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Introduction to Cosmology AA1053

Available as a University Certificate

and a module for CertHE, DipHE and BSc in Astronomy.

Sample Notes

These sample pages from the Course Notes for the module Introduction to Cosmology have been selected to give an indication of the level and approach of the course. They are not designed to be read as a whole, but are intended to give you a flavour of the syllabus, style, diagrams, images, equations, mathematical content and presentation. They are a subset of the colour, navigable on-line version of the learning materials. All enrolled students will be sent a CDRom with all the Course Notes and learning materials, in addition to having access to them via the course website.

• All sections of notes will be available in modest colour and basic navigation in pdf format suitable for downloading and printing at home. It is anticipated that most students will prefer to use the notes in the colour pdf files on the CDRom.

• B/w paper copies will be sent out on request to anyone who has

notified us that they have special educational needs. July 2008

U N I V E R S I T Y C E R T I F I C A T E I N C O S M O L O G Y 2 0 0 1 / 2 0 0 2

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Cosmology: Introduction Recent observational discoveries have pushed cosmology from the periphery of science into the focus of basic research and public interest. As we develop our observational technology, the resulting data are likely to introduce more questions than will be answered, thus fuelling our appetite for yet more information.

osmology, is the astrophysical study of the history, structure, and dynamics of the universe. The word cosmology is derived from Greek: kosmos meaning world, and logos meaning knowledge or science.

Cosmology, as a science, has undergone massive changes over the past centuries, evolving from an almost purely philosophical approach to a modern science relying on objective astronomical observations. It has evolved to become an extremely complex science, including the study of particles, on scales smaller than an atom, as well as the formation and evolution of stars, galaxies and structures on scales as large as the observable universe. Today, international research teams are exploring the universe in search of new information about its constituents and the physical laws governing their creation and evolution. Many fascinating discoveries in the past few decades have pushed cosmology from the periphery of science into the focus of basic research and public interest. For a professional approach, the complexity of cosmology demands excellent mathematical skills, but it can nevertheless be explored qualitatively, which makes it fascinating for amateur astronomers and members of the public. The best way to investigate the issues of current research in cosmology is to explore the questions the researchers want to answer.

The first question usually asked in cosmology is: when did the Universe start and how will it end? Only in the last century has it been possible for scientists to establish that the universe is expanding. The remaining challenges are to determine

1. how fast the universe is expanding;

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2. how long this expansion will last;

3. if the expansion will ever stop or even reverse.

There are two ways to do this: first, by measuring the average mass density of the universe and finding out whether there is sufficient mass for the gravitational force to stop the current expansion, or second, by observing the expansion velocities of galaxies at great distances, and therefore earlier times, to measure the rate at which the universe is slowing down (or perhaps even accelerating). The second method, beside the conventional distance ladder (the Hubble Space Telescope provided great improvements) includes such promising approaches as observations of supernovae and gravitational lenses. There are major problems, such as contamination by intrinsic velocities, caused by gravitational interactions with other objects. Both methods are being actively pursued, and results are hoped for in the near future. The expansion of the universe, in the present era, can be interpreted as indicating that at one point the universe must have been enormously dense and hot - so hot that it consisted almost entirely of radiation. As the universe expanded, it cooled. This idea is known as the Hot Big Bang model. For decades it remained untested and controversial. Today observational data like the Cosmic Background Radiation (CBR) and the abundance of light elements, such as helium, deuterium and lithium, are in good agreement with the predictions of the Hot Big Bang model.

Another topical and exciting question, at present the subject of extensive research, is what is dark matter and what cosmological role does it play? Details of the statistical properties of the expected structures (galaxies, clusters, superclusters in the universe) are dependent on the amount of mass present in the universe. There is strong observational evidence that apart from the luminous objects (that radiate enough energy to be detected using modern technology) there is a vast amount of invisible matter, the dark matter. Dark matter may account for more than 80% of the total mass in the universe. The nature and exact distribution of the dark matter are still a mystery, despite a range of proposed models. The most obvious candidate for this dark matter is ordinary matter in the form of old, burned-out stars or stars that are too small to shine (Jupiter-like objects). However, the predictions of the abundance of the conventional matter, obtained by the measurement of the amount of light elements, shows that it cannot account for all dark matter. Something other than ordinary matter must be present - exotic particles. Exotic particles would be a relic of some process in the very early universe. It is very important to identify this exotic dark matter, by direct search and by accelerator experiments, with particle theory providing guidance to develop the experiments. At present the best accepted candidates for the exotic dark matter are weakly interacting particles (WIMPs), axions, and neutrinos with finite mass. Of these, only neutrinos are known to exist.

