Transcript n NEUTRINOS AND COSMOLOGY m
NEUTRINOS AND COSMOLOGY
n e n t n m
STEEN HANNESTAD UNIVERSITY OF AARHUS ISAPP 2006 MUNICH
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
ds
2
g
mn
dx
m
dx
n
Reduces to
ds
2
dt
2
a
2 (
t
)
dr
2
S k
2 (
r
)
d
2
in a homogeneous and isotropic universe
S k
2 (
r
) sin sinh
r
2 2 2 ( (
r r
) )
k k k
1 0 1
a(t)
: Scale factor, only dynamical variable
The Einstein equation
G
mn
8
GT
mn
g
mn is a combination of 10 coupled differential equations since the involved tensors are explicitly symmetric
However, it reduces to an evolution equation for
a(t)
(The Friedmann equation) in a homogeneous and isotropic universe
H
2
a a
2 8
G
3 3
k a
2 8
G
TOT
3
k a
2
THE TOTAL ENERGY DENSITY THEN BY DEFINITION INCLUDES NON-RELATIVISTIC MATTER, RADIATION AND THE COSMOLOGICAL CONSTANT
TOT
=
MATTER +
RADIATION
HOWEVER, THESE TYPES OF ENERGY BEHAVE COMPLETELY DIFFERENTLY AS A FUNCTION OF TIME AND SCALE FACTOR
MATTER
n
m
a
(
t
)
3
RADIATION
n
1
Constant
a
(
t
)
4
FROM THE ABOVE EQUATION RADIATION ~
a
4 (
t
) AND THE FACT THAT RADIATION ~
T
4 IT CAN BE SEEN THAT THE EFFECTIVE ’TEMPERATURE’ OF RADIATION SCALES AS
T
a
1
A DEFINITION: A QUANTITY WHICH IS CONSISTENTLY USED IS THE REDSHIFT, DEFINED AS
1
z
a
0
a
FROM THE SCALING OF PHOTON ENERGY IT CAN IMMEDIATELY BE SEEN THAT THE OBSERVED WAVELENGTH OF A PHOTON IS RELATED TO THE SCALE FACTOR OF THE UNIVERSE WHEN IT WAS EMITTED OBSERVED EMITTED 1
z
EVOLUTION OF ENERGY DENSITY WITH SCALE FACTOR log( ) a eq R M present -4 log(a) 0
The geometry of the universe
The Friedmann equation can be recast in terms of the density parameter,
k H
2
a
2
M
R
1 TOT 1 8
G
3
H
2 Open Universe
k =
-1, < 1 Flat Universe
k =
0, 1 Closed Universe
k =
1, >1
An empty universe expands linearly with time
a
(
t
) ~
t
Matter acts to slow the expansion, for example
a
(
t
) ~
t
1 / 2 for
R
1 ,
M
0
a
(
t
) ~
t
2 / 3 for
R
0 ,
M
1 If M + R >1 then the universe eventually recollapses
0 M 2, 0 M 1, 0
A cosmological constant acts to accelerate the expansion.
a
(
t
) ~
e
t
for 1 ,
M
0 In general the pressure of an energy density component can be written as
P
w
For the cosmological constant,
w
= -1 Any component which has
w <
-1/3 leads to an accelerated expansion and is referred to as
dark energy
0 M 2, 0.1
M 1, 0
The total contribution to
from baryons
Stars: ~ 0.005
Interstellar gas: ~ 0.005
Hot gas in clusters: ~ 0.03
BARYON ~ 0 .
045
The net lepton or baryon asymmetry can be Expressed in terms of the parameter h h
n B
n B n
h for baryons can be found from the present baryon density
b h
2 0 .
0037 ( h / 10 10 ) According to observations h ~ 6 x 10 -10
NEUTRINO THERMODYNAMICS AND BIG BANG NUCLEOSYNTHESIS
Thermodynamics in the early universe In equilibrium, distribution functions have the form
f EQ
exp((
E
1 m ) /
T
1 ) ,
E
p
2
m
2 When
m
~ supression
T
particles disappear because of Boltzmann
f
f MB
e
(
m
m ) /
T e
p
2 / 2
mT
Decoupled particles: If particles are decoupled from other species their comoving number density is conserved. The momentum redshifts as
p
~ 1/
a
The entropy density of a species with MB statistics is given by
s
f
ln
f d
3
p
,
f
In equilibrium, m (
X
) m (
X
)
e
(
E
m ) /
T
This means that entropy is maximised when m (
X
) m (
X
) 0 In equilibrium neutrinos and anti-neutrinos are equal in number!
