lecture 1 - iihecdeclerc/astroparticles/2012-13/...• lightyear = ly = 0.946 x 1016 m • 1 year =...
TRANSCRIPT
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The expanding universe
Lecture 1
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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
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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
2011-12 Expanding Universe 3
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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
Is presently acceleratingIs presently accelerating
2012-13 Expanding Universe 4
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2012-13 Expanding Universe 5
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Expanding Universe 6© Rubakov
2012-13
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Part 1Part 1ΛCDM ingredientsg
Hubble expansion – redshiftp
Cosmological principle
Geometry of the universe Robertson Walker metricGeometry of the universe – Robertson-Walker metric
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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 ª Mpc• highest redshift observed : dwarf galaxy at z ª 11
2012-13 Expanding Universe 8
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Hubble law 1• Hubble(1929): spectral lines of distant galaxies are
redshifted fi 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
) SuperNovaeIa and II
eloc
ity(k
mVe
Cepeids
2012-13 Expanding Universe 9Distance (Mpc)
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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 1obsem
vz zc
λ β βλ
+ = ≈ + → ≈ =
• Hubble law becomes linear relation between velocity and Distance
em
H DDistance 0z c v H D≈ =
• Confirms theories of Friedmann and Lemaître2012-13 Expanding Universe 10
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Hubble law 3• H0 = Hubble constant today – present value (PDG 2012)
t today=( ) ( )0 0( ) 70.2 1.4 km/sec /MpcH H t= = ±
If H(t) i t ll ti t t d if
0t today=( ) ( )0 0( ) p0 100 / sec/ 0.702 0.014H km Mpc h h= → = ±i
• If H(t) is same at all times → constant and uniformexpansion
• 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)
( ) ( )( )( )
2
28
3NG c
R ttπ= −2H t ktotρ
2012-13 Expanding Universe 11
( )( )3 R t
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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 obst R t
zt R t
λλ
+ = = R(t)=Scale of universe at time t( ) ( )em t R tλ
2012-13 Expanding Universe 12
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Cosmological redshift - time • Redshift is the measured quantity – it is related to a given
time during the expansion
• Large z means small t , or early during the expansion
0t t t=0tz== ∞
0
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 expansion - in which 2 objects are at restmoving with expansion - in which 2 objects are at rest
2012-13 Expanding Universe 13
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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
nsion
2012-13 Expanding Universe 14
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Cosmic Microwave Background photons
WMAP 5 years dataUniform temperature up to ΔT/T ~ 10-5Uniform temperature up to ΔT/T 10
2012-13 Expanding Universe 15
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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
2012-13 Expanding Universe 16
Milky WayCOBE - radio
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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 with
light sources (stars) - total flux expected is μ rmaxThe 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 Dt/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
