Abstract
Chapter 5, entitled Cosmology and General Relativity: Mathematical Description of the Universe, provides a full-fledged introduction to Relativistic Cosmology. The chapter begins with a long mathematical interlude on the geometry of coset manifolds. These notions are necessary for the mathematical formulation of the Cosmological Principle, stating homogeneity and isotropy, but have a much wider spectrum of applications. In particular they will be very important in the subsequent chapters about Supergravity. Having prepared the stage with this mathematical preliminaries, the next sections deal with homogeneous but not isotropic cosmologies. Bianchi classification of three dimensional Lie groups is recalled, Bianchi metrics are defined and, within Bianchi type I, the Kasner metrics are discussed with some glimpses about the cosmic billiards, realized in Supergravity. Next, as a pedagogical example of a homogeneous but not isotropic cosmology, an exact solution, with and without matter, of Bianchi type II space-time is treated in full detail. After this, we proceed to the Standard Cosmological Model, including both homogeneity and isotropy. Freedman equations, all their implications and known solutions are discussed in detail and a special attention is given to the embedding of the three type of standard cosmologies (open, flat and closed) into de Sitter space. The concept of particle and event horizons is next discussed together with the derivation of exact formulae for read-shift distances. The conceptual problems (horizon and flatness) of the Standard Cosmological Model are next discussed as an introduction to the new inflationary paradigm. The basic inflationary model based on one scalar field and the slow rolling regime are addressed in the following sections with fully detailed calculations. Perturbations, the spectrum of fluctuations up to the evaluation of the spectral index and the principles of comparison with the CMB data form the last part of this very long chapter.
Ma sedendo e mirando, interminati
Spazi di là da quella, e sovrumani
Silenzi, e profondissima quiete
Io nel pensier mi fingo; ove per poco
Il cor non si spaura…
Giacomo Leopardi
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Notes
- 1.
See Sect. 3.3 of Volume 1 for the definition of the pull-back and of the push-forward.
- 2.
In higher dimensional gravity theories the ball moves in n-dimensions.
- 3.
In the mostly minus conventions we have ds 2=g μν  dx μ⊗dx ν and g μν U μ U ν=1.
- 4.
This choice has the following motivation. In presence of a generic potential for the scalar field, the cosmological constant is redundant. Indeed any constant contribution to V(φ) just plays the role of a cosmological constant.
- 5.
Any one of the three equations is actually a consequence of the other two as a legacy of the Bianchi identities which constrain Einstein equations.
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Frè, P.G. (2013). Cosmology and General Relativity: Mathematical Description of the Universe. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5443-0_5
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