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The expanding universe
Particle Astrophysics , chapter 5p y , p
Summary of lecture 1H bbl i l• Hubble expansion law: – receding velocity vs distance
( )0 0 72 3 km/sec/Mpcv H D H= = ±
• Friedman equationshomogenous universe
( ) ( )( ) ( )
( )( )
22
22
83 tot
R t G cH tR t R t
t kρπ⎛ ⎞≡ = −⎜ ⎟⎜ ⎟⎝ ⎠– homogenous universe
– perfect fluid
• Observations of CMB:
( ) ( )( )
2
4 33
R t
dR GRRd
Pt
ρπ
⎝ ⎠
⎛ ⎞⎛ ⎞= = − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠Observations of CMB:
– k=023dt c⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
2 2 38 5 4cGH G Vρπ• Critical density today
2 2 30
8 5.43 c
cGH c GeV mρπ ρ= ⇒ =
( )tρ
• Closure parameter( ) ( )
( ) 1c
tt
ρρ
Ω =
Ω Ω Ω Ω Ω Ωp
chapter 5 astro-particle physics 2009-10 2
( ) 1K m r KΛΩ +Ω = Ω +Ω +Ω +Ω =
Summary of lecture 1
• Obsverved energy densitiestoday (t=0)
0.010.042 0.004
lum
B lum dark
Ω =Ω = Ω +Ω = ±y ( )
0.24 0.030.76 0.05
m B CDM
Λ
Ω = Ω +Ω = ±Ω = ±
1.0 0.02total
Λ
Ω = ±
( )• Time-redshift relation• Expansion rate vs time
( ) ( )01R
R tz
=+
Expansion rate vs time
( ) ( )( ) ( )( ) ( ) ( )( )2 3 4 220 0 1 0 1 0 0 1m r KH t H z z zΛ⎡ ⎤= Ω + +Ω + +Ω +Ω +⎣ ⎦
• Age of universe: flat, Wm=0.24, WΛ=0.76 ( )13 95 0 4 l 9G G±WΛ 0.76
chapter 5 astro-particle physics 2009-10 3
( )0 013.95 0.4 comp. matter only 9t Gyr t Gyr= ± ≈
Lecture 2• Deceleration parameter in Friedman equation - dark
energygy• Cosmic Microwave Background (CMB) radiation –
photon freeze-out at kT ª 0.3eV, relic photonsphoton freeze out at kT 0.3eV, relic photons• Energy density of relativistic particles in early radiation
dominated (kT > eV) universedominated (kT > eV) universe• Neutrino freeze-out at kT ª MeV – relic neutrinos• Decoupling of matter and radiation at z = 1100• Summaryy
chapter 5 astro-particle physics 2009-10 4
Effect of vacuum energy on expansion rate
DECELERATION PARAMETEREffect of vacuum energy on expansion rate
chapter 5 astro-particle physics 2009-10 5
Deceleration parameter• Taylor expansion around t0=0 of expansion parameter
( ) ( ) ( )( ) ( )( )21⎛ ⎞( ) ( ) ( )( ) ( )( )
( )
20 0
10 0 02
1
R t R R t t R t t
R t
⎛ ⎞= + − + − +⎜ ⎟⎝ ⎠
⎛ ⎞( ) ( )( ) ( ) ( )22
0 0 0 0011
0 2R t
a t H t t H t tR
q⎛ ⎞≡ = + − − − +⎜ ⎟⎝ ⎠
• Deceleration parameter q: acceleration or decelerationof expansion related to energy density at time tp gy y
( ) ( ) ( )( ) ( )
22 2 2
4 33
R t R t Gq t c PR t c H t
π ρ⎡ ⎤
⎡ ⎤= − = +⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦( ) ( )
( ) ( ) ( ) ( )0
3
00 0 0
2m
r
R t c H t
q q Λ
⎢ ⎥⎣ ⎦Ω
= = +Ω −ΩEquation of state: see table
chapter 5 astro-particle physics 2009-10 6
( ) ( ) ( )0 2 rq q Λ
Summary
( )42 1radiation c zρ +∼ ( )( )32
2
1matter c z
t
ρ +∼
( )
2
22 1
vacuum c cst
curvature c z
ρ
ρ +
∼
∼
chapter 5 astro-particle physics 2009-10 7
Acceleration today• Today (t=0)
Expansion accelerates ( ) ( )0 0 0 0r m q RΩ Ω ⎫
⇒ < ⇒ >⎬Ω > Ω ⎭Expansion accelerates
C fi d b f SN i t t hi h
mΛΩ > Ω ⎭
• Confirmed by surveys of SN: expansion rate at high z (earlier universe) slower than at low z (today)
Log(distance)g( )
chapter 5 astro-particle physics 2009-10 8Redshift
Average temperatureg pEnergy densityAnisotropies
CMB - COSMIC MICROWAVEAnisotropies
BACKGROUND RADIATION
chapter 5 astro-particle physics 2009-10 9
CMB in Big Bang modell h t i di ti ( l ti i ti ti l ) d i t• early hot universe: radiation (relativistic particles) dominates
over matter (dust, non-relativistic particles) & vacuum energy isnegligiblenegligible
