Wednesday, 26 March 2014

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.

1. J. Hawley and K. Holcomb. Foundations of Modern Cosmology. Oxford University Press, USA, 1997. ISBN 9780195104974. URL
2. kdh1/cos/cos17.pdf
3 Weinberg

Monday, 24 March 2014

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
1. K. A. Olive. Primordial big bang nucleosynthesis. ArXiv Astrophysics e-prints, Jan. 1999. URL
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.

Sunday, 23 March 2014

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.

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

Press, USA, 1997. ISBN 9780195104974. URL
2. A. Phillips. The Physics of Stars. Manchester Physics Series. John Wiley &
Sons, 1994. ISBN 9780471941552. URL
3. K. A. Olive. Primordial big bang nucleosynthesis. ArXiv Astrophysics e-prints, Jan.
1999. URL

Tuesday, 18 March 2014

The Thermal History of the universe - 1

The Thermal History of the universe
As T = (1 + z)T0 the universe gets hotter when we go further back in time. The average energy of the particles increases with the higher temperature. There is a real chance of more interactions can happen. Therefore, at early times all possible particles were relativistic. If they were interacting strongly then the particles would have remained in thermal equilibrium. In the following figure the thermal history of the

universe is shown up to seconds.
(courtesy Imperial Collage - Cosmology Lectures 2012)

As mentioned earlier time equal to zero is accepted as the big bang. However, the earliest we can start talking where classical general relativity gains control of the known universe as a whole is cosmic time of

known as the Planck time. Hawley and Holcomb (1997) state that, the characteristic length-scale of the universe known as the Plank length at this time, was
There is nothing at the moment we can say about the time, from the beginning until the Plank time
which is called Planck epoch. It is accepted that all four fundamental forces of nature, namely gravitational, electromagnetic, weak nuclear interactions and strong nuclear interactions composed a single force during this time. We think at the end of this Planck epoch, the gravitons fell out from equilibrium with the other particles, and gravity decoupled from the other forces. It is believed that the cosmic background
of gravitational waves formed from the gravitons streamed out through the universe.

Hawley and Holcomb (1997) further explain that, the temperature was so high from the Planck time till about unified epoch. During this stage electromagnetism, the weak interaction, and the strong
and we have very little understanding of the nature of the matter under such conditions. This interval is called the interaction were uni ed, and they made up a single indistinguishable force. The theories support that notion are called grand uni ed theories (GUTs). There are other incomplete theories exist that apply to conditions during the uni ed epoch as well.

Cosmologists believe sometime before the end of the uni ed epoch, universe entered a period of exponential expansion called in ation. If that occurred, it must have taken place sometime around
 after the big bang according to Hawley and Holcomb (1997). As noted by Hawley and Holcomb (1997) the most signi cant remnant of the uni ed epoch is the excess matter remaining after the epoch end. However, it is assumed that the universe consisted of a brew of highly relativistic particles, including quarks and more exotic particles.

Hadrons and Leptons are elementary fermions. We can categorise the type of particles of the universe like this:
  • Leptons - fundamental particles which participate in weak interactions, electrons, muons, taons and neutrinos are leptons
  • Hadrons - made up of quarks which participate in strong interactions.
Hadrons have two subfamilies
  •  Baryons: Fermions made up of three quarks, for example Proton and neutron.
  • Mesons: Bosons made up of two quarks. Examples are pions and kaons.
Hawley and Holcomb (1997) noted further that particles created from pure energy in ordinary processes must always be created in matter anti-matter pairs known as pair production. When the particle and anti-particle collide, they destroy one another, converting their rest mass to photon energy. The conservation of baryon number rule should be observed here like baryons anti particle should be negative baryons. However in most grand uni ed theories this conservation rule no longer holds. Reactions like transforming quarks into leptons and vice versa can occur, thus violating the baryon conservation. These particular reactions can occur in such a way that tiny excess of matter can remain. This process by which matter was
preferred over antimatter is called baryogenesis which would have created the material that our current universe consist of.

We think the end of the unified epoch came atand this was followed may be by quark epoch where universe consisted of free quarks and gluons, other carrier particles of combined electromagnetic and weak force, more exotic heavy particles and anti-particles. The weak and electromagnetic forces were unified as the electroweak interaction during most of the quark epoch.

The four fundamental interactions:

1. Electromagnetic interaction: these are responsible for the forces between electrons and protons in atoms, and for the emission and absorption of electromagnetic radiation, such as light. A small leftover electromagnetic interaction of the electrons and protons in atoms allows atoms to bind together to make molecules and so is responsible for chemistry.

2 Strong interactions : these provide the (very) strong force between quarks inside protons and neutrons. A small residual strong interaction between quarks binds protons and neutrons together in the nuclei of atoms.

3 Weak interactions : these are responsible for processes, such as radioactive beta-decay, that involve both quarks and leptons.

4 Gravitational interactions : these make apples fall, maintain planets in their orbits around stars, and control the expansion of the Universe. However, they are negligible within the atom. But when matter aggregates into huge (and electrically neutral) lumps, such as planets and stars, gravity holds sway. You will see that it also holds some surprises, undreamt of by Newton.

