Development of structure in early universecdeclerc/astroparticles/2009-10/8.structures.pdf · •...

Development of structure in earlyDevelopment of structure in earlyuniverse

Particle Astrophysics , chapter 8p y , p

Standard cosmological model• ΛCDM model = concordance model = Standard Cosmological

Model (SCM)

• Ingredients:• Universe is uniform and homogenous over large distances

(cosmological principle) • Universe is perfect frictionless fluid• Started from hot Big Bang some 13.7 Gyr ago• Is made up of 3 components : radiation + luminous and dark

matter + dark energy• Radiation dominated early universe, then matter dominated• Decoupling of radiation and matter at z ≈ 1100 (380.000 year) -

di t li h t l ti CMBpredicts relic photon population = CMB

chapter 8 astro-particle physics 2009-10 2

Standard cosmological model• Agreement with observations:

Abundance of light elements ↔ primordial nucleosynthesisObservation of high redshift objects, up to z ≈ 8Existence of all pervading Cosmic Microwave Background p g gradiation – uniform over large angles – cooled down to 3KEnergy density today = 5% baryons + 1% neutrinos + 18% cold dark matter + 76% dark energyDark matter observed at all scalesHubble constant today ª 71 km/s/MpcUniverse presently accelerating → dark energy neededTotal density today equal to critical density ρc ª 5 GeV/m3

Universe is flat with k ª 0


Question: origin of structure formation?• Postulate of BB model : universe started as uniform and

homogenous fluidT d t t t ‘ ll’ l l i l t f• Today : structures seen at ‘small’ scale: galaxies, clusters of galaxies, voids ..

• CMB anisotropies at level 510T ρ −Δ Δ≈ ≈CMB anisotropies at level

Universe today is not homogenous & was not

10T ρ

≈ ≈

Universe today is not homogenous & was not homogenous at z ≈ 1100

• ?? What generated these structures / anisotropies ??

• Answer : processes before radiation – matter decoupling• Popular models : exponential expansion during inflation period• But : do they predict temperature anisotropies observed in CMB

and large scale structures ?and large scale structures ?


• Universe today is not homogenous & was nothomogenous & was not homogenous at z ≈ 1100


Inflation models• Origin of structures = quantum fluctuations in energy density in

very early universeFl i ‘f ’ d ‘i fl d’ d i i fl i i d• Fluctuations are ‘frozen out’ and ‘inflated’ during inflation period

• Inflation period : between Planck (1019GeV, 10-43 s) and GUT (1015G V 10 34 ) ti l th(1015GeV ,10-34 s) era – exponential growth

• Fluctuations then grow linearly with expansion during radiation dominated eradominated era

• Fluctuations = seeds for development of greater fluctuations in density during matter dominated period (stars )density during matter dominated period (stars …)

• Fluctuations collapse under gravity during matter dominated era• Primordial fluctuations could only start to grow if distributed• Primordial fluctuations could only start to grow if distributed

over distances O(galaxy clusters) = O( few Mpc)• Primordial fluctuations propagate inside horizon as soundPrimordial fluctuations propagate inside horizon as sound

waves in cosmic fluidchapter 8 astro-particle physics 2009-10 6

Overview lectureC t t i i t b f ti fi ld ?• Can structures originate because of magnetic fields?

• Horizon and flatness problems in SCMf C f• Inflation - Chaotic inflation model

• Quantum fluctuations & inflation - Spectrum of primordial fl t tifluctuations

• Gravitational collapse – Jeans radiusG th f t t i di i• Growth of structure in an expanding universe

• Evolution of fluctuations during radiation era - Link to neutrino massmass

• Growth of fluctuations in matter dominated eraTemperature fluctuations & anisotropies in CMB• Temperature fluctuations & anisotropies in CMB

• Structure of angular power spectrum of CMB – acoustic peaks


•Galactic and intergalactic magnetic fieldsGalactic and intergalactic magnetic fields•2 problems in SCM related to initial conditions required: horizon and flatness problems

SOME PRELIMINARIESrequired: horizon and flatness problems


Galactic and intergalactic B fields• Galactic and intergalactic magnetic fields cannot explain lumpy

structure of universe

• At small = stellar = scales : intense magnetic fields onlyimportant in later stages of star evolution

• At galactic scale : <B (Milky Way)> ~ 3µG • At intergalactic scale <B> of 10-7 – 10-11 G << G(galactic)• → intergalactic magnetic fields probably have very small role in

d l t f l l t tdevelopment of large scale structures

• Development of large scale structures mainly due to gravity• Electromagnetic interactions play subsidiary role