• When cooled down to kT ª few MeV : – p n e g n n n remain– p, n, e, g, ne, nm, nt remain– recombination of n,p,e to H
• era of BB nucleosynthesis untill cooled down to kTªeVe p H γ− + → +
• era of BB nucleosynthesis untill cooled down to kTªeV -formation of light atoms D,He,Li,Be
• When free electron density too small formation of H stops –When free electron density too small, formation of H stops photons decouple from matter
( )1 1100 d 380 000
• g’s = CMB - cool down with expansion - expect T ª few K today
( )1 1100 and 380.000decdecz t y+ ≈ ≈
g s CMB cool down with expansion expect T few K today
chapter 5 astro-particle physics 2009-10 10
What can we learn from CMB• Expect energy of g’s to be +- same in all directions• Expect uniform energy distribution of black body radiation with
T ª 3K or l ª 2mm• matter anisotropies in universe at time of decoupling leave
imprint (temperature anisotropies) in distribution of g’s today• Angular distance between 2 g directions depends on curvature
fof space• Fit sizes of anisotropies to
LCDM d l di tiLCDM model predictionsÆWm, Wb, WL, H0, …
chapter 5 astro-particle physics 2009-10 11
CMB discovery & historyDi d i 1965 b P i d Wil (B ll l b ) h• Discovered in 1965 by Penzias and Wilson (Bell labs) whensearching for radio emission from Milky WayUniform radio noise from outside Milky Way• Uniform radio noise – from outside Milky Way
• Cannot be explained by stars, radio galaxies etcE th b d b ti li it d t l th d t• Earth based observations: limited to cm wavelengths due to absorption of mm waves in atmosphere – spectrum compatible with T~3 5Kwith T 3.5K
• Prediction of Big Bang theory: expect uniform radiation at ~3K throughout universeg
• COBE satellite observations in 1990s: down to mm wavelengths – observe T=2.725Kg
• Discovery of small angle anisotropies Dq=7°• WMAP satellite (2003) detailed maps of T anisotropies and ( ) p p
polarisations Dq=0.2°chapter 5 astro-particle physics 2009-10 12
COBE measures black body spectrum• Plancks radiation law for
relativistic photon (Bose) gasIntensity Q
l=2mm 0.5mm
• Black body with temperatureT emits radiation with power Q at frequencies
Intensity Qmaximum
Q at frequencies w
( )3
2 2,4
Q T ωωω =
i t
( ) 2 2,4
1kTQ
ce
ωπ−
• maximum power at
3 3kTeω ω⎛ ⎞− =⎜ ⎟⎝ ⎠
w
• w at max fiT=2.725KkT⎜ ⎟
⎝ ⎠
CMB Spectrum by COBE satellite
Frequency n (cm-1)
• or E(photons)ª meVchapter 5 astro-particle physics 2009-10 13
p y(NASA, 1990)
CMB temperature 1• Early universe is radiation dominated - neglect curvature term
2 28 GR Rπ⎛ ⎞⎜ ⎟
2 2
3 rR Rρ⎛ ⎞= ⎜ ⎟⎝ ⎠
124 8R Gρ π ρ⎛ ⎞424 84
3r
rR GRR
ρ π ρρρ
− ⎛ ⎞∝ ⇒ = − = − ⎜ ⎟⎝ ⎠
• Integration 22
2
3 132r t
ccG
ρπ
= 1
• Stefan-Boltzmann law for photon gas in thermal equilibrium4
44 1gTσ ⎛ ⎞
=
( )42 42 3 3
4 12 15r
gTc kTc c
γσρ ππ
⎛ ⎞= = × ×⎜ ⎟
⎝ ⎠2
• gg = photon degrees of freedom = 2chapter 5 astro-particle physics 2009-10 14
CMB temperature 2 • Expect for radiation dominated expansion in early
universe (up to T=380.000y) that (1)=(2)( p y) ( ) ( )11 43 5 445 2 1 1.31c MeVkT kT
⎛ ⎞⎛ ⎞= × × ⇒ ≈⎜ ⎟⎜ ⎟ ⎜ ⎟ 1
21
23
101 52
3
1
2
0
kT k
K
TG t
T
g tγπ= × × ⇒ ≈⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
21
101.52 10rad domTt
− ×⇒ =
• Expect for t=14Gyr that TCMB ª 10K : higher than 2.7K measuredmeasured
• Explanation: radiation cooled more quickly in later, matter dominated era 2−matter dominated, era
chapter 5 astro-particle physics 2009-10 15
23
