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Outline

1 Introduction

Nucleosynthesis

Why Primordial?Helium Production in Stars

2 Thermal History of the Universe

Time Scale vs. Temperature

Thermal History of Early Universe

Interactions3 Helium Synthesis

Neutron-proton abundance ratio

Nucleii Abundances

4 BBN CodesReaction Rates

Simulation Parameters

5 Results

6 Conclusion

Rohin Kumar Big Bang Nucleosynthesis Codes
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Big Bang Nucleosynthesis

Introduction

Nucleosynthesis is the process of formation of nucleus.

Big Bang Nucleosynthesis is one of the successes of thestandard model of Cosmology.

The original work done by Gamow & Ralph Alpher

predicted the right amount of nuclide abundances.

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Introduction


Big Bang Nucleosynthesis is one of the successes of the

standard model of Cosmology.




Bi B N l h i
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Introduction







Bi B N l th i
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Introduction







O tli
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Outline

1 Introduction

Nucleosynthesis

Why Primordial?Helium Production in Stars






Nucleii Abundances

4 BBN CodesReaction Rates


5 Results

6 Conclusion


N l th i
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Nucleosynthesis

For any nuclear reaction to take place a particle of charge

has to penetrate the electrostatic repulsion. For example, if

nuclei of charge Z1 and Z2 have to come closer to a

distance r we need to overcome the coloumb barrier.

V =Z1Z2e

2

r

=1.44Z1Z2

r(fm)MeV

The average thermal energy of particles in the

Maxwell-Boltzmann distribution is kT = 8.62 108T keV

We need the temperatures of the order of 106 or higher to

cross this barrier for the nuclear reaction to take place.

Gamow later this idea was extended to develop the theory

for Big Bang Nucleosynthesis.


Nucleosynthesis

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Nucleosynthesis





V =Z1Z2e

2

r

=1.44Z1Z2

r(fm)MeV








Nucleosynthesis
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Nucleosynthesis





V =Z1Z2e

2

r

=1.44Z1Z2

r(fm)

MeV








Nucleosynthesis
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Nucleosynthesis





V =Z1Z2e

2

r

=1.44Z1Z2

r(fm)

MeV








Nucleosynthesis
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Nucleosynthesis





V =Z1Z2e

2

r

=1.44Z1Z2

r(fm)

MeV








Outline

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Outline

1 Introduction

Nucleosynthesis

Why Primordial?

Helium Production in Stars






Nucleii Abundances

4

BBN CodesReaction Rates


5 Results

6 Conclusion


Stellar vs Primordial
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Stellar vs. Primordial

There is a remarkable difference between the nucleosynthesisprocess that takes place in stars and the one that happens in

the early universe right after the big bang.

Primordial StellarTime Scale mins B.y

Type Adiabatic cooling Isothermic

Density 105 100g/cm3

Baryon-to-Photon ratio () 1010 1


Outline
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Outline

1 Introduction

Nucleosynthesis

Why Primordial?







Nucleii Abundances

4

BBN CodesReaction Rates


5 Results

6 Conclusion


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It is experimentally observed that the universe is mostly

dominated by two elements namely Hydrogen( 75%) andHelium( 25%).

This implies there is one neutron for every 7 protons (i.e.

np 1

7).

We can roughly calculate the order of Heproduced in stars

(for e.g sun) and it accounts for only 5%4

He is primordial!!!


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np 1

7).



He is primordial!!!



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np 1

7).



He is primordial!!!



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np 1

7).



He is primordial!!!


Basic Idea!
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Basic Idea!

To form Deuterium n+ p D+

Neutron Half-life (1/2(n)) 10min

Neutron capture in BBN should happen within 10 minutes

So nuclei from the hot big bang must have freezed out!


Basic Idea!

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Basic Idea!






Basic Idea!
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Basic Idea!






Outline
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1 Introduction

Nucleosynthesis

Why Primordial?