2012-13 Expanding Universe 17
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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 time2 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 distance2012-13 Expanding Universe 18
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Stretching of the universe scaleExample: closed universe
R(t1) R(t2)
© L B t ö
2012-13 Expanding Universe 19
© L. Bergström
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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 sin1
drds c dt r d dr
θ θ φ⎡ ⎤= − + +⎢ ⎥−⎣ ⎦k2R t
• 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
2012-13 Expanding Universe 20
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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 global mass-energy density of
universe 2kcuniverse ( ) ( ) ( )2 2 1energy matterkct
H t R t−Ω − =
• 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=2012-13 Expanding Universe 21
( ) ( )tot critt tρ ρ=
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C i d tiGlobal energy tCosmic destinyGlobal energydensity
geometry
k0 big crunch
2012-13 22Expanding Universe
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Excercise• D. Perkins, chapter 5• Oplossing meebrengen op het examenOplossing meebrengen op het examen
2012-13 Expanding Universe 23
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k from CMB observations
Curvature of universe
CMB photons
movie
z = 1100model
CMB photonst=380.000 y
k 0k>0
k
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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 expansion9
01 1 13.9 10( )
t t yearH t H
= ⇒ = = ×
• Horizon distance today in flat static universe (Hubble 0( )H t H
distance) ( )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 )
( ) ( )( )( )
22
28
3NG c
R ttπ= −H t ktotρ
2012-13 Expanding Universe 25
( )( )R t
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Matter dominated flat universe• Integration of Friedman equation with k=0 and only non-
relativistic matter
( )13 2
392
GMR t t⎛ ⎞= ⎜ ⎟⎝ ⎠
( )2⎜ ⎟⎝ ⎠
( )0 031 R t t 92 1 9 1 10t yr= = ×( )( )
0
0 0 2H R t= = 0
0
9.1 103
t yrH
= = ×
• 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)
2012-13 Expanding Universe 26
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Part 2Part 2ΛCDM ingredientsg
Dynamics of the expansion – Friedman Lemaître y pequations
Energy density of universeEnergy density of universe
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Dynamics of the expansion• Einstein field equations of general relativity
1 8R G Tℜ Λ82 N
R g G T g
g
μν μν μν μνπ− ℜ = −Λ
Λ⎛ ⎞ Cosmological8
8N N
gG T
Gμν
μνπ πΛ⎛ ⎞
= −⎜ ⎟⎝ ⎠
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
2012-13 Expanding Universe 28
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Friedmann-Lemaître equation• Time dependent evolution of universe = Friedman-Lemaître
equation( ) ( ) ( )
22
2 8 NR t G cπ⎛ ⎞⎜ ⎟k( ) ( )( ) ( ) ( )( )
22
83
NR t G cR t R t
tπ⎛ ⎞
= −⎜ ⎟⎜ ⎟⎝ ⎠
H t ktotρ≡
curvature term=2
2kcR
( ) 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= 70.2 1.4 /Mpc = Hubble constant todays
=
= ±0 0HH
H t
t
• energy density ρ is model dependent - ΛCDM model :
( ) ( ) ( ) ( )
( )s
2012-13 Expanding Universe 29
( ) ( ) ( ) ( )baryon coldDM rad DarkEntot t t t tρ ρ ρ ρ ρ= + + +
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Second Friedman equation• Consider universe = perfect fluid with energy density ρc2
• Conservation of energy in volume element dV ; P = pressure of fluid
dE PdV= − ( )2 23 RP c Rcρ ρ⎛ ⎞
= − + ⎜ ⎟⎝ ⎠
