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The expanding universe

Lecture 1

The early universechapters 5 to 8chapters 5 to 8

Particle Astrophysics , D. Perkins, 2nd edition, Oxford

5. The expanding universe6 Nucleosynthesis and baryogenesis6. Nucleosynthesis and baryogenesis7. Dark matter and dark energy components8. Development of structure in early universe

Expanding universe : content• part 1 : ΛCDM model ingredients: Hubble flow, cosmological principle, geometry of universe

• part 2 : ΛCDM model ingredients: dynamics of expansion, energy density components in universe

• Part 3 : observation data – redshifts, SN Ia, CMB, LSS, light element abundances ‐ ΛCDM parameter fits

• Part 4: radiation density, CMB• Part 5: Particle physics in the early universe, neutrino density

• Part 6: matter‐radiation decoupling• Part 7: Big Bang Nucleosynthesis• Part 8: Matter and antimatter

2013‐14 Expanding Universe lect1 3

The ΛCDM cosmological model• Concordance model of cosmology – in agreement with all observations = Standard Model of Big Bang cosmology

• ingredients:

Universe = homogeneous and isotropic on large scalesUniverse = homogeneous and isotropic on large scales

Universe is expanding with time dependent rate

Started from hot Big Bang, followed by short inflation period

Is essentially flat over large distances

Made up of baryons, cold dark matter and a constant darkade up o ba yo s, co d da atte a d a co sta t daenergy + small amount of photons and neutrinos

Dark energy is related to cosmological constant ΛDark energy is related to cosmological constant Λ

Lecture 3

Lecture 1re 2

Lectur

Lecture 3

Expanding Universe lect1 6© Rubakov

2013‐14

Lecture 4

Part 1Part 1ΛCDM ingredientsgHubble expansion – redshiftp

Cosmological principle

Geometry of the universe Robertson Walker metricGeometry of the universe – Robertson‐Walker metric

Some distances• Earth‐Sun = AU = 150 x 109 m = 150 x M km• Lightyear = Ly = 0.946 x 1016 m• 1 year = 31.5 106 s• parsec = pc = 3.3 Lyparsec pc 3.3 Ly• Mpc = 3.3 MLy• Radius Milky Way ª 15 kpc

ruler in cosmologyi O(M )• Radius Milky Way ª 15 kpc

• Sun to centre MW ª 8 kpcA d d l (M31) t th 800 k

is ~ O(Mpc)

• Andromeda galaxy (M31) to earth ª 800 kpc• Width Local Group of ª 30 galaxies ª 2 Mpc• Average inter‐galactic distance ªMpc• highest redshift observed : dwarf galaxy at z ª 11

Hubble law 1• Hubble(1929): spectral lines of distant galaxies are redshiftedfi galaxies move away from Earth

• Receding velocity increases with distance : Amount of shift Dl depends on apparent brightness (~distance D) of galaxyDl depends on apparent brightness ( distance D) of galaxy

Original Hubble plot

m/s) SuperNovae

Ia and II

elocity

Cepeids

2013‐14 Expanding Universe lect1 9Distance (Mpc)

Hubble law 2• Interpret redshift z as Doppler effect

• For relativistic objectsFor relativistic objects

( )11 11

obs emobs em em

βλ λ βλ λ λ βλ β− +

= = ⎯⎯⎯→ +Δλz =λ

• For close‐by objects, in non‐relativistic limit

( )1emλ β−λ

1 1obs

λ β βλ

+ = ≈ + → ≈ =

• Hubble law becomes linear relation between velocity and Distance

H DDistance0z c v H D≈ =

• Confirms theories of Friedmann and Lemaître2013‐14 Expanding Universe lect1 10

Hubble law 3• H0 = Hubble constant today – best direct estimate (HST,

WFC3) t today=( ) ( )0 0( ) 73.8 2.4 km/sec /MpcH H t= = ±

• If H(t) is same at all times→ constant and uniform

0t today=0 100 / sec/ 0.738 0.024H km Mpc h h= → = ±i

• If H(t) is same at all times → constant and uniformexpansion

Ob ti h th t i t t t d• Observations show that expansion was not constant and that Hubble ‘constant’ depends on time