Horizon problem in SCM• Particle horizon = distance over which one can observe a

particle by exchange of a light signal• Particle and observer are causally connected• Horizon distance in static universe of age t HD ct=

• Today : Hubble radius 0

04cct GpcH∼ ∼

• In flat universe with k=0 : on very large scale light travels ≈ in straight line

• In expanding universe : optical horizon today = distance between 2 points travelled by light during time t

• has expanded between t=0 (light emission) and t=t0 (today)

( ) ( )0 0

0t zcdt cdzD t R

=

= =∫ ∫chapter 8 astro-particle physics 2009-10 10

( ) ( ) ( ) ( )00

0Ht z

D t RR t H t= =∞

= =∫ ∫

Horizon problem in SCM• most cosmological models

– Radiation dominated : n = 1/2( ) nR t t∼

– Matter dominated : n = 2/3

→ optical horizon today ( ) 00 1H

ctD tn

=−

• fraction of universe (dimension R(t))

1 n−

( ) ( )( )

( )1H nD tF t t

R−

= ∝inside optical horizon increases with t

• Horizon distance at time of decoupling (z = 1100, tdec = 4 x 105y) as t d

( ) ( )R t

seen today ( ) ( ) ( )0 1H dec dec H dec decD t ct D t ct z≈ → ≈ +

• Angle subtended by that optical horizon today is small fraction of observable universe ( )

( )1

13

dec decdec

ct zθ

+°∼ ∼


( )03decdecc t t−

Horizon problem in SCM• Only particles inside horizon at tdec were in causal contact at

time of decoupling = time of CMB freeze-out• Why is CMB uniform (to within 1/105) over full sky , ie over

angles >> 1° ??• Origin of causal contact between CMB photons at large angles?


Flatness problem• universe is flat today,but has been much flatter in early universe• How come that curvature was so finely tuned to Ω=1 in very

early universe?• Total energy density, relative to critical density

( )( ) ( )

( )( ) ( )

2 2

2 2 2 21 0 1 1.02 0.020 0

tot

obskc kctR t H t R H

Ω = + →Ω = + ±=

• At GUT scale t ≈ 10-34s and kT ≈ 1015 GeV

( ) ( ) ( ) ( )0 0R t H t R H

suppose that curvature term( ) ( )

( )2

2 2 1kc OR t H t

=

• Then we expect today that( ) ( )

( ) ( ) ( ) ( )2 2 2 22 2

dec decR t R t H t H tkc kc ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟


( ) ( ) ( ) ( )( )( )

( )( )

( )( )

( )( )2 2 2 2 0 00 0

dec dec

dec decR t R H t HR H R t H t= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Flatness problem• So that

( ) ( ) ( ) ( )

2 2

2 2 2 252

0 010kc kc

R H R t H t= ×

• Argument turned around Very small !!!g52 341 today 1 10 at 10t s− −Ω = → Ω = ± =

• How could Ω have been so closely tuned to 1 in early universe?

• Requires thermal equilibrium outside scale of particle horizons• → Guth (1981) : preliminary stage of BB = rapid exponential Guth (1981) : preliminary stage of BB rapid exponential

expansion by huge factor = inflation• Several models , eg chaotic inflationg


Exponential expansion, chaotic inflationp p ,Scalar field in early universePrimordial density fluctuations

INFLATION MODELSPrimordial density fluctuations


Exponential expansion 1• Solution to horizon problem : accelerating phase = period of

inflation - prior to decelerating phase (SCM expansion)• Causal contact between particles exists at earlier times –

causal contact is lost during inflation• Comoving distance decreases with time • Friedman equation for dominant and constant rΛ• Inflation between t1 and t2

( )2

8 H t tR Gπ ρ⎛ ⎞ ( )2

1

12 1

1

83

H t tR GR

R R eπ ρΛ −⎛ ⎞= ⇒⎜ ⎟

⎝ ⎠=

12

0

128

3 c

GH H ρπ ρρ

Λ Λ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

=


c⎝ ⎠

Exponential expansion 2• At GUT time scale : particle horizon

( ) 34 2610 horizon distance 01t GUT s ct m−−= ⇒ ≈( )

( )( )

