NEUTRINOS AND COSMOLOGY

STEEN HANNESTAD AARHUS UNIVERSITY

11 OCTOBER 2010

e

Where do Neutrinos Appear in Nature?

AstrophysicalAstrophysicalAccelerators Accelerators Soon ?Soon ?

Big Bang Big Bang

(Today 330 (Today 330 /cm/cm33)) Indirect EvidenceIndirect Evidence

Nuclear ReactorsNuclear Reactors

Earth Crust Earth Crust (Natural (Natural Radioactivity)Radioactivity)

Particle AcceleratorsParticle Accelerators

Earth AtmosphereEarth Atmosphere(Cosmic Rays)(Cosmic Rays)

SunSun

SupernovaeSupernovae(Stellar Collapse)(Stellar Collapse)

SN 1987ASN 1987A

(2005)(2005)

Lecture 1 (today):

Basic cosmology – the Friedmann equation

Neutrino thermodynamics and decoupling in the early universe

Bounds on massive neutrinos

Neutrinos and BBN - Theory vs. Observations

Bounds on the number of neutrino species

Lectures 2 and 3:

Structure formation in the universe

Absolute value of neutrino masses

How many neutrinos are there?

Future observational probes

Friedmann-Robertson-Walker Cosmology

Line element

dxdxgds 2

Reduces to

222222 )()( drSdrtadtds k

in a homogeneous and isotropic universe

a(t): Scale factor, only dynamical variable

1)(sinh

0

1)(sin

)(2

2

2

2

kr

kr

kr

rSk

The Einstein equation

gGTG 8

However, it reduces to an evolution equation for a(t) (The Friedmann equation) in a homogeneous and isotropic universe

22

2

2

38

338

akG

akG

aa

H TOT

is a combination of 10 coupled differential equationssince the involved tensors are explicitly symmetric

TOT=MATTER + RADIATION

Constant

)(

)(41

3

tan

tamn

RADIATION

MATTER

THE TOTAL ENERGY DENSITY THEN BY DEFINITIONINCLUDES NON-RELATIVISTIC MATTER, RADIATIONAND THE COSMOLOGICAL CONSTANT

HOWEVER, THESE TYPES OF ENERGY BEHAVECOMPLETELY DIFFERENTLY AS A FUNCTIONOF TIME AND SCALE FACTOR

1 aT

FROM THE ABOVE EQUATION

)(~ 4RADIATION ta

AND THE FACT THAT

4RADIATION ~ T

IT CAN BE SEEN THAT THE EFFECTIVE ’TEMPERATURE’ OF RADIATION SCALES AS

A DEFINITION:

A QUANTITY WHICH IS CONSISTENTLY USED ISTHE REDSHIFT, DEFINED AS

a

az 01

FROM THE SCALING OF PHOTON ENERGY IT CANIMMEDIATELY BE SEEN THAT THE OBSERVEDWAVELENGTH OF A PHOTON IS RELATED TO THESCALE FACTOR OF THE UNIVERSE WHEN IT WAS EMITTED

z1EMITTED

OBSERVED

log(

log(a)-4 0

MR

aeq

present

EVOLUTION OF ENERGY DENSITY WITH SCALE FACTOR

The geometry of the universe

Open Universe k = -1, < 1 Flat Universe

k = 0, Closed Universe k =1, >1

11 TOT22 RMaH

k23

8

H

G

The Friedmann equation can be recast in terms of thedensity parameter,

An empty universe expands linearly with time

tta ~)(

Matter acts to slow the expansion, for example

0,1~)( 2/1 MRtta for

1,0~)( 3/2 MRtta for

If M+ R>1 then the universe eventually recollapses

M M

A cosmological constant acts to accelerate the expansion.

