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.

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. 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.

Reference:

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

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.

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.

 Reference:
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 http://books.google. co.uk/books?id=uUFVb-DHtCwC.
3. E. Kolb and M. Turner. The Early Universe. Frontiers in Physics. Westview Press, 1994. ISBN 9780201626742. URL http://books.google.co.uk/books id=Qwijr-HsvMMC.

The Hot Big Bang Model

The hot big bang model is currently the best explanation we have for the evolution of the universe, and it has become a key part of the standard model of cosmology. In this model we make an assumption that, the universe is homogeneous and isotropic on the largest scales.

We also make an assumption that the laws of physics, which have been verified in laboratory conditions are also valid in the early universe. Finally, we assume that the cosmological principle holds. With the assumption of homogeneity and isotropy the evolution of the universe is governed by the Friedmann equations obtained from General relativity.

 From these equations of motion, and our knowledge of the content of the universe today, a picture emerges in which a universe began in a hot dense state, and expanded and cooled into the one we see around us today. There are many observable relics from this hot dense origin for example the radiation we observe as the Cosmic Microwave Background (CMB). The best evidence we have for the isotropy of the observable universe is the uniformity of the temperature of the Cosmic Microwave Background Radiation (CMBR). Today the CMBR study reveals that we have microwave radiation photons with 2.75 K temperature throughout the universe.

Distributions of galaxies give us also direct evidence of homogeneity. As the universe cooled different physics was in operation and different particles were present. These different types of particles were baryons, electrons, photons, neutrinos, fermions and bosons and antiparticles. Baryon is comprised of three quarks and is not a fundamental particle. Baryons participate in strong interactions. The Electron is regarded as a fundamental particle. Historically neutrinos were thought to be massless particles and travel close to the speed of light. General acceptance is there are three types of neutrinos and they all interact weakly with other particles. Neutrinos denoted by symbol ν. However some recent experimental evidence indicates that neutrinos may have a mass.

The elementary particles are divided into two main groups depending on the amount of spin that they carry and they are referred to as Bosons and Fermions. All neutrinos and Baryons are fermions while the photon which has twice the spin of the electron is a boson. Standard model of the Big Bang Nucleosynthesis (SBBN) describes the approximately first twenty or so minutes of the evolution of the universe. It is the phase of evolution in which protons and neutrons, which previously had existed as separate particles combined to form atomic nuclei. Only once the energy scale has dropped sufficiently is this energetically favourable.

The abundance of light elements is an- other relic of the hot big bang, and armed with this theoretical knowledge astrophysicists and astronomers accumulated the abundances of the light elements thought to be produced just after the big bang. These elements are Helium 4 [4H e], Helium 3 [3He], Deuterium, Lithium 7 [7Li], Beryllium and Boron. With the predicted abundances at hand astrophysicists and astronomers aimed their telescopes to far ends of the universe and observed the abundances of those light elements and begin to compare the observed to predicted. The standard Big Bang Nucleosynthesis model achieved significant success in predicting the light-element abundances produced during the nucleosynthesis that agrees well with the observations. It also helps us constrain the parameters of the standard model of cosmology, in particular the number of baryons with respect to photons.

 New discovery announced today: http://www.bbc.co.uk/news/science-environment-26605974