Lecture 2 - IIHE

The expanding universe

Lecture 2

Last lecture

• Universe is flat

• Dynamics given by Friedman equation

• Cosmological redshift

• Closure parameter

• Energy density evolves with time

2010-11 Expanding Universe 2

2

2 8

3

NR t G

R ttH t totρ

0

01 0R t

z z tR t

c

tt

t

2 3 4 22

0 0 1 0 1 0 0 1m r kH t H z z z

Expanding universe part 2

• Densities of radiation and matter in early universe and today

– Cosmic Microwave Background photons

– Radiation domination era : particle and nuclear physics in the early universe

– Matter-radiation decoupling at z=1100 and t=380.000y

– Matter domination era : late universe

• Big Bang Nucleosynthesis in more detail

• What about antimatter?


© Rubakov

Ωrad

Ωbaryons

ΩCDM

Ωneutrino

2010-11 4Expanding Universe

Tod

ays

lect

ure

Cosmic Microwave Background photons

CMB expected in Big Bang model• In the early hot universe:

– radiation dominates over matter

– vacuum energy is negligible

• When kT few MeV formation of H, D and other light atoms starts

• baryons and photons are in thermal equilibrium

• When free electron density becomes too small, formation of H stops

• photons decouple from matter → evolve independently from otherparticles

• Photon temperature drops with expansion → expect today : few K, or few mm wavelength

• Should be seen today as uniform microwave background


e p H

CMB discovery in 1965• Discovered in 1965 by Penzias and Wilson (Bell labs)

when searching for radio emission from Milky Way

• Observed a uniform radio noise from outside the MilkyWay

• This could not be explained by stars, radio galaxies etc

• Use Earth based observatory: limited to cm wavelengths due to absorption of mm waves in atmosphere

• Observed spectrum was compatible with black body radiation with T = (3.5 1) K

• Obtained the Nobel Prize in 1978 (http://nobelprize.org/)


• To go down to mm wavelengths : put instruments on satellites

• COBE = COsmic Background Explorer (NASA) satellite observations in 1990s: mm wavelengths

• Large dipole anisotropy due to motion of solar system in universe, with respect to CMB rest frame

• After subtraction of dipole radiation was uniform up to 0.005%

• Has perfect black body spectrum with T = 2.725K

• Discovered small anisotropies over angular ranges =7

• WMAP = Wilkinson Microwave Anisotropy Probe (NASA, 2003)

• Produces fine graded maps of temperature anisotropies

COBE and WMAP


solar system 300 kmvs

COBE measures black body spectrum


Intensity Q

Frequency (cm-1)

=2mm 0.5mm

3

2 2,

41

2

k

Q Tc

e T

• Plancks radiation law for relativistic photon gas

• Black body withtemperature T emitsradiation with power Q at frequencies

COBE measures black body spectrum


Intensity Q

Frequency (cm-1)

=2mm 0.5mm

2.725 0.001

max 2

T CMB K

mm

0.235E kT meV

• CMB has ‘perfect’ black body spectrum

• Fit of data of differentobservatoria to black body spectrum gives

• Or

CMB energy density vs time

• In our model the early universe is radiation dominated

• For flat universe → Friedmann equation

• energy density of radiation during expansion

• Integration yields


2

2

8

3

N

rad

GR

R

1281 4

43

N radGd R

Rdt

22

2

3 1

32 N

rad

cc t

G t

4 41rad z R

CMB number density today 1

• CMB photons have black body spectrum today

• They also had black body spectrum when CMB was created

• But ! Temperature T in past was higher than today

• CMB = photon gas in thermal equilibrium

• → Bose-Einstein distribution : number of photons per unit volume in momentum interval [p,p+dp]


2

2 3 1E

k

p dpn p dp

e T

γg

2gγ = number of photon substates

Black body

CMB number density today 2


Nn n p dp

V

3

0 411n t cm

3

2

12.404

kTn

c

gγ=2

T=2.725K

CMB energy density today

• Energy density

• Equivalent mass density


2c n p dpE

42 4

2 3 3

1

2 15

gc t k

cT

2 3

0 0.261rc t MeV m

31 34.65 10r kg m 0

54.84 10rr

ct

CMB temperature vs time

• for t0 = 14Gyr expect TCMB (today) 10K !!!