However, the neutrino lepton number is not nearly as well constrained observationally as the baryon number It is possible that
n
n
n
n B n
~ 10 10
A small note on how to generate asymmetry (for leptons or baryons)
Conditions for lepto (baryo) genesis (Sakharov conditions)
L
-violation: Processes that can break CP-violation: Asymmetry between particles and antiparticles
X
l l
Non-equilibrium thermodynamics:
n
L n L
X
,
X L L CP
violation Non-equilibrium
L
violation Much more to come in the lecture by Michael Plümacher
Thermal evolution of standard, radiation dominated cosmology
H
2
a
2
a
2 8
G
3
B
F
2 30 7
gT
2
gT
8 30 4 for bosons 4 for fermions Total energy density
TOT
(
g B
7 8
g F
) 2 30
T
4 30 2
g
*
T
4
Temperature evolution of
g *
In a radiation dominated universe the time-temperature relation is then of the form
t
1 2
H
1 3 32
G
1 / 2
t s
2 .
4
g
* 1 / 2
T
2
MeV
The number and energy density for a given species,
X
, is given by the Boltzmann equation
f
t X
pH
f
p X
C e
[
f X
]
C i
[
f X
]
C e
[
f
]:
Elastic collisions
, conserves particle number but energy exchange possible (e.g. ) [scattering equilibrium]
C i
[
f
]:
Inelastic collisions X
X
i
i
, changes particle number [chemical equilibrium] Usually,
C e
[
f
] >>
C i
[
f
] so that one can assume that elastic scattering equilibrium always holds.
If this is true, then the form of
f
is always Fermi-Dirac or Bose-Einstein, but with a possible chemical potential.
Particle decoupling
The inelastic reaction rate per particle for species
X
is int
C i
3
X d
( 2
p
) 3
X
n X
v
In general, a species decouples from chemical equlibrium when int
H H
2
N
(
T
) 1 / 2
T
2
m Pl
The prime example is the decoupling of light neutrinos (
m
<
T D
)
weak
n
v
T
3
G F
2
T
2
T D
1 MeV After neutrino decoupling electron-positron annihilation takes place (at T~
m
e /3) Entropy is conserved because of equilibrium in the
e + e -
plasma and therefore
s i
s f
( 2 4 7 8 )
T i
3 2
T f
3
T T i f
11 1 / 3 4 The neutrino temperature is unchanged by this because they are decoupled and therefore
T
n ( 4 / 11 ) 1 / 3
T
0 .
71
T
(
after annihilati on)
There are small corrections to this because neutrinos still interact slightly when electrons and positrons annihilate (neutrino heating) n n
e
m , t / n 0 1 .
0083 1 .
0041 Additional small effect from finite temperature QED effect (
O
( a )) ~ 1% In total the neutrino energy density gets a correction of n n 0 0 .
04 Dicus et al. ’82, Lopez & Turner ’99 + several other papers
Upper limit on the mass of light neutrinos: For light neutrinos,
m
<<
T
dec , the present day density is n
h
2 3 3 4
T
n
T
3
n
m
n /
c
m
n 30 eV Assuming that the three active species have the same mass A conservative limit ( n
h
2 <1) on the neutrino mass is then
m
n 30 eV For any of the three active neutrino species
Mass limits on very massive neutrinos (
m
>>
T D
): If MB statistics is used one finds (as long as elastic scattering equilibrium holds) by integrating the Boltzmann equation
n
3
Hn
v
inelastic (
n
2 2
n eq
) This is the standard equation used for WIMP annihilation (e.g. massive neutrinos, neutralinos, etc) This equation is usually not analytically solvable (Riccati equation), but is trivial to solve numerically.
For a particle with standard weak interactions one finds that the species decouples from
chemical
equilibrium when
m T D
20
As long as a species is close to equilibrium
n
v
n eq
v
with
n eq
g mT
2 3 / 2
e
m
/
T
Comparing this to the Hubble rate
H
2
N
(
T
) 1 / 2
T
2
m Pl
yields ~
H
n
0 /
n
v
1
Dirac neutrinos
v D
G F
2
m
2 2 Majorana neutrinos
v M
G F
2
m
2 2
m i
2
m
2
G F
2
m i
2 2
m i
is the mass of the annihilation product (usually the most massive final state available) This means that n
h
2
m i
m
2 2 for for Dirac Majorana
This leads to a lower bound (the Lee-Weinberg bound) on very massive neutrinos
m
n 4 GeV (Dirac) 12 GeV (Majorana)
T D
Because of the stringent bound from LEP on neutrinos lighter than about 45 GeV
N
n 2 .
