The Cosmological Background Radiation 0521574374, 9780521574372

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THE COSMOLOGICAL BACKGROUND RADIATION

This is an introduction to the physics, astrophysics and cosmology of the cosmic microwave background radiation. The standard big bang model of

the universe is adopted at the outset. The topics then covered include the origin of the background, then intrinsic fluctuations, followed by the

universe and background radiation after recombination. Finally, measurement of the radiation and its anisotropies is presented, together with a review of the current status of results and experiments. The authors assume that the reader has a basic understanding of the central concepts of general relativity, but they avoid rigorous mathematical proofs and manipulations,

preferring instead to concentrate on the information needed by hands-on cosmologists and astrophysicists. In this first edition in the English language, the authors have completely revised the text of the French

edition. The level is ideally suited to final-year undergraduates in physics or astronomy. Marc Lachieze-Rey is research director at the Centre d’études de Saclay, France and Edgard Gunzig is at L’Université Libre de Bruxelles, Belgium.

THE COSMOLOGICAL BACKGROUND RADIATION MARC LACHIEZE-REY and EDGARD GUNZIG

..:-r CAMBRIDGE : a; UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 lRP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011—421 1, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia

© Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times 10/13pt, in 3B2 [KT] A catalogue recordfor this book is available from the British Library Library of Congress cataloging in publication data Lachiéze-Rey, Marc. The cosmological background radiation/ Marc Lachiéze-Rey and Edgard Gunzig.

p.

cm.

Includes bibliographical references.

ISBN 0 521 05215 7.—ISBN 0 521 05215 7 (pbk.) l. Cosmic background radiation. (Edgard) II. Title.

QB991.C64L33 1999 523.1—dc21 98-39471

2. Astrophysics.

CIP

ISBN 0 521 05215 7 hardback ISBN 0 521 05215 7 paperback

I. Gunzig, E.,

Contents

Preface Acknowledgement

1

2

page xi xiii

Introduction

1

1.1

The discovery of the background radiation

1

1.2

The origin of the background radiation

4

The standard big bang models 2.1 The space—time metric 2.2 The spatial properties of space—time 2.2.1 Co-ordinates, proper distance and expansion 2.2.2 2.2.3

2.3

2.4

2.5

The proper comoving distance Redshifts

7 8 10 10 12 12

The Friedmann—Lemaitre equations 2.3.1 The equivalence of matter and radiation

13 14

2.3.2 The critical density 2.3.3 The cosmological parameters 2.3.4 Length scales Cosmic chronology 2.4.1 The age of the universe

15 16 17 17 17

2.4.2

Time zero and the Planck time

18

2.4.3

Cosmological events

19

The horizon

19

2.5.1

21

The horizon as a function of cosmic time

2.6

Questions around the big bang 2.6.1 Curvature, antimatter and the cosmological constant 2.6.2 The problem of causality

21 23 27

2.7

Inflation

28

Contents

vi

2.7.1 2.7.2

3

The illusions of inflation Inflation and primordial fluctuations

Galaxy formation 3.1

3.2

Gravitational instability 3.1.1 Fluctuation scales The statistics of the fluctuations 3.2.1 The power spectrum

3.2.2

3.3

3.4

3.5

3.6

3.7

Velocities and the gravitational potential

3.2.3 The correlation function 3.2.4 The power law spectrum The initial fluctuations 3.3.1 Isocurvature and adiabatic fluctuations

3.3.2 Statistics and the initial spectrum The growth of fluctuations 3.4.1

Linear and non-linear evolution

3.4.2

Linear growth and the Jeans criterion

The growth in the fluctuations before recombination 3.5.1 The growth in the primordial power spectrum 3.5.2 The damping of fluctuations After recombination

3.6.1

The growth of fluctuations after recombination

3.6.2 3.6.3 3.6.4

Linear and non-linear scales Cosmic matter and its distribution The distribution of matter

Scenarios for the birth of galaxies and cosmic structures 3.7.1 Purely baryonic scenarios 3.7.2 3.7.3 3.7.4

The CDM scenario Other scenarios Constraints from the CMBR

The origin of the microwave background 4.1 The early universe 4.2 The interaction between matter and radiation

4.3 4.4

4.2.1 4.2.2

Therrnalisation in the early universe Compton scattering

4.2.3

The black-body distribution and quantum statistics

Stages in coupling Distortions in the early universe 4.4.1 Electron annihilation

3O 31 33 34 35 37 38 40 41 43 45 47 48 48 49 49 51 52 54 55 55 56 57 62 65 66 67 69 71 73 73 74 74 76 81 83 84

Contents

4.5

4.4.2 Nucleosynthesis Recombination 4.5.1 The surface of last scattering 4.5.2 Ionisation and recombination

4.5.3 4.6

The spectrum of the CMBR

The temperature after recombination 4.6.1 The temperature of radiation

4.6.2

The temperature of matter

Intrinsic fluctuations in the CMBR

5.1

The surface of last scattering 5.1.1

The homogeneity of the surface of last scattering

5.1.2

An absolute frame of reference?

5.2

Angular scales at recombination 5.2.1 Late re-ionisation?

5.3

Galaxy formation

5.4

5.3.1 The fluctuations at recombination The Sachs—Wolfe effect 5.4.1 Gravitational waves

5.5

The Doppler shift

5.6 5.7

The adiabatic component The observability of the intrinsic fluctuations

After recombination

6.1

Global distortions of the spectrum 6.1.1 Distortion by Comptonisation

6.2

The re-ionised universe 6.2.1 Total re-ionisation?

6.3

Other types of global distortion

6.4

6.3.1

The disintegration of massive particles

6.3.2

The scattering of radiation by dust

Fluctuations after recombination 6.4.1 The Sunyaev—Zel’dovich effect 6.4.2 The kinetic Sunyaev—Zeld’ovich effect

6.4.3 6.5

The importance of the Sunyaev—Zel’dovich effect

6.4.4 The Vishniac effect Gravitational effects

6.5.1

The early and late integrated Sachs—Wolfe effect

6.5.2 6.5.3

The effect of non-linear structures The effect of a statistical distribution

vii

85 86 87 88 90 91 91 92

95 95 95 96 97 98 98 99 101 103 104 106 106 109 109 110 114 115 117 117 118 119 119 122 123 124 125 127 128 129

Contents

viii

6.5.4 6.6

‘Exotic’ effects on the CMBR

6.6.1 6.6.2 6.7

Effects of gravitational lensing A universe in anisotropic expansion Is the universe rotating?

6.6.3 Gravitational waves Cosmic strings 6.7.1 Cosmic strings and primordial fluctuations

6.7.2

The effect of cosmic strings on the primordial fluctuations

Measuring the CMBR 7.1 Thermal deviations and distortions 7.2 Unwanted contributions

7.3

7.4

7.5

7.6

7.7

7.2.1

Physical phenomena

7.2.2

The various contributions

Telescopes and antennae 7.3.1 7.3.2 7.3.3

Submillimetre telescopes The antennae The Antarctic site

7.3.4

Space observations

Reception, detection and calibration 7.4.1 Absolute and difference measurements 7.4.2 The choice of the frequency of the observation

7.4.3

Sensitivity

7.4.4

Direct and heterodyne detection

7.4.5

Heterodyne detection

7.4.6 Incoherent detection Measurement using molecules 7.5.1 The CN molecule 7.5.2 The CH molecule 7.5.3 Carbon atoms Absolute measurements of the CMBR 7.6.1 A submillimetre excess? 7.6.2 Recent results: FIRAS and the Canadian rocket 7.6.3 Measurements at long wavelengths

7.6.4 The polarisation of the CMBR An overview of several recent missions and projects 7.7.1 7.7.2 7.7.3

129 130 130 131 132 133 134

Ground-based measurements Balloons Satellites

135

137 137 139 139 141 146 146 146 146 148 150 150 152 153 154 154 156 158 159 160 160 161 162 163 164 164 166 166 167 168

Contents

7.8

The COBE satellite 7.8.1

The differential microwave radiometer

170 171

7.8.2 7.8.3

The far-infra-red absolute spectrophotometer The infra-red photometer, DIRBE

173 174

Measurements of the anisotropy 8.1 Angular scales

8.2

8.1.1

Small and large scales

8.1.2

Ways of measuring the anisotropies

The analysis of angular fluctuations 8.2.1 The harmonic decomposition 8.2.2

The autocorrelation function

8.2.3

The autocorrelation function and spherical harmonics

8.3

Measurements on large scales (6 > 1°)

8.4

8.3. 1 Saskatoon 8.3.2 The ACME—HEMT experiment 8.3.3 The Python experiment 8.3.4 The Tenerife group Balloon measurements

8.4.1 8.4.2 8.5

8.6

8.7

8.9

The FIRS, MSAM and TopHat experiments The ULISSE experiment

8.4.3 The MAX mission Satellite measurements 8.5.1 The RELIKT programme and the PROGNOZ satellite 8.5.2 Measurements of the anisotropy by COBE Intermediate angular scales 8.6.1 A three—beam experiment 8.6.2 The observations at Owens Valley Results on small scales (0 < 1’) 8.7.1 Interferometric observations 8.7.2 The IRAM observations

8.7.3 8.8

ix

Anisotropies from SuZIE

The Sunyaev—Zel’dovich effect 8.8.1 Radiometry with a single antenna 8.8.2 Bolometric methods 8.8.3 Interferometric techniques 8.8.4 The kinetic Sunyaev—Zel’dovich effect The dipole and quadrupole

177 178 178 180 183 184 188 189 192 193 194 195 197 200 200 203 204 208 208 210 214 215 215 217 217 219 219 220 220 220 221 223 224

Contents

8.9.1 8.9.2 8.9.3 8.9.4 8.9.5 9

10

Definitions and notation

The Doppler effect an the expected dipole The first attempts to measure the dipole Present-day results for the dipole Cosmic velocities

Conclusions

9.1

The formation of galaxies and the cosmic background glow 9.1.1 Baryonic scenarios 9.1.2 Non-baryonic matter scenarios 9.1.3 Explosive scenarios

9.2

In conclusion

Bibliography 10.1 CMBR web sites 10.2 Recent review articles on the CMBR 10.3 Popular books 10.4 Technical works on the CMBR and related subjects

Index

224 225 227 229 229 233 235 235 236 236 237 239 239 240 240 241 243

Preface

The cosmic microwave background is an essential element of theoretical and observational cosmology and one of the foundation stones of the big bang models. It is, in fact, its discovery, which we describe in the introduction, that brought a large amount of support to these models at the end of the 19605.

In fact they go right back to 1931 when George Lemaitre formulated them. We present the general characteristics of the big bang models in the first

chapter. Since its first detection, the radiation has been amply observed. First of all it was a question of verifying the predictions of the big bang,

which was accomplished with remarkable accuracy. Yet it was also a question of better understanding the fundamental cosmological processes. How, for example, are galaxies and other structures that we observe today

formed? It was realised that the properties of the cosmic microwave background were crucial to understanding this fundamental question. This

is why we devote the second chapter entirely to this question. Having done this, in the third chapter we describe the radiation itself and discuss such

questions as how and when it came into being, as well as its properties. In fact, after its isotropy and thermal character had been established, astro-

physicists started to be interested in possible deviations from this behaviour. This is why in the next chapters we examine the theoretical predictions of such deviations; in chapter 4 the deviations that were produced right at the beginning of the existence of the microwave background and in chapter 5 those which came into being afterwards.

Following this first part, which is devoted to the theory, the second deals with the observations. Chapter 6 discusses the general principles, the

various methods used and the difficulties that have been encountered. We finish here with the absolute measurements of the microwave background. We describe the COBE space experiment which recently made a very

xii

Preface

important contribution to our understanding of the subject. Chapter 7 deals with a very active branch of observation, namely the observation of the fluctuations in the radiation. This provides limits on the various models for

the formation of galaxies described earlier. Here we also describe the main experiments and the results that have already been obtained from them.

Acknowledgement

Edgard Gunzig acknowledges

support by EU grants PSS-159-B,

PSS* 0992 and CIl*-CT97-0004 and by the OLAM, Fondation pour la Recherche Fondamentale, Bruxelles.

1 Introduction

In July 1965 the Astrophysical Journal (vol. 142, p. 419), an American periodical, announced the discovery of a background electromagnetic radiation. This radiation was isotropic and unpolarized, exhibited no seasonal variation and was of cosmic origin. The authors of the article were two physicists from the Bell Telephone Laboratory, Arno Penzias and

Robert Wilson. In the same issue of the journal (vol. 142, p. 414) four astrophysicists from the Institute for Advanced Study at Princeton, Robert Dicke, Jim Peebles, Peter Roll and David Wilkinson, suggested that this newly discovered radiation had in fact been emitted during a phase of the universe when it was very hot and dense. This was precisely what had been predicted by the so-called big bang model proposed by George Gamow, Ralph Alpher and Robert Herman fifteen years previously and following the work of Georges Lemaitre. The discovery of this radiation, now named the ‘cosmic background radiation’ was and still is considered a very powerful argument in favour of

the big bang model. In recognition of the capital importance of their discovery, Penzias and Wilson were awarded the Nobel Prize for Physics in 1978.

1.1 The discovery of the background radiation

In fact the discovery of the cosmological microwave background radiation was partly due to chance. At the end of the 19503 the laboratories of the

Bell Telephone Company had started to work on the problems of satellite communication. NASA was soon to launch the ECHO satellite in 1959. The expected signal from this satellite would be very weak, therefore it was necessary to develop a highly sensitive receiver. Two of the Bell 1

Introduction

2

Company laboratories were well placed to contribute to this programme. At one of these, Murray Hill, work was being carried out on detectors to improve their sensitivity. At the other site at Holmdel in New Jersey, only fifty kilometres from Princeton, people were perfecting a horn antenna. The geometrical form of such an antenna is particularly useful for the detection of weak signals because they are largely able to cut out ‘nuisance’ radiation from behind. This was why such a horn with a 3 m

opening was constructed in 1961 for the detection of the weak signals from ECHO.