matter domT t−
− ∝
Photon energy density today• CMB photons follow Bose-Einstein distribution• nb photons with ( )
2 gp dpN p dp γ⎛ ⎞⎜ ⎟nb photons with
momentum in [p,p+dp]( )
2 3 21E
kT
p pN p dpe
γ
π= ⎜ ⎟⎡ ⎤ ⎝ ⎠−⎢ ⎥⎣ ⎦
• Integration over momentum gives 3411N cmγ−=
• Energy density
γ
2 30.261rc MeV mρ −=Energy density
E i l t d it
rρ
31 34 65 10 kg mρ − −ו Equivalent mass density• So that closure parameter
4.65 10r kg mρ = ×
( ) 50 4 84 10rρ −Ω = = ×
chapter 5 astro-particle physics 2009-10 16
( )0 4.84 10rcρ
Ω ×
Anisotropies in CMB radiationCOBE 2 years data
• Discovered in 1992 by COBE• Small dipole anisotropy in
COBE 2 years data
temperature of radiation,O(10-3K) due to movement of solar system
l ti t di t t tt ( 370relative to distant matter (v=370 km s-1) – Doppler effect
• Galactic emission• Galactic emission• Faint temperature
fluctuations (Order 10-5 K) influctuations,(Order 10 K) in CMB (after subtraction of dipole& galactic emission)
• Imprints of density fluctuations in early universe, at surface of last scattering (chapter 8)
chapter 5 astro-particle physics 2009-10 17
Relativistic fermionsRelativistic bosonsEvolution of particle content down to kT ª MeV
RELATIVISTIC PARTICLES IN THE p
EARLY UNIVERSE
chapter 5 astro-particle physics 2009-10 18
relativistic particles in early universe• relativistic boson gas = photons, W and Z bosons …
(gb = nb of spin substates) 2 gp dp ⎛ ⎞(gb p )BE statistics ( )
2 3 21
bE
kT
gp dpN p dpeπ
⎛ ⎞= ⎜ ⎟⎡ ⎤ ⎝ ⎠−⎢ ⎥⎣ ⎦
• relativistic fermions (leptons & quarks) = fermion gas( b f i b t t )
⎢ ⎥⎣ ⎦
(gf = nb of spin substates)FD statistics ( )
2
2 3 21
fE
kT
gp dpN p dpeπ
⎛ ⎞= ⎜ ⎟⎜ ⎟⎡ ⎤ ⎝ ⎠+⎢ ⎥
• All contribute to energy density with
1kTeπ ⎝ ⎠+⎢ ⎥⎣ ⎦
All contribute to energy density with
( )*
42 4 ** 2 3 3
1 715 2 8b f
gc kT g g gρ π⎛ ⎞
= = +⎜ ⎟⎝ ⎠
∑ ∑
chapter 5 astro-particle physics 2009-10 19
( ) 2 3 315 2 8 fcπ ⎜ ⎟⎝ ⎠
∑ ∑
Degrees of freedom kT > 100 GeV• Assuming only particles from Standard Model of
particle physics * 728 90 106 75p p y
• If SuperSymmetry is correct at TeV and higher: g* x 2
* 728 90 106.758
g = + × =
• If SuperSymmetry is correct at TeV and higher: g x 2
chapter 5 astro-particle physics 2009-10 20
Cool down to kT ª GeV• Production of particles stops when• For example above 160 GeV (see LEP @ CERN)
2kT Mc
For example, above 160 GeV (see LEP @ CERN)
d G V ti le e W W+ − + −+ → +
• order GeV particles:( ) ( )2 280 91M c GeV M c GeV= =W Z( ) ( )( ) ( )2 115 50M c GeV M GeV> >H SUSY
• W and Z decay withP ti l l i h d h kT<<100G V
( ) 23, 10W Z sτ −≈
• Particles no more replenished when kT<<100GeV
chapter 5 astro-particle physics 2009-10 21
Cool down to kT ª 200 MeVL=0.2GeV• When kT << L(QCD) ª200 MeV:
quarks and gluons are no longer f k b d t t
confinement
free fi quark bound states = hadrons
as
M t h d d ithAsymptotic freedom( ) ( )8 2310 weak ints. 10 strong ints.s sτ − −= −
• Most hadrons decay with
q2 (GeV2)
( ) ( )
• Muon and tauon decay weakly
chapter 5 astro-particle physics 2009-10 22
( ) ( )6 152 10 319 10s sτ μ τ τ− −= × = ×
Cooldown to kT ª few MeV• Left are : p, n, e, g, ne, nm, nt
and their anti-particles and
106.75
p
* 7 432 10 10⎛ ⎞+ ⎜ ⎟2 10 108 4
g ⎛ ⎞= + = ≈⎜ ⎟⎝ ⎠
g*10
3 4• Below 1 MeVGeV MeV
3.4
mainly e, g, ne, nm, ntand anti particles
kT(GeV)TeV
and anti-particlesg* = 3.36
chapter 5 astro-particle physics 2009-10 23
Expansion rate vs Temperature• Early universe, down to kTª eV, is radiation dominated
2 1cρ ( ) ( ) ( ) 1R t tH
ρ4c
