particle production from coherent oscillation
DESCRIPTION
Particle Production from Coherent Oscillation. Hiroaki Nagao Graduate School of Science and Technology, Niigata University, Japan. DESY Theory Workshop, October, 1 st , 2009 . In collaboration with Takehiko Asaka. ( Niigata Univ.). Introduction. [ e x:A.D.Linde (‘82,‘83)]. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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)]
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Framework
• Particle production from coherent oscillation(Neglect expansion of our univ.)
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How are they produced?!
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・ So far,….
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[ex: M.S.Turner (‘83)]
When is this approximation valid?
: φ decay occurs
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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)]
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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
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Growth for mode k*
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Phase cancellation
・ The mode k* is ensured to grow!
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Analytical results
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◎Distribution function of scalar
◎Number density
◎Growing mode
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Evolution of occupation number
for
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Yield of produced scalar
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Number density
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Provide Good Approximation !
11Is this treatment valid forever ?
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Non-perturbative effect
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‘Bose condensation’
・ Effect of higher order corrections of coupling gS
・ Reflect the statistical property of χ
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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)]
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Analytical results
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◎Distribution function
◎Number density
Correspond to the energy conservation condition in non-rela. φ decay.
where
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Evolution of occupation number
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for
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Yield of produced fermion
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Non-perturbative effect
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‘Pauli blocking’
Effect of higher order corrections of coupling gF
Reflect the statistical property of ψ
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How to estimate this frequency ??
Averaging method!
Long periodic oscillation around 1/2
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Decay process of non-rela. φScalar Fermion
Decay processes are forbidden for
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Abundance of heavy particles
Heavy particles can be produced are induced at
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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
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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
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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)]
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・ Field operator
・ Hamiltonian density under the time dependent background
Eigenstate of Hamiltonian
Diagonalization of Hamiltonian
[ex: M.Peloso et al(‘00)]
Particle picture
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Diagonalization
◎Bogoliubov transformation
・ Commutation relation (Equal time)
◎Diagonalized Hamiltonian
Eigenstate of Hamiltonian
where
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Particle number・ Number operator
◎Number density of produced ψ
・ Distribution function in k space
Pauli exclusion principle
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Solution for mode function
◎Solution for
starts at
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Superposition of oscillation
only contain oscillating behavior??
・ Leading order contribution
Leading contribution for β
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Cause the phase cancellation at
Growth of Growth of occupation number
Grow!
Growing mode = Energy conservation in decay process
Growth of β
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Growth of occupation number
starts at
・ By taking
Growth of occupation number @
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Number density for scalar
◎ contribution
・ Definition of number density
・ Exchange the order of integration
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・ Expand in terms of and perform integration in time
( General hypergyometric function )
・ Integration in momentum space
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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/
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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!
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Averaging method
Originate from Dirac eq.
[ex:A.H.Nayfeh et.al (‘79)]
◎Variation of parameters
◎Averaging
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Averaged solution◎Averaged solution for fermion
Long periodic oscillation around 1/2
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Consistency◎We obtain following results by the method of averaging
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Evolution of number density
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・ Growth of number density would be stopped because of the absence of phase cancellation
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Distribution function in k space
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