• Expansion rate evolves as function of changing energy‐matter density of universe (see Friedman equations)

( ) ( )( )( )

R ttπ

= −2H t ktotρ

( )( )3 R t

Sources of redshift• Motion of object with respect to observer – Doppler effect(red and blue shifts): e.g. rotation of stars in galaxies – seegalaxy rotation curves (dark matter)

• Gravitational redshift : stretching of wavelength close toGravitational redshift : stretching of wavelength close to heavy object ‐ local effect – negligible over long distances

• Cosmological redshift : stretching of wavelength due to• Cosmological redshift : stretching of wavelength due to expansion of universe – dominant at high redshifts

( )( )

0 01 obs t R tz

t R tλλ

+ = = R(t)=Scale of universe at time t( ) ( )em t R tλ

Also called a(t)

Cosmological redshift - time • Redshift is the measured quantity – it is related to a giventime during the expansion

• Large z means small t , or early during the expansion

0t t t0tz== ∞

0t tz==

todayBigBang

• Actual (proper) distance D from Earth to distant galaxy attime t

• D can only be measured for nearby objects( ) ( )D t r= ⋅R t

ca o y be easu ed o ea by objects

• r = co‐moving coordinate distance – in reference frame co‐moving with expansionmoving with expansion

Cosmological principle - large scales• universe is isotropic and homogeneous at large scales

• There is no preferred position or direction

• Expansion is identical for all observers

• Is true at scale of inter‐galactic distances : O(Mpc)

•2dF quasar surveyq y•Two slices in declination and right Rig

ascention•Plot redshift vs right ascension

ght ascenascension•Each dot = galaxy

Equatorial coordinates

Cosmic Microwave Background photons

WMAP 5 years data ‐ 2.75°K skyUniform temperature up to ΔT/T ~ 10‐5Uniform temperature up to ΔT/T 10

Cosmic Microwave Background photonsPLANCK 4 years of data – 2.75°K skyUniform temperature up to ΔT/T ~ 10‐5

Structures - small scales• Over short distances universe is clumpy• Galaxy clusters O(10 Mpc) diameter• galaxies, eg Milky Way : 15 kpc COMA cluster: 1000 galaxies

20Mpc diameterM74 IR

p100 Mpc(330 Mly) from EarthOptical + IR

M74‐ IR

Milky WayCOBE ‐ radio

Olbers’ paradox• Olber (~1800): Why is the sky not bright at night?

• Suppose that universe is unlimited & filled uniformly withSuppose that universe is unlimited & filled uniformly withlight sources (stars) ‐ total flux expected is ~ rmax

The sky does not look dark because:The sky does not look dark because:

• Observable universe has finite age – for constant i li ht h f i O( t )expansion: light can reach us from maximum rmax = O(ct0 )

• Stars emit light during finite time Dt – flux reduced by factor Δt/t0

• Light is redshifted due to expansion ‐ eg red light Æ IR –finally undetectable flows

• CMB fills universe – dark matter does not light upg p

Robertson-Walker metric 1assume universe = homogeneous and isotrope fluid

• Distance between 2 ‘events’ in 4‐dimensional space‐timeDistance between 2 events in 4 dimensional space time

2 2 2 2 2 2ds c dt dx dy dz⎡ ⎤= − + +⎣ ⎦Co‐movingdistance dr

Universe is expanding

⎣ ⎦ distance dr

• scale factor R(t), identical at all locations, only dependenton time ( )22 2 2 2 2 2ds c dt dx dy dz⎡ ⎤= + +⎣ ⎦R ︵t ︶

• proper distance D from Earth to distant galaxy at time t

( )ds c dt dx dy dz⎡ ⎤= − + +⎣ ⎦R ︵t ︶• proper distance D from Earth to distant galaxy at time t

( ) ( )D t r= ⋅R t• r = co‐moving coordinate distance2013‐14 Expanding Universe lect1 20

Stretching of the universe scaleExample: closed universe

R(t1) R(t2)

Robertson-Walker metric 2space is not necessarily flat

• Introduce k = spatial curvature of universeIntroduce k spatial curvature of universe