0,1~)(

Mteta for

In general the pressure of an energy density componentcan be written as

wP For the cosmological constant, w = -1

Any component which has w < -1/3 leads toan accelerated expansion and is referred to asdark energy

M M

The total contribution to from baryons

Stars: ~ 0.005

Interstellar gas: ~ 0.005

Hot gas in clusters: ~ 0.03

045.0~BARYON

The net lepton or baryon asymmetry can be Expressed in terms of the parameter

According to observations ~ 6 x 10-10

n

nn BB

for baryons can be found from the present baryondensity

)10/( 0037.0 102 hb

NEUTRINO THERMODYNAMICSAND BIG BANG NUCLEOSYNTHESIS

Thermodynamics in the early universe

22

)1/)exp((

1mpE

TEfEQ

,

In equilibrium, distribution functions have the form

When m ~ T particles disappear because of Boltzmann supression

Decoupled particles: If particles are decoupled from other species their comoving number density is conserved. The momentum redshifts as p ~ 1/a

mTpTmMB eeff 2//)( 2

In equilibrium,

In equilibrium neutrinos and anti-neutrinos are equal in number!However, the neutrino lepton number is not nearly as wellconstrained observationally as the baryon number

The entropy density of a species with MB statistics is given by

)()( XX

TEefpdffs /)(3ln ,

This means that entropy is maximised when

0)()( XX

(true if processes like occur rapidly) XX

1010~

n

n

n

n BIt is possible that

Conditions for lepto (baryo) genesis (Sakharov conditions)

CP-violation: Asymmetry between particles and antiparticlese.g. has other rate than

L-violation: Processes that can breaklepton number (e.g. )

Non-equilibrium thermodynamics:In equilibrium always applies.

A small note on how to generate asymmetry(for leptons or baryons)

llX

LL nn

llX

llX

L L

XX ,

violation

violation

L

CPNon-equilibrium

Much more to come in the lecture by Michael Plümacher

Thermal evolution of standard, radiation dominated cosmology

3

82

22 G

a

aH

fermions for

bosons for

42

42

308

7

30

gT

gT

F

B

Total energy density

4*

24

2

3030)

8

7( TgTgg FBTOT

Temperature evolution of g*

In a radiation dominated universe the time-temperaturerelation is then of the form

22/1*

2/1

1 4.232

3

2

1

MeVs Tgt

GHt

The number and energy density for a given species, X, isgiven by the Boltzmann equation

][][ XiXeXX fCfCp

fpH

t

f

Ce[f]: Elastic collisions, conserves particle number but energy exchange possible (e.g. ) [scattering equilibrium]

Ci[f]: Inelastic collisions, changes particle number(e.g. ) [chemical equilibrium]

Usually, Ce[f] >> Ci[f] so that one can assume that elasticscattering equilibrium always holds.

If this is true, then the form of f is always Fermi-Dirac orBose-Einstein, but with a possible chemical potential.

iXiX

iiXX

The inelastic reaction rate per particle for species X is

Hint

Particle decoupling

vnpd

fC XX

Xi

3

3

)2( int

In general, a species decouples from chemicalequlibrium when

Plm

TTNH

22/1)(2

After neutrino decoupling electron-positron annihilationtakes place (at T~me/3)

Entropy is conserved because of equilibrium in thee+- e-- plasma and therefore

3/1

4

112)

8

742( 33

i

ffifi T

TTTss

The neutrino temperature is unchanged by this because they are decoupled and therefore

on)annihilati after (71.0)11/4( 3/1 TTT

The prime example is the decoupling of light neutrinos (m < TD)

MeV 1223 DFweak TTGTvn

There are small corrections to this because neutrinosstill interact slightly when electrons and positronsannihilate (neutrino heating)

0041.1

0083.1

/

,

0

e

Additional small effect from finite temperature QED effect(O()) ~ 1%In total the neutrino energy density gets a correction of

04.00

Dicus et al. ’82, Lopez & Turner ’99

+ several other papers

Upper limit on the mass of light neutrinos:

For light neutrinos, m << Tdec, the present day density is

eV 30/

4

33

3

2

m

mnT

Th c

Assuming that the three active species have the same mass

A conservative limit (h2<1) on the neutrino mass is then

eV 30mFor any of the three active neutrino species

If MB statistics is used one finds (as long as elastic scattering equilibrium holds) by integrating the Boltzmann equation

)(3 22eqnnvHnn

inelastic

This is the standard equation used for WIMP annihilation(e.g. massive neutrinos, neutralinos, etc)

This equation is usually not analytically solvable (Riccatiequation), but is trivial to solve numerically.