The third crucial question is how did the large-scale structure of matter form and how is it evolving? Maps of the distribution of galaxies in the current universe reveal a network of thin, filamentary structures between small and large central concentrations of galaxies, separated by quasi-spherical voids. When looking at objects at large distances we are observing the universe’s past (because

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luminosity of the object. Attributes that can fulfil this requirement are colour, spectral characteristics and variability.

Colour If the colours of the stars in the vicinity of the Sun are plotted on a graph against their luminosities, an interesting pattern develops (Figure 2.3). This diagram is called the Hertzsprung-Russell (HR) diagram. The majority of the stars lie in a narrow band, with the brightest stars being blue and the dimmest being red. This band is referred to as the main sequence and is occupied by stars during the hydrogen burning stage of their evolution.

Solar = 1

10

100

1/10

1/100

Sun

10 000Lu

min

osity

ColourBlue Red

Red Giants

InstabilityStrip

White Dwarfs

Main Sequence

See also Figures 5.5 and 5.6 in the recommended text, Foundations of Modern Cosmology

Hot stars are blue/white and generally very bright, while cool stars are red and generally dim. There are exceptions to this trend — red giants and white dwarfs for example, whose luminosities are significantly affected by their extreme sizes. However, these stars are relatively rare and their positions too can be defined on the HR diagram. As main sequence stars all lie within a narrow band there is good correlation between colour and luminosity. Consequently, from the HR diagram, we can use the colour of a main sequence star to find its luminosity. Measurement of the star’s apparent magnitude then allows calculation of the distance to the star (Equation 2.6). For isolated stars this method has poor accuracy. However, if all the stars in a group or cluster of stars, (i.e. all at the same distance) are observed, then comparison of the colours and apparent magnitudes of the group as a whole to the main sequence, permits calculation of the distance to the group. This method, called main sequence fitting, is particularly successful for the

Figure 2.3 The Hertzsprung –Russell diagram. Hydrogen burning stars lie on a line called the Main Sequence. Variable stars (e.g. Cepheid variables) lie within the instability strip.

C O L O U R S

In astronomy, colour is often measured using the UBV system. Here a colour is defined in terms of the ratio of fluxes in two narrow wavelength bands. The expression for the colour [B–V] is

[ ]

−=−

V

B

FF.VB log52

(similar to the expression for apparent magnitudes). where FB is the flux in the B band etc.



measurement of distances to globular clusters, both in our own Galaxy and in nearby galaxies in which individual stars can be resolved. The technique also permits an estimation of the age of the globular cluster, showing these objects to be some of the oldest in the universe.

Spectral Characteristics

Diffraction Grating

Detector

Star Light

Red Light

Blue Light

Relative amounts of lightat each wavelength (colour).

To measure the spectral characteristics of a candidate for a standard candle the light from the star must be passed through a diffraction grating before it is projected onto a detector (Figure 2.4). The effect of the diffraction grating is the same as that of a prism: the light is spread out into its constituent wavelengths or spectrum. It should be noted that the light emitted from stars is not confined to the visual band and useful spectral characteristics are present across the whole wavelength range. Fortunately, the process of producing spectra can be utilised at virtually any wavelength. The optical spectrum of a typical main-sequence star is shown in Figure 2.5. Figure 2.6 shows the spectrum of a white dwarf.

0

500

1000

1500

2000

2500

3000

400 420 440 460 480 500 520 540

Wavelength (nm)

Flux

0

200

400

600

800

365 415 465

Wavelength (nm)

Flux

The features in these stellar spectra make the spectral type, and hence, for instance, the colour, easy to estimate. Other types of objects can also be identified by spectral characteristics. In Figure 2.7 the spectrum of a supernova is shown while in Figure 2.8 the spectrum of a quasar is shown.

Figure 2.4. Diffraction grating has similar effect to a prism, splitting the light into its component wavelengths. Modern detectors can split the light in the visible wavelength range into thousands of individual bands.

Figure 2.5 (Left) Main sequence star spectrum. Note the sharp absorption lines. Figure 2.6. (Right) White Dwarf Spectrum. Note the broad absorption lines.