( ) ( )2 3 3d c R Pd Rρ ⋅ = −
• Differentiate 1st Friedman equation & substitute
Rc⎝ ⎠
43
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= + + = =
2012-13 Expanding Universe 30
8 NGπ
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Equations of state 11. Matter component: ‘dust’ of non-relativistic particles
(v
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Equation of state 2• Relation between pressure and energy - introduce w
P w=2 wcρ=
( ) R⎛ ⎞ R⎛ ⎞( )2 23 RP c Rcρ ρ⎛ ⎞
= − + ⎜ ⎟⎝ ⎠
( )2 23 1Rc w
Rcρ ρ ⎛ ⎞= − + ⎜ ⎟
⎝ ⎠
( ) ( ) ( )3 1 3 11w wconst R zρ − + += ∝ +
• For small t assume that ~R tβ ( ) ( )2
3 1 wR t t
+∼
1di ti1
2R130
radiation w
matter w
=
=
2
23
R t
R t
∼
∼
2012-13 Expanding Universe 32
1&vacuum w constρ= − = HtR e∼
-
( ) ( )3 11 wρ ++( ) ( )1 zρ ∝ +
1 0 2±2012-13 Expanding Universe 33
© J. Frieman1 0.2DarkEnergyw = − ±
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Critical density• flat universe (k=0) = equilibrium between open and close universe• Friedmann equation becomes
( ) ( )( )2 2
22
8 33 8
N
N
cR t G HH tR t G
πρπ
⎛ ⎞≡ = ⇒ =⎜ ⎟⎜ ⎟⎝ ⎠
cρ
• density ρc is called critical density
( ) 3 8 NR t Gπ⎝ ⎠
ρc• Value today is
( )2
27 300
3 9.6 108c
Ht kg mG
ρπ
− −= = ×
2 3 3
8
5.4 5 protons at rest per mN
c
G
c GeV m
πρ = ≈
• Observations →universe is geometrically flat, so present averagedensity of universe, away from galaxies, is a few protons per m3
2012-13 Expanding Universe 34
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closure parameter• Define closure parameter W
( ) ( )( )ctt
ρρ
=Ω t
• For general geometry with k≠0, Friedmann equation becomes
( )cρ
2 2 22
2
83
N
c c
G H H kcHR
π ρρ ρ
⎛ ⎞= ⇒ = −⎜ ⎟
⎝ ⎠
( )( ) ( ) ( ) ( )
2
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 tObservations:
2
2 2K
c
kcH R
ρρ
= = −kΩ ( ) ( ) 1kt tΩ +Ω =flat universeand Ω=1 at
ll ti
2012-13 Expanding Universe 35
cρ all times
-
pauze
2012-13 Expanding Universe 36
-
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 expansionPresent-day observations:
1. Radiation (from CMB energy density)
( )( )0 55 10radtρ
−Ω
2 Luminous baryonic matter : p n nuclei in stars gas dust
( )( )( )
50
0
5 10radradc
ttρ
−Ω = ≈ ×
2. Luminous baryonic matter : p,n,nuclei in stars, gas, dust
29 39 10 0.01lumlum lumkg mρ
ρρ
− −≈ × ⇒ Ω = ≈
2012-13 Expanding Universe 37
cρ
-
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 l ti t ti
28 3
y
4.0 10 0.042 0.004b
b bkg mρ− −= × ⇒ Ω = ±
4. Total matter density - from galactic rotation curves, gravitational lensing, etc (chapter 7) - is clumpy
in ΛCDM model this is composed of baryons and (mostlycold) dark matter
5. Neutrino density – relics of early universe
27 32.2 10 0.24 0.03m m B DMkg mρ− −= × ⇒ Ω = Ω +Ω = ±
0 004 0 02Ω
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Observed energy densities 35. total density: fit of ΛCDM cosmological model to large set
of observation data (CMB, SN, …) – PDG 2010
6 By subtraction we obtain the vacuum energy density
1 1.006 0.006ktotalΩ = −Ω = ± 0k ≈6. By subtraction we obtain the vacuum energy density
non clumpy – homogeneous effect – PDG 2010
Conclusions:0.726 0.015ΛΩ = ±
– 95% of energy content of universe is of unknown type– Today most of it is dark vacuum energyy gy– See further in chapter 7
2012-13 Expanding Universe 39
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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
( )( ) ( )
001 0z z tR t
+ = = ( ) ( )3 11 wzρ +∝ +
1. Radiation = photons and relativistic particles
( )2 23 3 41 1 1E hc cνρ ρλ= ⇒∼ ∼ ∼( )( ) ( ) ( ) ( ) ( )
3 3 4
2 44 40 21 1
radV R R Rc t R t
z c z
ρ ρλ
ρρ= = + ⇒ +∼
2 Non relati isti matter