984 0 .
008 This bound is mainly of academic interest. However, the same argument applies to any WIMP, such as the neutralino (very similar to Majorana neutrino).
BIG BANG NUCLEOSYNTHESIS
The baryon number left after baryogenesis is usually expressed in terms of the parameter h h
n B n
t
t
0 According to observations h ~ 10 -10 parameter and therefore the h 10
10
10 h is often used From h the present baryon density can be found as
b h
2 0 .
0037 h 10
Immediately after the quark-hadron transition almost all baryons are in pions. However, when the temperature has dropped to a few MeV (T <<
m
) only neutrons and protons are left In thermal equilibrium
n n n p
exp(
m
/
T
) ,
m
1 .
293 MeV However, this ratio is dependent on weak interaction equilibrium
n-p
changing reactions n
e
e n
n
e
n e
n
e
p
p
n
e p
Interaction rate (the generic weak interaction rate)
n
p
n
v
T
3
G F
2
T
2
T freeze
1 MeV After that, neutrons decay freely with a lifetime of t
n
886 0 .
8
s
However, before complete decay neutrons are bound in nuclei.
Nucleosynthesis should intuitively start when T ~ E b (D) ~ 2.2 MeV via the reaction
p
n
D
However, because of the high entropy it does not.
Instead the nucleosynthesis starting point can be found from the condition
production
(
D
)
destructio n
(
D
)
production
n B
destructio n
n
v
v e
E b
/
T
T BBN
E b
ln( h ) 0 .
2 MeV Since
t
(
T
BBN ) ~ 50 s << t n only few neutrons have time to decay
At this temperature nucleosynthesis proceeds via the reaction network 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 3
He
( 4
He
, ) 7
Be T
( 4
He
, ) 7
Li
ABUNDANCES HAVE BEEN CALCULATED USING THE WELL-DOCUMENTED AND PUBLICLY AVAILABLE FORTRAN CODE NUC.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 h
Helium-4: Essentially all available neutrons are processed into He-4, giving a mass fraction of
Y P
4
n He n N
n n
2
n n
n p T BBN
0 .
25 for
n n
/
n p
~ 1 / 7
n n n p T BBN
exp(
m
/
T weak
) 2 exp(
t
exp(
BBN
t
/
BBN
t /
n
t )
n
) ~ 1 / 7
Y
p depends on h because
T
BBN changes with h
T BBN
E
ln(
B
, h
D
)
D, He-3: These elements are processed to produce He-4. For higher h ,
T
BBN is higher and they are processed more efficiently Li-7: Non-monotonic dependence because of two different production processes Much lower abundance because of mass gap
Higher mass elements. Extremely low abundances
Confronting theory with observations
He-4:
He-4 is extremely stable and is in general always produced, not destroyed, in astrophysical environments The Solar abundance is
Y
= 0.28, but this is processed material The primordial value can in principle be found by measuring He abundance in unprocessed (low metallicity) material.
Extragalactic H-II regions
I Zw 18 is the lowest metallicity H-II region known
Olive, Skillman & Steigman
Izotov & Thuan Most recent values: Fields & Olive Izotov & Thuan :
Y
:
Y
0
.
238 0
.
002 0
.
005 0
.
244 0
.
002
Deuterium: Deuterium is weakly bound and therefore can be assumed to be only destroyed in astrophysical environments Primordial deuterium can be found either by measuring solar system or ISM value and doing complex chemical evolution calculations
OR
Measuring D at high redshift The ISM value of (
D
/
H
)
ISM
1 .
60 0 .
09 0 0 .
.
05 10 10 5 can be regarded as a firm lower bound on primordial D
1994: First measurements of D in high-redshift absorption systems A very high D/H value was found (
D
/
H
)
High
z
1 .
9 2 .
5 10 4 Carswell et al. 1994 Songaila et al. 1994 However, other measurements found much lower values (
D
/
H
)
High
z
2 .