• Distance between 2 events in 4‐dim space‐time

⎡ ⎤( ) ( )2

2 2 2 2 2 2 22 sin

1drds c dt r d d

rθ θ φ

⎡ ⎤= − + +⎢ ⎥−⎣ ⎦k

• k : curvature of space→ geometry – obtained from

⎣ ⎦

k : curvature of space geometry obtained fromobservations – can be ‐1,0,1

• R(t) : dynamics of expansion→ Friedman‐Lemaître• R(t) : dynamics of expansion → Friedman‐Lemaître equations – model dependent

Curvature k in FLRW model • k = +1, 0, ‐1 depends on space geometry

• Curvature is independent of timeCurvature is independent of time

• Curvature is related to total mass‐energy density of universe 2kuniverse

( ) ( ) ( )2

2 2 1totalct

H t R tΩ − =

• It affects the evolution of density fluctuations in CMB radiation – path of photon is different

• Therefore it affects pattern of the CMB radiation today

• For universe as a whole, over large distances, space‐timeFor universe as a whole, over large distances, space time seems to be flat: k=0 from CMB observations

( ) ( )t tρ ρ=

( ) ( )tot critt tρ ρ=

k from CMB observations

Curvature of universe

CMB photons

z = 1100model

CMB photonst=380.000 y

k 0k>0

k<0 z = 0

observe

C i d tiGlobal energy tCosmic destinyGlobal energydensity

geometry

k<0 expandsfor everfor ever

k=0 velocityasymptotically zero

k>0 big crunch

2013‐14 25Expanding Universe lect1

Excercise• D. Perkins, chapter 5

• Oplossing meebrengen op het examenOplossing meebrengen op het examen

Hubble time – Hubble distance• Expansion parameter H(t) has dimension of (time)‐1

• Hubble time = expansion time today for constant expansionHubble time expansion time today for constant expansion

1 13.6 10t yearH

= = ×

• Hubble distance = Horizon distance today in flat static

universe ( )0 0 4.2HD t c t Gpc= =

• But! Look back time and distance in expanding universedepend on dynamics in H(t) (see part 2) depe d o dy a cs (t) (see pa t )

( ) ( )( )( )

R ttπ

= −H t ktotρ

( )( )R t

Example: matter dominated flat universe• Integration of Friedman equation with k=0 and only non‐relativistic matter

( )13 2

GMR t t⎛ ⎞= ⎜ ⎟⎝ ⎠

( )2⎜ ⎟

⎝ ⎠

( )0 031 R t t 92 1 9 1 10t yr= = ×( )( )

0 0 2H R t= = 0

9.1 103

= = ×

• Dating of earth crust and old stars shows that age of universe is of order 14 x 109 yru e se s o o de 0 y

• → Need other energy components (see part 2)

Part 2Part 2ΛCDM ingredientsg

Dynamics of the expansion – Friedman Lemaître y pequations

Energy density of universeEnergy density of universe

Dynamics of the expansion• Einstein field equations of general relativity

1 8R G Tℜ Λ82 NR g G T g

μν μν μν μνπ− ℜ = −Λ

Λ⎛ ⎞ Cosmological8

μνππΛ⎛ ⎞

= −⎜ ⎟⎝ ⎠

Cosmologicalconstant

Geometry of space‐timeRobertson‐Walker metric

Energy‐momentumFunction of energy content of universegyIs model dependent11 3 -1 -26.67 10 m kg sNG −= ×

• Friedmann & Lemaître (1922) : Solutions for uniform and homogeneous universe behaving as perfect frictionlessfluidfluid

Friedmann-Lemaître equation• Time dependent evolution of universe = Friedman‐Lemaître equation

( ) ( ) ( )2

22 8 NR t G cπ⎛ ⎞

⎜ ⎟k( ) ( )

( ) ( )( )( )

NR t G cR t R t

tπ⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠H t k

totρ≡

curvature term=2

( ) energy density of universe globally, 'at large'=ttoρ t

( ) expansion rate = Hubble 'constant' at time t=H t( )( ) ( )( )expansion rate = Hubble constant at time t

km= 73.8 2.4 /Mpc = Hubble constant todays

= ±0 0HH

• energy density ρ is model dependent - ΛCDM model :

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )baryon coldDM rad DarkEntot t t t tρ ρ ρ ρ ρ= + + +

