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Friedmann’s Equation - A description of our Universe

Friedmann’s Equation - A description of our Universe. Alexander Merle (Munich University of Technology) (AlexanderMerle@web.de) & Katharina Hübner (Leibniz-Gymnasium Altdorf) (katharina.huebner@onlinehome.de). Contents:. 1. Mathematical Basics 2. Crash Course on General Relativity

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Friedmann’s Equation - A description of our Universe

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  1. Friedmann’s Equation - A description of our Universe Alexander Merle (Munich University of Technology) (AlexanderMerle@web.de) & Katharina Hübner (Leibniz-Gymnasium Altdorf) (katharina.huebner@onlinehome.de)

  2. Contents: 1. Mathematical Basics 2. Crash Course on General Relativity 3. Friedmann’s Equation 4. Different Solutions 5. The Parameters of our Universe

  3. 1. Mathematical BasicsWhat the hell is a tensor??? An ordered set of numbers, which is characterized by its behavior in coordinate transformations: new coordinates Sum convention!!!  sum over ‘i’ and ‘k’ old indices new indices old coordinates

  4. The metric tensor Easy and well-known: Pythagorean Theorem more advanced: coefficients depending on the coordinates (these are called ‘metric tensor’)

  5. Exemplum gratii: The Minkowski-Metric this gives a flat space (no dependence on the coordinates)  special relativity

  6. 2. Crash Course on General RelativityEinstein’s Field Equation ... looks complicated but is not! LHS: geometry g’s and R’s (= functions of the metric g)  gravitation in form of curvature RHS: PHYSICS!!!  T: ‘energy momentum tensor’ (contains the complete energy of the system)  every energy contributes to the gravitation

  7. Energy momentum tensor of an ideal fluid pressure density 4-velocity space components time component proper time (time in the rest system )

  8. 3. Friedmann’s EquationThe Robertson-Walker-Metric Assumptions: mass distribution in the universe is in average homogeneous and isotropic curvature (-1, 0 or +1) scale factor (‘world radius’)  put this into Einstein’s equation and combine it with the energy-momentum-tensor of an ideal fluid with time-independent pressure and density  ‘equation of motion’ for the whole universe

  9. further assumption: homogeneity & isotropy  density and pressure depend only on time time-component (i,k=0)  space-components (i,k=1,2,3)   ‘BIG’ problem: only 2 equations for 3 variables  expedient: another equation (equation of state) needed

  10. Separate pressure into two parts: pressure by matter and by radiation  two equations of state  two equations of state and  combine that with the two previous equations two ‘conservation laws’ constant quantities  combine that with the physical constant to get rid of the ‘disturbing’ physical units:

  11. COMBINATION OF ALL THE COMPLICATED FORMULAS  THEN AFTER A LOT OF WORK WE FINALLY GET FRIEDMANN’S EQUATION  this is a dynamic (?!?) “equation of motion” for the scale factor / world radius R(t) WHAT’S LEFT?????  POSSIBLE SOLUTIONS AND THE PARAMETERS FOR OUR UNIVERSE PRESENTED BY KATHARINA

  12. 4. Different Solutions (Examples) Parameters of Friedmann’s Equation: - : Einstein’s Cosmological Constant  has an influence on the development of the universe for high values of R(t) - k: curvature (+1, 0 or -1)  describes the geometrical properties of the whole space HERE: some examples  for all solutions we get: dR/dt  as R  0 BIG BANG!!!

  13. A closed universe  the universe expands to a certain point, stops and then starts collapsing  R  0 again

  14. Einstein’s static universe constant radius  static universe (Λ!), BUT: unstable!

  15. The de-Sitter-Universe de-Sitter-Universe  the universe expands, but the expansion is decelerated and the radius approaches a constant

  16. 5. The Parameters of our Universe nothing can stop the expansion flat space ‘Revival’ of Λ t0: NOW acceleration Hubble-constant

  17. What we believe at the moment  the universe started with the big bang and expands eternally and even accelerated

  18. References: Thorsten Fließbach, Allgemeine Relativitätstheorie, Spektrum Verlag 2003 (all pictures are from this book) Particle Data Group, Particle Physics Booklet, CERN Library, 2004 Langenscheid’s Taschenwörterbuch Englisch, Langenscheid, 1964 BOOKTIP: Kip S. Thorne, ‘Black Holes and Time Warps: Einstein’s Outrageous Legacy’ (German: ‘Gekrümmter Raum und verbogene Zeit: Einsteins Vermächtnis’) best popular book in this world about general relativity!

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