1. J. Hawley and K. Holcomb. Foundations of Modern Cosmology. Oxford University
Press, USA, 1997. ISBN 9780195104974. URL
2. How the universe works - Andrew Norton Page 62

Friedmann Equations and Hubble Parameter

Friedmann Equations:
Inserting both the stress energy tensor and the FLRW Metric and taking the  fluid to be comoving with the expansion of the universe, we arrive at equations of motion for scale factor a, known as the Friedmann
equation as shown in Olive (1999),
where a -dot is the first derivative of scale function a with respect to cosmological proper time t and is the density. The equation of acceleration a (Double dot)
The third equation is called the conservation equation and can also be derived from the stress energy tensor.
If pressure and density are non-negative which means ( ro + 3p) > 0 and the cosmological constant is negligible then the a is always negative. This implies is that _a(dot) must be either negative or positive and the universe must be either expanding or contracting. Assuming it is expanding the minus sign of the right hand side of second equation above implies that the expansion of the universe is slowing down with
time. However research papers published by Perlmutter et al. (1999) for the supernova Cosmology Project, Brian Schmidt from High-Z Supernova research Team and Riess (2000) cast doubt on this claim. They have received the Nobel prize for proving that the expanding universe accelerating and a small cosmological constant is needed to explain this. To discuss the evolution of the scale factor further, aided by above equations if we go back in time, _a must increase and so the curve a(t) must concave downwards towards the time (t) axis. As a result, there should be some nite time in the past, scale factor must have been equal to zero. The event when scale factor a towards 0 and consequently when the time is defined to be zero is
a singularity at which the density of the universe become in nite. This is often referred to as the big bang.

Hubble Parameter
In 1914 Vesto Slipher found that light from most galaxies is redshifted. By taking this work further Edwin Hubble measured the recession of speed of distant galaxies from their observed redshift and established their distance from earth. Then Hubble derived a law that said, the recession velocity, v, of a galaxy is directly proportional to its distance, d, from the earth. Therefore
where the constant of proportionality, H0 is known as Hubble's constant with the dimensions of [Time]-1 (per). For the present time H0 = 72 +/-  8 kms per second per Mpc (Mega Parsec). However there is an greater uncertainty in the value of H0 and it is conventional to parameterise it's value in terms of the parameter h de ned as follows;
Current observations indicate a range of values for h, 0:6 < h < 0:75. As we can see, then the rate of the expansion of the universe is determined by the Hubble parameter as
What the Hubble parameter has given us is the observational proof of the expansion of the universe. The observations of the universe continue to constrain the expansion rate and the universe is expanding with time as general relativity originally predicted.

1. K. A. Olive. Primordial big bang nucleosynthesis. ArXiv Astrophysics e-prints, Jan. 1999. URL
2. S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro,S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, and Supernova Cosmology Project. Measurements of Omega and Lambda from 42 High-Redshift Supernovae. ApJ, 517:565{586, June 1999. doi: 10.1086/307221.
3. A. G. Riess. The Case for an Accelerating Universe from Supernovae. PASP, 112: 1284{1299, Oct. 2000. doi: 10.1086/316624.

Saturday, 15 March 2014

Einstein Equations and FLRW Metric

Any modern treatment of cosmology, the study of the structure and evolution of the universe on the largest scales, begins with a number of assumptions. The first is that physical laws, which have been tested in our local environment, are universal, hold- ing at all times and in all places in the universe. Importantly this includes the theory of General Relativity, which is our best theory of gravity at the present time. The second major assumption is that we will need to assume some symmetries for the universe. These symmetries are motivated both by simplicity and by observations of the cosmos. The assumed symmetries are that the universe is is homogeneous and isotropic on the largest scales, which states that we do not occupy a special place in the universe. In Einstein gravity, spacetime is described by four-dimensional metric, gµν (χk ), that satisfies the Einstein equations.
where Rµν the Ricci tensor, R is the Ricci scalar, Tµν is the stress-energy tensor for all the fields present and Λ is the cosmological constant. FLRW Metric The current understanding we have of the evolution of the universe is based upon the Friedmannn-Leimatre-Robertson-Walker metric which has the symmetries of homogeneity and isotropy, motivated by observations. It’s validity supported by direct and indirect observational evidence extends back to the beginning of the epoch of primordial nucleosynthesis. The high degree of symmetry of the FLRW metric is considered as the cornerstone of the standard cosmological model. It depends on only one dynamical variable cosmic scale factor a(t) (in some literature this is symbolised as R(t)). The FLRW metric is given explicitly as
where spherical polar coordinates r, θ, and φ are comoving coordinates, t is the proper time, a(t) is the scale factor, k is the curvature parameter which can take values of +1, 0 and -1 for spaces of constant positive curvature, flat or negative spatial curvature, respectively. The scale factor a(t) has dimensions of length. If k = +1 then r ranges from 0 to 1. In below figure the two dimensional analogue of the three dimensional curvature manifolds of homogeneous and isotropic universe are illustrated.
An observer in a homogeneous and isotropic universe, moving so the universe is observed to be isotropic, would measure the stress-energy tensor to be
as stated in Peebles and Ratra (2003) where the diagonal form is a consequence of the symmetry and the diagonal components define the pressure and energy density. We assume that the fuid is comoving with the expansion of the universe. According to Kolb and Turner (1994), from the energy conservation
law for adiabatic expansion, the change in energy in a comoving volume element v = a3 is equal to minus the pressure times the change in volume i.e,
where p is the pressure and is the density. The parameter w is a constant and lies in the range of 0 <= w <=1, which is called the Zel dovich interval. This equation of state is the most general form for
the stress energy in a FLRW space-time.

1. P. J. Peebles and B. Ratra. The cosmological constant and dark energy. Reviews of Modern Physics, 75:559{606, Apr. 2003. doi: 10.1103/RevModPhys.75.559.
2. P. Coles and F. Lucchin. Cosmology: The Origin and Evolution of Cosmic Structure. John Wiley & Sons, 2002. ISBN 9780471489092. URL
3. E. Kolb and M. Turner. The Early Universe. Frontiers in Physics. Westview Press, 1994. ISBN 9780201626742. URL id=Qwijr-HsvMMC.