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Lecture-04Big-Bang Nucleosysthesis
http://power.itp.ac.cn/~hep/cosmology.htm
Ping HeITP.CAS.CN
2006.03.04
2
• H, He, Li, … Light-elements are produced by big-bang nucleosysthesis (BBN);
• Heavy metals (<Fe) are created in stars;• Super-heavy metals (>Fe) are generated in SNs.
Basic Ideas of Nucleosynthesis
3
A 12
A: mass number
Z: charge number (p)
A - Z: neutron number (n)
Z: C A=12, Z=6
4.0 Preliminaries
In nuclear physics
For pre-exponential factors:
n p
n p
N n p A A
m 939.566MeV, m 938.272MeV
Q=m m 1.293MeV
m m m m /
4
3/ 2
(Eq-4.1)exp( ), 2A A A
A A
m T mn g
T
4.1 Nuclear Statistical Equilibrium (NSE)
When thermal equilibrium, for nuclear species A, the number density is
NSE >H
5
Moreover, chemical equilibrium
(Eq-4.2)
Zp+(A-Z)n A+
( ) , A p nZ A Z
Eq-3.1 also applies to n, p, hence we have
A p n
3A/2
Z A-Z -Ap n p n
N
(Eq-4.3)
exp( /T)=exp[(Z +(A-Z) )/T]
2=n n 2 exp[(Zm +(A-Z)m )/T]
m T
6
(Eq-4.4)( ) , A p n AB Zm A Z m m
Definition of binding energy of the nuclear species A(Z)
Substituting Eq-3.3 into 3.1, the abundance of A is:
3( 1) / 2
3/ 2 -(Eq-4.5)
22 exp( / ),
A
A Z A ZA A p n A
N
n g A n n B Tm T
A 2 3 3 4 12
A
A
Z H H He He C
B (MeV) 2.22 6.92 7.72 28.3 92.2
g 3 2 2 1 1
Table-1
7
Define total nucleon density:
N n p A ii
AA i
iN
(Eq-4.6)
n =n +n + (An ) ,
AnX , species A mass fraction, X 1,
n
So Eq-3.5 becomes: 3(A-1)/2
A-1 (1-A)/2 (3A-5)/2 5/2A A
N
1 Z A-Z Ap n (Eq-4.7)
TX =g ς(3) π 2 A
m
B η X X exp( ),
TA
8 2 3NB γ 2
γ
Eq-4.8n ς(3)
η 2.68 10 (Ω h ), n gT , n π
Baryon-to-photon ratio
So in NSE, the mass fraction of species A, A AX X (η, T)
丰度:质量百分比
nB=nN
8
4.2 Initial Conditions (T>>1MeV, t<<1sec)
Key points: neutron-to-proton ratio
The balance of neutron and proton is maintained by the weak interactions:
-e
-e
+e (Eq-4.9)
n p+e +ν
ν +n p+e
e +n p+ν ,
If H, Chemical equilibrium
n v p e (Eq-4.10)μ +μ =μ +μ ,
9
So, we have:
n p n p (Eq-4.11)n / p n / n = X / X = exp [ - / T+( - ) / T] , eQ
n pwhere Q m m 1.293MeV
Based upon charge neutrality, we have:
10/ ( / ) ( / ) 10e e pT n n n n
Similarly:
/ 1T
10
nexp( / ), (Eq-4. 1 ) 2
pEQ
Q T
The equilibrium n/p ratio:
T → high
n/p → 1
11
Rates for interactions between neutrons and protons, for example
2 5 4
3 3 3
(Eq-4.13)
( )[1 ( )] (2 ) ( )
, 2 2 2
pe n e e pe n
e n
e n
f E f E M p e n
d p d p d p
E E E
2 2 2(Eq-4.14)(1 3 ), =1.26, F A AM G g g
In terms of neutron lifetime n2
-1 2 503
2 2 1/ 20 1
(Eq-4.15
where
)
(Eq-4.16)
(1 3 ) 2
( ) ( 1) 1.636,
Fn n pe A e
q
Gg m
d q
12
885.7 15minn s
Lifetime of neutron
0( ) n
t
n t n e
Since
1/ 20
1/ 2
1( ) / exp( )
2
( ) ln(2) 10.23minn
n
n t n
n
So half-life of neutron:
In fact:
1/ 2 ( ) 10.5 0.2minn
13
So, we have:2 2 1/ 2
10 (Eq-4.17)
( ) ( 1)( ) ,
[1 exp( )][1 exp(( ) )]
/ , / , / , /
pe n n q
e e e e e
qd
z q z
q Q m E m z m T and z m T
where
In high- and low-Temperature limits:
1 3
2 2 5 2 5760
(Eq-4.18)( / ) exp( / ) ,
(1 3 ) ,
n e epe n
A F F e
T m Q T T Q m
g G T G T T Q m
14
By comparing to the expansion rate,1/ 2 2 2*1.66 / 5.5 /pl plH g T m T m , we have:
