Particle Productionfrom

Coherent OscillationHiroaki Nagao

Graduate School of Science and Technology, Niigata University, Japan

In collaboration with Takehiko Asaka1(Niigata Univ.)

DESY Theory Workshop, October, 1st , 2009

Introduction

• Inflation ・ Solve the problems of Standard Big Bang Cosmology

・ Provide the origin of density fluctuation・ Supported by CMBR observation

• Reheating ?? ・ Coherent oscillation of scalar field・ Energy transfer into elementary particles

2SM , SUSY(?)…??

Our focus!

[ex:A.D.Linde (‘82,‘83)]

[ex:WMAP 5yr. (‘08)]

Framework

• Particle production from coherent oscillation(Neglect expansion of our univ.)

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How are they produced?!

・ So far,….

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[ex: M.S.Turner (‘83)]

When is this approximation valid?

: φ decay occurs

Our analysis

◎Use the method based on Bogolyubov transformation・ Solve E.O.M for mode function

・ Estimate distribution function

Find the behavior of5

e.g.)

e.g.)

In weak coupling limit to avoid the preheating effect

[ex:N.N.Bogolyubov(‘58)]

[ex:L.Kofman et al(‘94) M.Peloso et al(‘00)]

Perturbative expansion in coupling

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◎ Solution of [ex:Y.Shtanov et al(‘94) A.D.Dolgov(‘01) ]

E.O.M

starts at

starts at

Growth for mode k*

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Phase cancellation

・ The mode k*　 is ensured to grow!

Analytical results

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◎Distribution function of scalar

◎Number density

◎Growing mode

Evolution of occupation number

for

Yield of produced scalar

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Number density

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Provide Good Approximation ！

11Is this treatment valid forever ?

Non-perturbative effect

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‘Bose condensation’

・ Effect of higher order corrections of coupling 　 gS

・ Reflect the statistical property of χ

Q. How to estimate this exponent??

Much longer time scale 　　　 than period of coherent oscillation

Average over the oscillation period of φ

“Averaging method”!![ex:A.H.Nayfeh et.al (‘79)]

Analytical results

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◎Distribution function

◎Number density

Correspond to the energy conservation condition in non-rela. φ 　 decay.

where

Evolution of occupation number

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for

Yield of produced fermion

Non-perturbative effect

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‘Pauli blocking’

Effect of higher order corrections of coupling gF

Reflect the statistical property of ψ

How to estimate this frequency ??

Averaging method!

Long periodic oscillation around 1/2

Decay process of non-rela. φScalar Fermion

Decay processes are forbidden for

Abundance of heavy particles

Heavy particles can be produced are induced at

Summary• Particle production from coherent oscillation Neglect expansion Weak coupling limit• Obtain the exact distribution function up to by using Bogolyubov transformation → ・ Applicable in the beginnings of production ・ Imply the production of heavy particles• Higher-order correction is crucial in the later time ・ Provide the difference between χ and ψ ・ Can be estimated by the averaging method

Thank you for your attention.

Danke schön.

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BACKUP SLIDE

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Number density of coherent oscillation

Same dilution rate

Treat coherent oscillation as non-relativistic particles Approximation

・ Estimate 　　　 by decay of non-relativistic φ

Particle picture・ Field operator

・ Hamiltonian density under the time dependent background

Off-diagonal element!

Eigenstate of Hamiltonian Disable the particle picture

Diagonalization of Hamiltonian

[ex:M.G.Schmidt et.al(‘04)]

・ Field operator

・ Hamiltonian density under the time dependent background

Eigenstate of Hamiltonian

Diagonalization of Hamiltonian

[ex: M.Peloso et al(‘00)]

Particle picture

Diagonalization

◎Bogoliubov transformation

・ Commutation relation 　 (Equal time)

◎Diagonalized Hamiltonian

Eigenstate of Hamiltonian

where

Particle number・ Number operator

◎Number density of produced ψ

・ Distribution function in k space

Pauli exclusion principle

Solution for mode function

◎Solution for 　　　　　

starts at

Superposition of oscillation

only contain oscillating behavior??

・ Leading order contribution

Leading contribution for β

Cause the phase cancellation at

Growth of Growth of occupation number

Grow!

Growing mode = Energy conservation　　　　　　　　 in decay process

Growth of 　 β

Growth of occupation number

starts at

・ By taking

Growth of occupation number ＠

Number density for scalar

◎ contribution 　　　　

・ Definition of number density

・ Exchange the order of integration

・ Expand 　 in terms of and perform integration in time

( General hypergyometric function )

・ Integration in momentum space

Averaging method

◎Variation of parameterswhere

・ Remove the short-periodic oscillation・ Only contain the long periodic terms

◎Averaging [ex:A.H.Nayfeh et.al (‘79)]

w/

Later time behavior◎Averaged solution for scalar

◎Later time behavior of occupation number

Its exponent is consistent with the result of parametric resonance

[ex:M.Yoshimura(‘95)]

Exponential growth!

　　　　 Averaging method 　　

Originate from 　　　　 Dirac eq.

[ex:A.H.Nayfeh et.al (‘79)]

◎Variation of parameters

◎Averaging

Averaged solution◎Averaged solution for fermion

Long periodic oscillation around 1/2

Consistency◎We obtain following results by the method of averaging

Evolution of number density

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・ Growth of number density would be stopped because of the absence of phase cancellation

Distribution function in k space

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