A few years earlier the two radio astronomers Arno Penzias and Robert Wilson had joined the laboratory. In fact they had a special interest in this

antenna. It was sufficiently sensitive, once it had been equipped with an appropriate detector, to observe astronomical sources of small angular diameter. In fact, for sources of the right angular diameter for its beam, it

was the most sensitive radio telescope in existence. Because of its compactness and excellent directivity it would be able to measure its gain

accurately and identify all possible sources of nuisance noise. It thus presented the possibility of making absolute measurements rather than just the differential measurements to which radio astronomers are usually limited.

In 1963 the antenna lay unused and the two radio astronomers inherited some of it. After being used for the ECHO satellite it had been adapted for the TELSTAR satellite. To this end a MASER receiver operating at 7.3 cm,

that is at a frequency of 4.08 GHz, was installed, backed up by an amplification stage. The task of the two radio astronomers was to transform the

instrument into a radio telescope and then calibrate it properly. In order to make the most of its capabilities, they intended to use it for observations requiring absolute flux measurements, observe regions of our Galaxy and confirm the spectra of a number of radio sources. In fact they expected above all to show that no radiation came from the halo of our Galaxy at a wavelength of around 7 cm. As a follow up they intended to construct a

receiver working at 21 cm and thus, amongst other things, study the hydrogen present in clusters of galaxies. One of the first tasks of Penzias was to build a liquid helium cryostat to replace the liquid nitrogen cryostat and so ensure the effective cooling of

the detector. From their very first observations the two radio astronomers noticed that they were registering a higher flux than they had predicted. Radio astronomers often express measured fluxes as temperatures. Penzias

and Wilson had recorded ‘temperatures’ that were too high. So it was necessary to work out the contributions from the sky, the antenna itself, the

1.1 The discovery ofthe CMBR

3

wave-guide and the various parts of the apparatus. This excess temperature was not understood, but several explanations were still possible. First of all it was possible that emission from the atmosphere at these wavelengths

might be stronger than had previously been thought. However, the lack of any variation of the signal with direction appeared to rule out this explanation. It was also possible that man—produced interference affected the readings. In order to look at this possibility more closely, Penzias and Wilson undertook to sweep the horizon with the antenna. The observations excluded both this explanation and a possible origin from our Galaxy, the

Milky Way. The only other possibility was that discrete astronomical sources were responsible, but, given the properties of the best known of these, this seemed just as improbable.

All that remained was to check the radiation due to the antenna itself. After a very detailed and accurate calculation they concluded that this could not provide an explanation. In spring a couple of pigeons had nested in the shelter provided by the antenna. Was it possible that this

might have caused some bizarre electromagnetic effect? They got rid of the pigeons and carefully cleaned the antenna, but the problem remained. Faced with these repeated failures, they were almost prepared to give up

all hope of obtaining absolute measurements of the halo of our Galaxy. Nevertheless, one day they happened to speak with an astrophysicist

colleague (B. Burke) who had heard others mention a background radiation. Jim Peebles, an astrophysicist at Princeton, had in fact performed calculations that implied such predictions from big bang models. Burke advised them to contact the Princeton group under Dicke. A meeting was organised. The Princeton group confirmed, somewhat disappointedly, that they had arrived, albeit a bit late, at this conclusion and understood the nature of the discovery. Two articles were sent jointly to the Astrophysical Journal. In fact Penzias and Wilson initially were not really interested in cosmology. The Princeton group, on the other hand, had expected such observations to be made. Robert Dicke, its director, and Jim Peebles had carried

out calculations showing that, because of its expansion, the universe should be filled with such radiation]. Two of their colleagues, Peter Roll and David Wilkinson, had begun to design a radiometer capable of measuring it. Already for several months the group had been dedicating itself to building what was later to be called ‘Dicke’s Radiometer’ in order to measure the

1 In fact these calculations had been performed in the framework of an oscillating universe, in which cycles of collapse, rebounding and renewed expansion took place.

4

Introduction

cosmological (microwave) background radiation. (Henceforth we shall use

the abbreviation CMBR for the cosmological microwave background radiation.) Although the Princeton group was not aware of it at that time, the existence of this radiation had been predicted long before. Alpher and Herman, the collaborators of George Gamow, had in 1949 predicted its

existence at a temperature of a few degrees Kelvin. The USSR astrophysi-

cists Doroshkevich and Novikov had also independently predicted its existence in 1964. After the confirmation of the discovery the Princeton group began an

observational study of this background radiation. In 1965 Roll and Wilkinson carried out measurements at wavelength xi. = 3 cm. This measurement gave some idea of the spectrum of the radiation. Its thermal nature which had been predicted by the big bang model seemed to confirm the model. The temperature was estimated to be 3.0 i 0.5 K. Subsequently other measurements confirmed its thermal nature and by the middle of 1966 the spectrum from 2.6 mm to 21 cm had been established.

Moreover the radiation had been detected much earlier, around 1940, by two American astronomers, Adams and Dunham, at the Mount Wilson

Observatory, although they had not recognised it as such. They had discovered weak interstellar emission lines which were later identified with CH, CH4r and CN molecules. The radiation was produced by excited molecules and the temperature had been estimated by 1941 to be around 2.3 K (/1 = 2.64 mm; see section 7.5). Shortly after the 1965 observations several authors realised that these molecules were in fact excited by photons from the CMBR. These results not only confirmed the existence of the radiation but also provided a measurement at another wavelength. They

thus further confirmed that the radiation’s spectrum followed a black-body curve.

1.2 The origin of the background radiation

There are two very important characteristics of this radiation filling the entire universe: on the one hand its perfect isotropy (it has the same properties, most importantly its intensity, in every direction in the sky); and on the other hand its distribution in terms of wavelength, or in other words its spectrum, obeys extremely accurately what physicists call the black-

body law. As far as we know at present, only thermal processes, that is processes produced by a system in thermal equilibrium, are capable of

producing such radiation. On the other hand the isotropy strongly indicates that the processes involve the universe as a whole. The only way to

I .1 The discovery of the CMBR

5

understand the origin of such a phenomenon is to suppose that the entire universe went through a phase in which matter and electromagnetic

radiation were in thermal equilibrium. This is precisely what Gamow and his collaborators had predicted around 1940. The Princeton group had

made a similar prediction just before 1965. For both it was of prime importance to explain the relative abundances of the chemical elements in the universe as a whole within the framework of the newly formulated big bang model. By big bang we shall mean a scenario in which the universe passed through an extremely hot and dense primordial phase. This does not necessarily imply an initial singularityz, ‘birth’ or ‘creation’ of the uni-

verse3. The originality of these models stems from the idea that the primordial universe was sufficiently dense and hot for almost its entire contents to be

in thermodynamic equilibrium (in a sufficiently distant past). In this case the laws of thermodynamics or, more precisely, the laws of quantum statistics allow one to calculate the characteristics of the various popula— tions of particles and quanta present. Thus electromagnetic radiation behaved as black-body radiation since at this time the universe itself behaved as a black body. It is a long time since the universe was in thermal equilibrium. One of the main occurrences marking the end of this coupling between matter and radiation goes by the name recombination. It took place about 15 thousand million years ago, about half a million years after the beginning of our

phase of expansion, which is somewhat incorrectly called the ‘birth of the universe’. (The exact times depend on the particular cosmological model adopted.)

Before recombination matter was ionised and the electrons were free and very numerous. The photon density was very high. Frequent collisions between photons and electrons ensured complete equilibrium of matter and radiation. Because of this the universe was opaque and any information carried by a photon was rapidly lost during the continual scatterings with the free electrons. As a result, this optical and radio astronomy can reveal nothing whatsoever about this period. The CMBR dates from the epoch of

recombination, that is from the time when the universe became transparent.

2 The possible avoidance of an initial singularity was extensively discussed in ‘Self—consistent cosmology, the inflationary universe, and all that . . .’ (Gunzig E. & Nardone P., Fundamentals ofCosmic Physics, 1987, vol. 11, pp. 311—443). 3 For the possible avoidance of an initial singularity see, for example, Cosmology, A First Course, by Marc Lachieze—Rey, Cambridge University Press.

6

Introduction

Because of this fact it provides us with the earliest information that we can hope to receive about the universe, at least in the form of electromagnetic radiation. It can reveal to us the state of the universe in its earliest stages,

stages described by the big bang model.

2 The standard big bang models

The CMBR provides us with the most important evidence supporting the

big bang model. The rest of this book explains how and why this is so. In order to understand the physics and interpretation of the CMBR it is necessary to review the fundamental notions behind the model, notably the

chronology and the meaning of the main observables. Big bang models are based on the theory of general relativity and follow from a number of assumptions. These are the following.

I Homogeneity of space applies. Thus it is assumed that all points of space are equivalent and the properties associated with each point are the same, that is, the laws of physics are the same everywhere. This homogeneity is taken to apply to spatial scales greater than those of galaxies, clusters of galaxies and even superclusters i.e. greater than a few

hundred Mpcl. I Isotropy of space applies. This means that there is no privileged direction in space. (Again this refers to large scales.) These two assumptions make up what is called the cosmological principle.

I That the matter in the universe can be described very simply in terms of what is called a perfect fluid. In this case its properties are completely given by its density p and its pressure p. I That the laws of physics are the same everywhere. Models of the universe satisfying the above assumptions are called Friedmann—Lemaitre models and have a number of properties in common.

They are completely fixed by three parameters. Generally one chooses these to be Hubble’s constant, H0, which measures the present rate of expansion of the universe, the density parameter, 9, which measures 1 1 Mpc, or megaparsec, equals 1000 000 parsecs; 1 pc equals 3 X 1018 cm.

7

8

Standard big bang models

the mean mass density of the universe today, and the cosmological constant, A.

Sometimes it is helpful to replace one of the last two parameters by the deceleration parameter, qo, which can be expressed in terms of the latter (see below). Most cosmological calculations assume that A = 0, in which

case there are only two parameters. However, there is nothing to justify this assumption; in fact it seems now that it should be rejected. In particular there is one model, called the Einstein—dc Sitter model, in which it is

assumed that A = 0 and 9 =1. Although observations hint at lower values, this model is often taken as a prototype because it renders calcu-

lations particularly simple. Cosmological calculations are often done for this very specific case. For a long time cosmologists have been trying to determine which of these models is in the best agreement with the observed universe. To do this it is necessary to determine the matter content of the universe as well as to determine the cosmological constant. This provides one with enough information to fix the spatial geometry of the universe, described by the curvature parameter k, and its dynamics, described by the scale factor R(t). In the following sections we shall define and discuss the properties of these two observables. Now, there is a big hope that the fiJture CMBR observations will be of great use in helping us to determine the cosmic parameters and thus to decide what the right model to describe our universe is. The assumptions listed above would imply that the universe should necessarily be described by one of the Friedmann—Lemaitre models. The observed recession of the galaxies, determined through their redshifts, singles out those models in which the universe is expanding. In this case, at least if one does not make any further assumptions at odds with presentday physics, general relativity tells us that the only valid models have a big bang or in other words the scale factor R(t) took on an extremely small value in the finite past. As we shall see, this is precisely what observations of the CMBR indicate. In this sense, the CMBR is interpreted as a very convincing bit of evidence in support of the big bang theory.

2.1 The space—time metric

The fundamental idea on which modern cosmology is based comes from general relativity which tells us that the properties of the universe are geometrical in nature. By geometry one means not just the geometry of space, but the geometry of space—time. Thus this picture includes both the

structural geometrical properties of the universe and those describing its

2.1 The space—time metric

9

evolution. The metric is essential for the description of space—time within this framework. The fundamental idea at the heart of almost all cosmological models

and, in particular, the big bang or standard model is that space—time is homogeneous and isotropic. This cosmological principle implies that the space—time metric can be written in a very general form, called the Robertson—Walker (RW) metric: dr2

ds2 = dt2 — R2(t)(

+ r2(d¢92 + sin2 9d¢2)).

(2.1)

l—kr2 In this system of co—ordinates, the values of the spatial co—ordinates r, 6

and qfi (these are called spherical polar co—ordinates) of any point in space— time (and therefore of any physical object at this point) are constant. Thus

even if the universe is expanding, the values of these co-ordinates can be regarded as being attached to any cosmic object that is carried along (passively) by this expansion. For this reason they are called comoving coordinates. t is both the universal cosmic time and the proper time measured by any observer in ‘free fall’, such as for example by ourselves in our Galaxy (at least approximately). The ‘curvature parameter’, k, can only take one of the three values 1, 0 or —1. Its sign determines the spatial curvature of the universe, which is constant at every point (see the following section). The function R(t), called the scale factor, describes the way in which space grows in the course of time, that is the cosmic expansion. Models that satisfy the cosmological principle (and in which gravitation is described by a metric theory) can be described by the metric (2.1) and differ from one another in their values of k and the function R(t). Later we shall see that, if we assume the standard physical laws, k and R(t) are determined by the values

of H0, 9 and A discussed in the previous section. Observations indicate that the universe is expanding, which means that today the function R(t) is increasing. By measuring the redshifts of galax-

ies we have a direct handle on the values of R(t) in the past when the light was emitted from these galaxies. The fact that all measured redshifts are

positive, that is towards the red rather than the blue, implies that the function R(t) is increasing. Thus the universe is expanding. Hubble’s constant, H0, measures the present rate of change of the scale

factor. Thus H0 2 (1/R0)(dR/dt)0 = (R/R)0, where a dot represents differentiation with respect to time, t. In this work a subscript zero will

denote values of quantities in the present epoch. There is a theorem in

10

Standard big bang models

general relativity that states that, provided that a certain condition is satisfied (the positive-energy condition), the second derivative of R(t) is always negative (this condition is satisfied by ordinary matter and radiation). It then follows from some fairly elementary mathematical

arguments that there must always have been expansion in the past, unless there was some breakdown in the laws of physics as we know them. Consequently the scale factor, R(t), must have been extremely small in the finite past. Models that are in agreement with these assumptions and

arguments are called big bang models. If one does not make relatively exotic assumptions or invoke non-standard physics such as the spontaneous creation of matter and equations of state with negative pressure or attempt to incorporate quantum gravity, these models follow on quite naturally from the observations.

