Cosmology

114 Pages·2004·1.145 MB·English

by  Bartelmann M.

Cosmology
Matthias Bartelmann
Institut fu¨r Theoretische Astrophysik
Universita¨t Heidelberg
Contents
1 TheHomogeneousUniverse 4
1.1 GeometryandDynamics . . . . . . . . . . . . . . . . 5
1.1.1 Assumptions . . . . . . . . . . . . . . . . . . 5
1.1.2 Metric . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Redshift . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Dynamics . . . . . . . . . . . . . . . . . . . . 8
1.1.5 RemarkonNewtonianDynamics . . . . . . . 9
1.2 Parameters,AgeandDistances . . . . . . . . . . . . . 11
1.2.1 FormsofMatter . . . . . . . . . . . . . . . . 11
1.2.2 Parameters . . . . . . . . . . . . . . . . . . . 12
1.2.3 ParameterValues . . . . . . . . . . . . . . . . 14
1.2.4 AgeandExpansionoftheUniverse . . . . . . 15
1.2.5 Distances . . . . . . . . . . . . . . . . . . . . 17
1.2.6 Horizons . . . . . . . . . . . . . . . . . . . . 20
1.3 ThermalEvolution . . . . . . . . . . . . . . . . . . . 21
1.3.1 Assumptions . . . . . . . . . . . . . . . . . . 21
1.3.2 QuantumStatistics . . . . . . . . . . . . . . . 21
1.3.3 PropertiesofIdealQuantumGases . . . . . . . 23
1.3.4 AdiabaticExpansionofIdealGases . . . . . . 26
1.3.5 ParticleFreeze-Out . . . . . . . . . . . . . . . 26
1.4 RecombinationandNucleosynthesis . . . . . . . . . . 29
1.4.1 TheNeutrinoBackground . . . . . . . . . . . 29
1.4.2 PhotonsandBaryons . . . . . . . . . . . . . . 30
1
CONTENTS 2
1.4.3 TheRecombinationProcess . . . . . . . . . . 31
1.4.4 Nucleosynthesis . . . . . . . . . . . . . . . . 34
2 TheInhomogeneousUniverse 38
2.1 TheGrowthofPerturbations . . . . . . . . . . . . . . 39
2.1.1 NewtonianEquations . . . . . . . . . . . . . . 39
2.1.2 PerturbationEquations . . . . . . . . . . . . . 40
2.1.3 DensityPerturbations . . . . . . . . . . . . . . 41
2.1.4 VelocityPerturbations . . . . . . . . . . . . . 43
2.2 StatisticsandNon-linearEvolution . . . . . . . . . . . 45
2.2.1 PowerSpectra . . . . . . . . . . . . . . . . . 45
2.2.2 EvolutionofthePowerSpectrum . . . . . . . 46
2.2.3 TheZel’dovichApproximation . . . . . . . . . 48
2.2.4 NonlinearEvolution . . . . . . . . . . . . . . 49
2.3 SphericalCollapse . . . . . . . . . . . . . . . . . . . 52
2.3.1 CollapseofaHomogeneousOverdenseSphere 52
2.3.2 CollapseParameters . . . . . . . . . . . . . . 53
2.3.3 ThePress-SchechterMassFunction . . . . . . 55
2.4 HaloFormationasaRandomWalk . . . . . . . . . . . 57
2.4.1 Correct Normalisation of the Press-Schechter
MassFunction . . . . . . . . . . . . . . . . . 57
2.4.2 ExtendedPress-SchechterTheory . . . . . . . 58
2.4.3 HaloDensityProfiles . . . . . . . . . . . . . . 60
3 TheEarlyUniverse 63
3.1 StructuresintheCosmicMicrowaveBackground . . . 64
3.1.1 Simplified Theory of CMB Temperature Fluc-
tuations . . . . . . . . . . . . . . . . . . . . . 64
3.1.2 CMB Power Spectra and Cosmological Param-
eters . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.3 Foregrounds . . . . . . . . . . . . . . . . . . 71