5 10 5 Burles & Tytler 1996 Burles & Tytler
The discrepancy has been ”resolved” in favour of a low deuterium value of roughly (
D
/
H
)
High
z
3 .
4 0 .
5 10 5 Burles, Nollett & Turner 2001
Li-7: Lithium can be both produced and destroyed in astrophysical environments
Production
is mainly by cosmic ray interactions
Destruction
is in stellar interiors Old, hot halo stars seem to be good probes of the primordial Li abundance because there has been only limited Li destruction
Li-abundance in old halo stars in units of log( 7 Li / H ) 12 Spite plateau Molaro et al. 1995
Li-7 abundances are easy to measure, but the derivation of primordial abundances is completely dominated by systematics
There is consistency between theory and observations All observed abundances fit well with a single value of eta This value is mainly determined by the High-z deuterium measurements The overall best fit is h 5 .
1 0 .
3 10 10 Burles, Nollett & Turner 2001
This value of h into translates
b h
2 0 .
020 0 .
002 And from the HST value for
h h
0 .
72 0 .
08 One finds 0 .
028
b
0 .
054 luminous
m
0 .
3
0 .
01
BOUND ON THE RELATIVISTIC ENERGY DENSITY (NUMBER OF NEUTRINO SPECIES) FROM BBN The weak decoupling temperature depends on the expansion rate
H
8
G
3 2 3
GN
(
T
)
T
4 15 And decoupling occurs when int
G F
2
T
5
H
T D
N
(
T
) 1 / 6
N
(
T
) is can be written as
N
(
T
)
N
(
T
)
e
,
e
,
N
(
T
)
e
,
e
,
N
(
T
) n ,
SM
n extra
v
,
SM
N
(
T
) n ,
SM
( 3
N
n )
Since
n n n p BBN
exp(
m
/
T D
) The helium production is very sensitive to
N
n
N
n = 4
N
n = 3
N
n = 2
The actual bound on the number of neutrino species is dominated completely by systematic uncertainty From present observations it is possible to deduce roughly that
N
n 4 i.e. the presence of a sterile neutrino is not excluded by BBN at present Later we will see how this fits with bounds from structure formation.
FLAVOUR DEPENDENCE OF THE BOUNDS Muon and tau neutrinos influence only the expansion rate.
However, electron neutrinos directly influence the weak interaction rates, i.e. they shift the neutron-proton ratio
n-p
changing reactions n
e
e n
n
e
n e
n
e
p
p
n
e p
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 potential 2) Decay of massive particle into electron neutrinos
f
n exp 1
E
n
T
n 1 ,
f
n exp 1
E
n
T
n 1 , n m n
T
n
e
e n
n
e
n e
n
e
p
p
n
e p
Increasing that
N
n n e decreases
n/p
so can be much higher than 4 Yahil ’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: 0 .
06
e
1 .
1 , m , t 6 .
9
Allowed region n e = 0.2
n e = 0.1
Allowed for n e = 0
If oscillations are taken into account these bounds become much tighter: Oscillations between different flavours become important once the vacuum oscillation term dominates the matter potential at a temperature of
T osc
20 ( 2
m eV
cos 2 0 ) 1 / 6 MeV For n m n t this occurs at
T
~ 10 MeV >>
T
BBN , leading to complete equilibration before decoupling.
For n
e
n m,t the amount of equilibration depends completely on the Solar neutrino mixing parameters.
Dolgov et al., hep-ph/0201287 SH 02 If all flavours equilibrate before BBN the bound on the flavour asymmetry becomes much stronger because a large asymmetry in the muon or tau sector cannot be masked by a small electron neutrino asymmetry Lunardini & Smirnov, hep-ph/0012056 (PRD), Dolgov et al., hep-ph/0201287 Abazajian, Beacom & Bell, astro-ph/0203442, Wong hep-ph/0203180
For the LMA solution the bound becomes equivalent to the electron neutrino bound for all species
e
, m , t 0 .
15 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 species massive decaying particles quintessence ….
The interaction rates themselves neutrino degeneracy changing fine structure constant ….
PERSPECTIVES
Neutrino thermodynamics and decoupling is well understood in the standard model.
Big bang nucleosynthesis provides a powerful probe of neutrino physics beyond the standard model. However, many of the abundance measurements are dominated by systematics which need to be better understood Deuterium measurements provide by far the most sensitive probe of the baryon density and is (now) consistent with CMB. More on this later!