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

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

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

Oct Observation #1: CMB looks boring Observation #2: In fact it is quite interesting

Oct Thanks to so that we are here appreciating the beauty of earth 370,000 years old 13.7 billion years old

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

Oct Turn on quantum fluctuations Amplitude of quantum fluctuation determines density fluctuation!

Oct Current data constraints Stringent constraints require accurate discriminator

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

Oct 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

Oct 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.

Oct In the limit of constant ε and δ Define slow-roll parameters

Oct 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

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

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

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

Oct Proposed method

Oct

Oct The spectral index is The power spectrum is The running of the spectral index is The mode function is

Oct Results and comparisons

Oct Standard Proposed

Oct 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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