2.2 The spatial properties of space—time If space is taken to be homogeneous (in agreement with the cosmological

principle) then at any given instant its structural and geometrical properties must be the same at every point in space. Thus space has constant curvature and the sign of this curvature is given by k. Big bang models fall

into three classes according to the sign of k. Spatial curvature, which in Euclidean space J93 is zero, can also be positive (spherical space ya) or negative (hyperbolic space 17:43). Usually one assumes that space has standard topological properties and is simply connected. In this case the universe is closed (of finite volume and circumference) if the curvature is positive (k = 1). If the curvature is either zero (k = 0) or negative (k = — 1) then it is open (infinite in volume and extent).

For models in which the curvature is non-negative it is useful to identify the radius of curvature with the scale factor R( t) (whose scale is arbitrary).

This identification is allowed by virtue of the fact that both scales grow proportionately during the expansion. One should note that the spatial curvature, k/ R2, tends to zero with increasing R.

2.2.] Co-ordinates, proper distance and expansion The choice of the metric in the form (2.1) suggests that one should use the comoving co-ordinates r, 6 and ¢ to fix the position of objects. In cosmology one can define several different types of distance. The metric

distance, or proper distance, is defined as the integral of the spatial part of

2.2 Spatial properties ofspace—time

11

the metric at constant time between two objects. It is the distance that would be measured by a ruler placed between the two points. Of course in cosmology distances are never measured like this and the proper distance

has only a limited practical value. Any cosmological distance D(t) increases in proportion to the scale

factor R( t), which expresses the expansion of the universe. Thus

R(t) (2.2)

DU) : DUO) E a

where R0 = R(t0) and to is the present age of the universe. The separation

between two ‘fixed’ points, which might be taken to represent galaxies, is a particular case of this. Upon differentiating one immediately obtains Hubble’s law:

Ra)

Ra)

D0) = DUO) At any time t, the relative speed, DU), with which two spatial points

separate is proportional to the proper distance, D(t), between them. These points might be two galaxies or one might be ourselves as observers and

the other any given galaxy. The constant of proportionality

RU)

H“) =m is called Hubble’s constant. It is a function of time and we denote its present value (at t = to) by H0 2 H(to). Let us suppose that a galaxy has a comoving co-ordinate, r, which remains constant and that we are at the origin of the co-ordinate system, r = 0. The proper distance, dprope,(r, t), between us and the galaxy at any instant t depends on the curvature of space. One defines the function Sk(x) by Sk(x) = x, ifk = 0; Sk(x) = sinx, ifk = l; and Sk(x) = sinhx, ifk = —1.

Integration of the line element yields dproper(ra t) : R(t)X(r)a where

12

Standard big bang models

x(r) = J dr’ (1 — kr’2)_1/2 = s;1(r)

(2.4)

is the inverse function of Sk. Since both r and x( r) are constant one can see immediately that the distance increases linearly with R( t).

2.2.2 The proper comoving distance The proper comoving distance dproper(r, t) varies in time and in proportion

to the scale factor. It is therefore enough to know the value of the proper distance today, dproper(r, to), to reconstruct its value at any time in the past. We call dpmpe,(r, to) = d(r) the comoving proper distance. It is thus defined so that it remains constant throughout the cosmic expansion. The proper distance between us and a point with co-ordinate r is d( r) = R0)“ r). Henceforth, if not stated otherwise, the term distance should be taken to mean the comoving proper distance.

2.2.3 Redshifts The proper distance, whether or not comoving, cannot be measured directly. On the other hand it is possible to relate it to the redshift, which is physically measurable. Recall the definition of the redshift. A given light source, such as a galaxy, emits radiation of a certain wavelength hemmed. However, this radiation is received at a different wavelength, lobsemd, by an observer.

This arises because of the expansion that takes place between the time of emission and the time at which the radiation is received. One defines _ lobserved

1+2

3

lemitted

where z > 0 corresponds to a redshift and z < 0 to a blue shift. The radiation that we receive here (r = 0) and now (t = to) was emitted at time temined. Relativity states that light travels along null geodesics and satisfies ds2 = 0. Thus, if we put the expression for the line element in (2.1) equal to zero, we obtain the relation

R0 1 + z = \.

(2.5)

R(temitted)

This absolutely fundamental relation relates the redshift of a galaxy, or any cosmic object, to the value of the scale factor at the time of emission of the

2.3 The Friedmann—Lemaftre equations

13

radiation. It thus provides a physical interpretation of z. In an expanding universe, R0 > R(temmed)- This implies that z > 0 and that galaxies should

be observed to be redshifted, as indeed they are. 2 is the most easily measured observable for a cosmological object and provides us with the

principal space—time co-ordinate. The relation (2.5) allows one to label different stages of the universe

chronologically through their corresponding redshifts. In fact, because of the expansion, R(t) increases with time and so there is a one-to-one relation between t and R(t). Any instant t can be fixed by the value of R( t) or by the corresponding value of 2 given by the relation (2.5). Thus the redshift 2 can be used to fix a given cosmic time and, indeed, various

moments in cosmological evolution. The fin‘ther back in time the greater the redshift 2. Today 2 = 0. Recombination, which was responsible for the CMBR, corresponds to zrec = 1500. The earliest times imaginable (t —> 0) correspond to z ——> 00. One can also use the redshift to fix the distances of cosmological objects. In general one cannot calculate the distance of an observed galaxy or

quasar, but one can often establish its redshift. Rather than trying to translate this redshift into a distance, we usually denote the distance simply

by the redshift z. The further away a galaxy the greater its redshift. Although generally redshifts of galaxies are less than unity, galaxies and quasars have been observed at z > 4.

These two viewpoints are not of course contradictory. 2 represents both the time of emission and the distance at which a real, or imaginary, source must have emitted in the past the light that we receive today.

2.3 The Friedmann—Lemaitre equations The behaviour of R(t) and the curvature constant k are determined by

Einstein’s field equations of general relativity as functions of the matter— energy in the universe. Both also depend on another constant, the so-called cosmological constant A. If we assume that space is homogeneous and

isotropic and further assume that the cosmological constant is zero, then Einstein’s equations take the simple form

..

4

R = — 375 G(p + 3p)R, R2

87:

723 _ 3

(2.6)

k

__Gp__RE,

(2.7)

Standard big bang models

14

where G is the universal gravitational constant. Note that these equations are written in a system of units in which the speed of light, 0, takes the value 1. To retrieve the more familiar form for these equations it is sufficient to replace p by pcz. An equation expressing the conservation of energy follows directly from these two equations: d

3

__

501R ) —

E

3

pdt(R )-

(2-8)

In order to solve these equations we need to know how the mean density, p, and the pressure, p, of the universe vary with time. (As stated above, we

have put A = 0.) In the standard cosmological models the contents of the universe is expressed as a combination of matter and radiation. Thus p : pm + p,. We also know that the density of matter (assumed nonrelativistic) and of radiation fall off as R‘3 and R4, respectively. One need know only their present values in order to work out their values at any time

in the past or future.

2. 3.1 The equivalence of matter and radiation Today the energy density of electromagnetic radiation, p,, is much less than the density of matter, pm: we are in the epoch of domination by

matter. On the other hand the thermal pressure of matter is insignificant compared with pmcz. Another way of expressing this is to say that thermal

speeds are much less than c. This allows us to use the approximation p = 0 in Friedmann’s equations and gives us the very much simplified equations for the present epoch

431 R3 R = —7Gpm,oR—3, -

821

(2-9)

R3

R2 = 7 Gme—RQ — k,

(2.10)

where we have used the relation

p = pm = Pm.0(R/R0)_3, which holds for matter.

It is easy to calculate the time, called the equivalence of matter and radiation, at which radiation domination gives way to matter domination.

2.3 The Friedmann—Lemaitre equations

15

The ratio of matter to radiation density varies as R( t). Thus tequ, defined by p,(tequ) = pm(tequ), occurs at a redshift Zequ = pm,o/pr,0. (The latter ex-

pression may be written as Q/W; these parameters are defined in the following section.) We are in the epoch of matter domination and have been ever since the time of the equivalence of matter and radiation. Before this, radiation dominated and the dynamics, expressed through the behaviour of the scale factor, R(t), was different. The fact that the radiation pressure depends on the energy density of radiation as p, = §prc2 implies that the energy density falls off as p, oc R‘4. If we ignore matter in Friedmann’s equations, then they yield a solution that is close to the power law R or tl/Z.

2.3.2 The critical density

The Friedmann—Lemaitre equations give rise to 3H0 per-it:

G=1.——387>< M Q W lOQh .

This epoch marks the transition from radiation domination to matter domination. It plays an important role for the formation of galaxies and

cosmic structures and sets the scale for the power spectrum of the density fluctuations. This scale is equal to the value of the cosmic horizon (section 2.5) at the time of equivalence and is around Lfiqu m 10— 15 Mpc, roughly

the characteristic size of galaxy clusters. 2.4.3.2 Recombination

Recombination, which we discuss in detail in section 4.5, is, in some sense,

at the origin of the CMBR. It is defined as the moment when the temperature of radiation was around 4000 K. Since we know that radiation cools according to the law T, 0( 1/ R(t) (X 1 + z and since its present value is given by Tr,0 R: 2.7 K, we can argue that zrec z 1500. In most standard big

bang scenarios recombination takes place after matter—radiation equiva-

lence, during the period of matter domination. The universe before recombination we shall refer to as primordial, as

opposed to the recent universe. It should be noted that stars, galaxies and in fact all cosmic structures formed relatively late in the recent universe, that is at redshifts 2 no greater than about 10.

2.5 The horizon

All cosmic information comes to us in the form of electromagnetic radiation. In the big bang model, time cannot be pushed back beyond the beginning of the expansion period when R(t) = 0. No light could have been emitted before this period and we use 2‘ = 0 to denote this instant for convenience in the discussion. Since radiation propagates at a (maximum)

speed of c it cannot have travelled more than a given distance. This means that one cannot see objects beyond this distance. This limit is called the horizon. In a Newtonian universe it would correspond to a distance of cto,

Standard big bang models

20

where to is the time since the beginning of the period of expansion (i.e. the ‘age of the universe’). From the metric (2.1) we can write down the equation for the path of a photon that reaches us (r = 0) now (t = to), having been emitted at comoving co-ordinate re at time te: “ =

’9 :

dr

’0 dt

———=

—.

Xe iod" Junker/2 CLRU)

2.21

(

)

From this relation we can obtain the comoving co-ordinate, re, of the

source that emitted at time re a photon that reaches us today. The earliest time of emission, as we have said, corresponds to R = 0 or if you like 2 —> 00. If we replace re by its corresponding value (by the usual convention te = 0), we can obtain the maximum values for re or Xe- The

horizon is thus defined by the value rz 2 00 of the comoving co-ordinate r, or by the value x2 2 00 of the comoving co-ordinate x. Thus

=.. 962:“) — J0

dr

'0 dt

R0

dR

(1+ k72)1/2 — 40 W — 6J0 R(t)R(t)I

(222)

Today we can receive no information from regions of the universe more distant than rpm. From the previous relations we find for Friedmann— Lemaitre models

_

c

J1F(x)dx

12:00

9

ROHO 0

x

where F is given by (2.17). For most standard cosmological models (without inflation) the expansion law differs only slightly from a power law of the form R(t) o< ty. This implies that

xzzoo a: ctoRal/(I — y), where to = qu 1 is the present age of the universe (see section 2.4.1). In practice these give the dimensions of the region of the universe

causally connected to us. In most models of the universe this corresponds roughly speaking to the size of the observable universe. In fact this is not always the case. There is a distance, drec, beyond which the universe is opaque, at least for electromagnetic radiation. Since one cannot see beyond this distance it constitutes a practical limit on the observable universe. The corresponding redshift is the recombination redshift, 2m.

2.6 Questions around the big bang

21

As it happens, for the standard models in which the expansion is described by a power law, the values of r and x calculated for z —> oo differ only slightly from those calculated using zrec and for this reason we can use them interchangeably. However, this is no longer true for inflationary scenarios. In any case, in order to avoid any ambiguity, we shall define the size of the observable universe by the values of rm 0r Xrec, although of

course we know that with inflation these values are likely to be much less than the present horizon. With this definition the size of the observable

universe and the observables R0 and cHgl are of the same order of magnitude.

2.5.1 The horizon as a function of cosmic time With time the size of the causal region grows and more and more points

of the universe become causally linked to us. On the other hand at time t: 0 all points were causally unconnected. It is, moreover, one of the basic features of the big bang models that all points were from the beginning in the same state, since they are causally disconnected. For example, they have the same spatial curvature. In the same way that we defined the present horizon by the equation (2.22), we can define the

horizon at any cosmic time, t. It corresponds to the size of spatial regions of the universe that are causally linked at that epoch. The same reasoning yields rz=oo(t)

X22000) — J0

(17’

t

d l,

W — loft”)

(2.23)

As time unfolds, regions which are now beyond the horizon will enter the horizon. In big bang models the fimction R(t) is not very far off a power law, R oc ta. As a consequence the horizon’s size is roughly ct.

2.6 Questions around the big bang The standard models that we have just described are in very good agreement with most observations. Let us recall some of their successes.

I The fact that we do observe galaxies to have redshifts. There is every indication that these redshifts have a cosmological origin. For nearby

galaxies, these redshifts are, taking into account observational precision, proportional to their distance.