• BUT! 2.7K measured !


1143 5 4

132

45 2 1

32

ck

G gT

t

23

matter domT t

101.52 10rad dom

cstT

12t

22

2

3 1

32 N

rad

cc

G t

42 4

2 3 3

1

2 15

gc k

cT

CMB photons cooled more quickly in

later, matter dominated, era

Since t=380’000 years

Anisotropies in CMB radiation

• dipole anisotropy in temperatureof radiation,O(10-3K) due to movement of solar system relative to distant matter (v=300 km s-1) – Doppler effect

• Galactic emission

• Faint temperaturefluctuations,(Order 10-5 K) in CMB

• Imprints of density fluctuations in early universe, at surface of last scattering (chapter 8)


COBE 2 years data

Particle physics in early universe

relativistic particles in early universe

• In the early hot universe relativistic stable fermions and bosons contribute to the energy density

• Fermion gas = quarks, leptons

• Fermi-Dirac statistics

(gf = nb of substates)

• boson gas = photons, W and Z bosons …

• Bose Einstein statistics

(gb = nb of spin substates)


2

2 3 21

EkT

p dpn p dp

e

bg

2

2 3 21

EkT

p dpn p dp

e

fg

relativistic particles in early universe

• Bosons and fermions contribute to energy density with


2

2 3 21

EkT

p dpn p dp

e

bg 2

2 3 21

EkT

p dpn p dp

e

fg

42 4 *

* 2 3 3

1 7

15 2 8b fc t kT g g g

c

*g

2c n p dpE

Degrees of freedom for kT > 100 GeV


bosons per particle total

W+ W- 3 2 x 3 = 6Z 3 3gluons 2 8 x 2 = 16photon 2 2Higgs 1 1

total bosons 28

fermions per particle total

quarks 3 (color) x 2 (spin) 6 x 3 x 2 = 36antiquarks 36e,µ,τ 2 6 x 2 = 12neutrinos 1 3 x 1 = 3anti-neutrinos 1 3 x 1 = 3

total fermions 90

Degrees of freedom for kT > 100 GeV

• Assuming only particles from Standard Model of particlephysics

• Energy density in hot universe


* 728 90 106.75

8g

42 4

* 2 3 3

1

15 2c t kT

c

*g

what happens if there were particles from

theories beyond the Standard Model?

For instance : SuperSymmetry

• At LHC energies and higher : possibly SuperSymmetry

• Symmetry between leptons and bosons

• Consequence is a superpartner for every SM particle

• Double degrees of freedom g*


Neutralino = Dark Matter ?

• Neutral gaugino and higgsino fields mix to form 4 mass eigenstates

→ 4 neutralinos

• no charge, no colour, only weak interactions

• is Lightest Supersymmetric Particle – LSP - in R-parityconserving scenarios → stable

• Massive : Searches at LEP and Tevatron colliders

• may have been created in hot universe

• and survived till today

• To have right dark matter abundance


0

1

1 2

0 50m GeV c1 1

0 0e e

1 2

0 5m TeV c

Cool down to kT GeV

• Start from hot universe: plasma of leptons, quarks, gauge bosons, Higgs, exotic particles

• Temperature decreases with time

• Production of particles stops when

• For example, above 160 GeV (see LEP @ CERN)

• Most particles decay: W, Z, t, b, τ, ..

