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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information Relativistic Cosmology GEORGE F. R. ELLIS University of Cape Town ROY MAARTENS University of Portsmouth and University of the Western Cape MALCOLM A. H. MACCALLUM University of Bristol

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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sгo Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

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© Cambridge University Press 2012

This publication 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 2012

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Ellis, George F. R. (George Francis Rayner)

Relativistic cosmology / George Ellis, Roy Maartens, Malcolm MacCallum.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-521-38115-4

1. Cosmology. 2. Relativistic astrophysics. 3. Relativistic quantum theory.

I. Maartens, R. (Roy) II. MacCallum, M. A. H. III. Title.

QB981.E4654 2012

523.1dc23

2011040518

ISBN 978-0-521-38115-4 Hardback

Additional resources for this publication at www.cambridge.org/9780521381154.

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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

Contents

Preface Part 1 Foundations 1 The nature of cosmology 1.1 The aims of cosmology 1.2 Observational evidence and its limitations 1.3 A summary of current observations 1.4 Cosmological concepts 1.5 Cosmological models 1.6 Overview 2 Geometry 2.1 Manifolds 2.2 Tangent vectors and 1-forms 2.3 Tensors 2.4 Lie derivatives 2.5 Connections and covariant derivatives 2.6 The curvature tensor 2.7 Riemannian geometry 2.8 General bases and tetrads 2.9 Hypersurfaces 3 Classical physics and gravity 3.1 Equivalence principles, gravity and local physics 3.2 Conservation equations 3.3 The field equations in relativity and their structure 3.4 Relation to Newtonian theory Part 2 Relativistic cosmological models 4 Kinematics of cosmological models 4.1 Comoving coordinates 4.2 The fundamental 4-velocity 4.3 Time derivatives and the acceleration vector 4.4 Projection to give three-dimensional relations v © in this web service Cambridge University Press

page xi 3 3 5 9 17 20 23 25 26 28 31 34 35 37 39 51 53 56 56 61 64 69 73 73 74 75 76 www.cambridge.org

Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

vi

Contents

4.5 Relative position and velocity

79

4.6 The kinematic quantities

80

4.7 Curvature and the Ricci identities for the 4-velocity

86

4.8 Identities for the projected covariant derivatives

88

5 Matter in the universe

89

5.1 Conservation laws

90

5.2 Fluids

95

5.3 Multiple fluids

101

5.4 Kinetic theory

104

5.5 Electromagnetic fields

110

5.6 scalar fields

115

5.7 quantum field theory

117

6 Dynamics of cosmological models

119

6.1 The RaychaudhuriEhlers equation

119

6.2 Vorticity conservation

124

6.3 The other Einstein field equations

126

6.4 The Weyl tensor and the Bianchi identities

132

6.5 The orthonormal 1+3 tetrad equations

134

6.6 Structure of the 1+3 system of equations

139

6.7 Global structure and singularities

143

6.8 Newtonian models and Newtonian limits

147

7 Observations in cosmological models

153

7.1 Geometrical optics and null geodesics

153

7.2 Redshifts

156

7.3 Geometry of null geodesics and images

159

7.4 Radiation energy and flux

161

7.5 Specific intensity and apparent brightness

167

7.6 Number counts

170

7.7 Selection and detection issues

171

7.8 Background radiation

172

7.9 Causal and visual horizons

173

8 Light-cone approach to relativistic cosmology

180

8.1 Model-based approach

180

8.2 Direct observational cosmology

181

8.3 Ideal cosmography

186

8.4 Field equations: determining the geometry

187

8.5 Isotropic and partially isotropic observations

190

8.6 Implications and opportunities

194

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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