Second Friedman equation• Consider universe = perfect fluid with energy density ρc2

• Conservation of energy in volume element dV ; P = pressure of fluid

dE PdV= − ( )223 RP c

Rcρ ρ

⎛ ⎞= − + ⎜ ⎟

⎝ ⎠( ) ( )2 3 3d c R Pd Rρ ⋅ = −

• Differentiate 1st Friedman equation & substitute

Rc⎝ ⎠

NG RdRRdt

π⎛ ⎞⎛ ⎞= = − ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ 2

3Pρ +c

Equation of state

• ΛCDM model : Energy density consists of 3 components: matter, radiation (incl. relativistic particles), constant vacuum energy

3dt ⎝ ⎠⎝ ⎠ c

with 8matter vacuum vacradiation G

ρ ρ ρ ρ ρ ρπ ΛΛ

= + + = =

8 NGπ

Equations of state 11. Matter component: ‘dust’ of non‐relativistic particles

(v<<c) – matter pressure from kinetic energy2

2 23 3

knon rel

E vP c v cV c

⎛ ⎞⎛ ⎞ ⎛ ⎞= = × <<⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

0non relP − ≈

2. Radiation component: radiation & relativistic particles –di ti f id l

3 3V c⎝ ⎠ ⎝ ⎠ ⎝ ⎠

radiation pressure of ideal gas21

3 3relE cPV

3. Vacuum energy – negative pressure equivalent to

3 3rel V

3 acuu e e gy egat e p essu e equ a e t togravitational repulsion

2vacP c cstρ= − =

Equation of state 2• Relation between pressure and energy ‐ introduce w

P w=2 wcρ

( ) R⎛ ⎞ R⎛ ⎞( )223 RP c

Rcρ ρ

⎛ ⎞= − + ⎜ ⎟

⎝ ⎠( )2

23 1 Rc wRc

ρ ρ⎛ ⎞

= − + ⎜ ⎟⎝ ⎠

( ) ( ) ( )3 1 3 11w wconst R zρ

− + += ∝ +

• For small t assume that ~R tβ ( ) ( )2

3 1 wR t t

1di ti1

radiation w

matter w

1&vacuum w constρ= − = HtR e∼

( ) ( )3 11 w++( ) ( )1 zρ ∝ +

1 0 22013‐14 Expanding Universe lect1 35

Critical density• flat universe (k=0) = equilibrium between open and close universe

• Friedmann equation becomes

( ) ( )( )

8 33 8

cR t G HH tR t G

πρπ

⎛ ⎞≡ = ⇒ =⎜ ⎟⎜ ⎟⎝ ⎠

• density ρc is called critical density

( ) 3 8 NR t Gπ⎝ ⎠

• Value today is( )

227 30

03 9.6 10

8cHt kg mG

− −= = ×

5.4 5 protons at rest per mN

c GeV m

ρ = ≈

• Observations →universe is geometrically flat, so present averagedensity of universe, away from galaxies, is a few protons per m3

closure parameter• Define closure parameter W

( ) ( )( )c

• For general geometry with k≠0, Friedmann equation becomes

( )cρ

2 2 22

G H H kcHR

π ρρ ρ

⎛ ⎞= ⇒ = −⎜ ⎟

⎝ ⎠

( )( ) ( ) ( ) ( )

2 21t kcρ

= − ⇒2

2 2kc1 = Ω t -

H t R t

3 c c Rρ ρ⎝ ⎠

• Define curvature term

( ) ( ) ( ) ( )2 2c t H Rρ 2 2H t R t

Observations: 2

= = −kΩ ( ) ( ) 1kt tΩ +Ω =flat universeand Ω=1 at

cρ all times

Observed energy densities 1• Energy density is composed of: dust/matter (non‐rel), radiation (incl. rel particles), vacuum energy (L)

• Relative contributions unfluence dynamics of expansion( ) 1k kΩ+Ω = +Ω =m r ΛΩ + Ω + Ω

• Relative contributions unfluence dynamics of expansion

Present‐day observations:

1. Radiation (from CMB energy density)

( )( )0 55 10radtρ

2 Luminous baryonic matter : p n nuclei in stars gas dust

( )( )( )

