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Cosmological Models with No Big Bang

Cosmological Models with No Big Bang. 許文郁 Wun-Yi Shu Institute of Statistics National Tsing Hua University. Presentation at Institute of Physics National Chiao Tung University 2011/04/07.

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Cosmological Models with No Big Bang

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  1. Cosmological Models with No Big Bang 許文郁 Wun-Yi Shu Institute of Statistics National Tsing Hua University Presentation at Institute of Physics National Chiao Tung University 2011/04/07

  2. In the late 1990s, observations of Type Ia supernovae led to the astounding discovery that the universe is expanding at an accelerating rate. . . The explanation of this anomalous acceleration has been one of the greatest challenges of theoretical physics since that discovery. The current mainstream explanation of the accelerating expansion of the universe is dark energy— a mysterious force so named because researchers have never detected such thing. In general relativity, dark energy is represented as a cosmological constant. . We propose cosmological models that can explain the cosmic acceleration via the geometric structure of space-time, without introducing a cosmological constant into the standard field equation, negating the necessity for the existence of dark energy..

  3. There are four distinguishing features of these models: • the speed of light and the gravitational “constant” are not constant, but varies with the evolution of the universe, • time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity, • the spatial section of the universe is a 3-sphere, ruling out the possibility of a flat or hyperboloid geometry, and • the universe experiences phases of both acceleration and deceleration.

  4. Outline • Geometry • Cosmological Models • Dynamics of the Universe • Test of the Models • Conclusion

  5. 1. Geometry Paraboloid

  6. Geometry of Paraboloid Thedistance : On R2, (x, y) → (x+dx, y+dy) On R3, (x, y, x2+y2) → ( x+dx, y+dy, (x+dx)2 +(y+dy)2 ) .

  7. Geometry of Paraboloid

  8. Geometry of sphere

  9. Statistical Manifold Fisher information matrix The geometric properties (e.g. curvature) of M playimportant role in statistical inference.

  10. Statistical Manifold of Normal Distributions

  11. Hyperbolic Geometry

  12. Equal Distance Hyperbolic Geometry

  13. The Earth

  14. The world map

  15. Arc length in H

  16. Arc length in H

  17. Straight lines in H

  18. Straight lines in H

  19. Hyperbolic Geometry • Any two points can be joined by a straight line. • Any straight line segment can be extended indefinitely in a straight line. • Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. • All right angles are congruent. • Through a point not on a given straight line, more than one lines can be drawn that never meet the given line.

  20. Through a point not on a given straight line, More than one lines can be drawn that never meet the given line.

  21. 2. Cosmological models A cosmological model is defined by specifying: the spacetime geometry determined by a metric , the mass-energy distribution of the universe described in terms of a stress-energy-momentum tensor , and the interaction of the geometry and the mass-energy, which is depicted through a field equation.

  22. 2.1 Geometry of Space-time Henri-Emile-Benoit Matisse 1869/12/31~1954/11/03

  23. Woman with a hat.1905 (81×60cm). Private collection.

  24. Blue Nude IV.1952. The Matisse Museum, Nice.

  25. Le Bateau. 1953. The Museum of Modern Art, NY.

  26. The Museum of Modern Art hung the print upside-down for 47 days in 1961. In this period of time 11,600 people has passed through the gallery.

  27. The longest period of time for which a modern painting has hung upside down in a public gallery unnoticed is 47 days. This occurred to Le Bateau by Matisse in the Museum of Modern Art, New York City. In this time 11,600 people has passed through the gallery. . — The Guinness Book of Records, Guinness Superlatives, Ltd..

  28. A simple to ask, but hard to answer question: What are Space and Time ? The progress of science can be measured by revolutions that produce new answers to it.

  29. Newton’s View of Space and Time

  30. Newton’s View of Space and Time Absolute space, in its own nature, without relation to anything external, remains always similar and immovable, , Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external. . [From the scholium in the Principia]

  31. Einstein’s View of Space and Time Space is nothing apart from the things that exist; it is only an aspect of the relationshipsthat hold between things.. Time has no absolute meaning. There is no time apart from change. Time is described only in terms of change in the network of relationships that describes space. Time is nothing but a measure of change – it has no other meaning. .

  32. The spacetime in the neighborhood of the sun

  33. Geodesic curves → Geometry

  34. Einstein’s Field Equation Rab : Ricci curvature R : scalar curvature Space-time tells matter how to move, matter tells space-time how to curve. — John A. Wheeler

  35. Spacetime geometryin the neighborhood of the sun

  36. Spacetime geometryin the neighborhood of the sun

  37. 2.1.1 Geometry of the Universe Cosmological Principle: On the large scales, the universe is assumed to be homogeneous and isotropic. Expressed in the synchronous time coordinate and co-moving spatial Spherical/ hyperbolic coordinates , the line element of the spacetime metric takes the form: where the three options listed to the right of the left bracket correspond to the three possible spatial geometries: a 3-sphere, 3-dimensional flat space, and a 3-dimensional hyperboloid, respectively. .

  38. The spherical coordinate

  39. 2.1.1 Geometry of the Universe with Spatial Geometry S3

  40. 2.1.1 Geometry of the Universe with Spatial Geometry R3

  41. 2.1.1 Geometry of the Universe with Spatial Geometry Hyperboloid

  42. 2.1.2 Geometry of the Universe with Varying Speed of Light We view the speed of light as simply a conversion factor between time and space in spacetime. It is a Nature’s manifestation of the structure of spacetimegeometry. Since the universe is expanding, we speculate that the conversion factor somehow varies in accordance with the evolution of the universe, hence the speed of light varies with cosmic time. Denoting the speed of light as a function of cosmic time by c(t), we modify the metric as: .

  43. 2.1.2 Geometry of the Universe with Varying Speed of Light ( S3 )

  44. 2.1.2 Geometry of the Universe with Varying Speed of Light ( R3 )

  45. 2.1.2 Geometry of the Universe with Varying Speed of Light (Hyperboloid)

  46. 2.2 The Stress-energy-momentum Tensor The content of the universe is described in terms of a stress- energy-momentum tensor Tab . We shall take Tab to be the general perfect fluid form: ua: the 4-velocity of the cosmological fluid. . ρ: the proper average mass density. . P : the pressure as measured in the instantaneous rest . frame. .

  47. 2.2 The Stress-energy-momentum Tensor The components ofTab :

  48. 2.3 Einstein’s Field Equation Rab : Ricci curvature R : scalar curvature Space-time tells matter how to move, matter tells space-time how to curve. — John A. Wheeler

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