vii

Contents

Part 3 The standard model and extensions

9 Homogeneous FLRW universes

201

9.1 FLRW geometries

202

9.2 FLRW dynamics

210

9.3 FLRW dynamics with barotropic fluids

212

9.4 Phase planes

220

9.5 Kinetic solutions

225

9.6 Thermal history and contents of the universe

226

9.7 Inflation

238

9.8 Origin of FLRW geometry

246

9.9 Newtonian case

247

10 Perturbations of FLRW universes

249

10.1 The gauge problem in cosmology

250

10.2 Metric-based perturbation theory

251

10.3 Covariant nonlinear perturbations

262

10.4 Covariant linear perturbations

267

11 The cosmic background radiation

282

11.1 The CMB and spatial homogeneity: nonlinear analysis

282

11.2 Linearized analysis of distribution multipoles

287

11.3 Temperature anisotropies in the CMB

292

11.4 Thomson scattering

294

11.5 Scalar perturbations

295

11.6 CMB polarization

300

11.7 Vector and tensor perturbations

303

11.8 Other background radiation

303

12 Structure formation and gravitational lensing

307

12.1 Correlation functions and power spectra

307

12.2 Primordial perturbations from inflation

309

12.3 Growth of density perturbations

317

12.4 Gravitational lensing

330

12.5 Cosmological applications of lensing

339

13 Confronting the Standard Model with observations

345

13.1 Observational basis for FLRW models

346

13.2 FLRW observations: probing the background evolution

351

13.3 Almost FLRW observations: probing structure formation

355

13.4 Constraints and consistency checks

363

13.5 Concordance model and further issues

366

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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

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Contents

14 Acceleration from dark energy or modified gravity

370

14.1 Overview of the problem

370

14.2 Dark energy in an FLRW background

373

14.3 Modified gravity in a RW background

376

14.4 Constraining effective theories

390

14.5 Conclusion

391

15 `Acceleration' from large-scale inhomogeneity?

395

15.1 LemaоtreTolmanBondi universes

395

15.2 Observables and source evolution

399

15.3 Can we fit area distance and number count observations?

401

15.4 Testing background LTB with SNIa and CMB distances

403

15.5 Perturbations of LTB

406

15.6 Observational tests of spatial homogeneity

411

15.7 Conclusion: status of the Copernican Principle

415

16 `Acceleration' from small-scale inhomogeneity?

416

16.1 Different scale descriptions

416

16.2 Cosmological backreaction

421

16.3 Specific models: almost FLRW

423

16.4 Inhomogeneous models

426

16.5 Importance of backreaction effects?

432

16.6 Effects on observations

435

16.7 Combination of effects: altering cosmic concordance?

440

16.8 Entropy and coarse-graining

441

Part 4 Anisotropic and inhomogeneous models

17 The space of cosmological models

447

17.1 Cosmological models with symmetries

447

17.2 The equivalence problem in cosmology

452

17.3 The space of models and the role of symmetric models

453

18 Spatially homogeneous anisotropic models

456

18.1 KantowskiSachs universes: geometry and dynamics

457

18.2 Bianchi I universes: geometry and dynamics

458

18.3 Bianchi geometries and their field equations

462

18.4 Bianchi universe dynamics

467

18.5 Evolution of particular Bianchi models

474

18.6 Cosmological consequences

481

18.7 The Bianchi degrees of freedom

486

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Contents

19 Inhomogeneous models

488

19.1 LTB revisited

490

19.2 Swiss cheese revisited

491

19.3 Self-similar models

493

19.4 Models with a G3 acting on S2

495

19.5 G2 cosmologies

496

19.6 The SzekeresSzafron family

498

19.7 The StephaniBarnes family

501

19.8 Silent universes

501

19.9 General dynamics of inhomogeneous models

502

19.10 Cosmological applications

503

Part 5 Broader perspectives

20 Quantum gravity and the start of the universe

511

20.1 Is there a quantum gravity epoch?

511

20.2 Quantum gravity effects

512

20.3 String theory and cosmology

516

20.4 Loop quantum gravity and cosmology

526

20.5 Physics horizon

530

20.6 Explaining the universe the question of origins

532

21 Cosmology in a larger setting

535

21.1 Local physics and cosmology

535

21.2 Varying `constants'

539

21.3 Anthropic question: fine-tuning for life

542

21.4 Special or general? Probable or improbable?

546

21.5 Possible existence of multiverses

548

21.6 Why is the universe as it is?

554

22 Conclusion: our picture of the universe

555

22.1 A coherent view?

555

22.2 Testing alternatives: probing the possibilities

558

22.3 Limits of cosmology

559

Appendix Some useful formulae

561

A.1 Constants and units

561

A.2 1+3 covariant equations

563

A.3 Frequently used acronyms

565

References

566

Index

606

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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