22

Standard big bang models

I The fact that the distribution of cosmological sources in the sky appears to be, roughly speaking, isotropic.

I The agreement of the observed chemical abundances of helium, deuterium and lithium with the predictions of primordial nucleosynthesis.

I The experimental verification of the prediction by the big bang model that there can only be three types of neutrinos.

I The fact that the age of the universe is of the same order of magnitude as the age of the oldest stars.

I Last and most important, the existence of the CMBR and its properties, which of course form the subject matter of this book.

Of course these models do not provide the last and definitive word for physics and cosmology. From the point of view of their ability to describe the universe as we know it they have had a surprising amount of success, but they were not meant to explain everything. Their main deficiency, which they have in common with other models, is in explaining how galaxies and large-scale structures formed. The CMBR plays a very important role in relation to this and we shall treat the question in chapter 3.

Often in recent writings several questions are presented as ‘inadequacies’ of the big bang model. Take for example the ‘flatness problem’, or the homogeneity problem (sometimes referred to as the causality problem). One could no doubt add to these the questions of the cosmological constant, antimatter, topology and the singularity and so on. All these

questions, which we discuss in more detail later, are very interesting and we do not have all the answers. However, they are not problems of the big bang theory and in no way undermine its success. The big bang model has been prone to two types of misuse. Some

people would like the big bang model to explain everything. There are those for instance who, erroneously, liken the beginning of the era of

expansion, that is the initial singularity, to a ‘creation’ or a ‘beginning’ of the universe. They are filled with wonderment at the hand of God and are overjoyed that it cannot be seen. At the other extreme are those who criticise the model because it cannot answer the questions we just

mentioned. The latter position, which is somewhat fashionable these days, reveals a serious epistemological misconception, namely the confusion between a model and an explanation. The whole set of problems can be summed up by just one question: why does the big bang model, despite its extreme simplicity, succeed so well in describing our universe? There

is no fault in the model here. On the contrary it concerns its success. We

can hardly demand of a model that it should explain its own success. In

2.6 Questions around the big bang

23

any event, the answer to these questions will not be found in the big bang model itself, but will have to go beyond it. Even so, some original suggestions about how to get to grips with these questions have been put forward. We shall examine in the following sections the new physical theories that have been proposed and the modifications to the standard models that they imply. In the main these have been in the direction favoured by theoretical physicists, that is in the direction of a more unified physics. We shall give a very rapid overview of the ideas behind unified theories, inflation and quantum cosmology.

2. 6.1 Curvature, antimatter and the cosmological constant

The dynamics of the universe, as described by the Friedmann equations, involves several contributions: matter (the parameter 9), radiation ('10, and the cosmological constant (/1, which is formally identically to the so-

called vacuum energy). These three contributions can be compared with the contribution of the spatial curvature and these comparisons are fiindamental to the questions mentioned in the title of this section. 2.6.1.] The problem ofantimatter Because of the problem of ‘dark matter’, as well as for other reasons, we do not know the exact matter density today (2 = 0, x = 1) other than that

Q S 1. Nevertheless, we do know that it is at least 1000 times the density of radiation. This implies that the effect of the radiation density is entirely negligible today, that is '1’ 109 GeV) of

an individual baryon today and the mean energy of a CMBR photon

( oo, R(t) —> 0). The density of matter falls off as

R‘3, that of radiation as R’4, the cosmological constant as R0 and the curvature as R‘Z. In any event it is the density of radiation that dominates in the early universe, whereas the effects of matter, curvature and the cosmological constant are negligible. As an immediate consequence, still

considering the early universe, the Hubble constant H(t) varies as R”. On the other hand neither the amount of matter nor the curvature at that time had any effect on the behaviour of the early universe. One can define a critical density, 3H2(t)/(8JZG), as a function of time. However,

whether the density of matter is close to the critical density at that time has no cosmological importance. The whole discussion is entirely irrelevant, since, whatever the model, it is the density of radiation which dominates and, moreover, by an enormous factor. Furthermore, whichever

model is considered, the ratio of the matter density to this critical density

(analogously to .Q in the past) tends towards 0 and the ratio of the radiation density to this critical density (analogously to '1’ in the past) tends to 1. It is impermissible to talk of a coincidence here, since this happens in all versions of these models. (One should note also that the ‘classical’ reasoning about curvature often confuses the parameters 9 and '1’.) Statements of the ‘flatness problem’ often betray a poor understanding of the big bang model and its interpretation. In any case, this question has

nothing to do with the validity of the big bang model. On the other hand it is interesting and perfectly justifiable to discuss what general framework (e.g. quantum cosmology) might provide a theoretical justification of the

model. Thus it is legitimate to pose such questions as ‘Why do the Friedmann—Lemaitre models (and in particular the big bang models) rather than others describe our universe so well?’ and ‘Why one value of 9 rather

than another?’. These questions can of course only take on a meaning if one can compare the big bang models with other models that have different properties. We shall see (section 2.7) that the possible passage of the

universe through an inflationary phase might provide a possible clarification of this point.

26

Standard big bang models 2.6.1.3 The problem ofthe cosmological constant

We do not know the value of the cosmological constant A. Many think, as

did Einstein, that it ‘must be zero’. In fact a non-zero value implies that the properties of space—time are determined not solely by its material content, which contradicts Mach’s principle. However, the status of this principle is somewhat controversial and many cosmologists consider that the value of [I is a priori arbitrary. Moreover, recent discussion of the ‘age of the universe’, the formation of galaxies and the distribution of quasars favours a value of 0 100 and probably for z = 10 and maybe further in this direction. The epoch that follows, until today, can be

called non-linear. Recombination, for which the level of fluctuations is well below 1,

occurs during the linear era. As we shall see, this recombination marks a crucial stage in the evolution of the universe and brings about a profound change in the nature of the development of the fluctuations. It is at this moment that the CMBR appeared. The observation of the properties of the CMBR is extremely important for models of the formation of the universe, since the density fluctuations which are responsible for the formation of galaxies have also strongly influenced the CMBR. This has taken place either through the production, by relatively direct effects, of intrinsic

anisotropies during recombination (during which the radiation is formed) or later on by the propagation of the radiation through the density fluctuations, in a more or less advanced state of development, modifying its intensity and spectral distribution. Today the density fluctuations are well beyond the linear domain. On the scale of galaxies and clusters of galaxies 6(x, to) is very much greater than

1. The linear phase had thus evolved to the point at which it was no longer valid, when the density contrast attained its critical value of order 1. This

fundamental constraint, which obviously applies to all models of galaxy formation, can be expressed by the statement that the linear phase necessarily leads to a level of fluctuations for which (5 B l. Gravitation is the

driving force for the growth of condensations. It must overcome expansion, which tends to dilute matter, and pressure, which also opposes contraction.

The story of galaxy formation is to a large degree that of this competition.

3. 1. 1 Fluctuation scales

In order to study the evolution of the density contrast, it is necessary to

separate the various scale lengths. The (spatial) Fourier transform allows

36

Galaxyformation

one to decompose the fluctuation field into contributions corresponding to difl‘erent length scales, L = ZJI/ k, in precisely the same way as one decomposes light into its spectral parts. This treatment makes it possible to establish a link between the density fluctuations and the objects one observes today. We shall see that it is also extremely useful to analyse the

angular fluctuations of the CMBR, that is the difference in the intensity of the radiation observed in different directions. To each scale length, L, there can be associated a typical mass, M,

contained in the spatial region of the fluctuation. This mass remains the same—there is neither creation or destruction of matter — both under the effect of cosmic expansion and under the effect of gravitational contraction. These effects increase and decrease the proper spatial size of the fluctuation, respectively, but the total mass associated with the fluctuation remains the same. Using solar mass units the corresponding mass for star

clusters is approximately 106, that for galaxies is 10”, that for a galaxy cluster is 1012 and that for a supercluster is 1014. This method of describing the fluctuation in terms of the mass it contains rather than its spatial size is

very usefiil and frequently used. A fluctuation in the process of condensation corresponds to a fixed amount of matter, even if the corresponding volume shrinks, or at least grows slower than the overall Hubble expansion.

How can the mass associated with a given scale length of the fluctuation be estimated? To a first approximation, the true characteristic size of a fluctuation, lproper, follows the cosmic expansion and thus is proportional

to the scale factor R(t). The comoving length, L, stays the same by definition. (Remember that the two quantities are linked by the relationship L/Lproper 2 1+2.) By the same token, the comoving volume, 47tL3/3,

which we assume to be spherical, is constant and the true volume is

proportional to R( t)3 or 1/(1 + z)3 . The mass corresponding to this fluctuation of comoving size L will be given by M : (4J'tL3 /3)pm,0 where pm) is the mean matter density of the universe at the present time. Under the

effect of the contraction of the fluctuation, the expansion locally becomes a bit less rapid than that of the rest of the universe. Nevertheless, during the linear phase, the difference remains very small, which justifies the labelling of the fluctuation by a constant comoving scale length, L. Once the linear phase has terminated, the evolution deviates more and more from the law of expansion. The volume containing it grows less and less quickly compared with the cosmic expansion. It reaches a maximum radius or the turn-around point and finally begins to diminish, thus bringing about the real condensation of the fluctuation. As a consequence the

3.2 The statistics of thefluctuations

37

volume occupied by this condensation of matter (a galaxy for example) is much less than the size L associated with the fluctuation. The latter is by

definition the radius of the sphere that would contain this mass (of the galaxy) if no condensation had taken place; it is approximately the mean distance today between galaxies, say about 1 Mpc, whilst the dimension of a galaxy is at least ten times smaller.

3.2 The statistics of the fluctuations

It is impossible (and it would certainly be superfluous) to describe all the properties of the density field, 6(x, t). One restricts oneself to a statistical

description founded on probabilities and average or expectation values. The belief is that at each instant the information contained in this statistic

is enough to characterise the evolution of the fluctuations. This very farreaching assumption at least provides approximate results. Thus we define, for example, the probability

P[p(x1), ..., p(x1-), ~ . , p(xN)] that the density takes on the values p(x1), ..., p(x,-), ..., p(xN) at posi— tions x1, . . . , xi, . . . , xN respectively.

In general one assumes a simple statistical law. Many theories, for

example, assume that it is Gaussian. This means that the formulae giving the probability that the density takes any given value are Gaussian. In the case in point, the function takes the form of a ‘multivariate Gaussian’ in N variables, of the form

d M 4/2 P[P(x1)a . . . , pom] = 26%;? exp . (3.2) In this expression M is the ‘covariance’ matrix, which expresses the

pairwise correlation between the values of the density field at different points. The matrix element, M1-], is given by the correlation in the values of the deviation from the mean density at the two points considered, i.e. (p(xl-)p(xj)). (Here the average is an ensemble average.) M‘1 is the inverse matrix; det M is the determinant. If the distribution is taken to be Gaussian,

this formula contains all the necessary information. In fact, for a Gaussian distribution, once the two-point correlation has been defined, the three-

38

Galaxyformation

point, four—point and higher correlations can be derived. Thus the Gaussian

hypothesis enormously simplifies the calculations, since any statistical quantity describing the density distribution and its fluctuations can be derived from it.

Most currently accepted models for galaxy formation assume the initial fluctuation distribution to be Gaussian in (5. We should note, however, that this assumption can only be approximately true simply because the density

cannot take on negative values. In any case the original Gaussian character is quickly destroyed by the process of gravitational contraction. It is also important to note that the statistical properties of a field, such as the density field, can also be expressed in terms of the spatial Fourier transform

3.2. I The power spectrum The density fluctuation field 6(x) can be expanded in its spatial Fourier components, 6k, that is in plane waves. To the wave vector k of modulus k

there corresponds a scale length of L 2 23/ k. The Fourier modes 6,. are complex and defined by the expression

1 (5k = VJ d3x@ exp (ik - x).

(3.3)

V They can be inverted to give

6(x) = Z 6,, exp (ik - x).

(3.4)

k

Equation (3.4) is in fact equivalent to

6(x) =

Jd3 k 5,, exp (ik . x),

(3.5)

(27:)3 but in practice proves to be more practical. One should note that other authors use different norrnalisations, so the factors of Zn will appear

differently in the equations. This does not of course affect the results, but the calculations must be consistent. The factor V, the volume of the region of the universe over which the integral is taken, enters for mathematical correctness. However, this quantity generally cancels out when measurable quantities are calculated.

The isotropy of the universe implies that the mean values of the modes depend only on the absolute value, k = |k|, not on the orientation of the

3.2 The statistics of thefluctuations

39

wave vector k. One can characterise the mean level fluctuations of scale

length L by the power spectrum of the fluctuations

Me) = = jlkl_kd3k62,

(3.6)

the mean square of the fluctuations. The average is taken over all modes whose wave vectors k have modulus k = 23/ L. This spectrum is a fundamental quantity. If the statistics are not too complicated and in particular for the special case of a Gaussian distribution, it allows one to find almost all the properties of the density fluctuation field, such as the mean mass fluctuation in a volume of given

size and form (see the following sections). The power spectrum is often expressed in a slightly different form, in which it is made dimensionless by introducing the variance per unit In k:

412(k) E

V 47tk3P(k). (MP

(3.7)

From this formula one can calculate the mean density fluctuation (which is

occasionally written as 02) at time t:

=6(t)2:((,|6(x t)|2> =26,

(3.8)

or, alternatively,

1

(5(t)2= EEK P(k)k2 dk.