5 10radrad

tρ−Ω = ≈ ×

2. Luminous baryonic matter : p,n,nuclei in stars, gas, dust

0.01lumlum

Ω = <

Observed energy densities 23. total baryon density – from fit of the model of primordial

nucleosynthesis to measurements of light elementabundances (chapter 6)

30.26 baryons bn m−≈

4 T t l tt d it f fit t CMB d t i l

4.0 10 0.04b

b bkg mρ − −≈ × ⇒ Ω ≈

4. Total matter density – from fit to CMB data ‐ is clumpy

in ΛCDM model this is composed of baryons and (mostlycold) dark matter

0.31m DMbΩ = Ω +Ω =

5. Neutrino density – relics of early universe – from fit to CMB data 0 01Ω ≤

0.01νΩ ≤

Observed energy densities 35. total density: fit of ΛCDM cosmological model to large set

of observation data (CMB, SN, …) – PDG 2013

6 By subtraction we obtain the vacuum energy density

1 1.0005 0.0033ktotalΩ = −Ω = ± 0k ≈6. By subtraction we obtain the vacuum energy density

non clumpy – homogeneous effect – PDG 2013

Conclusions:0.692 0.010ΛΩ = ±

– 95% of energy content of universe is of unknown type

– Today most of it is dark vacuum energy0 31Ω

– See further in chapter 7 0.310.69

Ω ≈Ω ≈

Evolution of energy components 1• Cosmological redshift is related to the expansion rate

( ) ( )01 0R t

( ) ( )3 1 w+

1 R di ti h t d l ti i ti ti l

( )( ) ( )0

01 0z z tR t

+ = = ( ) ( )3 11 wzρ +∝ +

1. Radiation = photons and relativistic particles

( )2 23 3 4

1 1 1E hc cνρ ρλ

= ⇒∼ ∼ ∼( )( ) ( ) ( ) ( ) ( )

2 44 40 21 1

radV R R Rc t R t

ρ ρλ

ρρ= = + ⇒ +∼

2 Non relati isti matter l mp matter

( ) ( ) ( ) ( ) ( )2 40

1 1rad

z c zc t R t

= = + ⇒ +∼

2. Non‐relativistic matter – clumpy matter

( ) ( )32 23 3 3

1 1 1E mc c zρ ρ= ⇒ +∼ ∼ ∼ ∼

( ) ( )3 3 3 1m

c c zV R R R

ρ ρ⇒ +

Evolution of energy components 23. Vacuum energy is constant in our model

4. Curvature term( )2c cstρ

4. Curvature term

( ) ( )2

22 2 1 1kcc c zρ ρ−= ⇒ +∼ ∼( ) ( )2 2 2 1

kc c z

H R Rρ ρ= ⇒ +

• Dynamics of expansion can be rewritten

( ) ( ) ( ) ( ) ( )2 8 Gπ⎡ ⎤( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

83 m r k

GH t t t t t

H t t t t

π ρ ρ ρ ρΛ

⎡ ⎤= + + +⎣ ⎦

⎡ ⎤= Ω +Ω +Ω +Ω⎣ ⎦( ) ( ) ( ) ( )

( )( ) ( )( ) ( ) ( )( )0 0

4 220 01 1 1

m r kt

H t t t t

H z t z zt t

⎡ ⎤Ω +Ω +Ω +Ω⎣ ⎦⎡ ⎤= + + + + Ω+ +Ω Ω⎣Ω ⎦

⎣ ⎦

Summary

( )42 1radiation c zρ +∼ ( )( )32

1matter c z

ρ +∼

vacuum c cst

curvature c z

Lookback time

dz( )( )0

11 1 zdR dRHR dt R t

d dRR+

= ⇒ = =

( )1dzdtz H

= −⎡ ⎤+⎣ ⎦

( )0R dt R tH HR

( ) ( ) ( )00 2

R t R tR zd−= ⇒ =( ) ( )

( )0 21 1z z+ +

• Time elapsed between emission of a photon at expansion time tE by object at redshift zE and todayE E

( ) ( )