© in this web service Cambridge University Press

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Cambridge University Press 978-0-521-38115-4 - Relativistic Cosmology George F. R. Ellis, Roy Maartens and Malcolm A. H. Maccallum Frontmatter More information

Preface

This book provides a survey of modern cosmology emphasizing the relativistic approach. It is shaped by a number of Guiding Principles. · Adopt a geometric approach Cosmology is crucially based in spacetime geometry, because the dominant force shaping the universe is gravity; and the best classical theory of gravity we have is Einstein's general theory of relativity, which is at heart a geometric theory. One should therefore explore the spacetime geometry of cosmological models as a key feature of cosmology. · Move from general to special One can best understand the rather special models most used in cosmology by understanding relationships which hold in general, in all spacetimes, rather than by only considering special high symmetry cases. The properties of these solutions are then seen as specialized cases of general relations. · Explore geometric as well as matter degrees of freedom As well as exploring matter degrees of freedom in cosmology, one should examine the geometric degrees of freedom. This applies in particular in examining the possible explanations of the apparent acceleration of the expansion of the universe in recent times. · Determine exact properties and solutions where possible Because of the nonlinearity of the Einstein field equations, approximate solutions may omit important aspects of what occurs in the full theory. Realistic solutions will necessarily involve approximation methods, but we aim where possible to develop exact relations that are true generically, on the one hand, and exact solutions of the field equations that are of cosmological interest, on the other. · Explore the degree of generality or speciality of models A key theme in recent cosmological writing is the idea of `fine tuning', and it is typically taken to be bad if a universe model is rather special. One can, however, only explore the degree of speciality of specific models by embedding them in a larger context of geometrically and physically more general models. · Clearly relate theory to testability Because of the special nature of cosmology, theory runs into the limits of the possibility of observational testing. One should therefore pursue all possible observational consistency checks, and be wary of claiming theories as scientific when they may not in principle be testable observationally. · Focus on physical and cosmological relevance The physics proposed should be plausibly integrated into the rest of physics, where it is not directly testable; and the cosmological models proposed should be observationally testable, and be relevant to the astronomical situation we see around us. xi

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xii

Preface

· Search for enduring rather than ephemeral aspects We have attempted to focus on issues that appear to be of more fundamental importance, and therefore will not fade away, but will continue to be of importance in cosmological studies in the long term, as opposed to ephemeral topics that come and go. Part 1 presents the foundations of relativistic cosmology. Part 2 is a comprehensive discussion of the dynamical and observational relations that are valid in all models of the universe based on general relativity. In particular, we analyse to what extent the geometry of spacetime can be determined from observations on the past light-cone. The standard FriedmannLemaоtreRobertsonWalker (FLRW) universes are discussed in depth in Part 3, covering both the background and perturbed models. We present the theory of perturbations in both the standard coordinate-based and the 1+3 covariant approaches, and then apply the theory to inflation, the cosmic microwave background, structure formation and gravitational lensing. We review the key unsolved issue of the apparent acceleration of the expansion of the universe, covering dark energy models and modified gravity models. Then we look at alternative explanations in terms of large scale inhomogeneity or small scale inhomogeneity. Anisotropic homogeneous (Gцdel, Kantowski-Sachs and Bianchi) and inhomogeneous universes (including the Szekeres models) are the focus of Part 4, giving the larger context of the family of possible models that contains the standard FLRW models as a special case. In all cases the relation of the models to astronomical observations is a central feature of the presentation. The text concludes in Part 5 with a brief review of some of the deeper issues underlying all cosmological models. This includes quantum gravity and the start of the universe, the relation between local physics and cosmology, why the universe is so special that it allows intelligent life to exist, and the issue of testability of proposals such as the multiverse. The text is at an advanced level; it presumes a basic knowledge of general relativity (e.g. as in the recent introductory texts of Carroll (2004), Stephani (2004), Hobson, Efstathiou and Lasenby (2006) and Schutz (2009)) and of the broad nature of cosmology and cosmological observations (e.g. as in the recent introductory books of Harrison (2000), Ferreira (2007) and Silk (2008)). However, we provide a self-contained, although brief, survey of Riemannian geometry, general relativity and observations. Our approach is similar to that of our previous reviews, Ellis (1971a, 1973), MacCallum (1973, 1979), Ellis and van Elst (1999a) and Tsagas, Challinor and Maartens (2008), and it builds on foundations laid by Eisenhart (1924), Synge (1937), Heckmann and Schucking (1962), Ehlers (1961), Trьmper (1962, and unpublished), Hawking (1966) and Kristian and Sachs (1966). This approach differs from the approach in the excellent recent texts by Peacock (1999), Dodelson (2003), Mukhanov (2005), Weinberg (2008), Durrer (2008), Lyth and Liddle (2009) and Peter and Uzan (2009), in that we emphasize a covariant and geometrical approach to curved spacetimes and where possible consider general geometries instead of restricting considerations to the FLRW geometries that underlie the standard models of cosmology.