12

1GUT GUT mR t t⎛ ⎞

≈⎜ ⎟∼

• In SCM calculate back size of universe to GUT time scale in

( ) 001

tm

R ⎜ ⎟⎝ ⎠

radiation dominated universe• To inflate from 10-26m to 1m

in (t2-t1) need to satisfy

( )262 10 60R H t t≈ → >

• H ≈ constant

( )22 1

1

10 60H t tR

≈ → − >


Exponential inflation - flatness• Inflation yields flat universe – cfr inflate balloon• If curvature term at start of inflation is ~ 1 ( )

2

1kc O=

• Then at end of inflation curvature term is fine tuned to 1( ) ( )

( )2 21 1

1OR t H t

=

( ) ( ) ( )

2

22 21 522 2 12H t t

kcR tR t H −⎛ ⎞

⎜ ⎟( )

( )

( )( )

( )1 5222

22 2

1

2 1210

H t tR t H ekc R tR t H

−=⎜ ⎟⎜ ⎟⎝ ⎠∼ ∼

( )1


Scalar inflanton field• Postulate (Guth 1982): in early universe there exists matter in a

form that can be described by a scalar field : inflanton field fFi ld d ib l ti l f• Field describes scalar particles of mass m

• Analogy with Higgs mechanism in theories of fundamentalinteractionsinteractions

• Evolution of Universe goes through different phase transitions accompanied by spontanous symmetry breaking

• E.g. From Planck era (kT ~ 1019 GeV) to GUT era (kT ~ 1015

GeV) : spontaneous breaking of symmetry betweenelectroweak and strong forceelectroweak and strong force

• Spontanous symmetry breaking = scalar Higgs fields acquirenon-zero valuesnon zero values

• Inflation = phase transition → presence of scalar field in veryearly universe followed by spontaneous symmetry breaking


Scalar inflanton field• At t1 = Planck era kT ~ 1019 GeV – scalar field dominates• For kT >> mc2 field is in ground state with vacuum expectation

value <f> = 0 → false vacuum state• Expansion of early universe and cooling : for kT < kTc ~ mc2 :

spontaneous symmetry breaking occurs• Vacuum expectation value of field becomes <f> = fmin

• Coincides with end of inflation phase – around GUT era• Inflation = transition from metastable false vacuum to true

vacuum


Inflation startswhen systemi i f l t t

Inflation stops when systemi i t t tis in false vacuum state is in true vacuum state


Chaotic inflationP d b Li d t l I 1982 84 t l bl i G th• Proposed by Linde et al. In 1982-84 to solve problems in GuthmodelInflation starts at different times in different locations• Inflation starts at different times in different locations

• Quantum fluctuations at Planck time → start value of inflation field is random some regions attain condition of inflationfield is random – some regions attain condition of inflation earlier than others

Early start of inflationEarly start of inflation

L t t t f i fl tiLate start of inflation


Equation of motion of scalar field2⎡ ⎤• Lagrangian energy of inflanton field ( ) ( )2

3

2L T V R Vφφ φ

⎡ ⎤= − = −⎢ ⎥

⎣ ⎦

• Mechanics → equation of motion of scalar field

3 0dVHφ φ+ + =

• Similar to oscillator / pendulum motion – related to reheating

3 0Hd

φ φφ

+ + =

p g• ‘slow roll’ approximation : start from small kinetic energy

compared to potential energy – field ‘rolls slowly’ into potentialsuch than inflation can occur

( )0

0

0 small cstV φ

φ φ φ φφ ρ= → ≈ =

≈• Friedman equation becomes• → R grows exponentially

( )0 φφ ρ

( )202 8 8G VRH cstφπ ρ π φ⎛ ⎞

= = = =⎜ ⎟


23 3 PL

H cstR M⎜ ⎟

⎝ ⎠

SummaryPl k

• During reheatinglarge amount of

Planck era

gparticles are created

chapter 8 astro-particle physics 2009-10 24GUT era

Quantum fluctuations = seedQ t fl t ti i l i d ti f ti l• Quantum fluctuations in early universe → production of particle-antiparticle pairs eg e+e-Are at heart of anisotropies in early universe

E tΔ Δ ∼• Are at heart of anisotropies in early universe• During inflation particle moves away from antiparticle &

annihilation not probable → matterannihilation not probable → matter• Inflation end time differs

t φφΔ

Δ =

• amplitudes of fluctuations in CMB observed today are related to amplitudes of density fluctuations at end of inflation

