Particle Production from Coherent Oscillation Hiroaki Nagao Graduate School of Science and Technology, Niigata University, Japan In collaboration with Takehiko Asak 1 (Niigata Univ.) DESY Theory Workshop, October, 1 st , 2009
Feb 22, 2016
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.)
3
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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