(3.9)

The evaluation 03 of this quantity at the spatial scale of 8h‘1 Mpc is used

for the normalisation of the spectrum (see section 3.6.3) Similarly one can define higher order moments:

2 0

V E (2703

J: P(k)k2” dk

(3.10)

3.2.1.] The transferfunction

The fluctuation spectrum evolves both in form and in intensity during the course of cosmic history. Moreover, the evolution of the fluctuations varies

from one spatial scale to another. There can be amplification, stagnation or

40

Galaxyformation

damping. For this reason one defines a transfer fimction .7(k; 11 —> t2) given by

Paola, = mic; n a 12)P(k)lt=tl-

(3.11)

Some authors define it differently as

502)

P(k)|,=,, =(— —)7(k;t1—>t2)P(k)|z=zp 6(t ) where 6(t) is the mean value (3.1) of the fluctuation at time t. This function can be calculated for the various models.

3.2.2 Velocities and the gravitational potential One can calculate from the power spectrum the mean values of the fluctuations of various quantities as functions of the comoving scale length, L, or of the associated mass, M = (431/ 3)p0L3. The velocity field corre-

sponding to the fluctuations can also be expanded in Fourier components. In general the velocities are assumed to have zero vorticity in such a way that the Fourier modes give rise only to wave vectors parallel to the velocities: v(k) = k|v(k)| / k. One can also define the power spectrum of the velocity field, Pu(k) = ([U(k)|2), where the mean values are calculated over all the possible orientations of the vector k. One can also estimate the fluctuations in the gravitational potential, (p.

They correspond in Newtonian physics to the fluctuations of the metric in general relativity. The potential is obtained from the Poisson equation

41¢ = 4JtGp, so that one obtains, to an order of magnitude, for the dependence with

length scale

6M 0(5_p MQC p o —3

for large scales, but that n< —3 for small scales. Nevertheless it can always be decomposed into pieces having this type of functional depen-

dence, which gives rise to a polynomial dependence. In fact, n < —3 would imply that the universe was extremely inhomogeneous, right up to large scales, which would appear to be incompatible with the observations of the isotropy of the CMBR. On the other hand, if n > —3 the inhomogeneities at small scales would have been so important that black holes would have been likely to form during the early stages of development of the universe.

Their evaporation would have produced an intense X-ray emission far above the observed level.

If we assume a power law spectrum in the calculation of the mass fluctuation we obtain

(67)”) rmsOC M-a,

(324)

where a = (3 + n)/6 is called the variance index. This expression is completely rigorous for —3 < n < 1. For higher values of n, a z §. To each spatial scale, fixed by the value of k, there corresponds a value of n and a value of a, which, a priori, evolve with time. If one calculates the mass fluctuation, not in a fixed volume, but with the Gaussian selection function given by equation (3.20), one obtains (SM

n + 3

(V) max r< 2

)R"‘”+3V2.

(3.25)

The associated fluctuation in the gravitational potential has a scale depen-

dence 6(1) or R(1_")/Z. Several special values of n for the initial spectrum of fluctuations have been investigated in the literature.

I The ‘minimal’ spectrum, n = 4, a : g, which corresponds to the most ‘uniform’ way of choosing the fluctuations.

3.3 The initialfluctuations

45

I The ‘white noise spectrum’ (or Poissonian spectrum), n = 0, a = %, for

which each scale has the same power. This corresponds to a process, which was responsible for the appearance of the fluctuations, whereby the particles would be distributed randomly. However, this hypothesis leads to fundamental problems, of such a nature that this is not one of

the usually accepted scenarios. I ‘Particles in a box’: n = 2, a 2%. This form has had virtually no

success. I The

‘Harrison—Zel’dovich’,

or

‘scale—invariant’

spectrum:

n : 1,

a = %. This, the most popularly accepted spectrum, was introduced by Harrison and Zel’dovich in 1970. The interest of this choice becomes evident when one estimates the fluctuations of the metric or, if one

adopts a Newtonian View, the gravitational potential, (1). From the equation (3.12), taking into account the L3 dependence of the mass M of a fluctuation of scale length L, one can show that, for a spectrum like this, fluctuations of the potential (or of the metric) keep the same level, 65¢, at all scale lengths. It is for this reason that the Harrison— Zel’dovich spectrum is also called scale invariant. Without knowing

the processes originally responsible for the fluctuations it is reasonable to believe that there was no characteristic size at the time when they were produced. Moreover, according to general relativity, the curvature

is the fiandamental physical quantity, since it is this that determines the dynamics. In the absence of other information, this spectrum would seem to be the most natural one to take. Although it was proposed for reasons which seemed important in the 1970s, it has since found a double justification. On the one hand, the observational results, from COBE (see section 7.8) and other experiments, are at least consistent with this form. On the other hand, modern ideas about inflation (see section 2.7) or symmetry breaking in the early universe suggest processes that could have given rise to fluctuations with such

a spectrum.

3.3 The initial fluctuations

According to the Friedmann—Lemaitre models, the universe was extremely homogeneous, which is confirmed by the isotropy of the observed CMBR. However, one cannot explain the presence of galaxies and largescale structures without supposing that the fluctuations had been produced at some time. The processes that actually created these primordial fluctuations and the instant of their creation remain unknown to us. It is

Galaxyformation

46

possible that they reach back in time to the epoch of ‘quantum cosmology’. Less likely is the idea that ‘quantum vacuum fluctuations’ during the inflationary phase were responsible. Other hypotheses have been proposed. In any case, it is useful to examine the characteristics of these initial fluctuations in order to predict their later development. It is in this respect that the different cosmogenic models differ from each other (apart from the usual cosmological parameters). For this reason it is helpful to review the main respects in which the fluctuations might differ, namely the following. I The moment at which the fluctuations were created in cosmic history (see below).

0 Their isocurvature or adiabatic nature, which defines the way in which they might affect, as well as matter, electromagnetic radiation. - The nature of matter that arises from these processes, notably the following.

Baryonic matter only. Cold dark matter (CDM). This matter is primarily non-baryonic, and the larger part of the mass is not made up by protons or neutrons, as is the case of ordinary matter, but by some as yet unknown type of particle. These particles must interact weakly with ordinary matter and electromagnetic radiation. They must also have decoupled relatively early in the history of the universe (at a temperature > 1 keV, at redshift z>4 X 106). They must, as their name indicates, have a relatively low temperature. Hot dark matter (HDM), which is again made up of non-baryonic particles, similarly to CDM. The difference is in their velocity distributions. For HDM the velocity distribution is characterised by a

higher temperature and a higher root mean square velocity of the particles.

Any combination of these components. 0 The statistics of the fluctuations and whether this corresponds to Gaussian statistics or quasi-Gaussian statistics on the one hand or nonGaussian statistics on the other. The highly non-Gaussian statistics

corresponds to the case in which objects, such as for example cosmic strings, have already strongly condensed (see section 6.7). I The spectrum of fluctuations (see section 3.2.1), which is particularly important in determining the statistics. If, for example, Gaussian statistics hold, the power spectrum contains all the statistical information about the fluctuations.

3.3 The initialfluctuations

47

I The initial strength of the fluctuations This level is not normally specified by the cosmogenic model. It is adjusted afterwards from observations by choosing a particular normalisation (see section 3.6.3).

3.3.1 Isocurvature and adiabatic fluctuations

In the early universe, thus at the time at which the fluctuations appeared, the density of radiation was greater than that of matter by a factor greater than 2/ 10 000. It is thus very important to know the way in which radiation, as well as matter, is also affected by the fluctuations. We distinguish two

types of fluctuations, which do not have the same evolution. I For isocurvature modes, the fluctuations in radiation density adjust in such a way as to maintain the total density the same. One can thus write

(3.26)

5p : 6(pmatter + pradiation) : 0However,

since

pradiation >> pmatter,

it

is

immediately

clear

that

5.3mm LJeans % Lzzoo. Because of the extremely high pressure of radiation,

which itself is coupled to matter, the Jeans length is about the same size as the horizon length. One should of course remember that one is talking here about the comoving length scale. The growth of the real horizon distance Lzzoo / (1 + z), is proportional to the time, t. A fluctuation of comoving scalelength L is able to condense so long as L > L2:oo or, equivalently, M > MFOO. Consider for example a fluctuation

of the same mass as a galaxy of approximately 1011MQ. It remains greater than the Jeans mass for about 1 year after its creation when the mass

contained within the horizon reaches 101 1 M0. It thus condensed during the first year of cosmic history, up to the point at which it ‘entered the horizon’. The higher the fluctuation’s mass the longer the time it has had to condense. It is thus easy to know how long a fluctuation of a given mass

52

Galaxyformation

has had to condense, but this is not enough. We also need to know how

efficient this process is. I For an adiabatic fluctuation a relativistic linear calculation tells us that

the density contrast grows more or less linearly with time. Thus

a or toc 12(2‘)2 0( (1 + z)—2,

zzequ.

(3.29)

In the case of an Einstein—dc Sitter universe (9 = 1, A = 0), these relations are exact. Otherwise they are only approximate. I Isocurvature fluctuations undergo no amplification and in fact decrease slightly (whilst the scale length is greater than the horizon length) up to

the time of equivalence. With the onset of domination by matter, these modes are transformed into adiabatic modes (when their scale length is smaller than the horizon size) and, consequently, their evolution becomes identical to the evolution of adiabatic fluctuations.

3. 5.1 The growth in the primordial power spectrum

At some point in the linear phase the rate of growth is the same for all

fluctuations, independently of their scale size. However, this rate applies only if the fluctuation size is larger than the horizon. Hence fluctuations of different scale sizes do not condense during the same period of time. As a

consequence there is a differential amplification and necessarily a change in the form of the spectrum.

3.5 The growth before recombination

53

To get some idea of the order of magnitude, consider the Einstein—dc

Sitter case. Here the scale factor satisfies R( t) 0c tl/2 during the radiationdominated era (2 > zequ) and R(t) oc tz/3 during the matter-dominated era (2 < zequ). Consider the case of the growth of the adiabatic fluctuations, for

example. Between the moment tin when the fluctuations appear and the moment zequ of equivalence of matter and radiation diequ : min/gain ‘_’ tequ; M)

According to what we have said, the amplification factor, .26, depends on the scale of the fluctuation and is equal to the duration during which this scale mass is larger than the Jeans mass. Let us call tH(M) the time when the mass M enters the horizon, defined by M = Mhorizon[tH(M)]. We then have t JgUin —" tequ; M) =

M H:

)9

M< Mequ;

te Jgain _> tequ; M) = tqu’

M>Mequ-

Then, from zequ to 2m, -/g(tequ —’ tree; M) I 1:

M 106MQ) is unimportant. Such scenarios are similar to those based on isothermal baryonic fluctuations. Similarly one occasionally defines an intermediate case (‘warm’ particles) in which the damping mass

is about the mass of a galaxy, in such a way that the spectrum is cut offjust at these scales.

3.6 After recombination 3. 6.1 The growth offluctuations after recombination It is particularly important to know the level of fluctuations at the time of recombination. There are at least two reasons for this. The first is that their characteristics are imprinted on the radiation present. Since some of the properties of this radiation remain unchanged from this epoch until the present day, we should be able to observe today in the CMBR these

imprints of the fluctuations made at the time of recombination. This allows us to test the various scenarios. Secondly, recombination sets the initial conditions for the second phase of cosmic history, dominated by matter. Matter fluctuations start their linear growth which ends with the

formation of galaxies and other cosmic structures. The spectrum at recombination depends on the initial spectrum and the (possible) sort of

amplification and on the possibilities of damping described above. After recombination electrons recombine with ions. Matter becomes

almost completely uncharged and no longer interacts with radiation. Free of the effects of radiation pressure, matter is free to contract and a new phase of linear growth begins. All fluctuations grow slowly until their mass

56

Galaxyformation

scale exceeds the Jeans mass (which, at the time of recombination, is

around 106 MG). During the linear phase following recombination the rate of growth is roughly the same for all fluctuations of mass M > MJeans z 106MO. If it

were not for universal expansion this growth would be extremely rapid, in fact exponential, and galaxies would form quickly. However, expansion reduces the efficiency of this gravitational instability. To carry out these calculations we have to take into account that now matter dominates and

hence we can do everything within the framework of the Newtonian approximation. One obtains the result that the fluctuations grow in rough proportion to the scale factor, i.e.

5(0 0C R(t)0 0, since today the fluctuations are non-linear, this implies

(Sm >(1 + zm)_l > 10—3.

(3.32)

However, observations of the CMBR indicate that the level of the fluctua-

tions can hardly be of this order of magnitude and is probably less. In order to resolve this difficulty, more detailed calculations are obviously necessary, as well as additional assumptions. One needs also to study the

fluctuations before recombination and their effect at recombination on the CMBR. The important question to ask is that of whether fluctuations of the scale of galaxies, for example, have had the time since recombination to reach

the non-linear level and, if so, when this happened. For the model to be valid it is necessary for this not to have happened too late (thus at not too low a value of z) on the scales of galaxies and clusters, because we observe

these objects at values of z greater than 1.

3.6.3 Cosmic matter and its distribution

3. 6.3. 1 The link between objects and statistics

Calculations of the dynamics involve fluctuations in the mass (including any possible dark matter) density. If we want to understand galaxy for— mation and the formation of other condensed objects, we have to be able to

58

Galaxyformation

deduce the properties of these objects from those of the density fluctuations. Even if we are unable to follow all the physical processes at play, an

operational model must at least allow us to predict the distribution and the properties of a given type of astronomical object (such as a galaxy or cluster) from the power spectrum of the density.

First of all we clearly need to specify the type of object that we are considering — galaxies or clusters for example — and pick out the corresponding characteristic (comoving) scale, L (or alternatively the mean

mass M contained in a sphere of radius L). One should remember that L and M are constant for any given fluctuation during the course of time. After this it is useful to apply a ‘smoothing’ of the field with the aim of not taking into account fluctuations on smaller scales, which we suppose

have no influence on the formation of the given objects. This smoothing is done by convolution using a window function as described in section 3.2.3.

Having done this we would hope to have extracted from the density fluctuation field all the necessary information for the precise scale, for instance the galaxy scale, that interests us. However, the most difficult part of the problem is still to come.

