedas
D :=(L/4πF)1/2. (33)
L
Belowweshowthatthesethreedistancesarerelatedasfollows
D =(1+z)D =(1+z)2D . (34)
L M A
Itwillbeusefultointroduceon(Σ,γ)‘polar’coordinates(r,ϑ,ϕ),suchthat
dr2
2 2 2 2 2 2
γ = +r dΩ , dΩ =dϑ +sin ϑdϕ . (35)
1−kr2
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Ann. Phys. (Leipzig)15,No.10–11(2006) 709
r a(t ) t
e o o
0
re =
= r e
r = r
r
dt
e D
Fig.2 Spacetimediagramforcosmicdistancemeasures.
Oneeasilyverifiesthatthecurvatureformsofthismetricsatisfy(5).(Thisfollowswithoutdoinganywork
byusingin[1]thecurvatureforms(3.9)intheansatz(3.3)fortheSchwarzschildmetric.)
Toprove(34)weshowthatthethreedistancescanbeexpressedasfollows,ifr denotesthecomoving
e
radialcoordinate(in(35))ofthedistantobjectandtheobserveris(withoutlossofgenerality)atr =0.
a(t0)
DA =rea(te), DM =rea(t0), DL =rea(t0)a(t ). (36)
e
Oncethisisestablished,(34)followsfrom(30).
FromFig.2and(35)weseethat
D =a(t )r δ, (37)
e e
hencethefirstequationin(36)holds.
Toprovethesecondonewenotethatthesourcemovesinatimedt0apropertransversaldistance
a(t )
dD =V⊥dte =V⊥dt0a(t0e).
Usingagainthemetric(35)weseethattheapparentangularmotionis
dD V⊥dt0
dδ = = .
a(te)re a(t0)re
Insertingthisintothedefinition(32)showsthatthesecondequationin(36)holds.Forthethirdequation
wehavetoconsidertheobservedenergyflux.Inatimedt thesourceemitsanenergyLdt .Thisenergyis
e e
redshiftedtothepresentbyafactora(te)/a(t0),andisnowdistributedby(35)overaspherewithproper
area4π(rea(t0))2(seeFig.2).Hencethereceivedflux(apparentluminosity)is
a(t ) 1 1
F =Ldt e ,
ea(t0) 4π(rea(t0))2 dt0
thus
La2(t )
F = e .
4πa4(t0)re2
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710 N.Straumann:Cosmologicalperturbationtheory
4 Ω0 = 1, ΩΛ = 0 Ω0 = 0.2, ΩΛ = 0.8
D
3 prop
D
com
z) D
0, ang
D( 2 Dlum
)
c
/0
H Fig. 3 Cosmological distance measures as a
(
1 function of source redshift for two cosmological
models. The angular diameter distance D ≡
ang
D andtheluminositydistanceD ≡D have
A lum L
beenintroducedinthissection.Theothertwowill
0
beintroducedlater.
0 0.5 1.0 1.5 2.00 0.5 1.0 1.5 2.0
z
Insertingthisintothedefinition(33)establishesthethirdequationin(36).Forlaterapplicationswewrite
thelastequationinthemoretransparentform
L 1
F = . (38)
4π(rea(t0))2 (1+z)2
Thelastfactorisduetoredshifteffects.
Twoofthediscusseddistancesasafunctionofz areshowninFig.3fortwoFriedmannmodelswith
differentcosmologicalparameters.Theothertwodistancemeasureswillbeintroducedlater(Sect.3.2).
0.2 Luminosity-redshiftrelationforTypeIasupernovas
AfewyearsagotheHubblediagramforTypeIasupernovasgave,asabigsurprise,thefirstseriousevidence
foracurrentlyacceleratingUniverse.Beforepresentinganddiscussingcriticallytheseexcitingresults,we
developonthebasisoftheprevioussectionsometheoreticalbackground.(Forthebenefitofreaderswho
startwiththissectionwerepeatafewthings.)
0.2.1 Theoreticalredshift-luminosityrelation
We have seen that in cosmology several different distance measures are in use, which are all related by
simpleredshiftfactors.TheonewhichisrelevantinthissectionistheluminositydistanceD .Werecall
L
thatthisisdefinedby
D =(L/4πF)1/2, (39)
L
whereListheintrinsicluminosityofthesourceandF theobservedenergyflux.
Wewanttoexpressthisintermsoftheredshiftzofthesourceandsomeofthecosmologicalparameters.
IfthecomovingradialcoordinaterischosensuchthattheFriedmann-Lemaˆıtremetrictakestheform
(cid:4) (cid:5)
dr2
g =−dt2+a2(t) +r2dΩ2 , k =0,±1, (40)
1−kr2
thenwehave
1 1
Fdt0 =Ldte· 1+z · 4π(rea(t0))2.
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