3(Eq-4.19)Γ/H ~ (T/0.8MeV) ,
Thus when T>0.8MeV, n/p -> equilibrium value, from (Eq-3.12), T->high, n/p ->1
At T>1MeV, rates of nuclear reactions for building up the light elements are also high -->NSE
15
Consider the following light elements: n, p, D-2, He-3, He-4, C-12, in NSE, the mass fractions are:
n p
3/22 N 2 n p
3 2 23 N 3 n p
(Eq-4.20)
(Eq-4.21)
X /X = exp(-Q / T) ,
X = 16.3(T/m ) ηexp(B /T)X X ,
X = 57.4(T/m ) η exp(B /T)X X , 9/2 3 2 2
4 N 4 n p
5 33/2 11 6 612 N 12 n p
n p 2 3 4 12
(Eq-4.22)
(Eq-4.23)
(Eq-4.24)
X = 113(T/m ) η exp(B /T)X X ,
X = 3.22 10 (T/m ) η exp(B /T)X X ,
1 = X X X X X X ,
(Eq-4.25)
16
From Eq-3.7, when AX 1
1(Eq-4.26)
/( 1),
ln( ) 1.5ln( / )A
NUCN
B AT
m T
X Tnuc
(MeV)
D-2 0.07
He-3 0.11
He-4 0.28
C-12 0.25
Table-2
17
4.3 Production of the Light Elements: 1-2-3
The weak rates are much larger than the expansion rate H, so (n/p)=(n/p)eq~1, and light elements are also in NSE.
n p MeV
3/ 2 122 N MeV
3 2 233 N MeV
9/ 2 3 344 N MeV
MeVX , X = 0.5 T T/
X = 4.1(T/m ) η exp(2.22/T ) 6 10
X = 7.2(T/m ) η exp(7.72/T ) 2 10
X = 7.1(T/m ) η exp(28.3/T ) 2 10
33/ 2 11 126
12 N MeV (Eq-4.27) X = 79(T/m ) η exp(92.2 / T ) 2 10 ,
From Eq-3.20 to Eq-3.25
4.3.1 step 1 ( t= sec, T=10MeV)210
18
4.3.2 step 2 ( t= 1sec, T=TF=1MeV)
F (Eq-4.28)n 1
exp( Q/T ) , p 6
freeze out
The weak interactions that interconvert n and pfreeze out ( )H
• Not really constant due to residual weak interactions.• The deviation of n/p from its equilibrium value becomes signifi
cant by the time nucleosynthesis begins. (See Fig.4.1)• At this time, the light nuclei are still in NSE.
n p
12 232 3
28 1084 12 (Eq-4.29)
X 1/ 7, X 6 / 7
X 10 , X 10
X 10 , X 10 ,
19
4.3.3 step 3 ( t= 1 to 3 minutes, T=0.3 to 0.1 MeV)
Major nuclear reactions:2
2 3
2 2 3
2 2 3
3 3
3 2 3
3 2 4(Eq-4.30)
,
,
,
,
,
n p D
D p He
D D He n
D D T p
He n T p
T D He n
He D He p
/ 1/ 6 1/ 7n p due to occasional weak interactions
/ 1/ 74,
at T=0.3MeV
EQn p
20
4( ) AA He n v is very low, due to
a). low abundances for D-2, He-3, and H-3, their NSE values:
12 19 1910 , 2 10 , 5 10
The light-element bottleneck
Deuterium bottleneck: NSE
2(Eq-4.31), n p D
9 10Since 10 ~ 10 , that is, there are 109-1010 photons
around one nucleon.
So when T=0.1MeV, t=3min, not enough high-energy photons(E>2.2MeV) to disassociate D-2.
21
4 He
P (H)
42 1n He
4 / 2nn n
44
(Eq-4.32)
4( / 2) 2( / )41/ 4 0.25
1 ( / )
1 0.25 0.75,
n NUC
N n p NUC
H
n n pnX
n n n n p
X
( / ) 1/ 7, 0.1 MeVNUCn p T
b). Coulomb-barrier suppression:
1/3 2/3 -1/31 2 MeVexp[ 2 ( ) ] ,v A Z Z T 1 2 1 2/( )A A A A A
v : thermally-averaged cross section times relative velocity.
If abundances of D-2, He-3, H-3 1 at TNUC=0.1MeV
Bottleneck is broken
22
Li-7: An abundance of the order 10 910 10 , is predicted by:
4 3 7 -10
4 3 7 -10
7 7(Eq-4.33)
, for 3 10
, for 3 10
,
He H Li
He He Be
Be n Li p
• H/p and He-4 are in dominative amounts;• Nuclei of A=5 and 8 are unstable, and with high
Coulomb-barrier suppression, BBN is stopped at He-4, so that no heavier elements produced.
