COSMOLOGICAL MODELS CARGESE` LECTURES 1998 GEORGE FRELLIS∗ and HENK VAN ELST† Cosmology Group Department of Mathematics and Applied Mathematics University of Cape Town, Rondebosch 7701, Cape Town, South Africa September 2, 2008 Abstract The aim of this set of lectures is a systematic presentation of a 1 + 3 covariant approach to studying the geometry, dynamics, and observational properties of relativistic cosmological models. In giving (i) the 1 + 3 arXiv:gr-qc/9812046v5 2 Sep 2008 basic covariant relations for a cosmological fluid, the present lectures cover some of the same ground as a previous set of Carg`ese lectures [7], but they then go on to give (ii) the full set of corresponding tetrad equations, (iii) a classification of cosmological models with exact symmetries, (iv) a brief discussion of some of the most useful exact models and their observational properties, and (v) an introduction to the gauge-invariant and 1+3 covariant perturbation theory of almost-Friedmann–Lemaˆıtre–Robertson–Walker universes, with a fluid description for the matter and a kinetic theory description of the radiation. e-print arXiv:gr-qc/9812046v5 ∗Electronic address: [email protected] †Electronic address: [email protected] 1 CONTENTS 2 Contents 1 Basic relations 5 2 1+3 covariant description 6 2.1 Variables ....................................... ..... 6 2.1.1 Average4-velocityofmatter . ......... 6 2.1.2 Kinematicalquantities . ........ 7 2.1.3 Mattertensor .................................. .... 8 2.1.4 Maxwellfieldstrengthtensor. ......... 8 2.1.5 Weylcurvaturetensor . ...... 8 2.1.6 Auxiliaryquantities. ........ 9 2.2 1+3 covariantpropagationandconstraintequations . ............ 9 2.2.1 Ricciidentities ............................... ...... 10 2.2.2 Twice-contractedBianchiidentities . ............. 10 2.2.3 OtherBianchiidentities . ........ 11 2.2.4 Maxwell’sfieldequations . ....... 12 2.3 Pressure-freematter(‘dust’) . ............. 12 2.4 Irrotationalflow................................. ........ 12 2.5 Implications.................................... ....... 13 2.5.1 Energyequation ................................ .... 13 2.5.2 Basicsingularitytheorem . ........ 14 2.5.3 Relationsbetweenimportantparameters . ............. 15 2.6 Newtoniancase ................................... ...... 15 2.7 Solutions ....................................... ..... 16 3 Tetrad description 16 3.1 Generaltetradformalism . .......... 16 3.2 Tetradformalismincosmology. ........... 18 3.2.1 Constraints ................................... .... 18 3.2.2 Evolutionofspatialcommutationfunctions . .............. 19 3.2.3 Evolutionofkinematicalvariables . ............ 20 3.2.4 Evolutionof matter andWeyl curvaturevariables . ............... 20 3.3 Completeset ..................................... ..... 21 4 FLRW models and observational relations 21 4.1 Coordinatesandmetric . ......... 21 4.2 Dynamicalequations .............................. ........ 22 4.2.1 Basicparameters ............................... ..... 22 4.2.2 Singularityandages . ... .... .... .... .... ... .... ...... 22 4.3 Exactandapproximatesolutions . ............ 23 4.3.1 Simplestmodels ................................ .... 23 4.3.2 Parametricsolutions . ....... 24 4.3.3 Early-timesolutions . ....... 24 4.3.4 Scalarfield .................................... ... 24 4.3.5 Kinetictheory................................. ..... 25 4.4 Phaseplanes..................................... ...... 25 4.5 Observations .................................... ...... 25 4.5.1 Redshift...................................... ... 25 4.5.2 Areas ......................................... 26 4.5.3 Luminosityandreciprocitytheorem . ........... 27 CONTENTS 3 4.5.4 Specificintensity .............................. ...... 28 4.5.5 Numbercounts .................................. ... 28 4.6 Observationallimits. .......... 29 4.6.1 Smalluniverses................................ ..... 30 4.7 FLRWuniversesascosmologicalmodels . ............ 31 4.8 Generalobservationalrelations . .............. 32 5 Solutions with symmetries 33 5.1 Symmetriesofcosmologies. .......... 33 5.1.1 Killingvectorfields. ....... 33 5.1.2 Groupsofisometries . ...... 34 5.1.3 Dimensionalityofgroupsandorbits . ........... 34 5.2 Classificationofcosmologicalsymmetries . ................ 35 5.2.1 Space-timehomogeneousmodels . ........ 35 5.2.2 Spatiallyhomogeneousuniverses . .......... 37 5.2.3 Spatiallyinhomogeneousuniverses . ........... 37 5.3 Bianchi Type I universes (s =3)................................ 38 5.4 Lemaˆıtre–Tolman–Bondi family (s =2)............................ 40 5.5 Swiss-Cheesemodels. .... ... .... .... .... .... ... ... ........ 42 6 Bianchi models 44 6.1 ConstructingBianchimodels . ........... 44 6.2 DynamicsofBianchimodels . ......... 