BBN - Reactions Explained

Deuterium bottleneck - At high temperatures and densities, according to Hawley and Holcomb (1997) neutrons and protons can fuse directly to form deuterium (also called heavy hydrogen) nuclei, or deuterons.

Deuterium is the isotope of hydrogen, and it contains one proton and one neutron

in its nucleus. The reaction that formed the deuterium is shown here is on of the first reactions

of key fusion reactions . The in this reaction represents a photon.This reaction liberates the binding energy (energy equal to the energy liberated when a nucleus is created from other nucleons or nuclei -Wikipedia)  of the deuterium nucleus in the form of photon. Deuterium then fuses with a proton or another deuterium as in here shown in the second reaction, 

 to form the helium nucleus 3He or as shown in below 3rd reaction, 

fuses with neutron to create a tritium 3H. As shown in following reaction 4 

 both these nuclei (3He and 3H) then react with additional particles, 3He with a neutron or a deuteron, and the tritium with a proton or a deuteron, to form 4He. This is the most common isotope in the universe and almost all helium in the universe was created in this nucleosynthesis epoch, shortly after the big bang.

We can fi nd that the equilibrium Hydrogen density is proportional to exp(B/kT) where B = 13:6eV is the binding energy of the hydrogen atom. The most strongly bound light nucleus is 4He, with binding energy B4 =28:5MeV . So most of the nucleons end up as Helium in equilibrium and that's why we should have Helium abundances now. As the universe cooled down and expanded di fferent nuclear reactions froze out, leaving the relic abundances of the stable nuclei.

Key Fusion Reactions:

Below are the Key fusion reactions of Big Bang nucleosynthesis taken from lecture

from Steven Weinberg:

4He is a very stable nuclei with close to 28MeV binding energy. However, a nuclei with atomic number A = 5 is unstable. Therefore the further fusion is rare with lower binding energies. However this would be overcome and the production of 7Li will proceeds through. These further reactions with Li production are shown here:

The weak interaction rates responsible for n - p equilibrium freeze -out at 

T ~0:8MeV . The neutron to proton ratio at this is about 1/6. However when taking

into account the fact that free neutron decays prior to deuterium formation, this

ratio drops to n/p ~ 1/7. Then the 4He mass fraction is  ~ 0:25.

Reference:

1. J. Hawley and K. Holcomb. Foundations of Modern Cosmology. Oxford University Press, USA, 1997. ISBN 9780195104974. URL http://books.google.co.uk/

books?id=eBFawfP8ak8C.

2.http://star-www.st-and.ac.uk/ kdh1/cos/cos17.pdf

3 http://en.wikipedia.org/wiki/Steven Weinberg

Big Bang Nucleosynthesis (BBN) -Historical background

Historical background: The early universe behaved like a nuclear explosion or like a fusion bomb, creating the temperatures required for the creation of light elements. After that first minute with the temperature close to perhaps a billion kelvins nuclear reactions started. Approximately 180 seconds after the big bang, the temperature of the universe was  according to Hawley and Holcomb (1997). The content of the Universe consisted of a dilute gas of free streaming neutrinos , photons ,
 electron positron pairs
 and trace amount of nucleons(the protons and neutrons) as noted by Boesgaard and Steigman (1985). The temperature and densities were still very high, but dropped sufficiently so that the nuclei of atom could remain stable. The creation of atomic nuclei through nuclear reactions called nucleosinthesis, thought to be commenced at this point. Hence, this period in the big bang is known as the nucleosynthesis epoch. Olive (1999) noted that Big Bang Nucleosunthesis (BBN) is the theory explaining the origins of the light elements D;3 He;4 He and 7Li and their primordial abundances. Ellis (2011) commented that, the theoretical framework for BBN is based on Friedmann-Lemaitre-Robertson-Walker cosmology and a network of nuclear reactions.

BBN requires temperatures greater than 100keV and corresponds to time scales less than 200 seconds. It was necessary to achieve a density n  . The current density of visible matter is

 and we can estimate  the current temperature of the universe as  ~ 10K. 

In the early Universe at temperatures T < or ~ to 1MeV , conditions for the synthesis of the light elements were attained. Weak interactions were in equilibrium at higher temperatures. The following processes fix the ratio of number densities of neutrons to protons. 

(a neutron plus a positive electron (positron)  create  a proton and an anti-nuetrino and vice-versa, a neutron plus a nuetrino create a proton and electron,  from a  nuetron  a proton, electron and antinuerino is fixed).

