Structures in the Universe by Exact Methods: Formation, Evolution, Interactions

258 Pages·2009·3.44 MB·English

by  Krzysztof Bolejko

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STRUCTURES IN THE UNIVERSE BY
EXACT METHODS
Formation, Evolution, Interactions
AsthestructuresinourUniversearemappedoutoneverlargerscales,andwith
increasingdetail,theuseofinhomogeneousmodelsisbecominganessentialtool
for analysing and understanding them.
This book reviews a number of important developments in the application of
inhomogeneoussolutionsofEinstein’sfieldequationstocosmology.Itshowshow
inhomogeneous models can be used to study the evolution of structures such as
galaxy clusters and galaxies with central black holes, and to account for cos-
mological observations such as supernova dimming, the cosmic microwave back-
ground, baryon acoustic oscillations or the dependence of the Hubble parameter
on redshift within classical general relativity.
Whatever‘darkmatter’and‘darkenergy’turnouttobe,inhomogeneitiesexist
on many scales and need to be investigated by all appropriate methods. This
book will be of great value to astrophysicists and cosmologists, from graduate
students to academic researchers.
Krzysztof BolejkoisanAssistantProfessorattheNicolausCopernicus
Astronomical Center, Polish Academy of Sciences, Poland. His area of research
is cosmology, in particular applications of inhomogeneous cosmological models,
study of light propagation, and backreaction effects.
Andrzej Krasin´ski is an Associate Professor at Nicolaus Copernicus
AstronomicalCenter,PolishAcademyofSciences,Poland.Heiscurrentlydoing
research in gravitation theory, in particular exact solutions of Einstein’s equa-
tions and inhomogeneous cosmological models. He is also a member of the edi-
torial board of the GRG Journal, where he is the editor of the “Golden Oldies”
series of reprints of classical papers on relativity.
Charles Hellaby is an Associate Professor in the Department of Math-
ematics and Applied Mathematics, University of Cape Town, South Africa. His
research interests are in cosmology and relativity, particularly properties and
applicationsofinhomogeneousmetrics,andrelatingobservationstocosmological
models.
Marie-No¨elle Ce´le´rier is a Researcher at the Laboratoire Univers et
Th´eories (LUTH), Observatoire de Paris-Meudon, France. Her two main fields
of interest are scale relativity and inhomogeneous cosmology.
CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS
General editors: P. V. Landshoff, D. R. Nelson, S. Weinberg
S. J. Aarseth GravitationalN-BodySimulations: ToolsandAlgorithms
J.Ambjørn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheoryApproach
A. M. Anile RelativisticFluidsandMagneto-fluids:WithApplications inAstrophysicsand
PlasmaPhysics
J. A. de Azc´arraga and J. M. Izquierdo LieGroups,LieAlgebras,CohomologyandSome
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O. Babelon,D. Bernard and M. Talon IntroductiontoClassicalIntegrableSystems†
F. Bastianelliand P.van Nieuwenhuizen PathIntegralsandAnomaliesinCurvedSpace
V. Belinskiand E. Verdaguer Gravitational Solitons
J. Bernstein KineticTheoryintheExpandingUniverse
G. F. Bertsch and R. A. Broglia OscillationsinFiniteQuantumSystems
N. D. Birrelland P. C. W. Davies QuantumFieldsinCurvedSpace†
K. Bolejko, A. Krasin´ski,C. Hellaby and M.-N. C´el´erier StructuresintheUniversebyExact
Methods: Formation,Evolution,Interactions
M. Burgess ClassicalCovariantFields
E. A. Calzetta and B.-L. B.Hu NonequilibriumQuantumFieldTheory
