1. Oct 22 20101 Quantization of Inflation Models Shih-Hung (Holden) Chen Collaborate with James Dent

2. Oct 22 20102 Outline 1.Motivation 2.Standard procedure and its limitation 3.Proposed method 4.Results and comparisons 5.Summary

3. Oct 22 20103 Motivation Observation #1: The earth is beautiful Observation #2: It sits in a nonhomogeneous Universe

4. Oct 22 20104 Observation #1: CMB looks boring Observation #2: In fact it is quite interesting

5. Oct 22 20105 Thanks to 10 -5 so that we are here appreciating the beauty of earth 370,000 years old 13.7 billion years old

6. Oct 22 20106 How to produce primordial density fluctuation? Inflation: a period of time when the universe is accelerated expanding flatness, horizon, monopole… Fridemann Equations

7. Oct 22 20107 Turn on quantum fluctuations Amplitude of quantum fluctuation determines density fluctuation!

8. Oct 22 20108 Current data constraints Stringent constraints require accurate discriminator

9. Oct 22 20109 Review of Standard Procedure D. Lyth, E. Stewart Phys.Lett.B302:171-175,1993. Define gauge invariant comoving curvature perturbation The most general form of scalar linear perturbation Field redefinition Put background evolution on-shell Becomes…

10. Oct 22 201010 Quantization: condition on mode functions that need to be satisfied at all time Expand real operator u in terms of mode functions in Fourier space Require

11. Oct 22 201011 Define vacuum state e.o.m of u k Mukhanov Sasaki Equation 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.

12. Oct 22 201012 In the limit of constant ε and δ Define slow-roll parameters

13. Oct 22 201013 Mukhanov Sasaki Equation is exact solvable under this limit! The solutions are linear combinations of 1 st and 2 nd 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

14. Oct 22 201014 Require the mode function approaches the ground state of SHO with constant frequency at the asymptotic region Bunch-Davies vacuum α =1,β=0

15. Oct 22 201015 Limitation of the standard mthod There exist examples the standard method does not apply.

16. Oct 22 201016 Example#1 I. Bars, S.H. Chen hep-th/1004.0752 Example#2 J. Barrow Phys.Rev.D49:3055-3058,1994. Clearly there is something wrong using the green curve to fit the red curve!! c=64b

17. Oct 22 201017 Proposed method

18. Oct 22 201018

19. Oct 22 201019 The spectral index is The power spectrum is The running of the spectral index is The mode function is

20. Oct 22 201020 Results and comparisons

21. Oct 22 201021 Standard Proposed

22. Oct 22 201022 Summary 1.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!

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