φ

amplitudes of density fluctuations at end of inflation 2

horHH tρ δ

ρ φΔ

= = Δ ∼

2CMB obs 510m

ρ φ

ρ φ −⎛ ⎞⎛ ⎞Δ⎯⎯⎯⎯→⎜ ⎟⎜ ⎟∼


10PL PLM Mρ

→⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Spectrum of primordial fluctuations• Quantum fluctuations = matter oscillations in cosmic fluid – are

amplified during inflation – expected density contrast?• Inflation = very fast → distance between crests & troughs larger

than horizon scale – are no longer in causal contact• Fluctuations are ‘frozen’ – can be described classically• Density fluctuations are f(space,time) and lead to modification

f t t & Ei t i fi ld tiof energy-momentum tensor & Einstein field equations• → gravitational potential Φ

S h i l t i ( )22 G rr π ρ

Φ =• Spherical symmetric case ( )3

rΦ =

• 2 cases: globally over horizon distance and over small arbitraryscale λ

2

2 Gπ ρΦ = ( )

2

23

G λπ ρΔΦ = Δ


23H( )

3ρ

Spectrum of primordial fluctuations• Fractional perturbation 2 2H ρλ

ρΔΦ Δ

=Φ

• Universe is effectively in stationary state

t H tΔΦ1

2 2 1λ

ρδ ⎛ ⎞Δ= ⎜ ⎟ ∼

• Density fluctuations follow power law dependence on λ

cst H cstΔΦ ≈ ≈Φ2

rmsλ ρ

δλ⎜ ⎟

⎝ ⎠

• Density fluctuations = oscillations → Fourrier decomposition in terms of wave numer k=2π/λ

( )⎡ ⎤• define density contrast δ in point x ( )( )x

xρ ρ

δρ

⎡ ⎤−⎣ ⎦=

• correlation function ξ between 2points separated by distance r ( ) ( ) ( )r x r xξ δ δ= +

• Average over all points in given volumechapter 8 astro-particle physics 2009-10 27

Spectrum of primordial fluctuations• Assume random phases of fluctuations – Fourrier

decomposition ( ) ( )2 3.ik rr k e d kξ δ= ∫

• Power of fluctuation spectrum

( ) ( )r k e d kξ δ= ∫( ) ( ) 2

P k kδ=• Integration over θ,f

( ) ( )

( ) ( ) 2sin4 krP k k dkξ ⎡ ⎤⎢ ⎥∫

sin1 0krkr ⎡ ⎤>> → ≈⎢ ⎥( ) ( ) 24r P k k dkkr

ξ π ⎡ ⎤= ⎢ ⎥⎣ ⎦∫( ) ( )3 3

1 0

1 n

krkr

kr kr kk Pξ +

>> → ⎢ ⎥⎣ ⎦

< → ≈ ∼• CMB observations compatible with• mean density contrast observed is as predicted by inflation

( ) ( )0.95 0.02n = ±

• From CMB observations 4 2

21k kδξ λ→∼ ∼ ∼


0.003 0.010 independent of wave numberlndn

d k = ±

Gravitational collapse and Jeans massGrowth of structure in expanding universep gEvolution of fluctuations during radiation eraEvolution of fluctuations during matter dominated era

GROWTH OF STRUCTUREEvolution of fluctuations during matter dominated era


Introduction • Large scale structure observed today in CMB originate from tiny

fluctuations in cosmic fluid during inflationary phase• Fluctuations are treated as classical density perturbations• Start from density fluctuations in non-relativistic matter -

baryonic and cold dark matter• The perturbations grow in expanding universe• In radiation era : also relativistic particles

• Will primordial fluctuations of matter density survive the expansion?


Gravitational collapse of gas clouds• Jeans model : collapse of gas cloud under its own gravity →

model of star formation and evolutionM m

• Gas cloud M, particle m• Gravitational collapse of cloud: particle

M m

1looses gravitational PE and gains KE, or heat

• Time for free fall of m from r=r0 to r=0

123

32FFtGπρ

⎛ ⎞= ⎜ ⎟⎝ ⎠

• Heated cloud : excitation of atoms, dissociation of molecules, radiation of photons

• Gravitational energy absorbed and re-emitted• Equilibrium when heated gas pressure = gravitational pressure• Cloud condenses when

2 3kiki

GM MkTE E E E ⎛ ⎞>> ≈ = ⎜ ⎟


2grav kingrav kinE E E Er m

>> ≈ ⎜ ⎟⎝ ⎠

Jeans length = critical dimension• Start from upward fluctuation in density in gas cloud (eg

fluctuation at end of inflation)• When will this lead to condensation?• Critical size of cloud = Jeans length

