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Lecture-04Big-Bang Nucleosysthesis

http://power.itp.ac.cn/~hep/cosmology.htm

Ping HeITP.CAS.CN

2006.03.04

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• 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

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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

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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

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(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

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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

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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 ,

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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

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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

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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

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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

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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

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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)

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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

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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

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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 ,

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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

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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.

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4 He

P (H)

42 1n He

4 / 2nn n

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(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

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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:

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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.

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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

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(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

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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

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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

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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:

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(Eq-4.35)(3.6 0.6) 10D He

H

Also provides constraint to

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4.5.3 measurement of He-4

He-4 can also besynthesis in stars

Hence, low Z low Y Primordialabundance

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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

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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

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Problem?

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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

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* (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

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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.

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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.