3.2 CosmologicalInflation . . . . . . . . . . . . . . . . . 72
CONTENTS 3
3.2.1 Problems . . . . . . . . . . . . . . . . . . . . 72
3.2.2 Inflation . . . . . . . . . . . . . . . . . . . . . 74
3.3 DarkEnergy . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.1 ExpansionoftheUniverse . . . . . . . . . . . 81
3.3.2 ModifiedEquationofState . . . . . . . . . . . 82
3.3.3 ModelsofDarkEnergy . . . . . . . . . . . . . 83
3.3.4 EffectsonCosmology . . . . . . . . . . . . . 84
4 TheLateUniverse 87
4.1 GalaxiesandGas . . . . . . . . . . . . . . . . . . . . 88
4.1.1 EllipticalsandSpirals . . . . . . . . . . . . . 88
4.1.2 Spectra,Magnitudesand K-Corrections . . . . 89
4.1.3 LuminosityFunctions . . . . . . . . . . . . . 91
4.1.4 CorrelationFunctionsandBiasing . . . . . . . 93
4.1.5 InterveningGas . . . . . . . . . . . . . . . . . 95
4.2 GravitationalLensing . . . . . . . . . . . . . . . . . . 98
4.2.1 Assumptions,IndexofRefraction . . . . . . . 98
4.2.2 DeflectionAngleandLensEquation . . . . . . 99
4.2.3 LocalLensMappingandMassReconstruction 101
4.2.4 DeflectionbyLarge-ScaleStructures . . . . . 102
4.2.5 Limber’s Equation and Weak-Lensing Power
Spectra . . . . . . . . . . . . . . . . . . . . . 103
4.3 GalaxyClusters . . . . . . . . . . . . . . . . . . . . . 106
4.3.1 GalaxiesinClusters . . . . . . . . . . . . . . 106
4.3.2 X-RayEmission . . . . . . . . . . . . . . . . 108
4.3.3 GravitationalLensingbyGalaxyClusters . . . 110
4.3.4 Sunyaev-Zel’dovichEffects . . . . . . . . . . 111
4.3.5 ClustersasCosmologicalTracers . . . . . . . 112
4.3.6 ScalingRelations . . . . . . . . . . . . . . . . 112
Chapter 1
The Homogeneous Universe
4
CHAPTER1. THEHOMOGENEOUSUNIVERSE 5
1.1 Geometry and Dynamics
1.1.1 Assumptions
• cosmologyrestsontwofundamentalassumptions:
1. when averaged over sufficiently large scales, the observable
properties of the Universe are isotropic, i.e. independent of
direction;
it remains to be clarified what sufficiently large scales are;
nearby galaxies are very anisotropically distributed, distant
If the universe is isotropic about all
galaxiesapproachisotropy,themicrowavebackgroundisal-
points,itmustbehomogeneous.
mostperfectlyisotropic
2. ourpositionintheUniverseisbynomeanspreferredtoany
other(cosmologicalprinciple);
reflects Copernican revolution of the world model, when it
was realised that the Earth is not at the centre of the Uni-
verse;
by the second assumption, the first must hold for every observer
in the Universe; if the Universe is in fact isotropic around all of
its points, it is also homogeneous; thus, these two assumptions
The galaxy distribution is mani-
areoftenphrasedas
festlyanisotropic...
theUniverseishomogeneousandisotropic
• theseareboldassumptions,whichhavetobejustified;obviously,
an ideally homogeneous and isotropic universe would not allow
ustoexist;itneedstobecarefullystudiedhowanidealisedworld
model following from these two assumptions can accomodate
structures
... butthemicrowavebackgroundis
phantasticallyisotropic.