The geometry in an expanding universe We have established that the universe exists in an expanding spacetime, described by the general theory of relativity. Geometrical properties of the spacetime are described by a metric, the meaning of which will become clear a little later. From simple geometrical considerations, a very general spacetime metric to describe the universe has been derived, in which the cosmological principle is obeyed. The metric is used to describe a small increment of “distance” — usually expressed as ds or s∆ (pronounced a d-s or delta-s) — between two points in the spacetime. The notation ds means an infinitesimal increment of “distance” s and the notation

s∆ means a small but finite increment of s ; for our purposes the distinction will not be important. The mathematical expression for 2)ds( or 2)s( ∆ — written 2ds or

2s∆ for brevity — is the metric of the spacetime.

Although we normally think of coordinates to be in an orthogonal system such as height, depth and width it is more convenient to introduce a system based on spherical polar coordinates. The spherical polar coordinates are r, θ, φ where r is the radial coordinate from the origin and θ, φ are similar to the longitude-latitude system on the Earth (except that θ = 90 – latitude). Figure 3.7 shows a pictorial representation of the coordinates. The spherical polar coordinates r, θ, φ are comoving coordinates (recall that they remain fixed) and the proper distance between comoving points can be found from their coordinates and the scale factor at the epoch of interest.

y

θ

x

z

r

ϕ

P

∆ϕ

∆θ

∆r

Pythagoras showed that 222 yxs ∆+∆=∆ for a

right angled triangle. This expression is actually a metric for 2-D flat space.

Figure 3.8 Definition of polar coordinate system. The r, θ, φ coordinates are shown relative to the origin of of the familiar x, y, z. coordinate system.



The metric can be written using spherical polar coordinates as

( ) ( ) ( )

∆+∆+

−∆−∆=∆ θsinφθ

12222

2

2222 r

rkrtRtcs Equation 3.12

where c is the speed of light and t is the cosmic time.

This equation is the Robertson-Walker metric and only a few very special versions, for different values of k, are important in cosmology. Because of the coordinate system that we have chosen to use, this equation looks far more complicated than it actually is. The metric is used primarily in the calculation of proper distances and angular sizes of remote galaxies. First, consider the terms and variables in the equation. Figure 3.8 shows how the spherical polar coordinates are defined to describe the position of an object P. Small changes in the coordinates can be represented by ∆r, ∆θ, ∆φ. The term c ∆t has units of distance like all the other terms in the equation. k is the curvature parameter and is discussed below.

The Curvature Parameter An interesting variable or parameter introduced in the Robertson-Walker metric is the curvature parameter k. This curvature parameter specifies the curvature of the three-dimensional spatial part of the spacetime. There are three homogeneous, isotropic geometries: flat, spherical and hyperbolic (see Figure 3.9).

k = 0flat geometry

k = –1hyperbolic geometry

k = 1spherical geometry

These corresponds to three values of k:

1. k = 0 flat geometry;

2. k = +1 spherical geometry;

3. k = −1 hyperbolic geometry.

Robertson-Walker metric

In this form of the Robertson-Walker metric r is the comoving angular diameter distance coordinate. A form that looks different but is equivalent can be written in which r is the comoving radial distance coordinate.

Figure 3.9 Space geometries: a) flat k = 0; b) spherical k = 1; c) hyperbolic k = −1.



µ−− ν+ν+→µ ee

The τdecays into either an e

τ−− ν+ν+→τ ee

or into a µ

τµ−− ν+ν+µ→τ .

The decay of the τ to a µ is followed by the decay of the µ to an e as above.

You might want to check that charge and lepton numbers (Le, Lµ, Lτ) are conserved in the above reactions. The decay times of the µ and τ are typically ~ 106 seconds and ~ 1013 seconds respectively.

Neutrinos have for a long time been suspected of oscillating between types in transit. Such neutrino oscillations have apparently been confirmed in June 2001, after many years of inconclusive results. The solar neutrino problem is thus reported to be solved. Two-thirds of electron neutrinos from the Sun have changed to muon or tauon neutrinos by the time of arrival at the Earth.

In the standard model, neutrinos are mass-less, but oscillations require them to have mass. Neutrino oscillations thus require physics beyond the standard model. Furthermore, conservation of lepton flavour number is apparently violated, and the possibility of neutrino decay is raised. The experiments that confirm oscillations are sensitive to differences in mass-squared. We therefore know that neutrinos have mass but do not know their actual masses.

Interactions, mediators and Feynman diagrams The particles that mediate the strong, weak and electromagnetic interactions are bosons (i.e. integer spin). The particles that mediate the gravitational interaction are expected to be bosons too (gravitons).

The particles that mediate the four interactions are shown in Table 4.5. As mentioned previously, a quantum field theory of gravity has not yet been established and gravity is not considered to be a part of the standard model.

