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 spatial geometries, 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 of the 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 with general relativity, the book contains an extensive prologue putting the results into a more general context.
I Prologue 1: Introduction 2: The Initial Value Problem 3: The Topology of the Universe 4: Notions of Proximity 5: Observational Support 6: Concluding Remarks II Introductory Material 7: Main Results 8: Outline, General Theory 9: Outline, Main Results 10: References and Outlook III Background and Basic Constructions 11: Basic Analysis Estimates 12: Linear Algebra 13: Coordinates IV Function Spaces, Estimates 14: Function Spaces, Distribution Functions 15: Function Spaces on Manifolds 16: Main Weighted Estimate 17: Concepts of Convergence V Local Theory 18: Uniqueness 19: Local Existence 20: Stability VI The Cauchy Problem in General Relativity 21: The Vlasov Equation 22: The Initial Value Problem 23: Existence of an MGHD 24: Cauchy Stability VII Spatial Homogeneity 25: Spatially Homogeneous Metrics 26: Criteria Ensuring Global Existence 27: A Positive Non-Degenerate Minimum 28: Approximating Fluids VIII Future Global Non-Linear Stability 29: Background Material 30: Estimates for the Vlasov Matter 31: Global Existence 32: Asymptotics 33: Proof of the Stability Results 34: Models with Arbitrary Spatial Topology IX Appendices A: Pathologies B: Quotients and Universal Covering Spaces C: Spatially Homogeneous and Isotropic Metrics D: Auxiliary Computations in Low Regularity E: Curvature, Left Invariant Metrics F: Comments, Einstein-Boltzmann
Hans Ringström 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 Research Fellow, supported by a grant from the Knut and Alice Wallenberg Foundation, a position which lasted until 2012. In 2011, Ringström obtained an associate professorship at the Royal Institute of Technology.
Reviews for On the Topology and Future Stability of the Universe
This impressive new book is first and foremost an original and thought-provoking contribution to the study of cosmology in research monograph form, in the best tradition of the kind of deep mathematical work which has played a crucial role in the development of the subject. Mihalis Dafermos,Classical and Quantum Gravity