• ofthefourinteractions(strong,weak,electromagneticandgravi-
tational), strong and weak are limited to length scales typical for
elementary-particle interactions; electromagnetism is limited in
range by the shielding of opposite charges, although magnetic
fields can bridge very large scales; the remaining force relevant
forcosmologyisgravity
• gravity is described by general relativity; Newtonian gravity was
constructedforisolatedbodiesandhasfundamentaldifficultiesin
explainingspacefilledwithhomogeneousmatter
• general relativity describes space-time as a four-dimensional
manifold whose metric tensor g is a dynamical field; its dy-
µν
namics is governed by Einstein’s field equations which couple
themetrictothematter-energycontentofspace-time
CHAPTER1. THEHOMOGENEOUSUNIVERSE 6
• as the structure of space-time determines the motion of matter
and energy, which determine the structure of space-time, general
relativityisinevitablynon-linear(incontrasttoelectrodynamics);
solutionsofEinstein’sfieldequationsarethusthustypicallyvery
difficulttoconstruct
1.1.2 Metric
• duetosymmetry,the4×4tensorg hastenindependentcompo-
µν
nents, the time-time component g , the three space-time compo-
00
nentsg ,andthesixspace-spacecomponentsg
0i ij
• the two fundamental assumptions greatly simplify the metric;
phrasedinamorepreciselanguage,theyread
1. when averaged over sufficiently large scales, there exists a
mean motion of matter and energy in the Universe with re-
specttowhichallobservablepropertiesareisotropic;
2. all fundamental observers, i.e. imagined observers follow-
ing this mean motion, experience the same history of the
Universe,i.e.thesameaveragedobservableproperties,pro-
videdtheysettheirclockssuitably
• considertheeigentimeelementds,
ds2 = g dxµdxν (1.1)
µν
spatial coordinates attached to fundamental observers are called
comoving coordinates; in such coordinates, dxi = 0 for funda-
mentalobservers;requiringthattheireigentimeequalthecoordi-
natetimedt,wehave
ds2 = g dt2 = c2dt2 ⇒ g = c2 (1.2)
00 00
• isotropy requires that clocks can be synchronised such that g =
0i
0; if that was impossible, the components of g singled out a
0i
preferreddirectioninspace,violatingisotropy;thus
g = 0 (1.3)
0i
• thelineelementisthusreducedto
ds2 = c2dt2 +g dxidxj (1.4)
ij
thus, spacetime can be decomposed into spatial hypersurfaces of
constanttime,i.e.itpermitsafoliation;withoutviolatingisotropy
CHAPTER1. THEHOMOGENEOUSUNIVERSE 7
and homogeneity, the spatial hypersurfaces can be scaled by a
functiona(t)whichcanonlydependontime,
ds2 = c2dt2 −a2(t)dl2 (1.5)
where dl is the line element of homogeneous and isotropic three-
space; a special case of (1.5) is Minkowski space, for which dl is
theEuclideanlineelement
• isotropyrequiresthree-spacetohavesphericalsymmetry;wethus
introducepolarcoordinates(w,θ,φ)wherewistheradialcoordi-
nateand(θ,φ)arethepolarangles:
The space-time of the universe can
h i
dl2 = dw2 + f2(w) dφ2 +sin2θdθ2 = dw2 + f2(w)dω2 , (1.6) be foliated into flat or positively or
K K
negatively curved spatial hypersur-
where dω is the solid-angle element; the radial function f (w) is
K faces.
permitted because the relation between the radial coordinate w
andtheareaofspheresofconstantwisstillarbitrary
• the metric expressed by the line element (1.6) is manifestly
isotropic; it can be shown that homogeneity requires f (w) to be
K
trigonometric,hyperbolic,orlinearinw,

f (w) =  wK−1/2sin(K1/2w) ((KK >= 00)) (1.7)
K  |K|−1/2sin(|K|1/2w) (K < 0)
where K is a constant parameterising the curvature of spatial hy-
persurfaces; f (w)and|K|−1/2 havethedimensionofalength
K
• analternativeformofthelineelementdsisobtainedsubstituting
theradialcoordinatebyr for f (w),then
K
dr2
dl2 = +r2dω2 (1.8)
1−Kr2
this is often used, but has the disadvantage of becoming singular
for K > 0andr = K−1/2
• we thus arrive at the metric for the homogeneous and isotropic