S. Carlip QuantumGravityin2+1Dimensions†
P.Cartier and C. DeWitt-Morette FunctionalIntegration:ActionandSymmetries
J. C. Collins Renormalization: AnIntroductiontoRenormalization, theRenormalization Group
andtheOperator-Product Expansion†
M. Creutz Quarks,GluonsandLattices†
P.D. D’Eath SupersymmetricQuantumCosmology
F. de Felice and C. J. S Clarke RelativityonCurvedManifolds
B. DeWitt Supermanifolds,2nd edition†
P.G. O Freund Introduction toSupersymmetry†
J. A. Fuchs AffineLieAlgebrasandQuantumGroups:AnIntroduction, withApplicationsin
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J. Fuchs and C.Schweigert Symmetries,LieAlgebrasandRepresentations: AGraduateCourse
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Y. Fujii and K. Maeda TheScalar-TensorTheoryofGravitation
A. S. Galperin,E.A. Ivanov, V. I.Orievetsky and E. S. Sokatchev HarmonicSuperspace
R. Gambini and J. Pullin Loops, Knots,GaugeTheoriesandQuantumGravity†
T.Gannon MoonshinebeyondtheMonster:TheBridgeConnectingAlgebra,ModularFormsand
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M. Go¨ckeler and T. Schu¨cker DifferentialGeometry,GaugeTheoriesandGravity†
C. Go´mez,M. Ruiz-Altaba and G.Sierra QuantumGroupsinTwo-DimensionalPhysics
M. B. Green, J. H. Schwarz and E. Witten SuperstringTheoryVolume1:Introduction†
M. B. Green, J. H. Schwarz and E. Witten SuperstringTheoryVolume2:Loop Amplitudes,
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V.N.Gribov TheTheoryofComplexAngularMomenta:GribovLecturesonTheoreticalPhysics
S. W.Hawking and G.F. R.Ellis TheLargeScaleStructureofSpace-Time†
F. Iachello and A. Arima TheInteractingBosonModel
F. Iachello and P.van Isacker TheInteractingBoson–FermionModel
C. Itzykson and J. M. Drouffe Statistical FieldTheoryVolume1:FromBrownianMotionto
Renormalization andLatticeGaugeTheory†
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Methods, ConformalFieldTheoryandRandomSystems†
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P.S. JoshiGravitationalCollapseandSpacetimeSingularities
J. I. Kapusta and C. Gale Finite-TemperatureFieldTheory:PrinciplesandApplications,2nd
edition
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CorrelationFunctions†
M. Le Bellac ThermalFieldTheory†
Y. Makeenko MethodsofContemporaryGaugeTheory
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N. H. March LiquidMetals:ConceptsandTheory
I. Montvay and G. Mu¨nster QuantumFieldsonaLattice†
L. O’Raifeartaigh GroupStructureofGaugeTheories†
T. Ort´ın GravityandStrings†
A. M. Ozorio de Almeida HamiltonianSystems:ChaosandQuantization†
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J. PolchinskiStringTheoryVolume1:AnIntroduction totheBosonicString
J. PolchinskiStringTheoryVolume2:SuperstringTheoryandBeyond
V. N. Popov FunctionalIntegralsandCollectiveExcitations†
R. J. Rivers PathIntegralMethodsinQuantumFieldTheory†
R. G. Roberts TheStructureoftheProton:DeepInelasticScattering†
C. Rovelli QuantumGravity†
W. C. Saslaw GravitationalPhysicsofStellarandGalacticSystems†
M. Shifman and A. Yung SupersymmetricSolitons
H. Stephani,D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt ExactSolutionsof
Einstein’sFieldEquations,2nd edition
J. Stewart AdvancedGeneralRelativity†
T. Thiemann ModernCanonicalQuantumGeneralRelativity
D. J. Toms TheSchwingerActionPrincipleandEffectiveAction
A. Vilenkin and E. P.S. Shellard CosmicStringsandOtherTopological Defects†
R. S. Ward and R.O. Wells,Jr TwistorGeometryandFieldTheory†
J. R. Wilson and G. J. Mathews RelativisticNumericalHydrodynamics
† Issued as a paperback.