Nucleii Abundances

4 BBN Codes

Reaction Rates


5 Results

6 Conclusion


Temperature vs. Time Scale
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p

We know from standard big bang model that

a4

=

4a

a= 4

8G

3

1/2

t = 3

32G1/2

t =

c2

48GaT4

1/2

t = 1.09secs.

T1010K

2


Temperature vs. Time Scale contd...
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p

Temperature Time

1011K 0.01secs

1010K 1.07secs

109K 3 mins108K 5.3 hrs

4 103K 105 yrs


Outline
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1 Introduction

Nucleosynthesis

Why Primordial?







Nucleii Abundances

4 BBN Codes

Reaction Rates


5 Results

6 Conclusion


Background
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g

Assume that the early universe is hot & consider particles

in thermal equilibrium at that temperature.

particles outnumbered anti-particles causing

matter-anti-matter asymmetry at the beginning

Hence, only the particle distributions are taken into

account in calculating nuclear reactions.


Background
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g








Background
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T > 1012K: The soup consists of ,leptons,mesons,n,p,n,pthis era is difficult to study because of the strong

interactions of the quark gluon plasma. It should be noted

that the inflation time scale is of the order of 1033s and is

before the process of baryogenesis.T 1012K: constitution is ,+,,e,, s and smallcontamination of n, p & Nn Np

T < 1012K: +, annihilation happens. All s dissipateat T 1011K;

, s decouple from leptons.

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T > 1012K: The soup consists of ,leptons,mesons,n,p,n,pthis era is difficult to study because of the strong

interactions of the quark gluon plasma. It should be noted

that the inflation time scale is of the order of 1033s and is

before the process of baryogenesis.T 1012K: constitution is ,+,,e,, s and smallcontamination of n, p & Nn Np

T < 1012K: +, annihilation happens. All s dissipateat T 1011K; , s decouple from leptons.


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Below 1011K mass difference of n, p more protons thanneutrons. 5 109K (t 4 sec) e+,e annihilate and heat

up photons.n

p

1

5.

At around 109K ns & ps combine together to give

nucleii.

At 4000K electrons captured by nucleii. The photons

there after freely stream and are observed in the

microwave frequencies, forming the isotropic Cosmic

Microwave Background.


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up photons.n

p

1

5.


nucleii.






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up photons.n

p

1

5.


nucleii.






Outline
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1 Introduction

Nucleosynthesis

Why Primordial?







Nucleii Abundances

4 BBN Codes

Reaction Rates


5 Results

6 Conclusion


Basics of interactions
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Consider the equilibrium condition at high temperatures. If at

some temperature there are particles in thermal equilibrium.

no. density of ith species of particles with momentum between

q & q+ dq is

number density

ni(q) =giq

2dq

h34

1

exp

Ei(q)ikT

1

+ sign for fermions & for bosons. Ei(q) = (m2

i + q2

)1/2

i =chemical potential, it is additive & is conserved in all reactions


Outline
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1 Introduction

Nucleosynthesis

Why Primordial?

Helium Production in Stars2 Thermal History of the Universe





Nucleii Abundances

4 BBN Codes

Reaction Rates


5 Results

6 Conclusion



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Nucleons weakly interact by the following reactions:

n+ e p+ e

n+ e+

p+ e

n p+ e + e

n+ p+ e


10

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For T > 1010K

(p n)

(n p)= expQ

kT

with Q = mnmpFor equilibrium, the principles of detailed balance implies

(n p) neutron density = (p n) proton density

nn

np=

(p n)

(n p)= exp

Q

KT

Xn = nnnn + np

=

1 + eQ/kT1


Outline

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1 Introduction

Nucleosynthesis

Why Primordial?






