On the Topology and Future Stability of the Universe

Hardcover | August 25, 2013

byHans Ringstrom

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The standard starting point in cosmology is the cosmological principle; the assumption that the universe is spatially homogeneous and isotropic. After imposing this assumption, the only freedom left, as far as the geometry is concerned, is the choice of one out of three permissible spatialgeometries, and one scalar function of time. Combining the cosmological principle with an appropriate description of the matter leads to the standard models. It is worth noting that these models yield quite a successful description of our universe. However, even though the universe may, or may not, be almost spatially homogeneous and isotropic, it is clear that the cosmological principle is not exactly satisfied. This leads to several questions. The most natural one concerns stability: given initial data corresponding to an expanding model ofthe standard type, do small perturbations give rise to solutions that are similar to the future? Another question concerns the shape of the universe: what are the restrictions if we only assume the universe to appear almost spatially homogeneous and isotropic to every observer? The main purpose of the book is to address these questions. However, to begin with, it is necessary to develop the general theory of the Cauchy problem for the Einstein-Vlasov equations. In order to to make the results accessible to researchers who are not mathematicians, but who are familiar withgeneral relativity, the book contains an extensive prologue putting the results into a more general context.

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The standard starting point in cosmology is the cosmological principle; the assumption that the universe is spatially homogeneous and isotropic. After imposing this assumption, the only freedom left, as far as the geometry is concerned, is the choice of one out of three permissible spatialgeometries, and one scalar function of time. Co...

Hans Ringstrom obtained his PhD in 2000 at the Royal Institute of Technology in Stockholm. He spent 2000-2004 as a post doc in the Max Planck Institute for Gravitational Physics, also known as the Albert Einstein Institute. In 2004 he returned to Stockholm as a research assistant. In 2007 he became a Royal Swedish Academy of Sciences ...
Format:HardcoverDimensions:768 pages, 9.21 × 6.14 × 0.1 inPublished:August 25, 2013Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199680299

ISBN - 13:9780199680290

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Table of Contents

I Prologue1. Introduction2. The Initial Value Problem3. The Topology of the Universe4. Notions of Proximity5. Observational Support6. Concluding RemarksII Introductory Material7. Main Results8. Outline, General Theory9. Outline, Main Results10. References and OutlookIII Background and Basic Constructions11. Basic Analysis Estimates12. Linear Algebra13. CoordinatesIV Function Spaces, Estimates14. Function Spaces, Distribution Functions15. Function Spaces on Manifolds16. Main Weighted Estimate17. Concepts of ConvergenceV Local Theory18. Uniqueness19. Local Existence20. StabilityVI The Cauchy Problem in General Relativity21. The Vlasov Equation22. The Initial Value Problem23. Existence of an MGHD24. Cauchy StabilityVII Spatial Homogeneity25. Spatially Homogeneous Metrics26. Criteria Ensuring Global Existence27. A Positive Non-Degenerate Minimum28. Approximating FluidsVIII Future Global Non-Linear Stability29. Background Material30. Estimates for the Vlasov Matter31. Global Existence32. Asymptotics33. Proof of the Stability Results34. Models with Arbitrary Spatial TopologyIX AppendicesA PathologiesB Quotients and Universal Covering SpacesC Spatially Homogeneous and Isotropic MetricsD Auxiliary Computations in Low RegularityE Curvature, Left Invariant MetricsF Comments, Einstein-Boltzmann