For a particle with standard weak interactions one findsthat the species decouples from chemical equilibrium when

20DT

m

Mass limits on very massive neutrinos (m >> TD):

vnvn eq

Tmeq e

mTgn /

2/3

2

Plm

TTNH

22/1)(2

1

0 /~ vnnH

As long as a species is close to equilibrium

with

Comparing this to the Hubble rate

yields

Dirac neutrinos

2

22 mGv F

D

Majorana neutrinos

22

22

2

222iFiF

M

mG

m

mmGv

This means that

Majorana for

Dirac for 2

22

im

mh

mi is the mass of the annihilation product (usuallythe most massive final state available)

This leads to a lower bound (the Lee-Weinberg bound) on very massive neutrinos

(Majorana)GeV

(Dirac)GeV

12

4m

TD

Because of the stringent bound from LEP on neutrinoslighter than about 45 GeV

008.0984.2 N

This bound is mainly of academic interest. However, the same argument applies to any WIMP, such as the neutralino (very similar to Majorana neutrino).

The baryon number left after baryogenesis is usually expressed in terms of the parameter

According to observations ~ 10-10 and therefore theparameter

0tt

B

n

n

1010 10

is often used

From the present baryon density can be found as

102 0037.0 hb

BIG BANG NUCLEOSYNTHESIS

Immediately after the quark-hadron transition almost allbaryons are in pions. However, when the temperature hasdropped to a few MeV (T << m) only neutrons and protons are left

In thermal equilibrium

MeV , 293.1)/exp( mTmn

n

p

n

However, this ratio is dependent on weak interactionequilibrium

n-p changing reactions

Interaction rate (the generic weak interaction rate)

MeV 1223 freezeFpn TTGTvn

After that, neutrons decay freely with a lifetime of

sn 8.0886

e

e

e

pen

pne

pen

However, before complete decay neutrons are bound innuclei.

Nucleosynthesis should intuitively start when T ~ Eb (D) ~ 2.2 MeV via the reaction

However, because of the high entropy it does not.Instead the nucleosynthesis starting point can be found fromthe condition

Dnp

MeV 2.0)ln(/

bBBNTE

ndestructio

Bproduction ET

evn

vn

b

Since t(TBBN) ~ 50 s << n only few neutrons have time to decay

)()( DD ndestructioproduction

At this temperature nucleosynthesis proceeds via the reactionnetwork

The mass gaps at A = 5 and 8 lead to small production of mass numbers 6 and 7, and almost no production of mass numbers above 8

The gap at A = 5 can be spanned by the reactions

LiHeT

BeHeHe74

743

),(

),(

ABUNDANCES HAVE BEEN CALCULATEDUSING THE WELL-DOCUMENTED AND PUBLICLY AVAILABLE FORTRAN CODENUC.F, WRITTEN BY LAWRENCE KAWANO

The amounts of various elements produced depend on the physical conditions during nucleosynthesis, primarily the values of N(T) and

Helium-4: Essentially all available neutrons are processedinto He-4, giving a mass fraction of

7/1~/25.024

pn

Tpn

n

N

HeP nn

nn

n

n

nY

BBN

for

Yp depends on because TBBN changes with

)ln(,

DB

BBN

ET

7/1~)/exp(2

)/exp()/exp(

nBBN

nBBNweak

Tp

n

t

tTm

n

n

BBN

D, He-3: These elements are processed toproduce He-4. For higher , TBBN

is higher and they are processed more efficiently

Li-7: Non-monotonic dependence because of twodifferent production processesMuch lower abundance because of mass gap

Higher mass elements. Extremely low abundances

Confronting theory with observations

The Solar abundance is Y = 0.28, but this is processed material

The primordial value can in principle be found by measuringHe abundance in unprocessed (low metallicity) material.

He-4 is extremely stable and is in general always produced,not destroyed, in astrophysical environments

He-4:

Extragalactic H-IIregions

I Zw 18 is the lowest metallicity H-II region known

Aver et al. 2010

Deuterium: Deuterium is weakly bound and thereforecan be assumed to be only destroyed inastrophysical environments

Primordial deuterium can be found either by measuringsolar system or ISM value and doing complex chemical evolution calculations

OR

Measuring D at high redshift

The ISM value of

can be regarded as a firm lower bound on primordial D

505.010.0 1009.060.1)/(

ISMHD

1994: First measurements of D in high-redshift absorptionsystems

A very high D/H value was found

4105.29.1)/( zHighHD

Carswell et al. 1994Songaila et al. 1994

However, other measurementsfound much lower values

Burles & Tytler 1996

5105.2)/( zHighHD

Burles & Tytler

Pettini et al 2010

The discrepancy has been ”resolved” in favour of a low deuterium value of roughly