Interaction: Strong Weak Electro-magnetic Gravitational

mediator 8 gluons W+, W, Z0 photon graviton

charge 0 +1, 1, 0 0 0

mass 0 80-90 GeV 0 0

range infinite* 1017 m infinite infinite

Neutrino Oscillations

You might want to find out more about neutrino oscillations a very topical subject.

Table 4.5. Mediators of interactions. *See notes on the range of the strong interaction.



The idea that particles are the mediators of interactions will be illustrated first for the electromagnetic interaction. Figure 4.1 is one Feynman diagram for the scattering of two electrons. Time passes from the left of the diagram to the right (some texts will instead show Feynman diagrams with the time arrow pointing from bottom to top). However, the diagram (Figure 4.1) does not depict the motions of the particles in space. Read the diagram like this: two electrons enter from the left; a virtual photon passes between them (they scatter off each other); the electrons leave on the right. The diagram thus shows scattering by the repulsion of the two negative charges, with the interaction being mediated by an exchange of photons the electromagnetic interaction carrier (Figure 4.1). Note again that Feynman diagrams do not represent the trajectories of particles in space: the above diagram may represent repulsion but the apparent divergence of the two electrons on the right is irrelevant. The force does not arise from a dynamical effect of the exchange of photons that is, it is not like the repulsion of two ice skaters throwing a ball between them.

Electromagnetic repulsion

e

Photon

Time

ee

e

The attraction of an electron and a positron could have one of its Feynman diagram looking very similar see Figure 4.2. In this case a photon passes between the electron and the positron but the force is repulsive. The force does not arise as a dynamical effect of the photon exchange or there could never be an attractive force. Instead you should think of the mediators as messengers carrying instructions for the behaviour of the interacting particles.

e

e+

e

e+

Time

Note the convention for antiparticles in this case the positron (e+) the arrow is in the opposite direction to the time axis. In a Feynman diagram the real particles have free ends (the electrons and positrons here) and the virtual particles

Figure 4.1 Feynman Diagram. Electrostatic repulsion between two electrons is shown. We will use the convention that time passes from left to right. The diagram represents the interaction only. i.e. It does not show the motion of the particles in space

Figure 4.2 Feynman daigram for electromagnetic attraction.



Phase Transitions and Symmetry Breaking Our understanding of the universe is that there are four fundamental interactions. The fields associated with each of these interactions possess symmetries. Symmetry exists when some property of the field remains unchanged under certain abstract mathematical operations. If the fields have symmetries that are indistinguishable from one another then the interactions are unified and appear identical. Unification is possible when the universe is extremely hot. In the very first moments of the universe all four interactions are believed to have been unified in a single interaction.

The properties of many substances change with temperature. For example, ice melts to water at 0 °C. As energy is added to the melting ice the temperature remains constant at 0 °C and all the energy is absorbed by the process of changing from ice to water. Within physics this is known as a phase transition. When unification between the fundamental interactions is lost, the physical system undergoes a change that is mathematically very similar to a phase transition and there is an energy exchange at this point as the physical system reorganises itself.

As the universe expands the constituents cool sufficiently to remove the unification of one or more interactions. This process is called symmetry breaking. As symmetry breaking takes place there is a mixture of two interactions within the space. The vacuum in which the mediating particles travel has the unusual property that the energy density is not the lowest possible, and for this reason it is called a false vacuum. Mathematical models of the false vacuum allow us to make general predictions about the conditions in the early universe and how they evolve with time. These predictions can be used to test the accuracy of the theoretical model.

Although the details of the physical ideas are beyond the scope of this course we can describe some of the unusual behaviour that results. The false vacuum has a pressure that is negative, which results in a repulsive gravitational force. During the inflationary era the universe is in the false-vacuum state and expands exponentially at constant energy density, driven by the repulsive gravity. The energy that accumulates during this expansion at constant energy density is released when the universe completes the phase transition from the false vacuum to a lower energy state. The energy release heats the universe to ~ 1027 K. At this point the HBB model takes over. For more details, see Section 8 on inflation.

Evolution of the Universe The major time periods in the history of the universe were briefly described in Figure 1.2. They are:

♦ Planck ♦ GUT ♦ Inflation ♦ Quark ♦ Hadron

!

!



♦ Lepton ♦ Radiation ♦ Matter ♦ Vacuum

The temperature and scale variations in the universe with respect to time are shown in Figures 6.1 and 6.2. The first three arrows indicate the times when the four interactions separate or decouple from each other. This all happens within the first microsecond. All four interactions are as we observe them today well before the universe is one second old.