12

sJ vGπρ

λ ⎛ ⎞= ⎜ ⎟

⎝ ⎠Depends on density r• Meaning of Jeans length:

Gρ⎝ ⎠

Lt• Time for sound waves to cross cloud of size L• when L << λJ sound wave (= propagation of density

sLt v=

perturbation) travels to/fro in cloud• When L >> λJ sound cannot travel fast enough from one side to

other – cloud condenses around density perturbations• For non-relativistic matter (mass m) 1

25 kTπλ ⎛ ⎞= ⎜ ⎟Jeans length is


3J G mλ

ρ= ⎜ ⎟⎝ ⎠

Structure growth in expanding universeSt t f d d it fl t ti Δ i t ti h• Start from upward density fluctuation Δρ in static, homogenousand isotropic fluid of non-relativistic particlesIn expanding universe growth of density fluctuation = power law• In expanding universe growth of density fluctuation = power law

• Density contrast evolution (M=mass universe, R=scale)

( ) ( )( ) ( )

2 03 1310

R c R tR GM z

zρδρ

δδ

⎛ ⎞Δ= − ≈ →⎟

Δ⎜

⎠= ∝ +

⎝

• Between decoupling time z=1100 and now primordial fluctuations of order 10-5 would have grown by x 1000

( )ρ ⎠⎝

fluctuations of order 10-5 would have grown by x 1000• But ! Density fluctuations in material (see galaxy surveys) are

much larger todaymuch larger today• → need baryonic matter + non-relativistic (cold) dark matter


Evolution of fluctuations in radiation era• Before decoupling at z ≈ 1100 universe is radiation dominated:

photons and relativistic particles, eg neutrinos• Velocity of sound is relativistic

3scv ∼

• horizon distance (≈ ct) is of order of Jeans length → no collapse1 12 2 23 32 32c Gπ π π⎛ ⎞ ⎛ ⎞ ⎛ ⎞2

2

3 32 329r sJ

c Gc vt

ctG

π π πρρ

λ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= → = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⎝ ⎠

• Density fluctuations in matter inside horizon survive –fluctuations continue to grow as R(t)

• But: photon and neutrino components have damping effect


Damping by photons• Fluctuations are most probably

adiabatic: baryon and photon densities fluctuate together as in

M=1011 Msundensities fluctuate together, as in sound wave

• Photons have ~ light velocity –g ystream away from regions of highdensity to regions of low densityPh t d it l th• Photon energy density larger thanbaryon density

• → diffusion damping (Silk damping)

M=1013 Msun

→ diffusion damping (Silk damping)• But: scattering with electrons of

plasma → drag effect• This has mostly effect on

fluctuations over small λ scales, encompassing small masses

timeencompassing small masses


Damping by neutrinosN t i l ti i ti h t d k tt• Neutrinos = relativistic hot dark matter

• In early hot universe: same number of neutrinos as photons• Have only weak interactions – no interaction with matter as

soon as kT ~ 3 MeVSt f d d it fl t ti lli i l• Stream away from upward density fluctuations = collisionlessdamping

• Velocity close to c : stream up to horizon new perturbations• Velocity close to c : stream up to horizon – new perturbations coming into horizon are not able to grow – ‘iron out’ fluctuations

• Close to very large perturbations ~ 500Mpc (super clusters)Close to very large perturbations 500Mpc (super clusters) neutrinos cluster gravitationally like dark matter – no time to escape

• Analysis of fluctuation spectrum of large scale structures → upper limit on neutrino mass

( ) 20 42 V<∑chapter 8 astro-particle physics 2009-10 36

( ) 2, , 0.42m e µ eV cν τ <∑

Baryonic mass inside horizonB i i id h i i di ti d i t d• Baryonic mass inside horizon in radiation dominated era

( ) ( )( )3H bM t t ctρ ⎫

⎪⎪∼( ) ( )( )