5105.04.3)/( zHighHD

Li-7: Lithium can be both produced and destroyedin astrophysical environments

Production is mainly by cosmic ray interactions

Destruction is in stellar interiors

Old, hot halo stars seem to be good probes ofthe primordial Li abundance because there hasbeen only limited Li destruction

Molaro et al. 1995

Li-abundance in old halo stars in units of

12)/log(7 HLiLi

Spite plateau

There is consistency between theory and observations

All observed abundances fitwell with a single value of eta

This value is mainly determinedby the High-z deuteriummeasurements

The overall best fit is

10103.01.5

Burles, Nollett & Turner 2001

This value of translatesinto

002.0020.02 hb

And from the HST valuefor h

08.072.0 h

One finds

054.0028.0 b

3.0

01.0

m

luminous

BOUND ON THE RELATIVISTIC ENERGY DENSITY(NUMBER OF NEUTRINO SPECIES) FROM BBN

The weak decoupling temperature depends on the expansionrate

15

)(2

3

8 43 TTGNGH

And decoupling occurs when

6/152int )(TNTHTG DF

N(T) is can be written as

)3()()(

)()()(

,,,

,,,,

NTNTN

TNTNTN

SMee

SMvSMee

extra

N= 3

N= 4

N= 2

Since )/exp( D

BBNp

n Tmn

n

The helium production is very sensitive to N

n-p changing reactions

FLAVOUR DEPENDENCE OF THE BOUNDS

Muon and tau neutrinos influence only the expansion rate.However, electron neutrinos directly influence theweak interaction rates, i.e. they shift the neutron-protonratio

e

e

e

pen

pne

pen

More electron neutrinos shifts the balance towards more neutrons, and a higher relativistic energy density can be accomodated

Ways of producing more electron neutrinos: 1) A neutrino chemical potential2) Decay of massive particle into electron neutrinos

TTE

f

TE

f

, , 1exp

1

1exp

1

e

e

e

pen

pne

pen

Increasing e decreases n/p so

that N can be much higher than 4Yahil ’76, Langacker ’82, Kang&Steigman ’92, Lesgourgues&Pastor ’99, Lesgourgues&Peloso ’00, Hannestad ’00, Orito et al. ’00, Esposito et al. ’00, Lesgourgues&Liddle ’01, Zentner & Walker ’01

BBN alone: 9.61.106.0 , , e

e= 0.1

e= 0.2

Allowed region

Allowed fore

= 0

If oscillations are taken into account these bounds becomemuch tighter:

MeV 6/10

2 )2cos(20 eVosc mT

Oscillations between different flavours become importantonce the vacuum oscillation term dominates the matterpotential at a temperature of

For this occurs at T ~ 10 MeV >> TBBN, leading tocomplete equilibration before decoupling.

For e the amount of equilibration depends completelyon the Solar neutrino mixing parameters.

Lunardini & Smirnov, hep-ph/0012056 (PRD), Dolgov et al., hep-ph/0201287Abazajian, Beacom & Bell, astro-ph/0203442, Wong hep-ph/0203180

If all flavours equilibrate before BBN the bound on theflavour asymmetry becomes much stronger becausea large asymmetry in the muon or tau sector cannot bemasked by a small electron neutrino asymmetry

Dolgov et al., hep-ph/0201287 SH 02

For the LMA solution the bound becomes equivalentto the electron neutrino bound for all species

15.0,, e Dolgov et al. ’02

See also Serpico & Raffelt 05

Using BBN to probe physics beyond the standard model

Non-standard physics can in general affect either

Expansion rate during BBN extra relativistic speciesmassive decaying particlesquintessence….

The interaction rates themselvesneutrino degeneracychanging fine structure constant….

PERSPECTIVES

Neutrino thermodynamics and decoupling is well understood in the standard model.

Big bang nucleosynthesis provides a powerful probeof neutrino physics beyond the standard model. However,many of the abundance measurements are dominated bysystematics which need to be better understood

Deuterium measurements provide by far the most sensitiveprobe of the baryon density and is (now) consistentwith CMB. More on this later!