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!