After smoothing the resulting field contains only fluctuations of the scale chosen or larger. We still need to understand where galaxies form, what properties they have, how many of them there are and how they are distributed etc. Of course it is illusory to imagine that the information contained in the density fluctuation field is sufficient to answer these questions, for this would imply that phenomena other than gravitation played no part in the formation of galaxies and cosmic structures. Yet we know for certain that other processes besides purely gravitational contraction, for instance radiative cooling, do play an important role. This pro-

cedure, although it is widely used, thus can constitute at best a first and very crude approximation.

There are, however, even more serious problems. This approach supposes implicity that the formation of objects of a certain scale size depends solely on fluctuations of the same scale. It is of course true that during the phase of linear evolution of the fluctuations different spatial scales evolve independently. However, this linear phase ceases long before objects as we know them begin to form. In fact, the process depends not only on the statistics of the fluctuations but also on their respective positions. The formation of objects of a certain type, even if one neglects dissipation and non-gravitational processes, depends on the complete statistics of the fluctuations at every scale. Obviously it is impossible to treat this problem exactly.

3. 6 After recombination

59

3. 6.3.2 The approach ofPress and Schechter The simplified procedure corresponding to such an approach is called the

Press and Schechter formalism. As we describe it and as it is often used today, it constitutes a very powerful approximation, which is, however, not really justified. The main reasons for its popularity are the ease with which it can be implemented and its apparent success, although the latter is certainly illusory. It alows one to estimate a distribution function of massive objects such as galaxies and clusters of galaxies which resembles both the distribution of real objects and also the artificial distributions

which are obtained by numerical simulations that attempted to reproduce the processes responsible for the formation of these objects. (However, one can also raise certain objections in relation to these numerical solutions so that the argument is not really convincing.) In fact we are not really able today to make a clear link between the statistics of the density field and the

distribution of cosmic objects. Thus it is really futile to hope to be able to validate or invalidate a particular model by applying some criterion based on the correspondence between reality and predictions that do not rest on a solid foundation. Having expressed these reservations, we shall now describe this procedure, which in fact is the best one can do at present. The standard prescription states for example that galaxies or other objects in which we are interested are present in all regions of the universe where the density field p(x), after smoothing at the corresponding scale L,

exceeds a certain relative level. This somewhat arbitrary procedure has been modified and refined. However, the main idea remains the same and, if we take into account the enormous arbitrariness in the basic method,

these changes and prescriptions seem to be superficial sophistications. Despite the lack of any real justification, this formalism appears to give acceptable results and, in the absence of any better understanding of non-

linear condensation, it has provided the first reasonable approach. Thus, for example, the distribution of galaxies reproduces the distribution of ‘maxima’ of the density field smoothed on the scale of 1 Mpc. It is pointless, as we have already said, to hope to confirm or falsify a model on the basis of this procedure whose justification is so flimsy, even though it is very commonly used.

3.6.3.3 Normalisation and the biasingfactor The various models for galaxy or structure formation should be considered phenomenological descriptions rather than real physical models. They are, for example, based on the assumption that the statistics of the initial

60

Galaxy formation

fluctuations are extremely simple, usually Gaussian, and likewise that the initial power spectrum is extremely simple. It is very rare that the initial level of the fluctuations is specified. Hence it has to be adjusted a poster— iorz’ by a normalisation, which is decided on the basis of observational results. This normalisation must be the most direct possible and be based

on the fewest additional assumptions. Bearing in mind the uncertainty in the non-linear calculations, normalisation can really only be based on the linear results. This presents us with two possible routes. We should normalise either on the basis of the observations of the CMBR, since the CMBR

arose during an epoch when the fluctuations were linear, or else on the basis of results for the present epoch, but for scales at which the develop-

ment of the fluctuations can be still be regarded as linear. Up to the time of the COBE results, the only possible normalisation was based on observations of the distribution of galaxies. To do this there were obviously two immediate constraints: on the one hand our ability to determine this distribution observationally, which ruled out very large

scales; and on the other our ability to observe sufficiently large scales for the linear regime to be valid. A scale of 8h‘1 Mpc was chosen as a

compromise. Thus, a quantity as was defined as the average relative density fluctuation at the scale of 8h‘l Mpc. This quantity is evaluated from the distribution of visible matter, usually galaxies (but see below).

Then the obtained value is reported in the spectrum with an assumed shape, in order to normalise it. One cannot go beyond this scale and still be sure of the observations. Even so, it is necessary to bear in mind that the linear approximation should still be treated with caution and that the resulting normalisation cannot really be considered as precise. 3.6.3.4 The biasingfactor The most fundamental limitation of this method relates to the fact that one observes the distribution of galaxies, that is of luminous objects, which are

luminous in a very specific way, rather than the distribution of the total matter. We know from dynamical arguments that the visible objects such as galaxies, clusters etc. probably represent only a small fraction of the total matter in the universe. Dark matter exists and is probably distributed in a

very different way from the visible component. There are very few clues about the link between the distribution of the total mass and the distribution

of galaxies etc. The usual way out of this problem is to introduce a phenomenological parameter, the biasing parameter b which expresses the relationship between the two. Thus

3.6 After recombination

61

6Ivgalaxies N = agalaxies : b(Stotalgalaxies

This factor can and indeed must, in order to agree with observational constraints, depend on the spatial scale. Furthermore, one can define a bias for each component of visible matter. Thus the values of b corresponding to spiral galaxies, elliptical galaxies, clusters of galaxies, galaxies observed in the infrared, etc. are not necessarily the same.

This phenomenological description in fact masks the fact that we do not know the physical processes which determine the relation between the value of the total density in a region and the probability that a luminous object such as a galaxy should be there. The idea itself is not unreasonable,

but it is still a great simplification of a situation that is poorly understood. All models that are not purely baryonic rely to some extent on these biasing

arguments and so are called ‘biased’ models. The simplest version is to suppose that objects such as galaxies form in zones where the value (5(x) of the total density fluctuation exceeds b multiplied by the mean value of the fluctuation 0. The physical processes responsible for the formation of galaxies would have been effective in the corresponding regions only, so that galaxies would form in regions of high density only (or alternatively

their formation in regions of low density would have been inhibited). The most complicated versions take b to depend on the spatial scale. Others will no doubt appeal to other laws. To the extent that one forgets the physics, everything is allowed a priori and one simply introduces supplementary parameters to save appearances. All this only goes to emphasise the present

weakness of models for galaxy formation and that this extremely simplified description is merely a relatively poor first-order approximation. Formation scenarios based on cold dark matter (9 z 1) require a large bias factor, of the order of 2, which apparently varies with the scale. On the other hand, scenarios with hot dark matter require a value of b less than 1,

or what is called ‘anti-bias’. Until 1992 there was no choice in the matter. The only choice for

normalisation of the density field involved the distribution of luminous objects and therefore the bias parameter. In fact no intrinsic anisotropy had been detected in the CMBR. Only upper limits were known and these were useless for any normalisation. Two types of normalisation were thus applied: either

osz 4. The estimated

value of IT depends on many parameters such as the mean density of the universe, 9, the density in the form of baryons, QB, the relative fraction of helium present compared with hydrogen, the redshift, Zion, from which the

gas started to be ionised and so on. For most of the standard models Zion is weak, for example 910 < 0.1 and Zion % 5. In this case cosmic expansion means that the gas density has

l 16

After recombination

already fallen significantly, so the optical depth remains low. Typically IT < 0.3° pixel. Its HEMT detectors will observe in five selected bands of 22, 30, 40, 60 and 90 GHz, to facilitate the separation of Galactic foreground

signals from the cosmic background radiation. The strategy will be to measure differences in temperature between points on the sky with a fixed angular separation of 135°. PSI/FIRE is an American project to image the CMBR on the entire sky, with an angular resolution of 05° at several frequencies. No decision regarding the choice of HEMTs and bolometers has been taken.

The Far InfraRed Space Telescope (FIRST), a project of the ESA, is a large antenna that can be used at 8 m, functioning at wavelengths in the range 85—600 urn. The cooling is passive for the telescope and active for the high-resolution heterodyne spectrometers and photometers. In addition to studying the physics of early galaxies and protogalaxies, the interstellar medium, protostellar clouds and the formation of stars, it will

170

Measuring the CMBR

also possibly be used for measuring fluctuations of the CMBR on small

scales of 6 z 10' (at A > 350 um) and particularly the Sunyaev-Zel’dovich effect. The FIRST mission has a number of similarities to the Planck mission (see above). For this reason, the ESA is studying the

possibiity of combining the two projects. I The Large Deployable Reflector (LDR) is a NASA project. The 20 m mirror, operating at wavelengths in the range 0.1-1 mm, will be well

adapted to the observation of the CMBR. I There are a few space projects, which, although not specifically dedicated to measurement of the CMBR, will be more or less directly implicated: AELITA is a joint project of Rome University and Russia. It will cover the X-ray, ultraviolet and infrared domains; the Space InfraRed Telescope Facility (SIRTF) is a NASA project and so is the Submillimeter Wave AstrOnomy Satellite (SWAS).

7.8 The COBE satellite

The Cosmic Background Explorer (COBE) satellite was conceived and built in the Goddard Space Center of the NASA and exclusively designed

to observe the CMBR. Since its launch in 1989, it has produced spectacular results, even if they do not have quite the importance they have been assigned both verbally and in the press. (For a history of the COBE satellite see Ripples of Time, by G. Smoot (Little, Brown and Company, 1993)).

COBE was equipped with detectors that swept the sky in the infra-red down to centimetre wavelengths. The whole configuration of the various

detectors as well as the orbit were chosen so as to minimise systematic measurement errors. COBE carried on board three instruments which complemented each other, and to which three distinct tasks were assigned: I an assembly of differential microwave radiometers (Diffuse Microwave Radiometer or DMR), for finding possible anisotropies in the CMBR; I a Michelson interferometer (FIRAS), to verify whether the CMBR is

thermal and to look for various distortions in the spectrum; and I an infrared photometer (DIRBE), for the detection of cosmic infra—red

radiation as well as local astrophysical radiation. During this mission the DIRBE and FIRAS have had the time to map every part of the sky at least once. The DMR, which does not need cooling, has been able to continue after all the helium had been used up. The concept of this satellite dates from the 19705. The project was

7.8 The COBE satellite

171

concretised around 1976 and assembled ideas from various groups. The responsibility for the DMR was given to George Smoot, of Berkeley. John Mather, originally from New York, was given the responsibility for FIRAS and Mike Hauser, of the Goddard Space Center, had the responsibility for DIRBE. COBE is the size and mass of a large car. It has a large cryostat of liquid

helium, a folding panel for protection against nuisance radiation, an electrical generator, an orbit control system and so on. The panel protects

the detectors and the cryostat, which is similar to the one used on the IRAS satellite, against solar and terrestrial radiation and interference from radio frequencies. FIRAS and the DIRBE (which we shall describe a bit later)

are inside the cryostat and maintained at a temperature of less than 2 K. The cryostat contained enough liquid helium (6501) to cool the instruments for the entire duration of the mission of about 1 year, until 20

September 1990. The DMR is not cooled in the cryostat, but only protected from radiation by the panel. Solar panels produce electrical energy, except

during the short eclipse periods, when the electricity is provided by batteries. Although the project was initially intended to be launched by the space shuttle, the catastrophe of 1986 brought about a change in plans and a Delta rocket was used for the launch, which took place in California. Its orbit was chosen so that the three types of detector swept the entire sky.

They were protected from solar and terrestrial radiation in order to keep the thermal environment stable. This almost polar circular orbit is at 900 km altitude, which was in fact a good compromise to avoid on the one hand atmospheric contamination from the Earth and on the other the

charged particles in the high—altitude radiation belts. Its path traces out the line delimiting night and day on the Earth. A control system orientates the detectors perpendicular to the Sun and away from the Earth, in order to limit the effects of their radiation. The satellite turns on its own axis once a minute, which reduces systematic errors in the differential micro-

wave radiometers. The rotation also ensures that the solar radiation falls on the satellite uniformly and so reduces temperature gradients. The orbit and the directioning of COBE were designed to ensure complete coverage of the sky every 6 months.

7.8.1 The differential microwave radiometer

The DiffiJse Microwave Radiometer (DMR) was intended to map out accurately the microwave sky and to observe any possible anisotropies in

172

Measuring the CMBR

the CMBR on large angular scales. In order to avoid any confiision with galactic emissions, the DMR observed the sky in three different wavelengths, 3.3, 5.7 and 9.6 mm, corresponding to 90, 53 and 31.5 GHz and

close to the peak emission of the CMBR. The latter exceeds galactic emission at these frequencies by a factor of 1000, although this is not necessarily the case for fluctuations in the CMBR. By observing in several

different frequencies one can, up to a certain point, eliminate galactic contributions (see section 7.2).

The DMR is an improvement on the Dicke detector. Smoot’s group had already used it on U-2 (aircraft) flights. One of the main innovations in the case of COBE was the use of cooling by liquid helium. The DMR measures the difference in the antenna temperature between

regions of the sky separated by 60°. The detector was designed to detect an anisotropy of one part in 100 000. The radiometers in the detector alternately, at very high frequency, record the radiation from the two nearly identical antennae, which are oriented at 30° to the axis of rotation of the satellite, with a beam width of 7°. Just like with other differential measurements described (see section 8.1.2), it is from the differences between the energies recorded by the antennae that the computer on board can calculate the corresponding temperature differences. This is done on the one hand

with respect to the stable reference temperatures on board and on the other in relation to the Moon. One of the observational tactics used was to

combine the rotational motion of the satellite about its own axis (with a period of 75 s) with the period of the orbit (103 min) with the orbital

precession period (1° per day), in such a way that all pairs of points separated by 60° could be examined.