• Run out of heavy particles when kT<<100GeV2010-11 Expanding Universe 24

2kT Mc

23, 10W Z s

e e W W 2 Ws M

12

1~rad domTt

when

From GeV to kT 200 MeV


• Phase transition from Quark Gluon Plasma (QGP) to hadrons

• Ruled by Quantum Chromo Dynamics (QCD) of strong interactions

s

q2 (GeV2)

asymptotic freedomQuarks cannot be free at distances

of more than 1fm = 10-15m

200QCD MeVE

TFrom fit to data

Strong coupling constant

• free quarks and gluons are gone and hadrons have been formed

• Most hadrons are short lived and decay with

• Example

• Leptons : muon and tauon decay weakly

around kT 200 MeV


8 2310 weak ints. 10 strong ints.s s

15319 10

17%

.......

s

0

1115

uds

n

p

n e

p

e

62 10

ee

s

Stable or long lived

<< 1µs

• After about 1ms all unstable particles have decayed

• Most, but not all, nucleons annihilate with anti-nucleons

we are left with

e-, , e, , and their anti-particles

g*

kT(GeV)TeV

GeV MeV

106.75

10

3.4

Cooldown to kT few 10MeV

2010-11 27

* 7 432 10 10

8 4g

p p

42 4

* 2 3 3

1

15 2c t kT

c

*g

Expanding Universe

Neutrino freeze out at ≈ 3MeV

• around few MeV: mainly relativistic e, , e, , + anti-particles

• few protons & neutrons start primordial nucleosynthesis

• Formation of light elements

• Equilibrium between photons and leptons

• Weak interaction cross section


, ,i ie e i e Weak interaction

25 2 s CM energy 1.166 10

6F

F

G sG GeV

2

2

3

2.22

...........

n p MeV

H

H

n H

Neutrino freeze-out at t ≈ 1s

• e+e- collision rate interactions/sec

• relative

• During expansion T decreases

• when W << H or kT < 3MeV or t > 1s

→ Neutrinos are no longer created

• Neutrinos decouple and evolve independently

• neutrino freeze-out relic neutrinos2010-11 Expanding Universe 29

W vn

e+, e- number density(FD statistics) ~ T3

, ,i ie e i e

Cross section ~ s ~ T2

Relative velocity

2 H t T5 W T

Weak interaction

Cosmic Neutrino Background

• Relic neutrinos are oldest relic of early universe –decoupled at about 1s – before CMB photons

• Should be most abundant particles in sky after CMB photons

• Should populate unievrse today as Cosmic Neutrino Background CνB or cosmogenic neutrinos

• what are numbers density and temperature today?

• At few MeV there was thermal equilibrium betweenphotons and leptons

• Number density neutrinos number density photons


, ,i ie e i e

Relic neutrino temperature

• At decoupling energy density of neutrinos from energydensity of photons

• After neutrino decoupling photons get energy boost fromelectromagnetic interactions

• Expansion is adiabatic – we expect same ratio today

→ expected Temperature of neutrinos today

• Challenge : detection of meV neutrinos !


e e134

11T T

0( ) 1.95T t K

2 2c c

0( )E t meV

CνB number density

• for kT << 1 MeV

• Most protons and neutrons are trapped in atoms (BBN)

• relativistic particles left are mainly photons + neutrinos

• Photons got boost compared to neutrinos

• expected density of relic neutrinos today is about CMB density

• for given species (e, , )

• CνB could explain part of Dark Matter : weakly interacting, massive, stable – is Hot DM


4

7* 3.36

8

Tg g g

T

33113

11N N cmN

42 4

* 2 3 3

1

15 2c t kT

c

*g

0.01

Relativistic particles in early times


g*

kT(GeV)TeV

GeV MeV

106.75

10

3.4

Neutrino Decoupling andnucleosynthesis

Quarks confinedin hadrons

ep recombinationTransition to matter dominateduniverse

Run out of relativisticparticles

Radiation-matter decoupling

• At tdec 380.000 years, or z 1100, or T 3500K

• matter decouples from radiation and photons can move freely & remain as today’s CMB radiation

