The following notes are intended for our students following A-Level Modular Physics, Cosmology, UCLES. They are intended as notes to clarify certain aspects of the course. They are not intended to be a replacement for their own notes,or to take the place of standard textbooks.

HTML is not a nice medium for using Greek symbols, so I have substituted some of the standard symbols with appropriate English letters.

3. The Standard Model of the Universe

3.1 Line spectrum

When gases are heated they absorb energy. Electrons move to higher energy levels (greater potential energy) Electromagnetic radiation is emitted when an electron returns to a lower energy level. The energy emitted is equal to the difference in energy levels between the two states. This is the photon energy given by E = hf , and the corresponding wavelength is given by w = c/f .

When light from a heated gas is analysed using a spectrometer the different wavelengths are clearly visible as a set of distinct lines. This is the emission spectrum of that gas. Each gas is characterised by a unique set distinct lines (wavelengths) . This is a result of the differences in the atomic structure of each element. No single wavelength can identify an element , but the overall pattern clearly identifies the element (or compound) This is of immense importance in studying spectra from distant star.

All stars emit a continuous spectrum of all wavelengths through the electromagnetic spectrum. The relative intensity of these wavelengths is given by the black body distribution. Gases in the surrounding 'atmosphere' absorb amounts of electromagnetic energy which correspond to the differences between energy levels for the electrons . As electrons return to the lower state the wavelengths are emitted in all directions. Therefore very little of the original electromagnetic energy at these wavelengths actually reaches the observer. The continuous spectrum of a star appears to be missing these wavelengths . This pattern of missing wavelengths in a continuous spectrum is called an absorption spectrum. For a given gas the position of these absorption lines coincides exactly with the wavelengths of the emission spectrum for that gas. Therefore , the study of the spectra from distant stars can be used to identify gases surrounding the star, by looking for characteristic 'fingerprint' pattern of gases and elements

The absorption spectrum of a distant galaxy can be used to determine the velocity v at which the galaxy is moving relative to our galaxy. This is because each distinct pattern of wavelengths will have been 'red shifted'...a consequence of the Doppler effect.The degree of red shift is directly proportional to the recessional velocity.

3.2 The Doppler Effect

The change of wavelength due to the relative motion between the source and observer is known as the Doppler Effect.
We will make the assumption that v is much smaller than c .
Let w be the wavelength when there is no relative velocity between source and observer, The change of wavelength is directly proportional to the relative velocity v between the source and the observer.
The change of wavelength, = (v x w)/c
and the fractional change of wavelength = v/c
As v becomes larger this approximation is no longer valid, due to the greater influence of relativity.

3.3 Hubble's Law


One of the most important discoveries occured in 1929, when Edwin Hubble discovered that the Universe is expanding. He found that the red-shifts of a set of local galaxies were directly proportional to the distance of these galaxies from our galaxy. This has became known as Hubble's Law, and is given by

v = Hd

where H is known as Hubble's constant.
The value of H is measured in km/s per Megaparsec.
A value of 50 km/s per Mpc would mean that a galaxy 1Mpc away would be receding at 50km/s. A galaxy 10 Mpc away would be receding at 500 km/s.

This equation does not take account of the effect of gravity on the outward expansion. The presence of gravity will oppose the outward expansion,so the velocity of expansion reduces with time. The value of H is the value of H as measured today. It is sometimes called the Hubble Parameter...since its value is not constant with time. ( A graph of distance between two galaxies against time is not a straight line)

The value of H will always give a calculated value of the age of the Universe (1/H) which is higher than the actual value. An accurate determination of H depends on the accuracy to which we can determine distances to distant galaxies.Unfortunately, the greater the distance of a galaxy the greater the uncertainty in measurements of that distance.Most estimates give values between 40 and 80 km/s per Megapasec.