There are six radiometers. Each has a heterodyne receiver, whose opening is alternated (at 100 Hz) between the two antennae, which detects a well-determined wavelength. The sensitivity of the two radiometers detect-

ing the two shortest wavelengths is increased by cooling the apparatus to 140 K. They can detect a temperature difiference of about 0.025 K in l s of

observation. The third radiometer, which functions at the ambient temperature, detects the radiation of longest wavelength and is only half as

sensmve. We shall give the measurement results in section 8.5.2. From the very first year, the radiometers did find a difference in temperature and finally the level was established to be a few microkelvins. This is 300 times less

than the dipole variation in the CMBR due to the motion of our Galaxy (see the detailed discussion in section 8.9). The sensitivity of these measurements also allowed one to see the effects of the Earth’s motion around

7.8 The COBE satellite

173

the Sun. This motion corresponds to a speed of 30 kms‘l, in a direction that changes in the course of the day. The corresponding variation in temperature of AT % 0.3 uK provides one with an additional calibration that is very precise. An initial calibration (see section 7.4.1) was carried out before the

launch using known sources and then, during the flight, using the radiation from the Moon. Furthermore, each radiometer was provided with 3 reference signal one every 2 h.

7.8.2 The far-infra-red absolute spectrophotometer

The Far InfraRed Absolute Spectrophotometer (FIRAS) is a Michelson interferometer intended to measure the CMBR, as well as the general

background radiation, with great accuracy by comparing it with a reference spectrum. During the COBE mission, FIRAS swept the entire sky twice, at

wavelengths in the range 0.1—10 mm, separated into two runs. The run of low frequencies was aimed at finding the spectrum of the CMBR by comparison with a calibrating black-body spectrum. The high-frequency run provided the possibility of measuring the emission from gas and dust

in our Galaxy. The opening angle of the instrument is 7°, the same as for the DMR. The sensitivity of this instrument is 100 times better than had

previously been obtained from instruments used in this sort of observation. Unlike the other instruments, it points in the same direction as the axis of rotation of the satellite.

FIRAS recorded the spectrum of the CMBR in the wavelength range from 100 um to 1 cm in 100 regions of the sky. This allowed it to detect

deviations from the black-body spectrum of 1 part in 1000. In the same way as the first receiver described above, the interferometer is a differential

instrument. It compares the energy from the CMBR with an adjustable reference, fixed in the satellite. The subtlety of this operation should be

understood. The sensitivity of the receiver is due to a calibration source that can be placed in front of the collecting horn, which emits radiation differing by less than 0.01% from a black body spectrum. The temperature of this source is adjusted so that its spectrum is as similar as possible to that of the cosmic radiation collected by the receiver. The interferometer thus directly and very precisely measures any difference between the cosmic spectrum and that of the reference black body. By continually

adjusting the latter, one would completely eliminate the resultant signal if the CMBR spectrum were of precisely black-body nature. At the same time, the temperature of the CMBR is also determined.

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Measuring the CMBR

In practice, radiation enters the interferometer from the horn, which allows only radiation coming from a well-determined direction. At the output of the horn, the initial beam is divided into two beams, which

recombine after having been reflected by two movable mirrors. One then determines the cosmic spectrum from the interference between the two beams. The interferometer covers a field of 7°, centred around the axis of

rotation of the satellite. The measurement of the dipole showed that FIRAS has a stability better than to within 2 X 10‘4 over long periods of time. The data from FIRAS and DMR showed that the ratio of the galactic to the CMBR contribution is minimum for the range 60—90 GHz.

7.8.3 The infra-red photometer, DIRBE

The DIRBE photometer measures the intensity of the radiation in the infrared. It records the brightness of the sky in ten photometric bands and in the wavelength range 1—300 pm. It also carries out polarimetric measurements in the range 1—3.5 pm. It uses an absolute radiometer behind a 19 cm telescope and is cooled down to 2 K. It alternates between the sky and the reference at a frequency of 32 Hz. This experiment is dedicated to

the measurement of the infrared background, its spectrum and its angular distribution. Very distant astronomical sources, in particular the first generation of protogalaxies, galaxies and stars, contributed to this background.

Although this provides precious data that could enrich our understanding of a number of cosmological phenomena based on the analysis of the

CMBR, the COBE receiver was not directly conceived to carry out such observations and we shall not analyse it in detail here. 7.8.3.1 The results from DIRBE

The liquid helium used in the DIRBE instrument was used up by 20 September 1990. DIRBE had obtained absolute brightness maps for the sky in ten photometric bands: J (1.2 urn), K (2.3 mm), L (3.4 pm) and M (4.9 mm) as well as the four bands used by the IRAS satellite 12, 25, 60 and 100 um and two further bands at 120—200 and 200—300 mm. The polarisation was measured in bands J, K and L. DIRBE qualitatively confirmed what we expected to find in the infra-red sky. At 1.2 mm, it is emission from stars in the disc of the Galaxy and isolated stars at high altitude that dominates. Zodiacal light, that is light scattered by interplanetary dust, is also important. This Galactic emission and the Zodiacal light dominate up to 3.4 mm. At

7.8 The COBE satellite

175

these wavelengths, images essentially show the disc and the bulge of our Galaxy. The extinction is in fact much less at these wavelengths than it is at optical wavelengths. At 12 and 25 um, emission by interplanetary dust dominates. At 60 um and beyond, emission by interstellar matter dominates the Galactic con— tribution and the interplanetary dust is less important. The ‘infra-red cirrus’ discovered by IRAS appears clearly at all wavelengths. These results have importance for above all interplanetary matter, our Galaxy and the infrared background. 7.8.3.2 The analysis ofthe COBE

It is obvious that great precautions to eliminate all nuisance radiation need to be taken in order to detect such low levels of variation. This is why the

physicists in charge of the observation rejected all data taken whilst the telescope was pointing in directions too close to the Earth and the Moon

when the data taken on one day differed too much from those on other days. (One is interested in cosmological effects, which cannot vary on such

short time scales.) Neither did they include data taken when the position and attitude of the satellite were poorly known. The remaining data are

corrected for the influences of the Moon and Jupiter and for the Doppler effect due to the movement of the satellite (and the Earth around which it is in orbit). The effect of the Earth’s magnetic field was estimated and more or less corrected for; it is no more that a few microkelvins. Other sources

of error were possible, such as those in the microwave region and those due to the orbital environment such as the change of temperature and voltage.

Measurements of the anisotropy

Since the discovery of the CMBR, scientists have become particularly interested in the question of possible anisotropies. In fact, theoretical arguments predicted that they should carry important cosmological information and in particular information about the processes of cosmic structure formation. Up to the beginning of the 19803, observational surveys had been carried out on all angular scales greater than 2 minutes of arc. Apart from the dipole component, none had been recorded down to the

limits of 10—4—10‘3. This was the sort of variation that was expected by the cosmogonic scenarios popular at the time. In the following decade the sensitivity of detection was improved by about an order of magnitude, yet still no anisotropies were found. Thus most of the cosmogonic models then in vogue were eliminated because of this fact. More complex and refined models were then developed, predicting intrinsic anisotropies of the order

of 10-6—10-5. Since then sensitivities have been improved still further. On most angular scales, the uncertainty about the sources of emission is greater than the uncertainty due to experimental error. We do not know whether emission is by discrete sources or background emission by our own Galaxy

through dust, free—free or synchrotron radiation. The first detections took place only early in 1990. Now it is clear that anisotropies have been detected by different teams,

on various angular scales, giving first indications about their angular power spectrum. Preliminary results are compatible with the level, shape and

Gaussian character of the statistics expected from our general ideas about cosmology, the primordial universe and models of structure formation.

Some specific models for the formation of large-scale structures have already been ruled out.

There is at present a lot of activity aimed at measuring the characteristics 177

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Measurements of the anisotropy

of these anisotropies in more detail. The thresholds for sensitivity become

of the same order of magnitude as the predictions of the various scenarios for galaxy formation and the formation of large scale structures, over a wider range of the spectrum. The spectral dependence of the anisotropies begins to be established. More angular scales are being explored. Among the first tasks are the verification of the intrinsic character of the fluctua-

tions (through multiwavelength observations) and of the Gaussian nature of the statistics and the determination of the shape of the angular spectrum

and the level and position of the first Doppler peak. In the more distant future, refined measurements will allow precise measurements of the geometry of the universe and the values of the main cosmological parameters Q, 93, H0 and A and one will be able to check the physics of the early universe (topological defects, inflation, gravitational waves, fossils of quantum cosmology), to probe the scenarios for structure formation, in particular the initial conditions, and to learn about the ionization

history of the universe. This will clearly constitute a very important field of activity in cosmology. We describe below the most recent measurements and results. We start

with angular scales of a few degrees, the first scales on which positive detections have been announced, and then progressively descend towards the smallest scales which are accessible to observations.

8.1 Angular scales 8.1.] Small and large scales

Anisotropies on small and large angular scales are in general treated separately. On the one hand this is because they arise from different physical processes, as we have seen in the first few chapters, and on the other hand because the methods for observing them are not the same, as we

shall explain below. The boundary between the two domains lies at about one or a few degrees.

On high angular scales it is difficult to overcome atmospheric fluctuations. In general one uses small horns with a large opening angle, of the order of 10°. Because they are less cumbersome than antennae, this is the domain of observations by satellite, such as RELIKT and COBE. The

search for fluctuations on large angular scales is important because they could not have been washed out by the effects of re-ionisation, as is

probably the case for those on small angular scales. Intrinsic anisotropies on the scale of a minute of arc and smaller have

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179

certainly been washed out. If such fluctuations on small scales are seen, then they were created by later processes. They can be distinguished from fluctuations of other origin, non-resolved radio galaxies for exam-

ple, by their spectrum, which is thermal or at least not very difierent (for example Comptonised). On small and intermediate angular scales, the atmospheric effects are not very dramatic and it is possible to carry out measurements from the ground with the aid of a radio telescope or

to use balloons. Large antennae with narrow beam widths of about a minute of arc are used. There are two distinct methods of measurement. Beam switching (see section 8.1.2), using just one antenna, and carto—

graphy, in which an interferometer is used in order to attain sufficient resolution. The technology that is used for measuring anisotropies is essentially the same as that used for absolute measurements, which we discussed above.

However, the observational strategy is completely different. Measurement of the absolute intensity of the CMBR is so difficult because it requires an

internal source that is both absolute and at a stable temperature to provide calibration. Measurements of anisotropies are differential; one looks for the existence of small variations of intensity of the spectrum in different directions. Their main difficulty is, however, the extreme weakness of the

differences in intensity on all angular scales, relative to the contribution of our Galaxy. Several parameters need to be optimised when deciding on a strategy for observation:

I the choice of frequency and the pass band, I the choice of the detection system, I the angular scale,

I the number and the spatial distribution of the regions of the sky that one has chosen to analyse and I the statistical methods to be employed to extract the most information from the data. These will depend on the number and spatial distribution of the regions of the sky observed, as well as on the type of apparatus used.

All these parameters depend strongly on the others, as well as on other factors that we shall mention below. The choice of frequencies corresponds more or less to the same demands as in the case of the absolute

measurements. With regard to angular scales, the performance of a detector depends on the so-called window function (which was described above). The narrower and sharper this function the better the perform-

ance. The limits of this function give the sensitivity limits of the

Measurements of the anisotropy

180

equipment at small angles (generally because of the spread of the beam) and at large angles.

The choice of angular scales on which to look for anisotropies is crucial. This is directly related to the process for which one is looking (see the first chapters) and the theoretical model one wants to test. The choice of angular

scale strongly influences the detection system used. Of course, financial considerations also enter the formula.

8.1.2 Ways ofmeasuring the anisotropies 8.1.2.1 Cartography There are generally two possible ways to present data concerning the fluctuations of the CMBR. The first one is to calculate the angular power spectrum, as we explain below in detail. Also it is possible simply to draw

a map of the sky, by measuring the intensity of the radiation region after region. This allows one to compare different results easily and to check the Gaussian character of the statistics. Maps with angular resolutions going from a degree to an arc rninute have already been obtained by the COBE,

MAX and Tenerife experiments. The most sophisticated cartography, with good resolution, requires interferometric methods (described below). Although maps can be produced from single-beam experiments, astronomers have used methods involving two, or more, beams in order to remove

nuisance contributions.

8.1.2.2 Interferometry Interferometry offers many advantages for observing the CMBR. First, an

interferometer measures only the correlated signals of the separated telescopes. Any non-correlated sources, which produce noise for observations

at full aperture, do not perturb the observations. This strongly minimizes the systematic effects caused by atmospheric or ground emission. In

addition, interferometry allows one to image and thus to remove the point sources which would otherwise contaminate the observations. This advan-

tage is particularly important at frequencies for which discrete sources very often dominate the signal (see section 8.7.1). Interferometry is thus adapted for CMBR observations at small angular scales. In particular, in the aperture synthesis technique, several radio

telescopes ‘synthesise’ a large aperture by electronically linking up a number of detectors. These interferometric techniques allow one to form images of small regions of the sky with the aim of finding anisotropies in

8.1 Angular scales

181

the CMBR on angular scales of a few seconds of arc and also to observe

the Sunyaev—Zel’dovich effect.