( ) ( ) ( ) ( ) ( )2 40 1 1radz c zc t R t ρρ = = + ⇒ +∼
2. Non-relativistic matter
( ) ( )32 23 3 31 1 1E mc c zρ ρ= ⇒ +∼ ∼ ∼ ∼2012-13 Expanding Universe 40
( ) ( )3 3 3 1mc c zV R R Rρ ρ⇒ +
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Evolution of energy components 23. Vacuum energy is constant in our model
4. Curvature term( )2c cstρ Λ =
4. Curvature term
( ) ( )2
22 2 1 1kcc c zρ ρ−= ⇒ +∼ ∼( ) ( )2 2 2 1kc c zH R Rρ ρ⇒ +
• Dynamics of expansion can be rewritten
( ) ( ) ( ) ( ) ( )2 8 Gπ ⎡ ⎤( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2
20
83 m r k
k
GH t t t t t
H t t t t
π ρ ρ ρ ρΛ
Λ
⎡ ⎤= + + +⎣ ⎦
⎡ ⎤= Ω +Ω +Ω +Ω⎣ ⎦( ) ( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )0 0
0
30
4 220 01 1 1
m r k
m r kt
H t t t t
H z t z zt t
Λ
Λ
⎡ ⎤Ω +Ω +Ω +Ω⎣ ⎦⎡ ⎤= + + + + Ω+ +Ω Ω⎣Ω ⎦
2012-13 Expanding Universe 41
⎣ ⎦
-
Summary
( )42 1radiation c zρ +∼ ( )( )32
2
1matter c z
t
ρ
ρ
+∼
( )
2
22 1
vacuum c cst
curvature c z
ρ
ρ +
∼
∼
2012-13 Expanding Universe 42
-
Lookback time
dz( )( )0
11 1 zdR dRHR dt R t
tH H
d dRR+
= ⇒ = =
( )1dzdtz H
= −⎡ ⎤+⎣ ⎦
( )0R dt R tH HR
( ) ( ) ( )0 0 21d
R tR 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
0
0
0t
Edzt t dt− = = −
⎡ ⎤∫ ∫ ( ) ( )0 1EE
Et z
t t dtz H z⎡ ⎤+⎣ ⎦
∫ ∫
2012-13 Expanding Universe 43
-
age of the universe• Age of universe from integration
From Big Bang time ( t=0 and z=∞) to present time ( t=t0) 0and z=0)
00
0t
dzt dt− = = −∫ ∫• Example: flat universe (Wk=0) with Wm=0.24, WΛ=0.76
( ) ( )0 00
1t dt
z H t∞= =
⎡ ⎤+⎣ ⎦∫ ∫
Example: flat universe (Wk 0) with Wm 0.24, WΛ 0.76
( )13 95 0 4 G±( )0 13.95 0.4t Gyr= ±
• Uitwerking meebrengen op examen2012-13 Expanding Universe 44
-
Age of the universe
( ) 1m r kΛΩ +Ω +Ω +Ω =
0=rΩ00.24
==
k
m
ΩΩ
H t 1 0.76Λ = − =mΩ Ω0 0H t
( )0 0
0
1.02613.95 0.4
H tt Gyr
== ±( )0 y
Only matter
2012-13 Expanding Universe 45
Only matter
mΩ
-
Horizon 1 • Extent of region accessible via light signals to comoving
observer at given time t
• 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( )
22 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
2012-13 Expanding Universe 46
-
Horizon 2• Particle horizon for observer at time t0 : tE →0
td dt⎡ ⎤( ) ( ) ( ) ( )0 0 020 00
1H
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=
2012-13 Expanding Universe 47
( )0 0R t t⎝ ⎠
-
Horizon 3• For a more complex cosmological model
( )( ) ( )0 200 1
r cI zdr cdzR tH Hkr
= − =−∫ ∫
( )( )( ) ( )( ) ( ) ( )( )3 4 20
121 1 1
z dzI zt t t t
=⎡ ⎤Ω + +Ω + +Ω +Ω +∫
• For Wk=0, Wm=0.24, WΛ=0.76 and integration to z=∞
( )( ) ( )( ) ( ) ( )( )0 0 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
2012-13 Expanding Universe 48
-
Deceleration parameter 1• Taylor expansion of expansion parameter around t0
(present time)
( ) ( ) ( )( ) ( )( )20 0 0 0 012
R t R t R t t t R t t t⎛ ⎞= + − + − +⎜ ⎟⎝ ⎠
( )( ) ( )( )
( ) ( )( ) ( )( )
220 0 0 0
2
11R t
H t t t H t t t
⎝ ⎠⎡ ⎤⎛ ⎞= + − − +⎢ ⎥⎜ ⎟
⎝ ⎠−− 0 02
R t R t( ) ( )( ) ( ) ( )( )
( ) ( ) ( )
0 0 0 00
22
2
11
R t
R tH t t H t t
⎢ ⎥⎜ ⎟⎝ ⎠ ⎢ ⎥⎣ ⎦
⎛ ⎞+ +⎜ ⎟
20R t
q( )( ) ( ) ( )2
0 0 0 00
12
H t t H t tR t
= + − − +⎜ ⎟⎝ ⎠
− 0q
• Deceleration parameter q(t): acceleration or decelerationof expansion at time tof expansion at time t
2012-13 Expanding Universe 49
-
Deceleration parameter 2• Deceleration parameter q(t) is related to energy density at
time t – for flat universe (k=0)
( )2
2
RR R Rq tR R R
⎛ ⎞⎛ ⎞= − = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ( ) 0 0q t R deceleration> ⇒ < ⇒
( ) ( )22 24 133
R R R
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( ) ( ) ( ) ( )
2m
rt
q t t tΛΩ
= +Ω −Ω
2012-13 Expanding Universe 50
2
-
Acceleration or deceleration?