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Structures in the Universe by
Exact Methods
Formation, Evolution, Interactions
KRZYSZTOF BOLEJKO
N. Copernicus Astronomical Center, Polish Academy of Sciences, Poland
ANDRZEJ KRASIN´ SKI
N. Copernicus Astronomical Center, Polish Academy of Sciences, Poland
CHARLES HELLABY
University of Cape Town, South Africa
MARIE-NOE¨LLE CE´LE´RIER
Laboratoire Univers et Th´eories (LUTH), Observatoire de Paris, France
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CAMBRIDGE UNIVERSITY PRESS
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The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521769143
© K. Bolejko, A. Krasinski, C. Hellaby and M.-N. C´el´erier 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13 978-0-511-65822-8 eBook (NetLibrary)
ISBN-13 978-0-521-76914-3 Hardback
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.
Contents
Foreword page xi
Acknowledgements xiv
1 The purpose of this book 1
I THEORETICAL BACKGROUND
2 Exact solutions of Einstein’s equations that are
used in cosmology 11
2.1 The Lemaˆıtre–Tolman model 11
2.1.1 Basic properties 11
2.1.2 Origin conditions 13
2.1.3 Shell crossings, maxima and minima 14
2.1.4 L–T models of cosmic structures 15
2.1.5 Defining a model by initial and final profiles with Λ=0 16
2.1.6 Additional methods of specifying models 22
2.1.7 Defining a model by initial and final profiles with Λ>0 33
2.1.8 A unified new form of the L–T solution 39
2.2 The Lemaˆıtre model 41
2.3 The Szekeres solution 42
2.3.1 Definition and metric 42
2.3.2 General properties of the Szekeres solutions 45
2.3.3 Properties of the quasi-spherical Szekeres solution 47
3 Light propagation 55
3.1 Light propagation in Lemaˆıtre–Tolman models 55
3.1.1 Central observer 55
3.1.2 Off-centre observer. SC method 56
3.1.3 Off-centre observer. AA method 58
3.2 Apparent and event horizons in the L–T model 60
3.2.1 Apparent horizons 61
3.2.2 Event horizons 63
3.2.3 Locating the AH− during expansion in elliptic regions 63
3.2.4 AH− near an origin in elliptic regions 64
3.2.5 AH+ during collapse in elliptic regions 65
3.2.6 AH− in the parabolic limit 66
viii Contents
3.2.7 Apparent horizons in parabolic regions 66
3.2.8 Locating apparent horizons in hyperbolic regions 67
3.2.9 Locating the event horizon 67
3.2.10 An illustration – a simple recollapsing model 67
3.3 Null geodesics in the quasi-spherical Szekeres model 74
3.3.1 Constant (x,y) geodesics 75
3.3.2 The redshift formula for the axial geodesics 79
3.4 Junction of null geodesics in the Szekeres Swiss-cheese model 81
3.5 The apparent horizons in the Szekeres model 83
II APPLICATIONS OF THE MODELS IN COSMOLOGY
4 Structure formation 91
4.1 Initial conditions 91
4.1.1 Transforming scales in the background 91
4.1.2 Initial density and velocity fluctuations 92
4.2 The galaxy plus black hole formation model 93
4.2.1 A galaxy with a central black hole 94
4.2.2 The black hole interior 95
4.2.3 A collapsed body 96
4.2.4 A wormhole 97
4.2.5 The exterior galaxy model 97
4.2.6 Numerical evolution of the models 101
4.3 Modelling a rich galaxy cluster 105
4.3.1 The most realistic model 105
4.3.2 Results of some other numerical experiments 106
4.4 Modelling a void 109
4.4.1 The algorithm 109
4.4.2 Void formation 113
4.4.3 Void evolution 114
4.5 Impact of radiation on void formation 116
4.5.1 The algorithm 118
4.5.2 Results 120
4.6 Evolution of cosmic structures in different environments 126
4.6.1 The algorithm 126
4.6.2 Double structures 128
4.6.3 Connection to the large-scale structure of the Universe 134
5 The cosmological constant and coincidence problems 137
5.1 Constraints on model parameters from the SN Ia data 138
5.1.1 The ΛCDM cosmological model 140
5.1.2 L–T models with zero cosmological constant 142
5.1.3 Illustration: flat L–T (Λ=0) models 144
5.1.4 Apparent acceleration 146
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