Rρ ∼ ( ) ( )
( )( )( )
14 2
H tR t tt
ρρ
= = − =
22
2
3 132r
ccG t
ρπ
=10
12
1.52 10rad domt
KT − = ×
( ) ( )* 22
11.66 kTH gt =( ) ( ) 2PLM
gc
2NPL
cGM
=Planck mass: 1.2 x 1019 GeV/c2
Grand Unification
chapter 5 astro-particle physics 2009-10 24
Grand Unification
Neutrino freeze out• Below 1 MeV: mainly relativistic e, g, ne, nm, nt + anti-
particlesp• few protons & neutrons Æ primordial nucleosynthesis• Equilibrium between photons and leptons• Equilibrium between photons and leptons
( ) , ,i ie e i eν νγ μ τ+ −+↔ ↔ + = Weak interaction2
s CMS energy 6FG s W vσ ρσπ = =∼
• (weak) interaction rate W << 1/H(t) when kT ª 3 MeV, or t > 1sor t 1s
• Neutrinos decouple and evolve independentlyne trino free e o t relic ne trinos• neutrino freeze-out Æ relic neutrinos
chapter 5 astro-particle physics 2009-10 25
Relic neutrino density & temperature• Number density neutrinos ª number density photons• Energy density neutrinos π energy density photons• Energy boost to photons through reaction
with consequencee e γ γ+ −+ → +
1q 134 (0) 1.95
11T T T Kν γ ν
⎛ ⎞= ⇒ =⎜ ⎟⎝ ⎠
• so that ρc2 (neutrinos) < ρc2 (photons) • for kT << 1 MeV 4
7 T⎛ ⎞⎛ ⎞⎜ ⎟
• And the expected density of relic neutrinos today is
7* 3.368
Tg g gTν
γ νγ
⎛ ⎞= + =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
And the expected density of relic neutrinos today is
• But ! Detection of cosmogenic n!33 113
11N N cmν γ
−⎛ ⎞= =⎜ ⎟⎝ ⎠But ! Detection of cosmogenic n!
chapter 5 astro-particle physics 2009-10 26
11⎝ ⎠
Radiation-matter decoupling - 1• At tdec ª 380.000 years, or z ª1100, or T ª 3000K• matter decouples from radiation and photons canmatter decouples from radiation and photons can
move freely & remain as today’s CMB radiation• Matter evolves independently atoms & molecules are• Matter evolves independently - atoms & molecules are
formedS ll ti l t t i ti l i i t• Small spatial temperature variations leave imprint on CMB (see chapt 8)
• Before tdec universe is ionised and opaque • average time between collisions << age t of universeg g• particles are in thermal equilibrium as long as
1W N vσ=
chapter 5 astro-particle physics 2009-10 27
1W N v tσ=
Radiation-matter decoupling - 2• Up to t ª 100.000 y equilibrium of p,H,e,g
e p H γ− + ↔ +formation of neutral hydrogenionisation of hydrogen atom
p γ→←
• When kT < I=13.6eV (H ionisation potential) ionisation
ionisation of hydrogen atom←
probability reduces• Number density of free protons Np and of neutraly p p
hydrogen atoms NH (Ne = density of free electrons, m=electron mass ) as function of T )
2
321 2H
IkTpN N mk e
N N N hTπ+
−⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
chapter 5 astro-particle physics 2009-10 28
2H eHN N N h⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
Radiation-matter decoupling - 3• Rewrite in function of fraction of hydrogen atoms
which are ionised x = np/(np+nH)= np/(nB)p ( p H) p ( B)2
2
321 2 I
kTx mk eTπ −⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠21 Bx N h⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠
• strong reduction of x around kT ª 0.3eV, or T ª 3500K• fiionisation stops around 3500K = period of p p
recombination of e and p to hydrogen atoms• Reshift at decouplingReshift at decoupling
( ) ( )( ) ( )full calculation0
1 1250 1 1100decdec dec
R kTz z
kTR t+ = = ≈ ⎯⎯⎯⎯⎯⎯→ + =
chapter 5 astro-particle physics 2009-10 29
( ) 0dec dec
dec kTR t
Summary
artic
lepe
r pa T(K)
ergy
pEn
chapter 5 astro-particle physics 2009-10 30Time t(s)