Substantial amounts of both D-2 and He-3 are left:5 4
2,3 10 10X 42,3(2 / 3 ) ( )He X n v
4 2 34 2,3( ) ( , )X He X D He So:
23
So, T should not be too high, i.e., T<0.1MeV, t=3min
otherwise, photon disassociation
However, T shouldnot be too low, i.e., T>0.02MeV, t~1hr
otherwise, kinetic energy not highenough to penetrateCoulomb potential.
24
4.4 Primordial Abundances: Predictions
What affect primordial nucleosynthesis?
1/ 2 ( )n g
1/ 2(1) ( )n2 2 5
1/ 2(1 3 ) / ( )weak F AG g T n
1/ 2 ( ) 10.5 0.2minn
1/3 41/ 2 1/ 2( ) ( ( ) )weak F
F
nn T n He
p
nexp( / )
p F
freeze out
Q T
25
(2) g
*1/ 2 2H g T* *1/ 6 4
Fg H T g He
(3) 4
4
2 32,3
( )
( , )
X He
X D He
An accurate analytic fit for primordial mass fraction of He-4
10
*
1/ 2 (Eq-4.34)
0.230 0.025log( /10 )
0.0075( 10.75)
0.014( ( ) 10.6)
PY
g
n
Primordial He-4abundance
Li-7 production process-I
Li-7 production process-II
26
4.5 Primordial Abundances: Observations
Primordial nucleosynthesis: 3min1hrAge of the universe: 13.8 billion years
The difficulty of measurement: contaminants from astrophysical processes,such as stellar production and destruction.
Specifically:
4.5.1 measurement of D
a) via the UV absorption studies of the local interstellar medium (ISM) in the solar system.
Atmosphere of Jupiter : (DCO, DHO)5(1 ~ 4) 10D H 5( ) (2 1) 10preD H
Consistent with
Hard task
27
b) high-z QSO absorption line
410 ( 0.03)obsD H z 54 10 ( 3.09)obsD H z
Since deuteron is weakly-bound easy to be destroyed
Primordial NSE value of D/H < 10^(-13), only when “the deuterium bottleneck” is broken , deuteroncan be accumulated in great amount.
( )H
In a star, more dense, so in NSE D/H < 10^(-13)
See Fig-4.4, constrain : 5 9/ 1 10 10D H
8 2 22.68 10 0.037 0.20B Bh h
28
4.5.2 measurement of He-3
hotter interiors: He-3 is destroyed cooler outer layers: He-3 is preserved low mass star : new He-3 from hydrogen burning
Notice that in a star, the processes for He-3 more complicated:
a) measure of oldest meteorites:
3 35(1.4 0.4) 10
p
He He
H H
b) measure of solar wind:
35
(Eq-4.35)(3.6 0.6) 10D He
H
Also provides constraint to
29
4.5.3 measurement of He-4
He-4 can also besynthesis in stars
Hence, low Z low Y Primordialabundance
30
Predicted He-4 abundance
(Eq 4.36)
0.227 for 2
0.242 for 3
0.254 for 4P
N
Y N
N
Present observations suggestthat:
0.25pY
31
4.5.4 measurement of Li-7
[ ] 12 log( )
2.2 0.1
LiLi
H
Lithium abundances versus metallicity (from acompilation of stellar observations by V.V. Smith.)
10(1.6 0.4) 10Li
H
32
Problem?
33
4.6 Primordial Nucleosynthesis as a Probe
10 10
2
11 10
(Eq-4.37)
4(3) 10 7(10) 10
0.015(0.011) 0.026(0.037)
0.015(0.011) 0.16(0.21)
6(4) 10 7 1(1.4) 10 ,
B
B
B
h
n s
a) non-baryonic form of matter
From the concordance of D, He-3, He-4, Li-7 abundances, we derive
From dynamical determinations
0 0.2 0.1
Dark matter
34
* (Eq 4.38)4 or g ( ~ MeV) 12.5N T
*
4 4
new bosons new fermions
(Eq 4.39)
10.75 (std model; 3)
7( ) ( )
8i i i i
g N
g T T g T T
b) Number of light neutrino flavors
4 4
new bosons new fermions
(Eq 4.40)
1.75 1.75( 3)
7( ) ( )
8i i i i
N
g T T g T T
0.25pY present observation
4N or cold components
35
4.7 Final Words
• Primordial nucleosynthesis: agreement between theory and observation indicating the standard cosmology is valid back to 10-2sec, or T=10MeV;
• Works as a probe for cosmology (B), and particle physics (v), etc;
• More precise observations for D, He-3, He-4, Li-7 are of great importance.
36
References
• E.W. Kolb & M.S. Turner, The Early Universe, Addison-Wesley Publishing Company, 1993
• L. Bergstrom & A. Goobar, Cosmology and Particle Astrophysics, Springer, 2004
• M.S. Longair, Galaxy Formation, Springer, 1998
• 俞允强,热大爆炸宇宙学,北京大学出版社, 2001
• 范祖辉, Course Notes on Physical Cosmology, See this site.