Abstract
An introductory guide to mathematical cosmology is given, focusing on the issue of the genericity of various important results which have been obtained during the last 30 or so years. Some of the unsolved problems along with certain new and potentially powerful methods which may be used for future progress are also given from a unified perspective.
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References
Although we gave no references in the main text of this paper, here is a broad but very short list of references which is meant to indicate useful and/or indispensible sources for the prospective mathematical cosmologist. A basic mathematical reference for our subject is the superb two-volume treatise by Y. Choquet-Bruhat et al. Analysis, Manifolds and Physics, Vol. I: Basics, 2nd Ed. (North-Holland, 1982), Vol. II, Applications, 2nd Ed. (North-Holland, 2000). As background reading, we suggest the very nice book by R. Geroch, Mathematical Physics, (University of Chicago Press, Chicago, 1985). Each of the following sources discusses one of the fundamental problems of mathematical cosmology. Singularities The standard reference is of course
S.W. Hawking, G.F.R. Ellis, The Large-Scale Structure of Spacetime, (CUP, 1973). Singularity theory is discussed in V.I. Arnold, et al. Singularities of differentiable mappings, Vol. I (Birkhauser, 1985). A presentation of parts of Arnold’s theory more suitable for physicists is contained in a recent paper by J. Ehlers et al., J. Math. Phys. 41 (2000) 3244–3378. Cosmic topology For a recent review of the theoretical and observational aspects of cosmic topology see the special issue of Class. Quant. Grav. 15, September 1998, edited by G.D. Starkman. The theoretical problem has many components. See
Papers by Hosoya, Kodama, Barrow-Kodama in the gr-qc Los Alamos Archives. See also A.E. Fisher, V. Moncrief, The reduced hamiltonian of general relativity and the σ constant of conformal geometry, in Mathematical and Quantum Aspects of Relativity and Cosmology, S. Cotsakis, G.W. Gibbons (eds.), Lecture Notes in Physics, 537 (Springer, 1998), pp. 70–101. Asymptotic problem
J. Wainwright, G.F.R. Ellis, Dynamical Systems in Cosmology (CUP, 1997). The bulk of this beautiful book treats in depth the Bianchi/GR/Fluid cosmologies, but some members of the Inhomogeneous/GR family are also covered. An advanced but excellent discussion of bifurcation theory is given in V.I. Arnol’d, Geometrical Methods in the Theory of Ordinary Differential Equations, (Springer, 1983). The elements of the analytic structure of dynamical systems and their singularity patterns in the complex plane are beautifully presented in M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, 1989) Chap. 8. There is a recent book on this subject, A. Roy Chowdhuri, Painlevé analysis and its applications, (Chapman and Hall/CRC, 2000). For some applications to cosmology see: S. Cotsakis, P.G.L. Leach, Painlevé analysis of the Mixmaster universe, J. Phys. A27 (1993) 1625–1631; P.G.L. Leach, S. Cotsakis, J. Miritzis, Symmetries, singularities and integrability in complex dynamics IV: Painlevé integrability of isotropic cosmologies, Grav. Cosm. 6 (2000) 282–290. Gravity theories and the early universe For the issue of choosing a gravity theory for building a realistic early universe cosmology no single general reference exists, but research is scattered in virtually every mathematical cosmology paper. We give here a few important recent references to show the flavour of research in a number of different cosmologies.
FRW/GR cosmologies: S. Foster, Scalar field cosmologies and the initial space-time singularity, gr-qc/9806098.
FRW/ST cosmologies: S.J. Kolitch, D.M. Eardley, Behaviour of the FRW cosmological models in scalar-tensor gravity, gr-qc/9405016.
FRW/String cosmologies: A.P. Billyard, A.A. Coley, J.E. Lidsey, Qualitative analysis of string cosmologies, gr-qc/9903095.
FRW/Brane cosmologies: J. Khoury, P.J. Steinhardt, D. Waldram, Inflationary solutions in the brane-world and their geometrical interpretation, hepth/0006069.
FRW/M-theory cosmologies: A. Lucas, B.A. Ovrut, D Waldram, Cosmological solutions of Hořava-Witten theory, hep-th/9806022; A.P. Billyard, A.A. Coley, J.E. Lidsey, Dynamics of M-theory cosmology, hep-th/9908102.
Bianchi/ST cosmologies: A.A. Coley, Qualitative properties of scalar-tensor theories of gravity, astro-ph/9910395.
Bianchi/String cosmologies: J.D. Barrow, K.E. Kunze, Spatially homogeneous string cosmologies, hep-th/9608045; J.D. Barrow, M.P. Dabrowski, Is there chaos in low-energy string cosmology?, hep-th/9711049; A.P. Billyard, A.A. Coley, J.E. Lidsey, Qualitative analysis of isotropic curvature string cosmologies, hep-th/9911086. A guiding ‘principle’ based on the use of symmetries is discussed in
J. Lidsey, Class. Quant. Grav. 13 (1996) 2449–2456.
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Cotsakis, S., Leach, P.G.L. (2002). Is Nature Generic?. In: Cotsakis, S., Papantonopoulos, E. (eds) Cosmological Crossroads. Lecture Notes in Physics, vol 592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48025-0_1
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DOI: https://doi.org/10.1007/3-540-48025-0_1
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