Oct 22 2010 1

Quantization of Inflation Models

Shih-Hung (Holden) Chen

Collaborate with James Dent

Oct 22 2010 2

Outline

1.Motivation2.Standard procedure and its limitation3.Proposed method4.Results and comparisons5.Summary

Oct 22 2010 3

MotivationObservation #1:The earth is beautiful

Observation #2:It sits in a nonhomogeneousUniverse

Oct 22 2010 4

Observation #1: CMB looks boring

Observation #2: In fact it is quite interesting

Oct 22 2010 5

Thanks to 10-5 so that we are here appreciating the beauty of earth

370,000 years old 13.7 billion years old

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 ProcedureD. 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…

Oct 22 2010 10

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 22 2010 11

Define vacuum state

e.o.m of uk

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 conditionexcept for slow-roll models.

Oct 22 2010 12

In the limit of constant ε and δ

Define slow-roll parameters

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 frequencywhich 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

Bunch-Davies vacuum

α =1,β=0

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 

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

Oct 22 2010 17

Proposed method

Oct 22 2010 18

Oct 22 2010 19

The spectral index is

The power spectrum is

The running of the spectral index is

The mode function is

Oct 22 2010 20

Results and comparisons

Oct 22 2010 21

6 .6 6 .8 7 .0 7 .2 7 .4c on fo rm a l tim e

1 .0

0 .5

0 .0

0 .5

1 .0runn ing

6 .6 6 .8 7 .0 7 .2 7 .4c on fo rm a l tim e

1 .0

0 .5

0 .0

0 .5

1 .0runn ing

6 .6 6 .8 7 .0 7 .2 7 .4c on fo rm a l tim e

2

1

0

1

2n

6 .6 6 .8 7 .0 7 .2 7 .4c on fo rm a l tim e

2

1

0

1

2

3n

4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 7 .0c on fo rm a l tim e0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0a b sU 2

4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 7 .0c on fo rm a l tim e0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0a b sU 2

Standard Proposed

Oct 22 2010 22

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!

Oct 22 2010 23

6 .6 6 .8 7 .0 7 .2 7 .4c on fo rm a l tim e

1 .0

0 .5

0 .0

0 .5

1 .0r