The ratio of neutrons to protons at equilibrium at temperature T is given by a Boltzman factor: 

where Nn and Np are number densities of neutrons and protons, delta m

 is the neutron proton mass difference,1.3 MeV. Olive (1999) notes that, when the temperature  the ratio of neutron to proton was

.

Reference: 

1. K. A. Olive. Primordial big bang nucleosynthesis. ArXiv Astrophysics e-prints, Jan. 1999. URL http://arxiv.org/abs/astro-ph/9901231.

2. A. M. Boesgaard and G. Steigman. Big bang nucleosynthesis - theories and observations. ARA&A, 23:319{378, 1985. doi: 10.1146/annurev.aa.23.090185.001535.

3. G. F. R. Ellis. Inhomogeneity eff ects in cosmology. Classical and Quantum Gravity, 28(16):164001, Aug. 2011. doi: 10.1088/0264-9381/28/16/164001.

The Thermal History of the universe - 2

Continuing from previous post: about the Thermal history of the universe. When the time is around

and a temperature of

the weak interaction (see previous post)  thought to be decoupled from the electromagnetic force. Now all four forces mentioned earlier were separated. During the transition the carrier particles of the uni fied electroweak force were transformed (hypothetically) into 4 new particles.
Three of them are called bosons

which acquired mass and the other one is massless photon. 

To discuss this further, according to Phillips (1994), it is generally accepted that, within the first nano seconds the universe was filled with a gas of fundamental particles
like leptons, anti-leptons, quarks, anti-quarks, neutrinos, ant-neutrinos, gluons and photons. We assume that quarks, anti-quarks and gluons annihilated and transformed to less massive particles when the temperature fell below . However, the number of quarks very slightly exceeded the number of anti-quarks. The small number of quarks remaining were thought to be responsible for the present number of protons and neutrons of the universe. when the temperature decreased further the heavier leptons and anti-leptons were annihilated as well.

When the cosmic time was

quarks formed neutrons and protons while 

Therefore between a millisecond to a second after the big bang the universe was consisted of electrons, positrons, neutrons, protons, neutrinos, antineutrinos and photons. At about 1s when

neutrinos started to decouple.

Soon after this, all of the positrons and most of the electrons were removed by annihilation of electron-positron pairs. This seems to have occurred when cosmic time was approximately 4 seconds and

. Phillips (1994) further states that, when 

t~3min and 

neutrons combined with protons to form light nuclei - Helium and other light particles, which lead to a universe with approximately 75% of its mass consisting of hydrogen and 25% of helium. 

After around 300,000 years later

and the temperature was around 4000k, it was a low enough temperature for the formation of stable atoms, and photons to decouple. Hydrogen and helium nuclei combined with electrons and

formed neutral hydrogen and helium atoms which lead to photons stopping to interact strongly with matter. The universe became transparent to electro-magnetic radiation which cooled down to about 3k at present time because of the expansion of the universe. This is the so-called cosmic microwave background detected by Penzias and Wilson. Olive (1999) claimed that, the connection between the BBN and the CMB is a key test to the Standard Big Bang Model.

About Penzias and Wilson: The accidental discovery of cosmic microwave background radiation is a major development in modern physical cosmology. Although predicted by earlier theories, it was first found accidentally by Arno Penzias and Robert Woodrow Wilson as they experimented with the Holmdel Horn Antenna. The discovery was evidence for an expanding universe, (big bang theory) and was evidence against the steady state model. In 1978, Penzias and Wilson were awarded the Nobel Prize for Physics for their joint discovery. http://en.wikipedia.org/wiki/Discovery_of_cosmic_microwave_background_radiation

Reference: 

1. J. Hawley and K. Holcomb. Foundations of Modern Cosmology. Oxford University

Press, USA, 1997. ISBN 9780195104974. URL http://books.google.co.uk/

books?id=eBFawfP8ak8C.

2. A. Phillips. The Physics of Stars. Manchester Physics Series. John Wiley &

Sons, 1994. ISBN 9780471941552. URL http://books.google.co.uk/books?

id=4SZpQgAACAAJ.

3. K. A. Olive. Primordial big bang nucleosynthesis. ArXiv Astrophysics e-prints, Jan.

1999. URL http://arxiv.org/abs/astro-ph/9901231.