101510510–510–1510–2510–45 10–35

Cosmic time (seconds)

Today

Tem

pera

ture

(K)

1032

1010

Matter dominates

Gravity decouples

Strong nuclearforce decouples

Weak andelectromagneticforces decouple Recombination

Many of the key observable features in our universe today are a direct consequence of the decoupling of the four interactions. See Figure 6.1. Gravity is the first to decouple from the other interactions and this point marks the start of the theoretical model described in this section. The strong interaction decouples at the end of the inflationary era, and finally at 10–10 s the weak and electromagnetic interactions decouple.

Radiation Dominated Era (10–43 – 1011 s) Planck Era The Planck mass, Planck length and Planck time refer to three “naturally occurring’’ units of mass, length and time. They are “natural’’ because they can all be constructed from different combinations of the following constants in physics: Planck’s constant h (or π= 2hh ) ; the gravitational constant G; and the speed of light c. Their definitions are the following.

Figure 6.1 The temperature of the universe as a function of cosmic time showing the times when the interactions decouple from each other.

Observable features are discussed in Section 5.

& t < 10–43 s

! T > 1032 K



The accelerating universe

Supernovae We may recall from Section 2 that supernovae can be regarded as standard candles. Because they are so bright, they can be detected at very large distances. Consequently they can be used to measure distances on cosmological scales and so help in the determination of the cosmological parameters.

Supernovae are stars that suddenly increase their brightness by many orders of magnitude for a relatively short time. At their brightest, their light output is comparable with that from an entire galaxy. Supernovae are classified into two main types, type I and type II, by their observational characteristics, which correspond to different physical causes.

Type II supernovae occur when a high mass, young star has exhausted the supply of nuclear fuel in the stellar core. At this point the core starts to collapse rapidly. The implosion produces a neutron star, releasing a large amount of energy that causes the outer envelope to be ejected into space. A neutron star is composed almost entirely of tightly packed neutrons and is extremely dense, with a small radius of only 10 to 20 km. Although supernovae of this type are very luminous, they are not as luminous as type Ia supernovae and are therefore detectable only at relatively low redshifts. This is one of the reasons why they are not so important as cosmological distance indicators as the type Ia supernovae.

Type Ia supernovae occur in binary star systems and are caused by the explosion of a white dwarf, throwing large amounts of matter and energy into space. A white dwarf is a star that has reached the end of its life and has a radius that is comparable to that of a small planet similar to the Earth. Many low-mass stars end their life as a white dwarf with a mass of 0.6 to 1 M&. More than half of stars exist in stellar systems with two or more stars orbiting each other. In some cases they will be of the right type and sufficiently close together that their evolution will bring about a type Ia supernova explosion.

Figure 9.2 illustrates a white dwarf in a close binary system. The companion star of the white star is generally thought to be a red giant. Any expansion of the red giant in the normal course of its evolution will cause its outer layers to be stripped off by the strong gravitational field of the white dwarf. This material is attracted towards the surface of the white dwarf, but as the system is rotating, the material cannot fall directly on to the white dwarf but instead spirals in via an accretion disc. Eventually the matter accretes onto the white dwarf, thereby gradually increasing its mass. Now there is a theoretical upper mass limit of 1.44 M&, known as the Chandrasekhar limit, that a white dwarf cannot exceed. This limit is determined by the mass at which the gravitational forces within the white dwarf caused by its own mass exceed the pressure available to support the star. (In a white dwarf this is electron degeneracy pressure, in contrast to the normal gas pressure that maintains the equilibrium against gravity in main sequence stars.) The consequences of this within the white dwarf binary system are catastrophic. As the mass of the white dwarf approaches the Chandrasekhar limit due to the in-falling material from its

An order of magnitude is a factor of ten.

Type II supernovae

Type I supernovae



The Large-Scale Structure of the Universe The universe is highly structured: galaxies, clusters, superclusters, voids, filaments, great walls, great attractors, and large quasar groups. Some of the largest structures are more than 100 Mpc across. Much effort in cosmology today is directed to mapping these large-scale structures and to understanding their origins. The dark matter, a non-zero cosmological constant, and the density parameters ΩT, ΩM, and ΩΛ are particularly important factors in the development of structure. This section discusses the attempts to map and understand large-scale structure (LSS).

he overall properties of the universe appear to be homogeneous on very large scales. However, the universe is evidently highly structured on scales from single galaxies to clusters, superclusters and even larger.

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Figure 10.1 The Coma cluster of galaxies. Credits: AURA/NOAO/NSF.

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