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Quantization of Inflation Models

Quantization of Inflation Models . Shih-Hung (Holden) Chen Collaborate with James Dent. Outline. Motivation Standard procedure and its limitation Proposed method Results and comparisons Summary. Motivation. Observation #1: The earth is beautiful. Observation #2:

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Quantization of Inflation Models

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  1. Quantization of Inflation Models Shih-Hung (Holden) Chen Collaborate with James Dent Oct 22 2010

  2. Outline • Motivation • Standard procedure and its limitation • Proposed method • Results and comparisons • Summary Oct 22 2010

  3. Motivation Observation #1: The earth is beautiful Observation #2: It sits in a nonhomogeneous Universe Oct 22 2010

  4. Observation #1: CMB looks boring Observation #2: In fact it is quite interesting Oct 22 2010

  5. 13.7 billion years old 370,000 years old Thanks to 10-5 so that we are here appreciating the beauty of earth Oct 22 2010

  6. How to produce primordial density fluctuation? Inflation: a period of time when the universe is accelerated expanding flatness, horizon, monopole… Fridemann Equations Oct 22 2010

  7. Turn on quantum fluctuations Amplitude of quantum fluctuation determines density fluctuation! Oct 22 2010

  8. Current data constraints Stringent constraints require accurate discriminator Oct 22 2010

  9. Review of Standard Procedure D. Lyth, E. Stewart Phys.Lett.B302:171-175,1993. The most general form of scalar linear perturbation Define gauge invariant comoving curvature perturbation Field redefinition Put background evolution on-shell Becomes… Oct 22 2010

  10. Quantization: Expand real operator u in terms of mode functions in Fourier space Require condition on mode functions that need to be satisfied at all time Oct 22 2010

  11. e.o.m of uk Mukhanov Sasaki Equation Define vacuum state Due to the non uniqueness of mode functions Vacuum is not uniquely determined yet! Need to impose a physical boundary condition! It turns out not so simple to impose physically reasonable boundary condition except for slow-roll models. Oct 22 2010

  12. Define slow-roll parameters In the limit of constant ε and δ Oct 22 2010

  13. Mukhanov Sasaki Equation is exact solvable under this limit! The solutions are linear combinations of 1st and 2nd Hankel function Due to the property of the Hankel function and z’’/z The equation approaches SHO with constant frequency which we know how to quantize Oct 22 2010

  14. Require the mode function approaches the ground state of SHO with constant frequency at the asymptotic region α =1,β=0 Bunch-Davies vacuum Oct 22 2010

  15. Limitation of the standard mthod There exist examples the standard method does not apply. Oct 22 2010

  16. Example#1 I. Bars, S.H. Chen hep-th/1004.0752  c=64b Clearly there is something wrong using the green curve to fit the red curve!! Example#2 J. Barrow Phys.Rev.D49:3055-3058,1994. Oct 22 2010

  17. Proposed method Oct 22 2010

  18. Oct 22 2010

  19. The mode function is The power spectrum is The spectral index is The running of the spectral index is Oct 22 2010

  20. Results and comparisons Oct 22 2010

  21. Standard Proposed Oct 22 2010

  22. Summary • The standard procedure only apply to a limited class of inflation models 2. Without an accurate method, it is hard to determine whether a model is compatible with observational constraints or not 3. In order to test all the existing models, there is a need to develop new quantization method 4. Our method can be improved by using quartic polinomial to fit z’’/z Thank You! Oct 22 2010

  23. Oct 22 2010

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