( )3 33

2

1b HR T M t

Tt T

ρ −

−

⎪⎪∝ ∝ ⇒ ∝⎬⎪∝ ⎪

• At time of matter-radiation decoupling

t T∝ ⎪⎭

⎫( )( ) 17

3

13 18

1

10 10 10H d

dec

d

cb b

bM t

z

t M Ms

ρ ρ ⎫= Ω +⎪⎪≈ ⇒ Ω⎬ ∼ ∼( )10 10

0.10

04H dec sdec sun unb

b

M tt M Ms ⇒ Ω⎬⎪Ω = ⎪⎭

• Mass of galaxies ~ 1011 Msun – Mass of clusters ~ 1014 Msun

• At decoupling galaxies & clusters fit inside horizon : causal contact


Fluctuations in matter dominated era• Neutral atoms are formed – sound velocity and Jeans length

drop2

15kT⎛ ⎞ 2

2553 2 10

3s s

HM ckTv v

c c−⎛ ⎞

= → = = ×⎜ ⎟⎝ ⎠

• Gravitational collapse happens faster (Jeans length)• Matter structures can grow• first globular clusters with ~ 105 stars = oldest objects• Cold dark matter plays vital role in growth of structures• density contrast at decoupling ~ 10-5 (CMB observations =

photons at decoupling time) • Growth of density contrast during matter-dominated era with

only baryonic matter cannot explain today structures in galaxies d l tand clusters


Observations

CMB


From density contrast to temperature anisotropyy p pyAngular spectrum of anisotropiesExperimental observations

ANISOTROPIES IN THE CMBExperimental observations


CMB temperature map

3dipole 10T mKT−Δ ≈

510T µKT−Δ ≈


Contrast enhanced to make 10-5 variations visible!

temperature fluctuations• From density fluctuations of matter over length λ, or wave

number k, to temperature fluctuations in CMB

• CMB temperature maps• WMAP: Angular resolution ~ 10’

large dipole = Doppler shift due to motion of solar system withrespect to CMB – subtracted before making anisotropy mapGalactic emission – subtracted → keep only extra-galactich tphotons

Primary anisotropy : effects occuring at z=1100Secondary anisotropy : interaction of photons with interstellargas during during 14Gy journey - interaction with electronsgenerates polarizationgenerates polarization


Primary anisotropies• Adiabatic fluctuations : photons & matter fluctuate together• Photons keep imprint of matter density contrasts – pressure

waves → sound waves - propagate with velocity vs

• In denser regions photons are compressed and get hotter13

!AD

matterT

TDMρ ρ ρ

ρΔ Δ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠→

• Dominant at small angular separations today : acoustic peaks

• Sachs-Wolfe effect : in denser regions photons are cooledbecause of gravitational potential caused by dense matter –smearing effect

• Sachs-Wolfe effect dominates at large scales → CMB i t i l ti f 3° danisotropies over angular separations of 3° and more


Angular spectrum of anisotropies• Statistical analysis of the temperature variations in CMB map –

multipole analysis• Measure correlations between temperature fluctuations (with

respect to mean of 2.7K) in 2 directions (n,m) in sky separatedb l θ ll i ith i θby angle θ – average over all pairs with given θ

• Expand in Legendre polynomials

C d ib i f• Cℓ describes size of temperature fluctuations


Angular spectrum of CMB anisotropies• Coefficients Cℓ describe size of temperature fluctuations• Dependent on: Fit of C as function

primordial density fluctuationsBaryon/photon ratioA t f d k tt

Fit of Cℓ as functionof ℓ yieldscosmologicalAmount of dark matter

Hubble constant

cosmologicalparameters

…

• Legendre polynomials have ℓ zeros between 0 and πℓ zeros between 0 and π

200 degreesπθ ≈ ≈

• First acoustic peak at ℓ ≈ 200→causal contact within angle of 1°→causal contact within angle of 1


Angular spectrum of CMB anisotropies• Where do we expect peaks?• Primordial density fluctuations = superposition of modes with

different λ = 2π/k• during inflation : for wavelengths λ > horizon distance

perturbation k = frozen – no causal connection• When universe expands, λ increases - wave k enters horizon as

t di ti b t h i li itstanding acoustic wave between horizon limits

( ) ( ) ( ) 11 2orn nttt R tt n

λλ −= ⇒ ∝∝ ∝( ) ( ) ( )

or 2 3 H

tD

t R tt

t nλ ⇒ ∝∝ ∝

• Modes of smaller λ enter horizon earlier• → series of acoustic peaks in amplitude vs ℓ


Position and height of peaks• Position and heigth of acoustic peaks depend on curvature,

matter and energy content, other cosmological parametersFlat universeDependence on baryon contentDependence on curvature


Ph i f Physics of

i di l fl t tiPhysics of Young universe

primordial fluctuationsAcoustic oscillations

Silk damping

Sachs-Wolfe effect

Silk damping

1° = horizon at decoupling


Fit of ΛCDM model• fit CMB power spectrum + other data to ΛCDM model (PDG

2009)

1.006 0.006totΩ = ±


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