Particle Astroppyhysics , chapter 5

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


( )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


Effect of vacuum energy on expansion rate

DECELERATION PARAMETEREffect of vacuum energy on expansion rate


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


( ) ( ) ( )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 +

∼

∼


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


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


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, …


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


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 − −×• Equivalent mass density• So that closure parameter

4.65 10r kg mρ = ×

( ) 50 4 84 10rρ −Ω = = ×


( )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)


Relativistic fermionsRelativistic bosonsEvolution of particle content down to kT ª MeV

RELATIVISTIC PARTICLES IN THE p

EARLY UNIVERSE


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⎛ ⎞

= = +⎜ ⎟⎝ ⎠

∑ ∑


( ) 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


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


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


( ) ( )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


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


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


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!


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=


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+

−⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠


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+ = = ≈ ⎯⎯⎯⎯⎯⎯→ + =


( ) 0dec dec

dec kTR t

Summary

artic

lepe

r pa T(K)

ergy

pEn

chapter 5 astro-particle physics 2009-10 30Time t(s)

Particle Astroppyhysics , chapter 5

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