8.1.2.3 Long-baseline interferometry with the VLA As the name indicates, the (Very Large Array) (VLA) is a very large interferometric network formed by 27 mobile antennae, distributed in the

form of a Y. It is located in New Mexico, USA and operated by the National Radio Astronomy Observatory (NRAO). Each antenna is 25 m in diameter. Their (electronic) combination gives

the angular resolution of an antenna 36 km across, namely 0.04” at best. Its coherent detectors (heterodyne systems) allow one to observe at various bands in the range 300—50 000 MHz (90 to 0.7 cm). This method is

adapted to observations on small angular scales. The VLA has provided some of the most sensitive studies of the anisotropies (section 8.7.1). However, other interferometers have also been used,

such as the three dishes of the Institut de Radioastronomie Millimetrique see (IRAM; section 8.7.2), on the Plateau de Bure, in France, with an array of bolometers. The Nobeyama Millimetre Array (NMA) in Japan is made up of several 10 m dishes to carry out very-long-baseline interferometry

(VLBI). The MilliMeter Array (MMA) of the NRAO will link together 30 or 40

dishes with diameters of around 8 m. With its excellent spatial resolution, it would allow imaging of the Sunyaev—Zel’dovich effect. 8.1.2.4 Beam switching

A technique currently used to measure the fluctuations on angular scales of typically a few minutes of arc involves the rapid switching of the direction of the detector, at a constant frequency, usually in the range

5—50 Hz. This technique is also called ‘chopping’. The constraints that determine the choice of angular scale derive from the size of the

smallest beam width. Radiation is captured by the telescope, in a solid angle that depends on its aperture as well as on the frequency observed.

Because of diffraction due to the finite size of the telescope, the width of the beam in fact is given by 0f : 1.22/1/D, where A is the wavelength of the observed radiation and D is the diameter of the aperture, in this case the primary mirror of the telescope. In order to optimise the

experiment, the angular amplitude of the switching must be considerably greater than the width of the beam. Otherwise there will be too much correlation between the two beams and interference will take place. This

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Measurements of the anisotropy

would be manifested as a very narrow window function (see below). In practice, the separation between the two alternate beams reaching the telescope, that is the angular scale, must be greater than 5—10 times 0f.

To give an example, this technique is used at the European Observatory of Tenerife (see section 8.3.4), with an angular scale of around 5 minutes of arc.

On the other hand, large angular separations create other problems. For example, when it is used on Cassegrain-type telescopes, the beam switching consists of moving the secondary mirror between two positions that

are symmetrically placed in relation to the optical axis, at a frequency of about 5— 10 Hz. In each of the two positions, a definite part of the primary mirror is illuminated, which inevitably produces thermal and emissivity gradients and creates a distortion of the signal. It is possible to get rid of this negative interference by introducing a ‘nodding’ of the telescope,

which effectively compares the intensity from one region with the average of those from the other two placed symmetrically in relation to it. This procedure allows one to correct for a large part of the faults and also for atmospheric emission.

Several groups have used a very subtle technique, which is particularly well adapted to small angular scales. The telescope that receives the CMBR is fixed to the ground and is carried around by the Earth’s rotation. This corresponds to 15° per hour multiplied by the cosine of the latitude.

These observations are carried out at the highest latitude possible, that is as close as possible to the poles. For example, the Partridge group carried out their observations every 20 min, realigning the telescope each time to the original direction. The clear advantage of this procedure is the selection

of a given angular scale, whilst covering a large succession of pairs of directions, in a few hours of observation.

Despite all the precautions and ingenuity used in these observations, other sources of error still remain, such as the inhomogeneity and time variation of atmospheric emission. This variability obviously affects all ground-based experiments and is particularly important for wavelengths

less than a few millimetres. All this interference increases with angular scale and so has a deleterious effect on the sensitivity. The switching is in general square-wave switching: one measures the CMBR at clearly separated positions. An improvement has been introduced

whereby switching is carried out with a sinusoidal wave form. This was used in the MAX, ACME and Saskatoon experiments (described below). Although the analysis is considerably more complex, the results are only slightly different.

8.2 Analysis ofangularfluctuations

183

8.1.2.5 Double switching Measurement by double switching (or double differentiation, or the three-beam method) constitutes a supplementary means of perfecting the

technique. Its use (see sections 8.6.1 and 8.6.2) provided excellent limits on the anisotropies. It involves observing a central field as well as the two symmetrical fields. The estimated quantity is finally the central field minus half the sum of the two lateral fields. In the case of OVRO, for

example, each is 7’ from the central field. In other words one subtracts the mean of the temperatures of the two side fields from the central

temperature. One thus obtains T(0) — [T(+7’) + T(—7’)]/2. This tech— nique has the advantage that it is sensitive only to the second (or higher)

derivatives of the background brightness of the sky, eliminating the firstorder (linear) ones. It is also insensitive to the fluctuations of the receiver. The data from MSAM (see section 8.4.1) were analysed in this

way.

8.2 The analysis of angular fluctuations The CMBR appears to us as a distribution of radiation over the surface of a

sphere (seen from the inside). In other words, it appears as a distribution of temperature

T0?) = To + ATUI) = To[1 + (AT/T)(21)], as a fiinction of angular direction 21 on the sky. Each direction is fixed by the unit vector (1 corresponding for example to the polar co-ordinates 6 (galactic longitude l) and q) (latitude b or colatitude Jr/2 — b). It is the

same irrespective of whether one considers the statistical distribution of the temperature, T, its absolute variation AT, or its relative fluctuation,

AT/ T . In general the results are expressed in terms of the relative fluctuations AT/ T. Their mean value is by definition zero. Their standard deviation, C0 = (A T/ T)rms, plays a very important role because in practice it contaminates all measurements of the fluctuations on non-zero angular

scales, as will be shown by the formulae which relate the measured quantities as functions of the theoretical quantities. One has to be carefiil because various authors use different notations. In particular, the introduced below may concern the absolute (AT) or the relative (AT/ T) fluctuations. In the following, we will consider coefficients and moments concerning (A T/ T), except for the presentation of the

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Measurements ofthe anisotropy

COBE results, where it is explicitly mentioned that they concern AT (see section 8.5.2).

Of course one can represent the fluctuations by a map on the celestial sphere. However, the models are able to predict only the statistics of the fluctuations, not their exact distribution. Statistical studies to compare these predictions with the observations are therefore necessary. In the

following paragraphs we shall describe the methods that are used in this task. This consists of decomposition in terms of spherical harmonics and the autocorrelation function.

8.2.] The harmonic decomposition

In general it is useful to expand any quantity, in our case AT/ T, that is defined over the surface of a sphere in terms of spherical harmonics,

according to the formula AT

A

00

m2!

A

7M) = Z Z aszzm(q),

(8.1)

[=2 mz—l

where the coefficients AT A

A

aim =J d9 —(q>Y/m(q) 4.7!

T

express the power of the spherical harmonic Ylm(f1). Note that the spherical harmonics satisfy the normalisation condition

JdQ Ylm Yl'm' : all’émm’a

where the integral is taken over the entire celestial sphere. Remember also the addition theorem which states that

21+ 1 Z ern(fi)Yln1(q):

4”

P

((008 06),

where P, is a Legendre polynomial and cosa = j) - i] is the cosine of the angle between the two directions considered. The statistics of AT/T is completely determined by the coefficients alm. One could have expanded the temperature itself (or its absolute fluctuation, although this comes down

8.2 Analysis ofangularfluctuations

185

to the same thing since the zeroth-order term is simply the average value over the sky and does not enter the summation) but the usual practice is to

write it in the form above. The dipole term, I = 1, is not usually included in the summation either. In fact it plays a somewhat special role. The corresponding component is much greater than all the others and its origin

is different (see section 8.9). The formula thus applies only to anisotropies on angular scales less than 180°. The study of anisotropies of the CMBR introduces the differences in temperature in directions of a given angular separation, say a. By carrying out an expansion in terms of the modes we see that to talk of an angular scale of a is more or less equivalent to talking

of a particular mode I. The latter can be considered as a measure of the reciprocal of the angle of separation, in radians—1. In other words, a mode 1 corresponds approximately to an angular separation of (60°) / 1. Both notations are used.

The homogeneity of the universe means that the CMBR is globally isotropic, which means that the statistics are rotationally invariant. Moreover, for Gaussian statistics, the coefficients are independent, with random phases. Because of this we define the relative coefficients for each mode I

by

C1: a,:2——11+1";|a1m|2= (101ml)

(8.2)

The squared RMS rotation-invariant multipole moment I is given by

2

T”

ATZl

21+ 1

T3

4%

1

(

)

Most often, observational results are expressed in terms of the variance per logarithmic interval of the angular spectrum,

2 [(21 + nay/(42:).

(8.4)

This is for instance the case for the Saskatoon data (see section 8.3.1). Note, however, a possible confusion in the notations used by various

authors. This implies that (almapmr) : éurémmrai where ( ) is an ensemble average, that is over different (fictitious) universes corresponding to different statistical realisations. The moments of the multipole a] are also

invariant with respect to rotation and the coefficients C1 define the angular power spectrum, whose determination is one of the fundamental goals of the harmonic analysis of the CMBR. In the case of Gaussian statistics, the

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Measurements ofthe anisotropy

latter provide all the information that one can obtain. One should be aware of the fact that not all authors use the same notation. Some use definitions

with additional factors of 4:! or 1/(21+ 1). Also, sometimes the coefficients a 1 or C1 are used to refer to the absolute fluctuations rather than to

the relative fluctuations as we do here. The RMS temperature anisotropy is given by

_ °° (21+4”no

6T T ms

(8.5)

[:2

For an ideal observation of the sky, the variance for measuring C1 is 2C1/(21 + 1). It comes from the fact that the values of C; have a statistical distribution (a x2 distribution) with 2] +1 degrees of freedom in the volume of the universe. This cosmic variance cannot be avoided, since one

observes only one realisation of the universe, which does not reflect the ensemble-averaged value! It affects the comparison between theory and

observation and does not depend on the quality of the experiment performed. According to the formula above, the variance is more significant for the low-l multipoles. For instance, it limits the knowledge of the power spectrum from the COBE results, at l S 20.

The component I = 1 defines the dipole and l = 2 defines the quadrupole (see section 8.9). The latter is written as

2 erms : (AT2)2 : szJOTZ = 5—612—

’

43

3

(86)

The measurements of DMR and COBE gave Qrms = 10.7 :1: 7 uK at the 95% confidence level (see section 8.5.2).

Good angular resolution means that one can obtain the higher order modes. For example, an angle of 5° gives information about the first 10—20 multipoles.

The values of the coefficients a; can be predicted by models of the galaxy and large-scale-structure formation, or rather from the statistical characteristics, in particular the power spectrum, P(k), of the fluctuations at the time of recombination. For models of galaxy formation in which the statistics of the inhomogeneities are Gaussian (cold matter for example) the statistics of the coefficients at,” can be calculated. In fact they are

distributed with a Gaussian distribution, with mean (am) 2 0. The fact that one can calculate the coefficients of any given model allows one to draw up a synthetic map of all the fluctuations of the CMBR, for which the

8.2 Analysis ofangularfluctuations

187

statistics are representative of ‘reality’ such as it is predicted by these models. Most often one is interested in the power spectrum of the anisotropies.

Since most cosmogonic models assume a power law for the density fluctuations, this will also be the case for the CMBR, at least within a

certain domain. For example, Gaussian statistics and a power law for the density fluctuations of the form P(k) o< k" lead to a form of angular

spectrum for the anisotropies given by

2

F[(2[ + n —1)/2]

a, Km,

(8.7)

or rather

a, _ 47rQ2 r[(21+ n — 1)/2]r[(9 — n)/2] (8.8) ’ _ 5T3 r[(2l + 5 — n)/2]r[(3 + n)/2] in the normalised quadruple form. When the quadrupolar moment is used, as in the above formula, to normalise the spectrum, the value referred to, called gas, is not the

measured value of the quadrupole but rather a conventional value that is used for normalisation and expressed by convention on the angular scale of

the quadrupole (see below). These formulae apply to the lowest moments (I S 30), i.e., those at which the Sunyaev—Zel’dovich effect dominates. For

a Harrison—Zel’dovich spectrum, that is one that is scale invariant with n = 1, one obtains

_ 6C2 _24Jr(Q/T0)2

1

C,

(8.9)

_1(l+1)” 51(l+1) OC1(1+1)‘ We recall that the value of Q quoted in this formula is in fact gas. This spectrum corresponds to fluctuations of the gravitational potential that are independent of the spatial scale and is adopted in cold dark matter models,

for example. More refined calculations have led to different formulae,

however. A power law spectrum, denoted by its spectral index n, needs to be normalised. This is done, unfortunately in a way that can often be confusing, by giving the value of the coefficient of the quadrupole in the

spectrum. This is what is called Qrms__ps (PS for power spectrum), or, in a shorter form, gas. It depends on the value of n and should not be confused

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Measurements of the anisotropy

with the real or measured value of the quadrupole, Qms, which, a priori, is completely independent, although they can be expected to have the same order of magnitude. Thus, for example, the COBE observations lead to values of about 10 uK for Qruns but about 15 uK for Qrms_ps. The statistics

that one adopts predict that, if the quadrupole were measured in the fictitious universes, that is for different statistical realisations, then the inferred values of Q would have an ensemble average of Qrms_ps.

8.2.2 The autocorrelation function

We often deal with the autocorrelation function of the temperature fluctuations of the CMBR, as a function of the angle 6. This is defined as

C(19): , T T

(8-10)

and depends only on cos 6 : €11 - in, where 211 and 212 are the unit vectors corresponding to the two directions of observations. The notation 5(6) is sometimes used instead of C(19), but it is better to reserve this for the density correlation function. Some authors designate the autocorrelation function for absolute temperature fluctuations rather than that for the relative fluctuations C(6). The averaging is taken over all accessible direc— tions on the sky with the angle of separation fixed at 6. From the correlation function we can define the power spectrum, Wr(k), which is given by its Hankel transformation. In the general case, knowing W7(k) is not sufficient for determining the statistical distribution, unless

one assumes that one has a Gaussian distribution. A Gaussian distribution can arise from a similar statistics in the distribution of density fluctuations. In any case, it is often assumed that it is so, in order to carry out an analysis

of the observations. Furthermore, most models do predict approximately Gaussian statistics. In the first analysis, ignoring the width of the beam, the measurement of the angular fluctuations for two beams yields the quantity

AT AT 7(6) =7(6>

([T(‘T)1)— T(‘?2)]2>l/2 To

nns

1/2

AT

AT A

AT A

22+