MODERN COSMOLOGY

References 107 ——1980The Large-Scale Structure of the Universe(Princeton, NJ: Princeton University Press) ——1982Astrophys. J. 26 ...

Chapter 3 Cosmological models George F R Ellis Mathematics Department, University of Cape Town, South Africa 3.1 Introduction Th ...

Introduction 109 (and indeed, in consequence of this remark) many of the ways of estimating the model parameters depend on model ...

110 Cosmological models whereλis the cosmological constant andκthe gravitational constant. Here Tij(with traceT =Taa)is the tota ...

Introduction 111 radiation, e.g. the cosmic background radiation (‘CBR’),pr=μr/ 3 ,.However, in more complex cases there will be ...

112 Cosmological models The description of matter and radiation in a cosmological model must be sufficiently complete to determi ...

1 + 3 covariant description: variables 113 (see equation (3.20)). The projected time and space derivatives ofUab,haband ηabcall ...

114 Cosmological models Applying this to the energy densityμshows that ifωaμ ̇=0inanopenset then∇ ̃aμ=0 there, so non-zero vor ...

1 + 3 Covariant description: equations 115 3.2.5 Weyl tensor In analogy to Fab,theWeyl conformal curvature tensor Cabcd defined ...

116 Cosmological models One can approximate ordinary matter in this way, with 1≤γ ≤2inorder that the causality and energy condit ...

1 + 3 Covariant description: equations 117 Singularity theorem. [21, 26, 28] In a universe where the active gravitational mass i ...

118 Cosmological models (2) The metric of the orthogonal 3-spacest =constant formed by meshing together the tangent spaces ortho ...

1 + 3 Covariant description: equations 119 3.3.3 Bianchi identities The third set of equations arises from theBianchi identities ...

120 Cosmological models 3.3.4 Implications Altogether, we have six propagation equations and six constraint equations; considere ...

Tetrad description 121 3.4 Tetrad description The 1+3 covariant equations are immediately transparent in terms of representing r ...

122 Cosmological models It follows (apply this relation to the coordinatexi) that in terms of the tetrad components, γabc(xi)=ea ...

Tetrad description 123 of vanishing torsion the relations for an orthonormal tetrad that are the analogues of the usual Christof ...

124 Cosmological models a Fermi-propagated (physically non-rotating) basis along the fundamental flow lines. Finally, the quanti ...

Models and symmetries 125 3.5.1.1 Killing vectors A space or spacetimesymmetry,orisometry, is a transformation that drags the me ...

126 Cosmological models 3.5.1.2 Groups of isometries The isometries of a space of dimensionnmust be a group, as the identity is ...