Edzt t dt− = = −

⎡ ⎤∫ ∫ ( ) ( )0 1EE

t t dtz H z⎡ ⎤+⎣ ⎦

∫ ∫

age of the universe• Age of universe from integrationFrom Big Bang time ( t=0 and z=∞) to present time ( t=t0) 0and z=0)

dzt dt− = = −∫ ∫• Example: flat universe (Wk=0) withWm=0.24, WΛ=0.76

( ) ( )00

t dtz H t∞

= =⎡ ⎤+⎣ ⎦

∫ ∫Example: flat universe (Wk 0) withWm 0.24, WΛ 0.76 (WMAP results)

( )0 13.95 0.4t Gyr= ±

• Uitwerking meebrengen op examen• Uitwerking meebrengen op examen

Age of the universe

( ) 1m r kΛΩ +Ω +Ω +Ω =

0=rΩ00.24

H t 1 0.76Λ = − =mΩ Ω0 0H t

( )0 0

1.02613.95 0.4

H tt Gyr

= ±( )0 y

Only matter

Particle horizon 1 • Maximum distance accessible via light signals to an observer at given time t0 ?0

• In static flat universe

• In an expanding FLRW universe for a photon following a

( )0 0 4.2static flatHD t ct Gpc− = =

• In an expanding FLRW universe, for a photon following a line with fixed θ and φ

2 2 2 ⎡ ⎤dr2 0ds = ( )2 2 2

21c R t

kr⎡ ⎤⎢ ⎥−⎣ ⎦

drdt =

• Proper distance travelled by photon emitted at time tE and comoving position rE ( ) ( )0 0ED t r R t= ⋅E

• Observed at time t0 and position r=0

( ) ( )0 0ED t r R t

Particle horizon 2• Particle horizon for observer at time t0 : tE→0

td dt⎡ ⎤( ) ( ) ( ) ( )0 0 020 0

Er t cD dr dt R t R tR tkr

t⎡ ⎤= =⎢ ⎥

−⎣ ⎦∫ ∫i

• Expanding, flat, radiation dominated universe1

( )( )0 0

12R t t

R t t⎛ ⎞

= ⎜ ⎟⎝ ⎠

( )0 02HD t ct=

• Expanding, flat, matter dominated universe( )0 0R t t⎝ ⎠

( )( )0 0

23R t t

R t t⎛ ⎞

= ⎜ ⎟⎝ ⎠

( )0 03HD t ct=

( )0 0R t t⎝ ⎠

Particle horizon 3• For a more complex cosmological model

( )( ) ( )0 2

r cI zdr cdzR tH Hkr

= − =−

∫ ∫

( )( )( ) ( )( ) ( ) ( )( )3 4 20

121 1 1

z dzI zt t t t

=⎡ ⎤Ω + +Ω + +Ω +Ω +∫

• ForWk=0,Wm=0.24,WΛ=0.76 and integration to z=∞

( )( ) ( )( ) ( ) ( )( )00 0 0 01 1 1m r Kt z t z t t zΛ

⎡ ⎤Ω + +Ω + +Ω +Ω +⎣ ⎦

For Wk 0, Wm 0.24, WΛ 0.76 and integration to z

03.3 3.3 14BBexpansionH

cD ct GpcH

=∼ ∼0H

Deceleration parameter 1• Taylor expansion of expansion parameter around t0(present time)

( ) ( ) ( )( ) ( )( )20 0 0 0 0

R t R t R t t t R t t t⎛ ⎞= + − + − +⎜ ⎟⎝ ⎠

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

( )( )( )22

0 0 0 0

H t t t H t t t

⎝ ⎠⎡ ⎤⎛ ⎞= + − − +⎢ ⎥⎜ ⎟

⎝ ⎠−− 0 0

R t R t( ) ( )( )

( )( )( )

( ) ( ) ( )

0 0 0 00

R tH t t H t t

⎢ ⎥⎜ ⎟⎝ ⎠ ⎢ ⎥⎣ ⎦

⎛ ⎞+ +⎜ ⎟

q( )( ) ( ) ( )2

0 0 0 00

H t t H t tR t

= + − − +⎜ ⎟⎝ ⎠

− 0q

• Deceleration parameter q(t): acceleration or decelerationof expansion at time tof expansion at time t

Deceleration parameter 2• Deceleration parameter q(t) is related to energy density attime t – for flat universe (k=0)