The Fate of the Universe



The gravitational attraction between masses slows down the outward expansion of the Universe. If we knew the mass of the Universe with a high degree of accuracy , we would be able to predict confidently just what the future of the Universe will be. Unfortunately much of the mass of the Universe appears missing...or at least it hasn't been found yet. This missing mass, which would enable the density of the Universe to be calculated , is difficult to observe, unlike stars which emit their own light. Yet an accurate determination is absolutely essential if we are to predict with some accuracy the fate of the Universe.


3.5 The Cosmological Principle


According to the Cosmological Principle there are no preferred places in the Universe. Measurements of the Universe made from Earth, disregarding local irregularities, can be considered to be identical to those made in any other part of the Universe. A classic illustration of this is in the way in which the expansion of the Universe, according to Hubble's Law, occurs.

Take a length of elastic.Mark dots at even intervals to indicate galaxies. Gradually pull the elastic apart.At any time the distances from a galaxy will vary according with Hubble's Law, irrespective of which galaxy (dot) is chosen as the reference point. A similar pattern can be observed by using a series of dots on a balloon which is gradually inflated.

Therefore, irrespective of where we are in the Universe, we should observe the same expansion properties , and the same laws of physics in operation. Earth has therefore no special significance, other than to humans, on a cosmic scale !



The Universe before t=0.01s : Some Problems to Solve !


The early Universe,according to the Big Bang, involved extremely high temperatures. So high are these temperatures, that it is extremely difficult to reach them in scientific experiments. Fundamental investigations into particles involve accelerating particles at very high speeds, which would normally be associated with particles at very high temperatures. It is possible to reach speeds in particle accelerators which correspond to 10^15 K , which would occur about 10^-15 s after the Big Bang. For times earlier than this, we need to rely on theoretical models,used to explain our present observations ,and 'extrapolate' these backwards to predict the mechanisms which occur in the very early stages of the Big-Bang. This is an incredible difficult task, because a number of complex interactions which take place . Protons and neutrons are thought to consist of smaller particles, called quarks At temperatures above 10^7 these quarks cannot be held together as protons and neutrons. At higher temperature the four fundamental forces become almost indistinguishable from one another . Different 'messenger' particles are believed to be responsible for these four different forces. Above 10^5 K photons, massless particles, are being converted into particles,and their anti-particles.The reverse process is also happening . The theories are therefore very different to theories involving everday experiences of forces,and the synthesis of the four fundamental forces makes them extraordinarily complex theories. These are known as Grand Unified Theories (GUTs). These attempt to explain the physical behaviour of particles and forces during the very early stages of the Universe, in a single set of equations.The more recent theories which try to reach the goal of TOE (Theory of Everything) are called String Theories. These have gained some success in providing a desription of gravity, but the theory itself appears to some scientists as a mathematical instrument or 'toy', with very little foundation based on physics (as most of us know it !)

At time t=0 a singularity is predicted, where all laws as we know them break down.Space and time are infinitely distorted. The uncertainty principle of quantum mechanics prevents any accurate predictions to be made for times less than 10^-45 s.

Microwave Background Radiation

In 1965 , Arno Penzias and Robert Wilson discovered , quite accidently, the existence of a low level microwave radiation. This was found to be virtually of the same intensity no matter which way they looked with their radio telescope. This was one of the greatest discoveries for the support of the Big Bang Theory. After billions of years the great fireball would have cooled to give a black-body peak temperature of approximately 2.7 K. Furthermore, this temperature would be associated with a peak wavelength of 1.07mm . Analysis of the radiation supported this prediction. Until recently, the uniformity of the background radiation was of some concern to many cosmologists.The apparent uniformity of the background radiation was not consistent with the requirements for the formation of galaxies. Small variations in the density are required in order that galaxies can be formed. This is due to the effect of gravitational attraction between particles of gases in these higher density regions. Therefore, this should be reflected in slight variations in the background radiation. The measurement of such variations was performed by Cosmic Background Explorer (COBE) in 1992.These measurements amounted to variations of just 30 millionths of a degree Kelvin.