• Matter evolves independently - atoms & molecules are formed → stars, galaxies, …

• If spatial temperature variations are present → leaveimprint on CMB (see chapt 8)

• Before tdec universe is ionised and opaque

• average time between collisions << age t of universe

• particles are in thermal equilibrium as long as


1W n vt

Protons and neutral hydrogen

Up to t 100.000 y thermal equilibrium of p, H, e,

When kT < 13.6 eV (ionisation potential of H) ionisation probability reduces

• 2 processes compete - number density of free protons Np

and of neutral hydrogen atoms NH as function of T


formation of neutral hydrogen

ionisation of hydrogen atom

e p H

2

321 2H

H

p

H

kN N mk

eN N he

TI

N

TNe = density of free electronsm=electron mass

Depends on densitiesNe and Np

Radiation-matter decoupling

• Rewrite in function of fraction x of ionised hydrogen atoms

• strong drop of x between kT 0.35 - 0.25 eV

• or T between 4000 – 3000 K

• ionisation stops around 3500K

• period of recombination of e and p to hydrogen atoms

• Stops when electron density too small


2

2

321 2

1 B

Ikx mk

ex N h

TTp

p H

p

B

NNx

N N N

e p H

Decoupling time

• Reshift at decoupling

• Full calculation

• When electron density is too small there is no H formation anymore

• → Photons freeze out as independent population = CMB

• start of matter dominated universe

• We are left with atoms, CMB photons and relic neutrinos

• + neutralinos if SuperSymmetry describes nature at high T2010-11 Expanding Universe 37

0

0

35001 1270

2.75

decdec

dec

R t kTz

kTR t

1 1100dec

z 53.7 10dect y

Era of matter-radiation equality

• since

• Density baryons = density photons when

• Density matter (baryons + Dark Matter) = density photons + neutrinos


3

baryonic matter T 4

photons T

0

0

11

1

bar bar

phot phot

t t

t t z1 870 1

decz z

1 3130z0

0

11

1.58 1

matter m

rphot neut

t t

t t z

Summary


Ene

rgy

per

par

ticl

e

T(K)

Time t(s)

Primordial nucleosynthesis BBN

Overview

• Primordial (Big bang) nucleosynthesis of light elements

• expected and observed abundances of light elements

• Expected and observed baryon/photon ratio


Big Bang or primordial nucleosynthesis

• Fusion processes occuring between kT 1 MeV (neutrino

decoupling) & kT0.3 eV (no ionised hydrogen left)

• Before t380.000y, in radiation dominated universe

• Period of fusion processes : synthesis of light elements:

• It is NOT

the synthesis of elements in stars

taking place during star formation

and evolution

in the matter dominated universe

after 380.000 y


2 3 4 7 7, , , ,H He He Be Li

neutrons and protons freeze-out

• At kT < 100MeV all hadrons have decayed

• Most nucleons and antinucleons annihilate – not all!

• Tiny fraction of nucleons is left – no antinucleons anymore

• Plus e, , e, , and their antiparticles

• Equilibrium between weak interactions

And neutron decay

• Interactions stop when W << H → neutron & proton freeze-out


p p

en p e

e

e

n e p

p e n

2 H t T5 W t n v T

neutron / proton ratio

• When kT=0.8MeV and t ≈ 1s weak interactions are too slow

• As soon as kT << 1 GeV nucleons are non-relativistic

• Probablity that proton is in

energy state in [E,E+dE]

• During equilibrium between

weak interactions

• at freeze-out time tFO

• During whole process neutrons

can decay with = (885.7 0.8)s


2

expn pn

p

M M cN

N kT

0.20n FO

p FO

N t

N t

2

pkT M c

2

expp

proton

EkT

M cP e

kT

0.20exp

1.2 0.20exp

n

p

N t t

N t t

Neutron/proton ratio

• Freeze-out of nucleons is not abrupt but stretched

• Due to competition between weak interaction and expansion rate

2010-11t(s)

n

p

N t

N t

T(keV)

t(s)