RR R Rq tR R R

⎛ ⎞⎛ ⎞= − = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ( ) 0 0q t R deceleration> ⇒ < ⇒

( ) ( )22 2

Gq t c Pc Hπ ρ

⎝ ⎠⎝ ⎠⎡ ⎤ ⎛ ⎞= − − + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

( ) 0 0q t R acceleration< ⇒ > ⇒

• Energy‐pressure relation: equations of state for

( )3c H⎢ ⎥⎣ ⎦ ⎝ ⎠

– Dust/matter

– Radiation/relativistic particlesp

– Vacuum energy( ) ( ) ( ) ( )

q t t tΛΩ

= +Ω −Ω

Acceleration or deceleration?

( ) ( ) ( ) ( )2

tq t t tΛ

Ω= +Ω −Ω

• A universe dominated by matter decelerates due to gravitational collapse :gravitational collapse :

( ) ( ) 0 0mq t t RΩ > ⇒ <∼

• A universe dominated by vacuum energy accelerates

( ) ( ) 0 0q t t RΛ≈ −Ω < ⇒ >

• Observations show that today( ) ( )

( ) ( )0 00 0r m q t R tΛ

Ω Ω ⎫⇒ < ⇒ >⎬Ω > Ω ⎭

mΛΩ > Ω ⎭

Part 3Observation dataObservation dataRedshift measurementsRedshift measurementsLuminosity distance

S N ISuperNovae IaCosmic Microwave Background

Large galaxy surveysLight element abundances

Measuring redshift• Atoms emit light at certain wavelengths – is measuredin laboratory (atom at rest)

• Atoms in quasars emitRestframe wavelength (Å)

2000 4000 6000

Atoms in quasars emitsame spectrum but light isredshifted when reaching

Restframe wavelength (Å)

CIV redshifted when reachingEarth

• Comparison of observed• Comparison of observedspectrum with labspectrum gives redshift zspectrum gives redshift z

examples

( )( )

01 obs tz

+ =( )em tλ

Magnitude and distance• Luminosity distance DL to star with intrinsic luminosity L

• F = power measured at Earth LF =

• Absolute magnitude M = magnitude at 10pc distance

• Effective magnitude m of star with absolute magnitude M

2.5logM L cst= − +

( ) 10

5log 251Mpc

z M⎛ ⎞

− = +⎜ ⎟⎝ ⎠

• Large negative magnitude = bright star: e.g. m(Sun)=‐26 ; m(Sirius,

1Mpc⎝ ⎠

g g g g g ( ) ; ( ,brightest star) = ‐1.7

• Far away object is dimmer and has larger (=smaller negative) m

SN Ia as standard candlesS I b i ht li ht f di t t SN b• Supernovae Ia are very bright – light from very distant SN can beobserved – up to z=1.7

• Are hydrogen poor• Are hydrogen poor

• All have roughly the same luminosity curve

M101 at 20 MlySN2011feSN2011fe Flashed on 18 december 2011

SN Ia as standard candles1995 (H Phili ) l ti b t k l i it d d li• 1995 (Hamuy, Philips): relation between peak luminosity and declinerate

• Measure Δm > obtain M e• Measure Δm15 ‐> obtain M

magnitude

solute

SN Ia – Nobel Prize 2011• S. Perlmutter ‐ Supernova Cosmology Project (SCP)

http://supernova.lbl.gov/

• A. Riess & B. Schmidt – High‐z Supernova Search Team (HZSNS) http://www.cfa.harvard.edu/supernova//HighZ.html

• High statistics SN monitoring– Distance: observed magnitude

18 December 1998

+ absolute magnitude from lightcurve

– Redshift from spectrak h hk h h f hf h

Redshift from spectra

• 1998: discover accelerated

osmi e pansion

BreakthroughBreakthrough of the of the yearyear

cosmic expansion

• Probe history at z < 2

Deceleration in past & acceleration nowThe principlePerlmutter, 2003

0% matter

00%100% matter

High density

Low densityAccelerated expansion

e D(z)

High densityDecelerated expansion

istanc

Redshift z pastnow

2013‐14 Expanding Universe lect1 60( ) ( )01 0.5R z R t= = ×

Recent survey of SuperNovae

arXiv:1105.3470v1

Abundances of light elements• Model of Big Bang Nucleosynthesis predicts abundances of light nuclei: H, D, He, Li , ..