45 6.2.1 Chaosintheseuniverses? . ....... 45 6.2.2 Horizonsandwhimpersingularities . ........... 47 6.2.3 Isotropisationproperties . .......... 47 6.3 Observationalrelations . ........... 48 6.4 Dynamicalsystemsapproach . .......... 48 6.4.1 Reduceddifferentialequations . ........... 48 6.4.2 Equationsandorbits . ... .... .... .... .... ... .... ...... 50 6.4.3 Equilibriumpoints and self-similar cosmologies . ................ 50 6.4.4 Phaseplanes ................................... ... 51 7 Almost-FLRW models 52 7.1 Gaugeproblem .................................... ..... 52 7.1.1 Keyvariables.................................. .... 53 7.2 Dynamicalequations .............................. ........ 54 7.2.1 Growthofinhomogeneity . ...... 54 7.3 Dust............................................ ... 55 7.3.1 Otherquantities............................... ...... 56 7.4 Perfectfluids .................................... ...... 56 7.4.1 Second-orderequations. ........ 57 7.4.2 Harmonicdecomposition. ....... 58 7.5 Implications.................................... ....... 59 7.5.1 Jeansinstability.... ... .... .... .... .... ... .... ....... 59 7.5.2 Short-wavelengthsolutions. .......... 60 7.5.3 Long-wavelengthsolutions. ......... 60 7.5.4 Changeofbehaviourwithtime. ........ 60 7.6 Othermatter..................................... ...... 61 7.6.1 Scalarfields................................... .... 61 7.6.2 Multi-fluidsandimperfectfluids . .......... 61 CONTENTS 4 7.6.3 Magneticfields ................................. .... 62 7.6.4 Newtonianversion .............................. ..... 62 7.6.5 Alternativegravity . ....... 62 7.7 Relationtootherformalisms . ........... 62 8 CBR anisotropies 63 8.1 Covariantrelativistickinetictheory. ................. 63 8.2 Angularharmonicdecomposition. ............ 64 8.3 Non-linear 1+3 covariantmultipoleequations . 65 8.4 Temperatureanisotropymultipoles . .............. 69 8.5 Almost-EGS-Theoremanditsapplications . ............... 71 8.5.1 Assumptions ................................... ... 71 8.5.2 Provingalmost-FLRWkinematics . ......... 72 8.5.3 Provingalmost-FLRWdynamics. ........ 74 8.5.4 Findinganalmost-RWmetric . ....... 74 8.5.5 Result........................................ .. 74 8.6 OtherCBRcalculations. ......... 74 8.6.1 Sachs–Wolfeandrelatedeffects . .......... 74 8.6.2 Othermodels................................... ... 75 9 Conclusion and open issues 76 9.1 Conclusion ...................................... ..... 76 9.2 Openissues ...................................... ..... 77 References 78 1 BASIC RELATIONS 5 1 Basic relations A cosmological model represents the Universe at a particular scale. We will assume that on large scales, space-time geometry is described by Einstein’s general theory of relativity (see, e.g., d’Inverno [1], Wald [2], Hawking and Ellis [3], or Stephani [4]). Then a cosmological model is defined by specifying [5]–[7]: * the space-time geometry represented on some specific averaging scale and determined by the metric µ gab(x ), which — because of the requirement of compatibility with observations — must either have some expanding Robertson–Walker (‘RW’) geometries as a regular limit (see [8]), or else be demonstrated to have observational properties compatible with the major features of current astronomical observations of the Universe; * the matter present, represented on the same averaging scale, and its physical behaviour (the energy- momentum tensor of each matter component, the equations governing the behaviour of each such component, and the interaction terms between them), which must represent physically plausible matter (ranging from early enough times to the present day, this will include most of the interactions described by present-day physics); and * the interaction of the geometry and the matter — how matter determines the geometry, which in turn determines the motion of the matter (see e.g. [9]). We assume this is through Einstein’s relativistic gravitational field equations (‘EFE’) given by1 G R 1 Rg = T Λ g , (1) ab ≡ ab − 2 ab ab − ab which, because of the twice-contracted Bianchi identities, guarantee the conservation of total energy- momentum Gab =0 T ab =0 , (2) ∇b ⇒ ∇b provided the cosmological constant Λ satisfies the relation Λ=0, i.e., it is constant in time and space. ∇a Together, these determine the combined dynamical evolution of the model and the matter in it. The description must be sufficiently complete to determine * the observational relations predicted by the model for both discrete sources and background radiation, implying a well-developed theory of structure growth for very small and for very large physical scales (i.e., for light atomic nuclei and for galaxies and clusters of galaxies), and of radiation absorbtion and