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xiii

Preface

A further feature of our presentation is that although it is solidly grounded in relativity theory, we recognize the usefulness of Newtonian cosmological models and calculations. We detail how the Newtonian limit follows from the relativistic theory in situations of cosmological interest, and make clear when Newtonian calculations give a good approximation to the results of the relativistic theory and when they do not. It is not possible to cover all of modern cosmology in depth in one book. We present a summary of present cosmological observations and of modern astrophysical understanding of cosmology, drawing out their implications for the theoretical models of the universe, but we often refer to other texts for in-depth coverage of particular topics. We are relatively complete in the theory of relativistic cosmological models, but even here the literature is so vast that we are obliged to refer to other texts for fuller details. In particular, the very extensive discussions of spatially homogeneous cosmologies and of inhomogeneous cosmologies in the books by Wainwright and Ellis (1997), Krasinґski (1997), and Bolejko et al. (2010) complement and extend our much shorter summaries of those topics in Part 4. Our guiding aim is to present a coherent core of theory that is not too ephemeral, i.e. that in our opinion will remain significant even when some present theories and observations have fallen away. Only the passage of time will tell how good our judgement has been. We have given numerical values for the key cosmological parameters, but these should be interpreted only as indicative approximations. The values and their error bars change as observations develop, so that no book can give definitive values. Furthermore, there are inherent limitations to parameter values and error bars which depend on the particular observations used, on the assumptions made in reducing the observational data, on the chosen theoretical model needed to interpret the observations, and on the type of statistical analysis used. In the text we have two kinds of interventions apart from the usual apparatus of footnotes and references: namely, exercises and problems. The Exercises enable the reader to develop and test his or her understanding of the main material; we believe we know the answers to all the exercises, or at least where the answer is given in the literature (in which case an appropriate reference is provided). By contrast, the Problems are unsolved questions whose solution would be of some interest, or in some cases would be a major contribution to our understanding. We are grateful to numerous people who have played an important role in developing our understanding of cosmology: we cannot name them all (though most of their names will be found in the reference list at the end), but we would particularly like to thank John Barrow, Bruce Bassett, Hermann Bondi,1 Marco Bruni, Anthony Challinor, Chris Clarkson, Peter Coles, Rob Crittenden, Peter Dunsby, Ruth Durrer, Jьrgen Ehlers,1 Henk van Elst, Pedro Ferreira, Stephen Hawking, Charles Hellaby, Kazuya Koyama, Julien Larena, David Matravers, Charles Misner, Jeff Murugan, Bob Nichol, Roger Penrose, Felix Pirani, Alan Rendall, Wolfgang Rindler, Tony Rothman, Rainer Sachs, Varun Sahni, Misao Sasaki, Bernd Schmidt, Engelbert Schucking, Dennis Sciama,1 Stephen Siklos, John Stewart, Bill Stoeger, 1 deceased

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xiv

Preface

Reza Tavakol, Manfred Trьmper, Christos Tsagas, Jean-Philippe Uzan, John Wainwright and David Wands for insights that have helped shape much of what is presented here. We thank the FRD and NRF (South Africa), the STFC and Royal Society (UK), and our departments, for financial support that has contributed to this work. George F. R. Ellis Roy Maartens Malcolm A. H. MacCallum

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