Transcript Ingen diastitel - Aarhus Universitet

Thermodynamics in the early universe
In equilibrium, distribution functions have the form
f EQ
1

, E
exp((E   ) / T  1)
p 2  m2
When m ~ T particles disappear because of Boltzmann
supression
f  f MB  e
( m ) / T  p 2 / 2mT
e
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  e  ( E   ) / T
In equilibrium,  ( X )    ( X )
(true if processes like XX   occur rapidly)
This means that entropy is maximised when
 ( X )   ( 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
nB
10

~ 10
n
n
Leptogenesis (more to come in the last lecture)
Conditions for lepto (baryo) genesis (Sakharov conditions)
L-violation:
Processes that can break
lepton number (e.g. X  ll )
CP-violation: Asymmetry between particles and antiparticles
e.g. X  ll has other rate than X  l l
Non-equilibrium thermodynamics:
In equilibrium nL  nL always applies.
X, X
L
L
CP  violation
Non-equilibrium
L  violation
Different possibilities:
Electroweak baryogenesis
does not require new physics, does not work!
SUSY electroweak baryogenesis
requires new physics, does not work in the MSSM
GUT baryogenesis
requires plausible new physics, it works
possible problems with reheating
Baryogenesis via leptogenesis
requires ”plausible” new physics, it works
Thermal evolution after the end of inflation
2

a
8G
2
H  2 
a
3
B 
2
gT 4 for bosons
30
72
F 
gT 4 for fermions
8 30
Total energy density
TOT
7
2 4 2
 ( g B   g F ) T 
N (T )T 4
8
30
30
Temperature evolution of N(T)
In a radiation dominated universe the time-temperature
relation is then of the form
1/ 2
1 1  3 

t  H  
2
 32G 
 t s  2.4N (T )
1 / 2
2
TMeV
The number and energy density for a given species, X, is
given by the Boltzmann equation
f X
f X
 pH
 Ce [ f X ]  Ci [ f X ]
t
p
Ce[f]: Elastic collisions, conserves particle number but
energy exchange possible (e.g. X  i  X  i )
[scattering equilibrium]
Ci[f]: Inelastic collisions, changes particle number
(e.g. X  X  i  i )
[chemical equilibrium]
Usually, Ce[f] >> Ci[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
3
d pX
int   Ci  f X 
 nX v
3
(2 )
In general, a species decouples from chemical
equlibrium when
int  H
H  2 N (T )
1/ 2
2
T
mPl
The prime example is the decoupling of light neutrinos (m < TD)
weak  n v  T 3GF2T 2  TD  1MeV
After neutrino decoupling electron-positron annihilation
takes place (at T~me/3)
Entropy is conserved because of equilibrium in the
e+- e--  plasma and therefore
si  s f
1/ 3
T
7 3
11

f 
 ( 2 4 )Ti 2T f3 
 
8
Ti  4 
The neutrino temperature is unchanged by this because
they are decoupled and therefore
T  (4 / 11) T  0.71T (after annihilati on)
1/ 3
BIG BANG NUCLEOSYNTHESIS
The baryon number left after baryogenesis is usually
expressed in terms of the parameter h
nB
h
n
t t 0
According to observations h ~ 10-10 and therefore the
parameter
h10  1010 h
is often used
From h the present baryon density can be found as
b h2  0.0037h10
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
nn
 exp(m / T ) , m  1.293MeV
np
However, this ratio is dependent on weak interaction
equilibrium
n-p changing reactions
e  n  e  p

e  n e  p

n  e  p  e

Interaction rate (the generic weak interaction rate)
n p  n v  T G T  Tfreeze  1 MeV
3
2
F
2
After that, neutrons decay freely with a lifetime of
 n  886 0.8 s
However, before complete decay neutrons are bound in
nuclei.
Nucleosynthesis should intuitively start when
T ~ Eb (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)  destruction ( D)
production  nB v
destruction

Eb

 TBBN  
 0.2 MeV
 Eb / T 
ln(
h
)
 n v e


Since t(TBBN) ~ 50 s << 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( 4He,  ) 7Be
T ( 4He,  ) 7Li
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
4nHe
2nn
YP 

nN
nn  n p
nn
np
TBBN
 0.25 for nn / n p ~ 1 / 7
TBBN
exp(t BBN /  n )
 exp(m / Tweak )
~ 1/ 7
2  exp(t BBN /  n )
Yp depends on h because TBBN changes with h
TBBN  
EB , D
ln(h )
D, He-3:
These elements are processed to
produce He-4. For higher h, TBBN
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 : Y  0.238  0.002  0.005
Izotov & Thuan : Y  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
5
( D / H ) ISM  1.60  0.0900..05

10
10
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
Li  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 translates
into
b h2  0.020 0.002
And from the HST value
for h
h  0.72  0.08
One finds
0.028 b  0.054
luminous  0.01
m  0.3
BOUND ON THE RELATIVISTIC ENERGY DENSITY
(NUMBER OF NEUTRINO SPECIES) FROM BBN
The weak decoupling temperature depends on the expansion
rate
8G
2 3GN (T )T 4
H
3

15
And decoupling occurs when
int  GF2T 5  H  TD  N (T )1/ 6
N(T) is can be written as
N (T )  N (T ) e  ,e  ,  N (T ) , SM
  extra
 v , SM
 N (T ) e  ,e  ,  N (T ) , SM (3  N )
Since
nn
np
 exp(m / TD )
BBN
The helium production is very sensitive to N
N = 4
N = 3
N = 2
The bound using most recent deuterium measurements
(Abazajian 2002)
N  3.8 (2 )
Sterile neutrinos:
If there are A-s oscillations in the early universe
at T > Tdec, then N>0
However, N<1 always
(corresponding to 1 extra
neutrino flavour)
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
e  n  e  p

e  n e  p

n  e  p  e

n-p changing reactions
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 
1
E

exp      1
T

e  n  e  p

e  n e  p

n  e  p  e

, f 
1
E

exp      1
T

,  

T
Increasing e decreases n/p so
that N 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 ,
  ,  6.9
Allowed region
e= 0.2
e= 0.1
Allowed for
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
2
Tosc  20(meV
cos 2 0 )1/ 6 MeV
For  this occurs at T ~ 10 MeV >> TBBN, leading to
complete equilibration before decoupling.
For e, 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 ,  ,  0.15
Dolgov et al. ’02
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
….
Example:
A changing fine-structure constant a
Many theories with extra dimensions predict 4D-coupling
constants which are functions of the extra-dimensional
space volume.
Webb et al.:
Report evidence for a change in a at the
a/a ~ 10-5 level in quasars at z ~ 3
BBN constraint:
BBN is useful because a changing a would change
all EM interaction rates and change nuclear
abundance
Bergström et al:
a/a < 0.05 at z ~ 1012