Only weak interactions in equilibriumNeutrons disappearno star formation

0.8 MeV

True variation

45Expanding Universe

Nucleosynthesis onset

• Non-relativistic neutrons also form atoms through fusion: formation of deuterium

• Photodisintegration of 2H stops when kT ≈ 60 KeV

• Then free neutrons are gone

• And deuterium freeze-out


2

2

2

2.22

formation of

desintegration of

n p H MeV

H

H

n

p

N

N

• Chain of fusion reactions

• ΛCDM model predicts values of relative ratios of light elements

• We expect the ratios to be constant over time

• Comparison to observed abundances today allows to test the

standard cosmology model

Nuclear chains

2 3

2

2 2

3 2 4

4 3

7

2

3

7

2.22H

He

B

n p MeV

H n H

H H

H H

H H He n

He He

Be n p

e

4

7

He

Li


• helium mass fraction

• Is expected to be constant with time – He in stars (formedafter BBN) has only small contribution

• model prediction at onset of BBN (kT 80keV, Nn/Np0.13 )

• Observation today in clusters, gas clouds …

He mass fraction


24

4 1 1

n p

n p

N NM He yY

M He H y N N

0.25predY

He

H

Ny

N

0.249 0.009obsY

Abundances of light elements

• Standard BB nucleosynthesis theory predicts abundancesof light elements today – example 7Li

• Observations today

• Abundances depend on baryon/photon ratio2010-11 Expanding Universe 49

7101.23 0.01 10

Li

HBBN StartskT80keV

1010

7Li H

Baryons and photons

• ratio of baryon and photon number densities

– Baryons = atoms

– Photons = CMB radiation

• Ratio constant since matter-radiation decoupling at z=1100

• Fluctuations in distribution of baryons and photons atdecoupling are ‘frozen’ → anisotropies seen today

• Observations :

– abundances of light elements, He mass fraction

– CMB anisotropies from WMAP


10

10 10baryon

photon

N

N

Abundances and baryon density


He mass fraction

abundances

Bh2

η10

Observations Of light elementsMeasure

Model PredictionsDepend on η10 Bh2

CMB observations with WMAP measure Bh2

η10

Best fit results PDG 2010


10

0.044 0.005

6.1 0.6 10

B

BN

N

5

10

0.249 0.009

/ 2.82 0.21 10

/ 1.7 0.06 0.44 10

pY

D H

Li H

pdg.lbl.gov

What about antimatter ?

• Antiparticles from early universe have disappeared!

• Early universe: expect equal amount of particles & antiparticles because all interactions conserve CP

• Observation of primary charged galactic cosmic rays: nucleiand no antinuclei

• Annihilation of matter with antimatter in galaxies wouldyield intense X-ray and -ray emission – not observed

• Few positrons and antiprotons fall in on Earth atmosphere : in agreement with pair creation in inter stellar matter

• Antiparticles also produced in showers in Earthsatmosphere = secundary cosmic rays


Baryon and antibaryons

• Baryon number conservation = strict law in laboratory

• If no B conservation proton decay allowed

• Assume net baryon number = 0 in early universe

• Assume equilibrium reactions

• At freeze out we expect

• observations


ep e

p p

1810B BNN

N N

10 9

4

6.1 0.6 10 10

10

B

B

B

N

N

N

N

Baryon-antibaryon asymmetry

• Is model wrong?

• Fundamental conditions for asymmetry in baryon-antibaryon content (Zacharov criteria):

– Baryon number violating interactions

– Non-equilibrium situation

– CP and C violation

• Eg in Grand Unified Theories proton can decay

• Search at colliders for violation of B and CP conservatinginteractions


Lecture 2 - IIHE

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