Nucleii Abundances

4 BBN Codes

Reaction RatesSimulation Parameters

5 Results

6 Conclusion


Nucleii Abundances-Theory
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ni =

ni(q)dq =4gi

h3 q2dq

exp

Ei(q)ikT

1

For every non-relativistic nuclei, (very good approximation), the

1 is ignorable

Ei(q) mi + q2

2mi

ni =4gi

h3exp

imi

kT

0

q2dqexp

q2

2mikT

= gi

2mikT

h2

3/2

exp

imi

kT


Theory
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Let there be a nucleus i of Zi ps & (Ai Zi) ns in equilibrium

i = Zip + (Ai Zi)n

Xi =niAi

nN; Xn =

nn

nN; Xp =

np

nNWhere nN= total no. density of nucleons (bound or free)

nN = nN0

a0a

3=

N0mN

a0a

3


The expressions for np,nn,ni are2 kT

3/2

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np = 2

2mpkT

h2

3/2exp

pmp

kT

nn = 2

2mnkTh2

3/2exp

nmn

kT

ni = gA

2mAkT

h2

3/2exp

imi

kT

From

exp(i/kT) = exp[(Zimp + (Ai Zi)mn)/kT]

nZpnAZn

2

mNkT

3A/22Aexp[(Zimp + (Ai Zi)mn)/kT]

but Bi = Zimp + (Ai Zi)mnmA

nAi = gAA3/22A

2

mNkT

3(A1)/2nZip n

AiZn exp(BA/T)


For closure density c =3H2

8Gfor H = h 100km/sec/Mpc &
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B = Bc = 2.68 108Bh

2

nB = BcmN

and n = 2(3)3

kTc

3

Xi =nAinN

= gAA3/22A

2

mNkT

3(A1)/2X

Zip X

AiZin (nN)

Ai1exp[BA/T]

Next use expression for n to get

Xi = gifA5/2

kT

mn

3(A1)/2A1XZp X

AZn e

Bi/kT

f = (3)A1(1A)/22(3A5)/2

where =nNn

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BBN Codes

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All the reactions shown in the previous figure there are

these coupled differential equations to be solved.

The whole network of reactions is solved by two-step

Runge-Kutta method with evolving time/temperature

parameter.

The network has to be simulated in a computer numerically

to find out the nuclei abundances.

The first and the most popular of the nucleosynthesis

codes is the Kawano code(NUC123) written in Fortran 77

based on the paper by Wagoner.

It is extremely user-friendly with menu driven interface.There are others based on the same algorithm such as

AlterBBN (written in C)


BBN Codes

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parameter.









BBN Codes

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parameter.









BBN Codes
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parameter.









BBN Codes
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parameter.









BBN Codes

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parameter.









Outline1 Introduction

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Introduction

Nucleosynthesis

Why Primordial?




Interactions

3 Helium Synthesis


Nucleii Abundances

4 BBN Codes


5 Results

6 Conclusion


Reaction Rates
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The total rate of change of abundance of nucleus i is given bythe following first-order differential equation

1

Ai

dXidt

=

j

Xj

Ajk(j)

jk

Xj

Aj

XkAk

[jk]jk l

Xj

Aj

XkAk

XlAl

[jkl]

Here k is the reaction rate of the kth species. Similarly [jk] isthe rate of reaction between jth and kth species calculated from

the thermally averaged cross-section.

Both Resonant and Non-Resonant reaction rates are taken into

account.


Outline1 Introduction
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Nucleosynthesis

Why Primordial?




Interactions

3 Helium Synthesis


Nucleii Abundances

4 BBN Codes


5 Results

6 Conclusion


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As NUC123 is menu-driven application, one can set the

parameters in the code by choosing relevent options in the

menu.

Computational Parameters

initial time step, time-step limiting constant, initial and final

temperatures, non-zero initial abundances of the nucleii

abundance below which the nucleii can be ignoredModel Parameters

gravitational constant, neutron lifetime, no. of neutrino

species, final Baryon-to- photon ratio, cosmological

constant and neutrino degeneracies

One can also vary the model parameters linearly and do

multiple runs.



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menu.









multiple runs.


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menu.









multiple runs.


Nuclide Abundances vs. Time
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Figure: Nucleii Abundances vs. Time (Log)


Mass Fractions vs. Time

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Figure: Helium, Hydrogen & Neutron mass fractions vs. Time (Log)

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Thank You

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bbn_du

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