universe,
h i
ds2 = c2dt2 −a2(t) dw2 + f2(w)dω2 (1.9)
K
with f (w)givenby(1.7);thisiscalledRobertson-Walkermetric
K
1.1.3 Redshift
• spatialhypersurfacescanexpandorshrinkcontrolledbythescale
functiona(t);thisleadstoared-orblueshiftofphotonspropagat-
ingthroughspace-time
CHAPTER1. THEHOMOGENEOUSUNIVERSE 8
• considerlightemittedfromacomovingsourceattimet reaching
e
a comoving observer at w = 0 at time t ; since ds = 0 for light,
o
themetric(1.9)requires
c|dt| = dw (1.10)
where the modulus on the left-hand side indicates that time can
run with or agains w, depending on whether w is measured to-
wardsorfromtheobserver
• thecoordinatedistancebetweensourceandobserveris
Z Z
to to cdt
w = dw = = const. (1.11)
eo
a(t)
te te
thusthederivativeofw withrespecttotheemissiontimet must
eo e
vanish
dw 1 dt 1 dt a
eo = o − ⇒ o = o (1.12)
dt a(t )dt a(t ) dt a
e o e e e e
• timeintervalsdt atthesourcearethuschangeduntiltheyarriveat
e
the observer in proportion to changes in the scale of the universe
betweenemissionandabsorption
• letdt = ν−1 bethecycletimeofalightwave,then
ν λ λ −λ a(t )
e = o = 1+ o e = 1+z = e (1.13)
ν λ λ a(t )
o e e o
thus, light is red- or blueshifted by the same amount as the Uni-
verseexpandedorshrunkbetweenemissionandobservation
1.1.4 Dynamics
• thedynamicsofthemetric(1.9)isreducedtothedynamicsofthe
scale factor a(t); differential equations for a(t) now follow from
Einstein’sfieldequations,whichread
8πG
G = T +Λg (1.14)
αβ αβ αβ
c2
Λ is the cosmological constant originally introduced by Einstein
inordertoallowstaticcosmologicalmodels
• G is the Einstein tensor constructed from the curvature tensor,
αβ
whichdependsonthemetrictensoranditsfirstandsecondderiva-
tives
• T isthestress-energytensorofthecosmicfluid,whichmustbe
αβ
of the form of the stress-energy tensor of a perfect fluid, charac-
terised by pressure p and (energy) density ρ, which can only be
functionsoftimebecauseofhomogeneity,
p = p(t) , ρ = ρ(t) (1.15)
CHAPTER1. THEHOMOGENEOUSUNIVERSE 9
• when specialised to the metric (1.9), Einstein’s equations (1.14)
reducetotwodifferentialequationsforthescalefactora(t):
(cid:18)a˙(cid:19)2 8πG Kc2 Λ
= ρ− +
a 3 a2 3
!
a¨ 4πG 3p Λ
= − ρ+ + (1.16)
a 3 c2 3
these are Friedmann’s equations; a Robertson-Walker metric
whose scale factor satisfies (1.16) is called Friedmann-Lemaˆıtre-
Robertson-Walkermetric;thescalefactorisuniquelydetermined
onceitsvalueatafixedtimet ischosen;weseta = 1today;
• the Friedmann equations can be combined to yield the adiabatic
equation AlexanderFriedmann
d (cid:16) (cid:17) d (cid:16) (cid:17)
a3ρc2 + p a3 = 0 (1.17)
dt dt
which intuitively states energy conservation: the left-hand side is
the change in internal energy, the right-hand side is the pressure
work; this is the first law of thermodynamics in absence of heat
flow(whichwouldviolateisotropy)
• since energy conservation (1.17) follows from the Friedmann
equations(1.16),anytwoequationsfrom(1.16)and(1.17)canbe
used equivalently to all three of them; we follow common prac-
tise and use the first-order equation from (1.16), which we will
calltheFriedmannequationhenceforth,and(1.17)whereneeded
1.1.5 Remark on Newtonian Dynamics
GeorgesLemaˆıtre
• note that (1.16) can also be derived from Newtonian gravity, ex-
cept for the Λ term; the argument runs like this: in a homoge-
neousandisotropicuniverse,asphericalregionofradiusRcanbe
identifiedaroundanarbitrarypoint,thematterdensitywithinthat
sphere must be homogeneous; the matter surrounding the sphere
cannot have any influence on its dynamics because it would have
to pull into some direction, which would violate isotropy; thus,
thesizeofthesphereisarbitrary
• suppose now a test mass m is located on the boundary of the
sphere;it’sequationofmotionis
!
G 4π 4πG
r¨ = − r3ρ = − rρ (1.18)
r2 3 3
thisisalreadythesecondeq.(1.16)exceptforthepressureterm