Relativistic cosmology

Tags: Malcolm A. H. Maccallum, Cambridge University Press, Roy Maartens, observations, Cosmology George F. R. Ellis, Relativistic Cosmology, cosmological models, FLRW, universe, George F. R. Ellis, the universe, general relativity, cosmological observations, George Francis Rayner, Bianchi, General dynamics, Background radiation, field equations, University Press, spacetime geometry, George Ellis, Relativistic Cosmology Cosmology, null geodesics, Bianchi identities, University of the Western Cape, Roy Maartens Malcolm A. H. MacCallum, Peter Dunsby, John Wainwright, relativity theory, astronomical observations, modified gravity models, dark energy models, David Wands, Jean-Philippe Uzan, web service, symmetric models, STFC and Royal Society, Christos Tsagas, Preface Reza Tavakol, degrees of freedom
Content: 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 Cosmology has been transformed by dramatic progress in high-precision observations and theoretical modelling. This book surveys key developments and open issues for graduate students and researchers. Using a relativistic geometric approach, it focuses on the general concepts and relations that underpin the Standard Model of the Universe. Part 1 covers foundations of relativistic cosmology, whilst Part 2 develops the dynamical and observational relations for all models of the Universe based on general relativity. Part 3 focuses on the standard model of cosmology, including inflation, DARK MATTER, dark energy, Perturbation theory, the cosmic microwave background, structure formation and gravitational lensing. It also examines modified gravity and inhomogeneity as possible alternatives to dark energy. Anisotropic and inhomogeneous models are described in Part 4, and Part 5 reviews deeper issues, such as quantum cosmology, the start of the universe and the multiverse proposal. Colour versions of some figures are available at www.cambridge.org/9780521381154. George F. R. Ellis FRS is Professor Emeritus at the University of Cape Town, South Africa. He is co-author with Stephen Hawking of The Large Scale Structure of Space-Time. Roy Maartens holds an SKA Research Chair at the University of the Western Cape, South Africa, and is Professor of Cosmology at the University of Portsmouth, UK. Malcolm A. H. MacCallum is Director of the Heilbronn Institute at Bristol, and is President of the International Society on General Relativity and Gravitation.
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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 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.
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
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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
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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
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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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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
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
© 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
© in this web service Cambridge University Press
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
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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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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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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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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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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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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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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