( ) ( ) ( ) ( )2
mr
tq t t tΛ
Ω= +Ω −Ω
• A universe dominated by matter decelerates due to gravitational collapse :gravitational collapse : ( ) ( ) 0 0mq t t RΩ > ⇒
• Observations show that today( ) ( )
( ) ( )0 00 0r m q t R tΛ
Ω Ω ⎫⇒ < ⇒ >⎬Ω > Ω ⎭
2012-13 Expanding Universe 51
mΛΩ > Ω ⎭
-
Deceleration in past & acceleration nowPhysics Nobel prize 2011
2012-13 Expanding Universe 52
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Part 3Observation dataObservation data
Redshift 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
2012-13 Expanding Universe 54
-
examples
( )( )
01 obst
zt
λλ
+ = ( )em tλ
2012-13 Expanding Universe 55
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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
24 LF
Dπ
• Effective magnitude m of star with absolute magnitude M2.5logM L cst= − +
( ) 10
g
5log 251Mpc
z M⎛ ⎞
− = +⎜ ⎟⎝ ⎠
LDm
• 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
2012-13 Expanding Universe 56
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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
2012-13 Expanding Universe 57
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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
em
agni
tude
solu
tem
Abs
Δm15
2012-13 Expanding Universe 58
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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
g
+ absolute magnitude from lightcurve
– Redshift from spectrak h hk h h f hf h
Redshift from spectra
• 1998: discover acceleratedosmi e pansion
BreakthroughBreakthrough of the of the yearyear
cosmic expansion
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Deceleration in past & acceleration nowPerlmutter, 2003
0% matter
00%100% matter
t di hth id
Lage dichtheidVersnelde expansie
e D
(z)
grote dichtheidVertraagde expansie
ista
ncd
Redshift z pastnow
2012-13 Expanding Universe 60( ) ( )01 0.5R z R t= = ×
-
CMB in Big Bang modell h i di i ( l i i i i l ) d i• early hot universe : radiation (relativistic particles) dominates over
matter (dust, non-relativistic particles) & vacuum energy is negligible
• When cooled down below kT few MeV :• When cooled down below kT ª few MeV : – Free p, n, e, g (+neutrino’s)
Pl f H+ d H ++ l i– Plasma of H+ and He++ nuclei– Matter-radiation thermal equilibrium
Wh E V (i i ti t ti l ) bi ti t t
e p H γ− + ↔ +
• When E < eV (ionisation potentials) : recombination to atoms
e p H γ− + → +
( ) 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• Observed 3K background photon radiation with black body spectrum2012-13 Expanding Universe 61
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A i t i iStructure Anisotropies in atom distribution
are ‘frozen’ in CMB
Structure formation in
early universe? are frozen in CMBy
hCMB photonsare released
© Univ Oregon
2012-13 Expanding Universe 62
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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 dark matter & dark energy duringexpansion
• Angular distance between 2 photondirections depends on curvature
of space
• CMB observations probe universeat z ~1100
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Large Scale Structures• Redshift measurements of large samples of galaxies • Sloan Digital Sky Survey (SDSS 2000-2014): million ofSloan 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 at
time of decoupling are expected to leave imprint in distribution of baryons today
• Baryon Acoustic Oscillations (BAO) observations probe universe at small z – are sensitive to dark matter
2012-13 Expanding Universe 64
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2dF quasar redshift survey 2003
SDSS survey of local skyLow z – recent universe
2012-13 Expanding Universe 65
Baryon Acoustic Oscillations
-
Abundances of light elements• Model of Big Bang Nucleosynthesis predicts abundances of
light nuclei: H, D, He, Li , ..
• They were formed at t ~3min when kT ~ 3MeV• Abundances are measured today – probe expansion model• Abundances are measured today – probe expansion model
before decoupling
All t b d it 0 042 0 004Ω ±• Allow to measure baryon density 0.042 0.004bΩ = ±
2012-13 Expanding Universe 66
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Fit ΛCDM model to observationsPdg.lbl.gov
ΛΩSN Ia distance-redshift surveys
ΛΩ
CMB anisitropiespsurvey by WMAP
Anisotropies in l
2012-13 Expanding Universe 67mΩ
galaxy surveys
-
Fit ΛCDM model to observationsPdg.lbl.govg g
2012-13 Expanding Universe 68
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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
2011-12 Expanding Universe 69