• They were formed at t ~20min when kT ~ KeV and z>>1100

• Abundances are measured today• Abundances are measured today

• probe expansion model at redshift z >> 1100, beforeti f CMBcreation of CMB

• Allow to measure baryon density 0.044 0.005BΩ = ±

• See lecture 2See ectu e

Big Bang Nucleosynthesis( )• when kT ~ 3 MeV , left with ( )

ν ν μ

τ−+

• Neutrinos decouple, anti‐particles are annihilated

F t f b ? Bi B N l th i d l

,, , , ppγ

• Fate of baryons? → Big Bang Nucleosynthesis model

• Protons and neutrons in equilibrium due to weakinteractions

• Neutrons are ‘saved’ by binding to protons → deuteronse nepν ++ ↔ + eepn ν−→ + +

• At kT ~ 80 KeV (20 mins)

2.22n p D MeVγ+ ↔ + +

2.22n p D MeVγ+ → + +• At kT 80 KeV (20 mins)

• Formation of light nuclei

2.22n p D MeVγ+ → + +

2 3 43, , ,, ..H He HeH• Fractions of elements predicted by BBN model2012‐13 Expanding Universe 63

Cosmic Microwave Backgroundf f i f li h l i (BBN d l) d 20 i• after formation of light nuclei (BBN model) around t=20 mins

• Plasma of H+, He++ , … nuclei, plus e‐ and photons – in thermal equilibriumequilibrium

• At recombination (380’000 y) , when E < eV (ionisation potentials) nuclei + e‐→ atoms + CMB photonsnuclei + e → atoms + CMB photons

( )1 1100 d 380 000 d T 3000K

CMBe p H γ− + → +

• Matter radiation decoupling

( ) dec1 1100 and 380.000 and T 3000decdecz t y K+ ≈ ≈ ≈

• Matter‐radiation decoupling

• Expect to see γ’s today as uniform Cosmic Microwave Background

Ob d 3K b k d h t di ti ith bl k b d t• Observed 3K background photon radiation with black body spectrum

• Probe history at z = 1100

Cosmic Microwave Background

Matter

photonsare released

are released

Baryons/nuclei and Photons decouple/freeze‐outBaryons/nuclei and photons in thermal equilibrium

Photons decouple/freeze outDuring expansion they cool

downExpect to see today a uniform

γ radiation which behaves like a bl k b d di ti

2012‐13 Expanding Universe 65

black body radiation

A i t i iStructure Anisotropies in atom distributionare ‘frozen’ in CMB

Structure formation in

early universe? are frozen in CMBy

hCMB photonsare released

What can we learn from CMB?• Density (pressure) anisotropies in photon‐baryon fluid at time of

decoupling should leave imprint in distribution of g’s today• observe temperature anisotropies O(10‐5)

• Growth of anisotropies depends on evolution of matter, radiation and dark energy during expansion

• Angular distance between 2 photon

directions depends on curvature

of space

Large Scale Structures• Redshift measurements of large samples of galaxies

• For ex.: Sloan Digital Sky Survey (SDSS 2000‐2014): millionFor ex.: Sloan Digital Sky Survey (SDSS 2000 2014): million of galaxies being recorded

• On average uniform population• On average uniform population

• Galaxies = baryonic matter

• Density (pressure) anisotropies in photon‐baryon fluid attime of decoupling are expected to leave imprint in distribution of galaxies today

• ‐> Baryon Acoustic Oscillations (BAO) observations

• probe universe at z <~ 5 – or evolution during last 7Gyearsy

2dF quasar redshift survey 2003

SDSS survey of local skyLow z – recent universe

Baryon Acoustic Oscillations

Fit ΛCDM model to observationspdg.lbl.gov

ΛΩSN Ia distance‐redshift surveys

CMB anisitropiespsurvey by WMAP

Anisotropies in l

2013‐14 Expanding Universe lect1 70mΩ

galaxy surveys

Fit ΛCDM model to observationsPdg.lbl.govg g

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